Subgroups — C₄×#540×C₆₇ #30713344

Order 17152.

2518 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#540×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2335	C₂	1	4	yes	maximal, complemented
3	C₆₇×#2333	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2330	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2328	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2324	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2319	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2314	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#127×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#125×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#123×C₆₇	C₂	1	4	yes	maximal, complemented
12	C₄×#121×C₆₇	C₂	1	2	yes	maximal, complemented
13	C₄×#121×C₆₇	C₂	1	4	yes	maximal, complemented
14	C₄²×D₄×C₆₇	C₂	1	2	yes	maximal, complemented
15	C₄×#117×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
16	#540×C₆₇	C₄	1	16	yes	direct product factor
17	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#123×C₆₇	C₂²	1	4	yes	—
20	C₂×#121×C₆₇	C₂²	1	2	yes	—
21	C₂×#121×C₆₇	C₂²	1	4	yes	—
22	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
23	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
25	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
26	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
27	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
29	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
30	C₂²×C₄²×C₆₇	C₂²	1	4	yes	—
31	#390×C₆₇	C₂²	1	2	yes	—
32	#390×C₆₇	C₂²	1	4	yes	—
33	#390×C₆₇	C₂²	1	2	yes	—
34	#390×C₆₇	C₂²	1	4	yes	—
35	#389×C₆₇	C₂²	1	2	yes	—
36	#389×C₆₇	C₂²	1	4	yes	—
37	#389×C₆₇	C₂²	1	2	yes	—
38	#388×C₆₇	C₂²	1	4	yes	—
39	#388×C₆₇	C₂²	1	4	yes	—
40	#388×C₆₇	C₂²	1	4	yes	—
41	#388×C₆₇	C₂²	1	4	yes	—
42	#388×C₆₇	C₂²	1	4	yes	—
43	#387×C₆₇	C₂²	1	4	yes	—
44	#387×C₆₇	C₂²	1	4	yes	—
45	#387×C₆₇	C₂²	1	4	yes	—
46	#386×C₆₇	C₂²	1	4	yes	—
47	#386×C₆₇	C₂²	1	4	yes	—
48	#386×C₆₇	C₂²	1	4	yes	—
49	#386×C₆₇	C₂²	1	4	yes	—
50	#385×C₆₇	C₂²	1	4	yes	—
51	#384×C₆₇	C₂²	1	1	yes	characteristic
52	#382×C₆₇	C₂²	1	1	yes	characteristic
53	#381×C₆₇	C₂²	1	4	yes	—
54	#380×C₆₇	C₂²	1	2	yes	—
55	#380×C₆₇	C₂²	1	4	yes	—
56	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
57	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
58	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
59	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
60	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
61	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
62	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
63	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
64	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
65	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
66	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
67	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
68	#376×C₆₇	C₂²	1	1	yes	characteristic
69	#376×C₆₇	C₂²	1	1	yes	characteristic
70	C₄³×C₆₇	C₂²	1	1	yes	characteristic
71	C₄³×C₆₇	C₂²	1	1	yes	characteristic
72	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
73	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
74	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
75	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
76	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
77	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
78	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
79	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
80	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
81	#127×C₆₇	C₂×C₄	1	8	yes	complemented
