Subgroups — C₂×C₆₇×#2353 #30713367

Order 17152.

3374 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2353	C₂	1	16	yes	maximal, direct product factor
3	C₂×#388×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₂×#387×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#386×C₆₇	C₂	1	2	yes	maximal
6	C₂×#383×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#381×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#380×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×C₂²⋊C₄×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#376×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#95×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
15	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₆₇	C₄	1	16	yes	complemented
18	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	—
19	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
22	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
24	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
26	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
27	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
28	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
29	#388×C₆₇	C₂²	1	16	yes	complemented
30	#387×C₆₇	C₂²	1	16	yes	complemented
31	#386×C₆₇	C₂²	1	16	yes	—
32	#383×C₆₇	C₂²	1	8	yes	complemented
33	#381×C₆₇	C₂²	1	8	yes	complemented
34	#380×C₆₇	C₂²	1	8	yes	complemented
35	C₄×C₂²⋊C₄×C₆₇	C₂²	1	16	yes	complemented
36	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	complemented
37	#376×C₆₇	C₂²	1	8	yes	complemented
38	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×#95×C₆₇	C₂²	1	2	yes	—
41	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
42	C₂×#95×C₆₇	C₂²	1	2	yes	—
43	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
44	C₂×#95×C₆₇	C₂²	1	2	yes	—
45	C₂×#95×C₆₇	C₂²	1	2	yes	—
46	C₂×#95×C₆₇	C₂²	1	2	yes	—
47	C₂×#95×C₆₇	C₂²	1	8	yes	complemented
48	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
49	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
50	C₂²×D₄×C₆₇	—	2	4	no	—
51	C₂²×D₄×C₆₇	—	2	4	no	—
52	C₂²×D₄×C₆₇	C₂×C₄	1	8	yes	—
53	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
54	C₂³×C₄×C₆₇	—	2	8	no	—
55	C₂³×C₄×C₆₇	—	2	8	no	—
56	C₂³×C₄×C₆₇	—	4	4	no	—
57	C₂³×C₄×C₆₇	—	4	4	no	—
58	C₂³×C₄×C₆₇	—	4	4	no	—
59	C₂³×C₄×C₆₇	C₂×C₄	1	8	yes	—
60	C₂³×C₄×C₆₇	C₂×C₄	1	8	yes	complemented
61	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
62	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
63	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
64	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
65	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
71	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
72	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
73	C₄×D₄×C₆₇	C₂×C₄	1	64	yes	complemented
74	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
75	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	complemented
76	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
77	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
78	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
79	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
80	C₂×C₂²⋊C₄×C₆₇	—	4	4	no	—
81	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
82	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	8	yes	complemented
83	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
84	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
85	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
86	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
87	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
88	C₂×C₄²×C₆₇	—	2	8	no	—
89	C₂×C₄²×C₆₇	—	4	4	no	—
90	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	complemented
91	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
92	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
93	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
94	#95×C₆₇	C₂³	1	4	yes	—
95	#95×C₆₇	C₂³	1	4	yes	—
96	#95×C₆₇	C₂³	1	8	yes	—
97	#95×C₆₇	C₂³	1	4	yes	—
98	#95×C₆₇	C₂³	1	8	yes	—
99	#95×C₆₇	C₂³	1	4	yes	—
100	#95×C₆₇	C₂³	1	8	yes	—
101	#95×C₆₇	C₂³	1	8	yes	—
102	#95×C₆₇	C₂³	1	8	yes	—
103	C₂⁴×C₆₇	—	2	4	no	—
104	C₂⁴×C₆₇	—	2	4	no	—
105	C₂⁴×C₆₇	—	2	4	no	—
106	C₂⁴×C₆₇	—	2	2	no	—
107	C₂⁴×C₆₇	—	2	2	no	—
108	C₂⁴×C₆₇	—	2	4	no	—
109	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
110	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
111	C₂⁴×C₆₇	#41	1	4	yes	—
112	C₂⁴×C₆₇	#41	1	4	yes	—
113	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
114	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
115	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
116	C₂×D₄×C₆₇	—	2	32	no	—
117	C₂×D₄×C₆₇	—	2	8	no	—
118	C₂×D₄×C₆₇	—	2	8	no	—
119	C₂×D₄×C₆₇	C₂²×C₄	1	16	yes	—
120	C₂²×C₄×C₆₇	—	2	4	no	—
121	C₂²×C₄×C₆₇	—	2	4	no	—
122	C₂²×C₄×C₆₇	—	2	2	no	—
123	C₂²×C₄×C₆₇	—	2	2	no	—
124	C₂²×C₄×C₆₇	—	2	4	no	—
125	C₂²×C₄×C₆₇	—	2	4	no	—
126	C₂²×C₄×C₆₇	—	2	4	no	—
127	C₂²×C₄×C₆₇	—	2	4	no	—
128	C₂²×C₄×C₆₇	—	2	4	no	—
129	C₂²×C₄×C₆₇	—	2	4	no	—
130	C₂²×C₄×C₆₇	—	2	4	no	—
131	C₂²×C₄×C₆₇	—	2	4	no	—
132	C₂²×C₄×C₆₇	—	2	16	no	—
133	C₂²×C₄×C₆₇	—	2	4	no	—
134	C₂²×C₄×C₆₇	—	2	4	no	—
135	C₂²×C₄×C₆₇	—	2	4	no	—
136	C₂²×C₄×C₆₇	—	2	4	no	—
137	C₂²×C₄×C₆₇	—	2	4	no	—
138	C₂²×C₄×C₆₇	—	2	16	no	—
139	C₂²×C₄×C₆₇	—	4	16	no	—
140	C₂²×C₄×C₆₇	—	4	16	no	—
141	C₂²×C₄×C₆₇	—	4	8	no	—
142	C₂²×C₄×C₆₇	—	4	8	no	—
143	C₂²×C₄×C₆₇	—	4	8	no	—
144	C₂²×C₄×C₆₇	—	4	32	no	—
145	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
146	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
147	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
148	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
149	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
150	C₂²×C₄×C₆₇	C₂²×C₄	1	16	yes	—
151	C₂²×C₄×C₆₇	C₂²×C₄	1	16	yes	complemented
152	C₂²×C₄×C₆₇	#41	1	8	yes	—
153	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
154	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
155	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
156	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
157	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
158	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
159	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
160	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
162	C₄⋊C₄×C₆₇	—	4	8	no	—
163	C₄⋊C₄×C₆₇	C₂²×C₄	1	16	yes	complemented
164	C₂²⋊C₄×C₆₇	—	2	16	no	—