82	#125×C₆₇	C₂×C₄	1	8	yes	complemented
83	#123×C₆₇	C₂×C₄	1	32	yes	complemented
84	#121×C₆₇	C₂×C₄	1	32	yes	complemented
85	#121×C₆₇	C₂×C₄	1	16	yes	complemented
86	C₄×D₄×C₆₇	—	2	8	no	—
87	C₄×D₄×C₆₇	—	2	8	no	—
88	C₄×D₄×C₆₇	—	4	8	no	—
89	C₄×D₄×C₆₇	—	4	8	no	—
90	C₄×D₄×C₆₇	C₂×C₄	1	16	yes	complemented
91	#117×C₆₇	—	2	4	no	—
92	#117×C₆₇	C₂×C₄	1	8	yes	complemented
93	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
94	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
95	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
97	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
98	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
99	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
102	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
103	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
104	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
105	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
106	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
107	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
108	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
109	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
110	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
111	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
112	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
113	C₂×C₄²×C₆₇	—	2	4	no	—
114	C₂×C₄²×C₆₇	—	4	8	no	—
115	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
117	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
119	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
120	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
122	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
123	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
124	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
125	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
126	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
129	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
130	#95×C₆₇	C₂³	1	2	yes	—
131	#95×C₆₇	C₂³	1	2	yes	—
132	#95×C₆₇	C₂³	1	2	yes	—
133	#95×C₆₇	C₂³	1	2	yes	—
134	#95×C₆₇	C₂³	1	4	yes	—
135	#95×C₆₇	C₂³	1	4	yes	—
136	#95×C₆₇	C₂³	1	4	yes	—
137	#95×C₆₇	C₂³	1	4	yes	—
138	#95×C₆₇	C₂³	1	4	yes	—
139	#95×C₆₇	C₂³	1	4	yes	—
140	#95×C₆₇	C₂³	1	4	yes	—
141	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
142	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
143	C₂×D₄×C₆₇	—	2	4	no	—
144	C₂×D₄×C₆₇	—	2	4	no	—
145	C₂×D₄×C₆₇	—	4	4	no	—
146	C₂×D₄×C₆₇	—	4	4	no	—
147	C₂×D₄×C₆₇	—	4	4	no	—
148	C₂×D₄×C₆₇	—	4	4	no	—
149	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
150	C₂×D₄×C₆₇	C₂²×C₄	1	16	yes	—
151	C₂×D₄×C₆₇	C₂²×C₄	1	16	yes	—
152	C₂²×C₄×C₆₇	—	2	2	no	—
153	C₂²×C₄×C₆₇	—	2	2	no	—
154	C₂²×C₄×C₆₇	—	2	2	no	—
155	C₂²×C₄×C₆₇	—	2	2	no	—
156	C₂²×C₄×C₆₇	—	2	8	no	—
157	C₂²×C₄×C₆₇	—	2	8	no	—
158	C₂²×C₄×C₆₇	—	2	8	no	—
159	C₂²×C₄×C₆₇	—	2	4	no	—
160	C₂²×C₄×C₆₇	—	2	8	no	—
161	C₂²×C₄×C₆₇	—	4	4	no	—
162	C₂²×C₄×C₆₇	—	4	4	no	—
163	C₂²×C₄×C₆₇	—	4	2	no	—
164	C₂²×C₄×C₆₇	—	4	4	no	—
165	C₂²×C₄×C₆₇	—	4	4	no	—
166	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
167	C₂²×C₄×C₆₇	C₂²×C₄	1	16	yes	—
168	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
169	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
170	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
171	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
172	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
173	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
174	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