165	C₂²⋊C₄×C₆₇	—	2	16	no	—
166	C₂²⋊C₄×C₆₇	—	4	8	no	—
167	C₂²⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
168	C₂²⋊C₄×C₆₇	C₂²×C₄	1	16	yes	complemented
169	C₄²×C₆₇	—	2	16	no	—
170	C₄²×C₆₇	—	4	8	no	—
171	C₄²×C₆₇	C₂²×C₄	1	16	yes	complemented
172	C₂³×C₆₇	—	2	8	no	—
173	C₂³×C₆₇	—	2	8	no	—
174	C₂³×C₆₇	—	2	16	no	—
175	C₂³×C₆₇	—	2	16	no	—
176	C₂³×C₆₇	—	2	4	no	—
177	C₂³×C₆₇	—	2	4	no	—
178	C₂³×C₆₇	—	2	8	no	—
179	C₂³×C₆₇	—	2	8	no	—
180	C₂³×C₆₇	—	2	8	no	—
181	C₂³×C₆₇	—	2	4	no	—
182	C₂³×C₆₇	—	2	2	no	—
183	C₂³×C₆₇	—	2	2	no	—
184	C₂³×C₆₇	—	2	4	no	—
185	C₂³×C₆₇	—	4	8	no	—
186	C₂³×C₆₇	—	4	8	no	—
187	C₂³×C₆₇	#117	1	16	yes	—
188	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
189	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
190	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
191	C₂³×C₆₇	C₂×#41	1	4	yes	—
192	C₂³×C₆₇	C₂×#41	1	4	yes	—
193	C₂³×C₆₇	C₂×#41	1	2	yes	—
194	C₂³×C₆₇	C₂×#41	1	2	yes	—
195	C₂³×C₆₇	C₂×#41	1	4	yes	—
196	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
197	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
198	D₄×C₆₇	—	2	32	no	—
199	C₂×C₄×C₆₇	—	2	8	no	—
200	C₂×C₄×C₆₇	—	2	8	no	—
201	C₂×C₄×C₆₇	—	2	8	no	—
202	C₂×C₄×C₆₇	—	2	8	no	—
203	C₂×C₄×C₆₇	—	2	8	no	—
204	C₂×C₄×C₆₇	—	2	16	no	—
205	C₂×C₄×C₆₇	—	2	4	no	—
206	C₂×C₄×C₆₇	—	2	4	no	—
207	C₂×C₄×C₆₇	—	2	2	no	—
208	C₂×C₄×C₆₇	—	2	2	no	—
209	C₂×C₄×C₆₇	—	2	4	no	—
210	C₂×C₄×C₆₇	—	2	4	no	—
211	C₂×C₄×C₆₇	—	2	4	no	—
212	C₂×C₄×C₆₇	—	2	4	no	—
213	C₂×C₄×C₆₇	—	2	4	no	—
214	C₂×C₄×C₆₇	—	2	4	no	—
215	C₂×C₄×C₆₇	—	2	4	no	—
216	C₂×C₄×C₆₇	—	2	4	no	—
217	C₂×C₄×C₆₇	—	2	4	no	—
218	C₂×C₄×C₆₇	—	2	4	no	—
219	C₂×C₄×C₆₇	—	2	4	no	—
220	C₂×C₄×C₆₇	—	2	4	no	—
221	C₂×C₄×C₆₇	—	2	4	no	—
222	C₂×C₄×C₆₇	—	4	16	no	—
223	C₂×C₄×C₆₇	—	4	16	no	—
224	C₂×C₄×C₆₇	—	4	32	no	—
225	C₂×C₄×C₆₇	—	4	8	no	—
226	C₂×C₄×C₆₇	—	4	8	no	—
227	C₂×C₄×C₆₇	—	4	8	no	—
228	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
229	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
230	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
231	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
232	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
233	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
234	C₂²×C₆₇	—	2	32	no	—
235	C₂²×C₆₇	—	2	4	no	—
236	C₂²×C₆₇	—	2	4	no	—
237	C₂²×C₆₇	—	2	8	no	—
238	C₂²×C₆₇	—	2	8	no	—
239	C₂²×C₆₇	—	2	4	no	—
240	C₂²×C₆₇	—	2	4	no	—
241	C₂²×C₆₇	—	2	4	no	—
242	C₂²×C₆₇	—	4	16	no	—
243	C₂²×C₆₇	—	4	4	no	—
244	C₂²×C₆₇	—	4	4	no	—
245	C₂²×C₆₇	C₂×#117	1	8	yes	—
246	C₂²×C₆₇	C₂×#117	1	4	yes	—
247	C₂²×C₆₇	C₄×#41	1	4	yes	—
248	C₂²×C₆₇	#518	1	4	yes	—
249	C₂²×C₆₇	#551	1	8	yes	—
250	C₂²×C₆₇	#553	1	8	yes	—
251	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
252	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
253	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
254	C₂²×C₆₇	C₂²×#41	1	2	yes	—
255	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
256	C₄×C₆₇	—	2	4	no	—
257	C₄×C₆₇	—	2	4	no	—
258	C₄×C₆₇	—	2	4	no	—
259	C₄×C₆₇	—	2	4	no	—
260	C₄×C₆₇	—	2	4	no	—
261	C₄×C₆₇	—	2	8	no	—
262	C₄×C₆₇	—	4	4	no	—
263	C₄×C₆₇	—	4	4	no	—
264	C₄×C₆₇	—	4	4	no	—
265	C₂×#2353	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
266	C₂×C₆₇	—	2	8	no	—
267	C₂×C₆₇	—	4	4	no	—
268	C₂×C₆₇	#2353	1	8	yes	direct product factor
269	C₂×C₆₇	C₂²×#117	1	1	yes	characteristic
270	C₂×C₆₇	C₂×C₄×#41	1	1	yes	characteristic
271	C₂×C₆₇	C₂×#518	1	1	yes	characteristic
272	C₂×C₆₇	C₂×#551	1	2	yes	—
273	C₂×C₆₇	C₂×#553	1	2	yes	—
274	C₂²×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic
275	#2353	C₂×C₆₇	1	16	yes	direct product factor
276	C₂×#388	C₂×C₆₇	1	2	yes	complemented
277	C₂×#387	C₂×C₆₇	1	2	yes	complemented
278	C₂×#386	C₂×C₆₇	1	2	yes	—
279	C₂×#383	C₂×C₆₇	1	1	yes	characteristic, complemented
280	C₂×#381	C₂×C₆₇	1	1	yes	characteristic, complemented
281	C₂×#380	C₂×C₆₇	1	1	yes	characteristic, complemented
282	C₂×C₄×C₂²⋊C₄	C₂×C₆₇	1	2	yes	complemented
283	C₂×C₄×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
284	C₂×#376	C₂×C₆₇	1	1	yes	characteristic, complemented
285	C₂²×#95	C₂×C₆₇	1	1	yes	characteristic, complemented
286	C₆₇	C₂×#2353	1	1	yes	characteristic, direct product factor, Sylow
287	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
288	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
289	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
290	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
291	C₂×C₄×D₄	C₄×C₆₇	1	16	yes	complemented
292	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	—
293	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
294	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
295	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
296	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
297	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
298	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
299	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
300	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
301	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
302	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
303	#388	C₂²×C₆₇	1	16	yes	complemented
304	#387	C₂²×C₆₇	1	16	yes	complemented
305	#386	C₂²×C₆₇	1	16	yes	—
306	#383	C₂²×C₆₇	1	8	yes	complemented
307	#381	C₂²×C₆₇	1	8	yes	complemented
308	#380	C₂²×C₆₇	1	8	yes	complemented
309	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	complemented
310	C₄×C₂²⋊C₄	C₂²×C₆₇	1	16	yes	complemented
311	#376	C₂²×C₆₇	1	8	yes	complemented
312	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
313	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
314	C₂×#95	C₂²×C₆₇	1	2	yes	—
315	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
316	C₂×#95	C₂²×C₆₇	1	2	yes	—
317	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
318	C₂×#95	C₂²×C₆₇	1	2	yes	—
319	C₂×#95	C₂²×C₆₇	1	2	yes	—
320	C₂×#95	C₂²×C₆₇	1	2	yes	—
321	C₂×#95	C₂²×C₆₇	1	8	yes	complemented
322	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
323	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
324	C₂²×D₄	—	2	4	no	—
325	C₂²×D₄	—	2	4	no	—
326	C₂²×D₄	C₂×C₄×C₆₇	1	8	yes	—
327	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
328	C₂³×C₄	—	2	8	no	—
329	C₂³×C₄	—	2	8	no	—
330	C₂³×C₄	—	4	4	no	—
331	C₂³×C₄	—	4	4	no	—
332	C₂³×C₄	—	4	4	no	—
333	C₂³×C₄	C₂×C₄×C₆₇	1	8	yes	—
334	C₂³×C₄	C₂×C₄×C₆₇	1	8	yes	complemented
335	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