176	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
177	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
178	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
179	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
180	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
181	C₄⋊C₄×C₆₇	—	2	4	no	—
182	C₄⋊C₄×C₆₇	—	2	4	no	—
183	C₄⋊C₄×C₆₇	—	2	4	no	—
184	C₄⋊C₄×C₆₇	—	4	4	no	—
185	C₄⋊C₄×C₆₇	—	4	4	no	—
186	C₄⋊C₄×C₆₇	—	4	4	no	—
187	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
188	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
189	C₄⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
190	C₂²⋊C₄×C₆₇	—	2	4	no	—
191	C₂²⋊C₄×C₆₇	—	2	8	no	—
192	C₂²⋊C₄×C₆₇	—	2	8	no	—
193	C₂²⋊C₄×C₆₇	—	4	4	no	—
194	C₂²⋊C₄×C₆₇	—	4	4	no	—
195	C₂²⋊C₄×C₆₇	—	4	4	no	—
196	C₂²⋊C₄×C₆₇	—	4	4	no	—
197	C₂²⋊C₄×C₆₇	—	4	4	no	—
198	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
199	C₂²⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
200	C₂²⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
201	C₄²×C₆₇	—	2	2	no	—
202	C₄²×C₆₇	—	2	2	no	—
203	C₄²×C₆₇	—	2	2	no	—
204	C₄²×C₆₇	—	2	2	no	—
205	C₄²×C₆₇	—	2	4	no	—
206	C₄²×C₆₇	—	2	8	no	—
207	C₄²×C₆₇	—	2	4	no	—
208	C₄²×C₆₇	—	4	4	no	—
209	C₄²×C₆₇	—	4	4	no	—
210	C₄²×C₆₇	—	4	2	no	—
211	C₄²×C₆₇	—	4	2	no	—
212	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
213	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
214	C₄²×C₆₇	#41	1	4	yes	—
215	C₄²×C₆₇	#41	1	4	yes	—
216	C₂³×C₆₇	—	2	1	no	—
217	C₂³×C₆₇	—	2	4	no	—
218	C₂³×C₆₇	—	2	4	no	—
219	C₂³×C₆₇	—	2	1	no	—
220	C₂³×C₆₇	—	2	4	no	—
221	C₂³×C₆₇	—	2	2	no	—
222	C₂³×C₆₇	—	2	4	no	—
223	C₂³×C₆₇	—	2	4	no	—
224	C₂³×C₆₇	—	2	2	no	—
225	C₂³×C₆₇	—	2	4	no	—
226	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
227	C₂³×C₆₇	C₂³×C₄	1	8	yes	—
228	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
229	D₄×C₆₇	—	2	16	no	—
230	D₄×C₆₇	—	4	16	no	—
231	D₄×C₆₇	—	4	16	no	—
232	C₂×C₄×C₆₇	—	2	1	no	—
233	C₂×C₄×C₆₇	—	2	1	no	—
234	C₂×C₄×C₆₇	—	2	2	no	—
235	C₂×C₄×C₆₇	—	2	2	no	—
236	C₂×C₄×C₆₇	—	2	4	no	—
237	C₂×C₄×C₆₇	—	2	2	no	—
238	C₂×C₄×C₆₇	—	2	2	no	—
239	C₂×C₄×C₆₇	—	2	2	no	—
240	C₂×C₄×C₆₇	—	2	2	no	—
241	C₂×C₄×C₆₇	—	2	2	no	—
242	C₂×C₄×C₆₇	—	2	4	no	—
243	C₂×C₄×C₆₇	—	2	8	no	—
244	C₂×C₄×C₆₇	—	2	1	no	—
245	C₂×C₄×C₆₇	—	2	1	no	—
246	C₂×C₄×C₆₇	—	2	2	no	—
247	C₂×C₄×C₆₇	—	2	4	no	—
248	C₂×C₄×C₆₇	—	2	4	no	—
249	C₂×C₄×C₆₇	—	2	4	no	—
250	C₂×C₄×C₆₇	—	2	2	no	—
251	C₂×C₄×C₆₇	—	2	4	no	—
252	C₂×C₄×C₆₇	—	2	2	no	—
253	C₂×C₄×C₆₇	—	2	4	no	—
254	C₂×C₄×C₆₇	—	2	2	no	—
255	C₂×C₄×C₆₇	—	2	2	no	—
256	C₂×C₄×C₆₇	—	2	4	no	—
257	C₂×C₄×C₆₇	—	4	16	no	—
258	C₂×C₄×C₆₇	—	4	4	no	—
259	C₂×C₄×C₆₇	—	4	16	no	—
260	C₂×C₄×C₆₇	—	4	4	no	—
261	C₂×C₄×C₆₇	—	4	16	no	—
262	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
263	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
264	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
265	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
266	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
267	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
268	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
269	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
270	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
271	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
272	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
273	C₂²×C₆₇	—	2	2	no	—
274	C₂²×C₆₇	—	2	8	no	—
275	C₂²×C₆₇	—	2	8	no	—
276	C₂²×C₆₇	—	2	4	no	—
277	C₂²×C₆₇	—	2	8	no	—
278	C₂²×C₆₇	—	4	4	no	—
279	C₂²×C₆₇	—	4	1	no	—
280	C₂²×C₆₇	—	4	1	no	—
281	C₂²×C₆₇	—	4	4	no	—
282	C₂²×C₆₇	—	4	4	no	—
283	C₂²×C₆₇	—	4	2	no	—
284	C₂²×C₆₇	—	4	4	no	—
285	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
286	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
287	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
288	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
289	C₂²×C₆₇	#585	1	2	yes	—
290	C₄×C₆₇	—	2	2	no	—
291	C₄×C₆₇	—	2	4	no	—
292	C₄×C₆₇	—	2	8	no	—
293	C₄×C₆₇	—	2	4	no	—
294	C₄×C₆₇	—	2	2	no	—
295	C₄×C₆₇	—	2	4	no	—
296	C₄×C₆₇	—	4	1	no	—
297	C₄×C₆₇	—	4	4	no	—
298	C₄×C₆₇	—	4	1	no	—
299	C₄×C₆₇	—	4	2	no	—
300	C₄×C₆₇	—	4	4	no	—
301	C₄×C₆₇	C₄×#41	1	4	yes	—
302	C₄×C₆₇	#540	1	4	yes	direct product factor
303	C₄×#540	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
304	C₂×C₆₇	—	4	2	no	—
305	C₂×C₆₇	—	4	8	no	—
306	C₂×C₆₇	C₂×C₄×#41	1	1	yes	characteristic
307	C₂×C₆₇	C₄×H(5,𝔽₂)	1	2	yes	—
308	C₂×C₆₇	C₂×#540	1	1	yes	characteristic
309	C₂×C₆₇	#3455	1	1	yes	characteristic
310	C₂×C₆₇	#3460	1	2	yes	—
311	C₂×#540	C₂×C₆₇	1	1	yes	characteristic
312	#2335	C₂×C₆₇	1	4	yes	complemented
313	#2333	C₂×C₆₇	1	4	yes	complemented
314	#2330	C₂×C₆₇	1	1	yes	characteristic, complemented
315	#2328	C₂×C₆₇	1	1	yes	characteristic, complemented
316	#2324	C₂×C₆₇	1	2	yes	complemented
317	#2319	C₂×C₆₇	1	2	yes	complemented
318	#2314	C₂×C₆₇	1	1	yes	characteristic, complemented
319	C₄×#127	C₂×C₆₇	1	1	yes	characteristic, complemented
320	C₄×#125	C₂×C₆₇	1	1	yes	characteristic, complemented
321	C₄×#123	C₂×C₆₇	1	4	yes	complemented
322	C₄×#121	C₂×C₆₇	1	2	yes	complemented
323	C₄×#121	C₂×C₆₇	1	4	yes	complemented
324	C₄²×D₄	C₂×C₆₇	1	2	yes	complemented
325	C₄×#117	C₂×C₆₇	1	1	yes	characteristic, complemented
326	C₆₇	C₄×#540	1	1	yes	characteristic, direct product factor, Sylow
327	#540	C₄×C₆₇	1	16	yes	direct product factor
328	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
329	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
330	C₂×#123	C₂²×C₆₇	1	4	yes	—
331	C₂×#121	C₂²×C₆₇	1	2	yes	—
332	C₂×#121	C₂²×C₆₇	1	4	yes	—
333	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
334	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
335	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
336	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
337	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
338	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
339	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
340	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
341	C₂²×C₄²	C₂²×C₆₇	1	4	yes	—
342	#390	C₂²×C₆₇	1	2	yes	—
343	#390	C₂²×C₆₇	1	4	yes	—
344	#390	C₂²×C₆₇	1	2	yes	—
345	#390	C₂²×C₆₇	1	4	yes	—
346	#389	C₂²×C₆₇	1	2	yes	—
347	#389	C₂²×C₆₇	1	2	yes	—
348	#389	C₂²×C₆₇	1	4	yes	—
349	#388	C₂²×C₆₇	1	4	yes	—
350	#388	C₂²×C₆₇	1	4	yes	—
351	#388	C₂²×C₆₇	1	4	yes	—
352	#388	C₂²×C₆₇	1	4	yes	—
353	#388	C₂²×C₆₇	1	4	yes	—
354	#387	C₂²×C₆₇	1	4	yes	—
355	#387	C₂²×C₆₇	1	4	yes	—
356	#387	C₂²×C₆₇	1	4	yes	—
357	#386	C₂²×C₆₇	1	4	yes	—
358	#386	C₂²×C₆₇	1	4	yes	—
359	#386	C₂²×C₆₇	1	4	yes	—
360	#386	C₂²×C₆₇	1	4	yes	—
361	#385	C₂²×C₆₇	1	4	yes	—
362	#384	C₂²×C₆₇	1	1	yes	characteristic
363	#382	C₂²×C₆₇	1	1	yes	characteristic
364	#381	C₂²×C₆₇	1	4	yes	—
365	#380	C₂²×C₆₇	1	2	yes	—
366	#380	C₂²×C₆₇	1	4	yes	—
367	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
368	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
369	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
370	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
371	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
372	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
373	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
374	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