336	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
337	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
338	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
339	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
340	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
341	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
342	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
343	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
344	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
345	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
346	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
347	C₄×D₄	C₂×C₄×C₆₇	1	64	yes	complemented
348	C₂×C₄⋊C₄	—	4	4	no	—
349	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	8	yes	complemented
350	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
351	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
352	C₂×C₂²⋊C₄	—	2	8	no	—
353	C₂×C₂²⋊C₄	—	2	8	no	—
354	C₂×C₂²⋊C₄	—	4	4	no	—
355	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	8	yes	—
356	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	8	yes	complemented
357	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
358	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
359	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
360	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
361	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
362	C₂×C₄²	—	2	8	no	—
363	C₂×C₄²	—	4	4	no	—
364	C₂×C₄²	C₂×C₄×C₆₇	1	8	yes	complemented
365	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
366	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
367	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
368	#95	C₂³×C₆₇	1	4	yes	—
369	#95	C₂³×C₆₇	1	4	yes	—
370	#95	C₂³×C₆₇	1	8	yes	—
371	#95	C₂³×C₆₇	1	4	yes	—
372	#95	C₂³×C₆₇	1	8	yes	—
373	#95	C₂³×C₆₇	1	4	yes	—
374	#95	C₂³×C₆₇	1	8	yes	—
375	#95	C₂³×C₆₇	1	8	yes	—
376	#95	C₂³×C₆₇	1	8	yes	—
377	C₂⁴	—	2	4	no	—
378	C₂⁴	—	2	4	no	—
379	C₂⁴	—	2	4	no	—
380	C₂⁴	—	2	2	no	—
381	C₂⁴	—	2	2	no	—
382	C₂⁴	—	2	4	no	—
383	C₂⁴	C₂²×C₄×C₆₇	1	4	yes	—
384	C₂⁴	C₂²×C₄×C₆₇	1	4	yes	—
385	C₂⁴	#41×C₆₇	1	4	yes	—
386	C₂⁴	#41×C₆₇	1	4	yes	—
387	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
388	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
389	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
390	C₂×D₄	—	2	32	no	—
391	C₂×D₄	—	2	8	no	—
392	C₂×D₄	—	2	8	no	—
393	C₂×D₄	C₂²×C₄×C₆₇	1	16	yes	—
394	C₂²×C₄	—	2	4	no	—
395	C₂²×C₄	—	2	4	no	—
396	C₂²×C₄	—	2	2	no	—
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	—	2	4	no	—
399	C₂²×C₄	—	2	4	no	—
400	C₂²×C₄	—	2	4	no	—
401	C₂²×C₄	—	2	4	no	—
402	C₂²×C₄	—	2	4	no	—
403	C₂²×C₄	—	2	4	no	—
404	C₂²×C₄	—	2	4	no	—
405	C₂²×C₄	—	2	4	no	—
406	C₂²×C₄	—	2	16	no	—
407	C₂²×C₄	—	2	4	no	—
408	C₂²×C₄	—	2	4	no	—
409	C₂²×C₄	—	2	4	no	—
410	C₂²×C₄	—	2	4	no	—
411	C₂²×C₄	—	2	4	no	—
412	C₂²×C₄	—	2	16	no	—
413	C₂²×C₄	—	4	16	no	—
414	C₂²×C₄	—	4	16	no	—
415	C₂²×C₄	—	4	8	no	—
416	C₂²×C₄	—	4	8	no	—
417	C₂²×C₄	—	4	8	no	—
418	C₂²×C₄	—	4	32	no	—
419	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
420	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