375	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
376	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
377	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
378	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
379	#376	C₂²×C₆₇	1	1	yes	characteristic
380	#376	C₂²×C₆₇	1	1	yes	characteristic
381	C₄³	C₂²×C₆₇	1	1	yes	characteristic
382	C₄³	C₂²×C₆₇	1	1	yes	characteristic
383	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
384	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
385	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
386	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
387	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
388	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
389	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
390	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
391	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
392	#127	C₂×C₄×C₆₇	1	8	yes	complemented
393	#125	C₂×C₄×C₆₇	1	8	yes	complemented
394	#123	C₂×C₄×C₆₇	1	32	yes	complemented
395	#121	C₂×C₄×C₆₇	1	32	yes	complemented
396	#121	C₂×C₄×C₆₇	1	16	yes	complemented
397	C₄×D₄	—	2	8	no	—
398	C₄×D₄	—	2	8	no	—
399	C₄×D₄	—	4	8	no	—
400	C₄×D₄	—	4	8	no	—
401	C₄×D₄	C₂×C₄×C₆₇	1	16	yes	complemented
402	#117	—	2	4	no	—
403	#117	C₂×C₄×C₆₇	1	8	yes	complemented
404	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
405	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
406	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
407	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
408	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
409	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
410	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
411	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
412	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
413	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
414	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
415	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
416	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
417	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
418	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
419	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
420	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
421	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
422	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
423	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
424	C₂×C₄²	—	2	4	no	—
425	C₂×C₄²	—	4	8	no	—
426	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
427	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
428	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
429	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
430	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
431	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
432	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
433	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
434	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
435	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
436	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
437	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
438	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
439	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
440	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
441	#95	C₂³×C₆₇	1	2	yes	—
442	#95	C₂³×C₆₇	1	2	yes	—
443	#95	C₂³×C₆₇	1	2	yes	—
444	#95	C₂³×C₆₇	1	2	yes	—
445	#95	C₂³×C₆₇	1	4	yes	—