421	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
422	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
423	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
424	C₂²×C₄	C₂²×C₄×C₆₇	1	16	yes	—
425	C₂²×C₄	C₂²×C₄×C₆₇	1	16	yes	complemented
426	C₂²×C₄	#41×C₆₇	1	8	yes	—
427	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
428	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
429	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
430	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
431	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
432	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
433	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
434	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
435	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
436	C₄⋊C₄	—	4	8	no	—
437	C₄⋊C₄	C₂²×C₄×C₆₇	1	16	yes	complemented
438	C₂²⋊C₄	—	2	16	no	—
439	C₂²⋊C₄	—	2	16	no	—
440	C₂²⋊C₄	—	4	8	no	—
441	C₂²⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
442	C₂²⋊C₄	C₂²×C₄×C₆₇	1	16	yes	complemented
443	C₄²	—	2	16	no	—
444	C₄²	—	4	8	no	—
445	C₄²	C₂²×C₄×C₆₇	1	16	yes	complemented
446	C₂³	—	2	8	no	—
447	C₂³	—	2	8	no	—
448	C₂³	—	2	16	no	—
449	C₂³	—	2	16	no	—
450	C₂³	—	2	4	no	—
451	C₂³	—	2	4	no	—
452	C₂³	—	2	8	no	—
453	C₂³	—	2	8	no	—
454	C₂³	—	2	8	no	—
455	C₂³	—	2	4	no	—
456	C₂³	—	2	2	no	—
457	C₂³	—	2	2	no	—
458	C₂³	—	2	4	no	—
459	C₂³	—	4	8	no	—
460	C₂³	—	4	8	no	—
461	C₂³	#117×C₆₇	1	16	yes	—
462	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
463	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
464	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
465	C₂³	C₂×#41×C₆₇	1	4	yes	—
466	C₂³	C₂×#41×C₆₇	1	4	yes	—
467	C₂³	C₂×#41×C₆₇	1	2	yes	—
468	C₂³	C₂×#41×C₆₇	1	2	yes	—
469	C₂³	C₂×#41×C₆₇	1	4	yes	—
470	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
471	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
472	D₄	—	2	32	no	—
473	C₂×C₄	—	2	8	no	—
474	C₂×C₄	—	2	8	no	—
475	C₂×C₄	—	2	8	no	—
476	C₂×C₄	—	2	8	no	—
477	C₂×C₄	—	2	8	no	—
478	C₂×C₄	—	2	16	no	—
479	C₂×C₄	—	2	4	no	—
480	C₂×C₄	—	2	4	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	4	no	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	2	4	no	—
486	C₂×C₄	—	2	4	no	—
487	C₂×C₄	—	2	4	no	—
488	C₂×C₄	—	2	4	no	—
489	C₂×C₄	—	2	4	no	—
490	C₂×C₄	—	2	4	no	—
491	C₂×C₄	—	2	4	no	—
492	C₂×C₄	—	2	4	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	4	16	no	—
497	C₂×C₄	—	4	16	no	—
498	C₂×C₄	—	4	32	no	—
499	C₂×C₄	—	4	8	no	—
500	C₂×C₄	—	4	8	no	—
501	C₂×C₄	—	4	8	no	—
502	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
503	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
504	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
505	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
506	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
507	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
508	C₂²	—	2	32	no	—
509	C₂²	—	2	4	no	—
510	C₂²	—	2	4	no	—
511	C₂²	—	2	8	no	—
512	C₂²	—	2	8	no	—
513	C₂²	—	2	4	no	—
514	C₂²	—	2	4	no	—
515	C₂²	—	2	4	no	—
516	C₂²	—	4	16	no	—
517	C₂²	—	4	4	no	—
518	C₂²	—	4	4	no	—
519	C₂²	C₂×#117×C₆₇	1	8	yes	—
520	C₂²	C₂×#117×C₆₇	1	4	yes	—
521	C₂²	C₄×#41×C₆₇	1	4	yes	—