446	#95	C₂³×C₆₇	1	4	yes	—
447	#95	C₂³×C₆₇	1	4	yes	—
448	#95	C₂³×C₆₇	1	4	yes	—
449	#95	C₂³×C₆₇	1	4	yes	—
450	#95	C₂³×C₆₇	1	4	yes	—
451	#95	C₂³×C₆₇	1	4	yes	—
452	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
453	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
454	C₂×D₄	—	2	4	no	—
455	C₂×D₄	—	2	4	no	—
456	C₂×D₄	—	4	4	no	—
457	C₂×D₄	—	4	4	no	—
458	C₂×D₄	—	4	4	no	—
459	C₂×D₄	—	4	4	no	—
460	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
461	C₂×D₄	C₂²×C₄×C₆₇	1	16	yes	—
462	C₂×D₄	C₂²×C₄×C₆₇	1	16	yes	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	2	no	—
466	C₂²×C₄	—	2	2	no	—
467	C₂²×C₄	—	2	8	no	—
468	C₂²×C₄	—	2	8	no	—
469	C₂²×C₄	—	2	8	no	—
470	C₂²×C₄	—	2	4	no	—
471	C₂²×C₄	—	2	8	no	—
472	C₂²×C₄	—	4	4	no	—
473	C₂²×C₄	—	4	4	no	—
474	C₂²×C₄	—	4	2	no	—
475	C₂²×C₄	—	4	4	no	—
476	C₂²×C₄	—	4	4	no	—
477	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
478	C₂²×C₄	C₂²×C₄×C₆₇	1	16	yes	—
479	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
480	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
481	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
482	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
483	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
484	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
485	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
486	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
487	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
488	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
489	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
490	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
491	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
492	C₄⋊C₄	—	2	4	no	—
493	C₄⋊C₄	—	2	4	no	—
494	C₄⋊C₄	—	2	4	no	—
495	C₄⋊C₄	—	4	4	no	—
496	C₄⋊C₄	—	4	4	no	—
497	C₄⋊C₄	—	4	4	no	—
498	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
499	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
500	C₄⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
501	C₂²⋊C₄	—	2	4	no	—
502	C₂²⋊C₄	—	2	8	no	—
503	C₂²⋊C₄	—	2	8	no	—
504	C₂²⋊C₄	—	4	4	no	—
505	C₂²⋊C₄	—	4	4	no	—
506	C₂²⋊C₄	—	4	4	no	—
507	C₂²⋊C₄	—	4	4	no	—
508	C₂²⋊C₄	—	4	4	no	—
509	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
510	C₂²⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
511	C₂²⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
512	C₄²	—	2	2	no	—
513	C₄²	—	2	2	no	—
514	C₄²	—	2	2	no	—
515	C₄²	—	2	2	no	—
516	C₄²	—	2	4	no	—
517	C₄²	—	2	8	no	—
518	C₄²	—	2	4	no	—
519	C₄²	—	4	4	no	—
520	C₄²	—	4	4	no	—
521	C₄²	—	4	2	no	—
522	C₄²	—	4	2	no	—
523	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
524	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
525	C₄²	#41×C₆₇	1	4	yes	—
526	C₄²	#41×C₆₇	1	4	yes	—
527	C₂³	—	2	1	no	—
528	C₂³	—	2	4	no	—
529	C₂³	—	2	4	no	—
530	C₂³	—	2	1	no	—
531	C₂³	—	2	4	no	—
532	C₂³	—	2	2	no	—
533	C₂³	—	2	4	no	—
534	C₂³	—	2	4	no	—
535	C₂³	—	2	2	no	—
536	C₂³	—	2	4	no	—
537	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
538	C₂³	C₂³×C₄×C₆₇	1	8	yes	—
539	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
540	D₄	—	2	16	no	—
541	D₄	—	4	16	no	—
542	D₄	—	4	16	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	4	no	—
554	C₂×C₄	—	2	8	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	4	no	—
559	C₂×C₄	—	2	4	no	—
560	C₂×C₄	—	2	4	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	4	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	4	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	4	no	—
568	C₂×C₄	—	4	16	no	—
569	C₂×C₄	—	4	4	no	—
570	C₂×C₄	—	4	16	no	—
571	C₂×C₄	—	4	4	no	—