522	C₂²	#518×C₆₇	1	4	yes	—
523	C₂²	#551×C₆₇	1	8	yes	—
524	C₂²	#553×C₆₇	1	8	yes	—
525	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
526	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
527	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
528	C₂²	C₂²×#41×C₆₇	1	2	yes	—
529	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
530	C₄	—	2	4	no	—
531	C₄	—	2	4	no	—
532	C₄	—	2	4	no	—
533	C₄	—	2	4	no	—
534	C₄	—	2	4	no	—
535	C₄	—	2	8	no	—
536	C₄	—	4	4	no	—
537	C₄	—	4	4	no	—
538	C₄	—	4	4	no	—
539	C₂	—	2	8	no	—
540	C₂	—	4	4	no	—
541	C₂	C₆₇×#2353	1	8	yes	direct product factor
542	C₂	C₂²×#117×C₆₇	1	1	yes	characteristic
543	C₂	C₂×C₄×#41×C₆₇	1	1	yes	characteristic
544	C₂	C₂×#518×C₆₇	1	1	yes	characteristic
545	C₂	C₂×#551×C₆₇	1	2	yes	—
546	C₂	C₂×#553×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂³×D₄×C₆₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2353	C₂	direct (N × H)
3. C₂×#388×C₆₇	C₂	semidirect
4. C₂×#387×C₆₇	C₂	semidirect
6. C₂×#383×C₆₇	C₂	semidirect
7. C₂×#381×C₆₇	C₂	semidirect
8. C₂×#380×C₆₇	C₂	semidirect
9. C₂×C₄×C₂²⋊C₄×C₆₇	C₂	semidirect
10. C₂×C₄×C₂²⋊C₄×C₆₇	C₂	semidirect
11. C₂×#376×C₆₇	C₂	semidirect
12. C₂²×#95×C₆₇	C₂	semidirect
17. C₂×C₄×D₄×C₆₇	C₄	semidirect
29. #388×C₆₇	C₂²	semidirect
30. #387×C₆₇	C₂²	semidirect
32. #383×C₆₇	C₂²	semidirect
33. #381×C₆₇	C₂²	semidirect
34. #380×C₆₇	C₂²	semidirect
35. C₄×C₂²⋊C₄×C₆₇	C₂²	semidirect
36. C₄×C₂²⋊C₄×C₆₇	C₂²	semidirect
37. #376×C₆₇	C₂²	semidirect
47. C₂×#95×C₆₇	C₂²	semidirect
60. C₂³×C₄×C₆₇	C₂×C₄	semidirect
73. C₄×D₄×C₆₇	C₂×C₄	semidirect
75. C₂×C₄⋊C₄×C₆₇	C₂×C₄	semidirect
82. C₂×C₂²⋊C₄×C₆₇	C₂×C₄	semidirect
90. C₂×C₄²×C₆₇	C₂×C₄	semidirect
151. C₂²×C₄×C₆₇	C₂²×C₄	semidirect
163. C₄⋊C₄×C₆₇	C₂²×C₄	semidirect
168. C₂²⋊C₄×C₆₇	C₂²×C₄	semidirect
171. C₄²×C₆₇	C₂²×C₄	semidirect
265. C₂×#2353	C₆₇	direct (N × H)
268. C₂×C₆₇	#2353	direct (N × H)
275. #2353	C₂×C₆₇	direct (N × H)
276. C₂×#388	C₂×C₆₇	semidirect
277. C₂×#387	C₂×C₆₇	semidirect
279. C₂×#383	C₂×C₆₇	semidirect
280. C₂×#381	C₂×C₆₇	semidirect
281. C₂×#380	C₂×C₆₇	semidirect
282. C₂×C₄×C₂²⋊C₄	C₂×C₆₇	semidirect
283. C₂×C₄×C₂²⋊C₄	C₂×C₆₇	semidirect
284. C₂×#376	C₂×C₆₇	semidirect
285. C₂²×#95	C₂×C₆₇	semidirect
286. C₆₇	C₂×#2353	direct (N × H)
291. C₂×C₄×D₄	C₄×C₆₇	semidirect
303. #388	C₂²×C₆₇	semidirect
304. #387	C₂²×C₆₇	semidirect
306. #383	C₂²×C₆₇	semidirect
307. #381	C₂²×C₆₇	semidirect
308. #380	C₂²×C₆₇	semidirect
309. C₄×C₂²⋊C₄	C₂²×C₆₇	semidirect
310. C₄×C₂²⋊C₄	C₂²×C₆₇	semidirect
311. #376	C₂²×C₆₇	semidirect
321. C₂×#95	C₂²×C₆₇	semidirect
334. C₂³×C₄	C₂×C₄×C₆₇	semidirect
347. C₄×D₄	C₂×C₄×C₆₇	semidirect
349. C₂×C₄⋊C₄	C₂×C₄×C₆₇	semidirect
356. C₂×C₂²⋊C₄	C₂×C₄×C₆₇	semidirect
364. C₂×C₄²	C₂×C₄×C₆₇	semidirect
425. C₂²×C₄	C₂²×C₄×C₆₇	semidirect
437. C₄⋊C₄	C₂²×C₄×C₆₇	semidirect
442. C₂²⋊C₄	C₂²×C₄×C₆₇	semidirect
445. C₄²	C₂²×C₄×C₆₇	semidirect
541. C₂	C₆₇×#2353	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₄×C₆₇	2
C₆₇×#2353	2
C₂×#388×C₆₇	2
C₂×#387×C₆₇	2
C₂×#386×C₆₇	2
C₂×#383×C₆₇	2
C₂×#381×C₆₇	2
C₂×#380×C₆₇	2
C₂×C₄×C₂²⋊C₄×C₆₇	2
C₂×#376×C₆₇	2
C₂²×#95×C₆₇	2
C₂×#2353	67

Subgroups — C2×C67×#2353 #30713367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2353 #30713367