572	C₂×C₄	—	4	16	no	—
573	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
574	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
575	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
576	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
577	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
578	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
579	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
580	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
581	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
582	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
583	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
584	C₂²	—	2	2	no	—
585	C₂²	—	2	8	no	—
586	C₂²	—	2	8	no	—
587	C₂²	—	2	4	no	—
588	C₂²	—	2	8	no	—
589	C₂²	—	4	4	no	—
590	C₂²	—	4	1	no	—
591	C₂²	—	4	1	no	—
592	C₂²	—	4	4	no	—
593	C₂²	—	4	4	no	—
594	C₂²	—	4	2	no	—
595	C₂²	—	4	4	no	—
596	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
597	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
598	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
599	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
600	C₂²	#585×C₆₇	1	2	yes	—
601	C₄	—	2	2	no	—
602	C₄	—	2	4	no	—
603	C₄	—	2	8	no	—
604	C₄	—	2	4	no	—
605	C₄	—	2	2	no	—
606	C₄	—	2	4	no	—
607	C₄	—	4	1	no	—
608	C₄	—	4	4	no	—
609	C₄	—	4	1	no	—
610	C₄	—	4	4	no	—
611	C₄	—	4	2	no	—
612	C₄	C₄×#41×C₆₇	1	4	yes	—
613	C₄	#540×C₆₇	1	4	yes	direct product factor
614	C₂	—	4	2	no	—
615	C₂	—	4	8	no	—
616	C₂	C₂×C₄×#41×C₆₇	1	1	yes	characteristic
617	C₂	C₄×H(5,𝔽₂)×C₆₇	1	2	yes	—
618	C₂	C₂×#540×C₆₇	1	1	yes	characteristic
619	C₂	C₆₇×#3455	1	1	yes	characteristic
620	C₂	C₆₇×#3460	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂⁵
prime radical	C₂×#540×C₆₇	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₆₇×#2335	C₂	semidirect
3. C₆₇×#2333	C₂	semidirect
4. C₆₇×#2330	C₂	semidirect
5. C₆₇×#2328	C₂	semidirect
6. C₆₇×#2324	C₂	semidirect
7. C₆₇×#2319	C₂	semidirect
8. C₆₇×#2314	C₂	semidirect
9. C₄×#127×C₆₇	C₂	semidirect
10. C₄×#125×C₆₇	C₂	semidirect
11. C₄×#123×C₆₇	C₂	semidirect
12. C₄×#121×C₆₇	C₂	semidirect
13. C₄×#121×C₆₇	C₂	semidirect
14. C₄²×D₄×C₆₇	C₂	semidirect
15. C₄×#117×C₆₇	C₂	semidirect
16. #540×C₆₇	C₄	direct (N × H)
81. #127×C₆₇	C₂×C₄	semidirect
82. #125×C₆₇	C₂×C₄	semidirect
83. #123×C₆₇	C₂×C₄	semidirect
84. #121×C₆₇	C₂×C₄	semidirect
85. #121×C₆₇	C₂×C₄	semidirect
90. C₄×D₄×C₆₇	C₂×C₄	semidirect
92. #117×C₆₇	C₂×C₄	semidirect
302. C₄×C₆₇	#540	direct (N × H)
303. C₄×#540	C₆₇	direct (N × H)
312. #2335	C₂×C₆₇	semidirect
313. #2333	C₂×C₆₇	semidirect
314. #2330	C₂×C₆₇	semidirect
315. #2328	C₂×C₆₇	semidirect
316. #2324	C₂×C₆₇	semidirect
317. #2319	C₂×C₆₇	semidirect
318. #2314	C₂×C₆₇	semidirect
319. C₄×#127	C₂×C₆₇	semidirect
320. C₄×#125	C₂×C₆₇	semidirect
321. C₄×#123	C₂×C₆₇	semidirect
322. C₄×#121	C₂×C₆₇	semidirect
323. C₄×#121	C₂×C₆₇	semidirect
324. C₄²×D₄	C₂×C₆₇	semidirect
325. C₄×#117	C₂×C₆₇	semidirect
326. C₆₇	C₄×#540	direct (N × H)
327. #540	C₄×C₆₇	direct (N × H)
392. #127	C₂×C₄×C₆₇	semidirect
393. #125	C₂×C₄×C₆₇	semidirect
394. #123	C₂×C₄×C₆₇	semidirect
395. #121	C₂×C₄×C₆₇	semidirect
396. #121	C₂×C₄×C₆₇	semidirect
401. C₄×D₄	C₂×C₄×C₆₇	semidirect
403. #117	C₂×C₄×C₆₇	semidirect
613. C₄	#540×C₆₇	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#540×C₆₇	2
C₆₇×#2335	2
C₆₇×#2333	2
C₆₇×#2330	2
C₆₇×#2328	2
C₆₇×#2324	2
C₆₇×#2319	2
C₆₇×#2314	2
C₄×#127×C₆₇	2
C₄×#125×C₆₇	2
C₄×#123×C₆₇	2
C₄×#121×C₆₇	2
C₄²×D₄×C₆₇	2
C₄×#117×C₆₇	2
C₄×#540	67

Subgroups — C4×#540×C67 #30713344

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#540×C₆₇ #30713344