Subgroups — C₂×C₆₇×#2813 #30715901

Order 17152.

3322 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#123×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#121×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2813	C₂	1	16	yes	maximal, direct product factor
4	C₂×#400×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#399×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#396×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#394×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#394×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂×#394×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#392×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#392×C₆₇	C₂	1	2	yes	maximal, complemented
12	C₂×#380×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
15	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
17	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#123×C₆₇	—	2	8	no	—
19	C₂×#123×C₆₇	C₂²	1	8	yes	complemented
20	C₂×#121×C₆₇	—	2	8	no	—
21	C₂×#121×C₆₇	C₂²	1	8	yes	complemented
22	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
26	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
27	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
28	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
29	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
30	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
31	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
32	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
33	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
34	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
35	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
36	#400×C₆₇	C₂²	1	8	yes	complemented
37	#399×C₆₇	C₂²	1	8	yes	complemented
38	#396×C₆₇	C₂²	1	16	yes	complemented
39	#394×C₆₇	C₂²	1	16	yes	complemented
40	#394×C₆₇	C₂²	1	16	yes	complemented
41	#394×C₆₇	C₂²	1	8	yes	complemented
42	#392×C₆₇	C₂²	1	8	yes	complemented
43	#392×C₆₇	C₂²	1	16	yes	complemented
44	#380×C₆₇	C₂²	1	8	yes	complemented
45	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
46	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
47	C₂×#95×C₆₇	C₂²	1	2	yes	—
48	C₂×#95×C₆₇	C₂²	1	2	yes	—
49	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
50	C₂⁵×C₆₇	C₂³	1	2	yes	—
51	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
52	C₂²×D₄×C₆₇	—	2	8	no	—
53	C₂²×D₄×C₆₇	—	2	4	no	—
54	C₂²×D₄×C₆₇	—	2	4	no	—
55	C₂²×D₄×C₆₇	—	4	4	no	—
56	C₂²×D₄×C₆₇	—	4	4	no	—
57	C₂²×D₄×C₆₇	—	4	4	no	—
58	C₂²×D₄×C₆₇	—	4	4	no	—
59	C₂²×D₄×C₆₇	—	4	4	no	—
60	C₂²×D₄×C₆₇	—	4	4	no	—
61	C₂²×D₄×C₆₇	—	4	4	no	—
62	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
63	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
64	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
65	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
66	C₂³×C₄×C₆₇	—	4	4	no	—
67	C₂³×C₄×C₆₇	D₄	1	8	yes	complemented
68	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
70	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
72	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
74	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
76	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
77	#123×C₆₇	—	2	32	no	—
78	#121×C₆₇	—	2	32	no	—
79	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
80	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
81	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
82	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
83	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
84	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
85	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
86	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
87	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
88	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
89	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
90	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
91	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
92	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
93	C₂×C₂²⋊C₄×C₆₇	—	4	4	no	—
94	C₂×C₂²⋊C₄×C₆₇	—	4	4	no	—
95	C₂×C₂²⋊C₄×C₆₇	—	4	4	no	—
96	C₂×C₂²⋊C₄×C₆₇	—	4	4	no	—
97	C₂×C₂²⋊C₄×C₆₇	—	4	4	no	—
98	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
99	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
100	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
101	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
102	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
103	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
104	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
105	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
106	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
107	#95×C₆₇	C₂³	1	4	yes	—
108	#95×C₆₇	C₂³	1	4	yes	—
109	#95×C₆₇	C₂³	1	8	yes	—
110	#95×C₆₇	C₂³	1	8	yes	—
111	C₂⁴×C₆₇	—	2	2	no	—
112	C₂⁴×C₆₇	—	2	4	no	—
113	C₂⁴×C₆₇	—	2	4	no	—
114	C₂⁴×C₆₇	—	2	2	no	—
115	C₂⁴×C₆₇	—	2	2	no	—
116	C₂⁴×C₆₇	—	2	4	no	—
117	C₂⁴×C₆₇	—	2	4	no	—
118	C₂⁴×C₆₇	—	2	4	no	—
119	C₂⁴×C₆₇	—	2	4	no	—
120	C₂⁴×C₆₇	—	2	4	no	—
121	C₂⁴×C₆₇	—	2	4	no	—
122	C₂⁴×C₆₇	—	2	4	no	—
123	C₂⁴×C₆₇	—	2	4	no	—
124	C₂⁴×C₆₇	—	2	2	no	—
125	C₂⁴×C₆₇	—	2	2	no	—
126	C₂⁴×C₆₇	—	2	4	no	—
127	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
128	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
129	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
130	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
131	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
132	C₂×D₄×C₆₇	—	2	16	no	—
133	C₂×D₄×C₆₇	—	2	8	no	—
134	C₂×D₄×C₆₇	—	2	8	no	—
135	C₂×D₄×C₆₇	—	4	8	no	—
136	C₂×D₄×C₆₇	—	4	8	no	—
137	C₂×D₄×C₆₇	—	4	8	no	—
138	C₂×D₄×C₆₇	—	4	8	no	—
139	C₂×D₄×C₆₇	—	4	8	no	—
140	C₂×D₄×C₆₇	—	4	8	no	—
141	C₂×D₄×C₆₇	—	4	8	no	—
142	C₂×D₄×C₆₇	—	4	16	no	—
143	C₂×D₄×C₆₇	—	4	16	no	—
144	C₂×D₄×C₆₇	—	4	32	no	—
145	C₂×D₄×C₆₇	—	4	32	no	—
146	C₂²×C₄×C₆₇	—	2	4	no	—
147	C₂²×C₄×C₆₇	—	2	4	no	—
148	C₂²×C₄×C₆₇	—	2	2	no	—
149	C₂²×C₄×C₆₇	—	2	2	no	—
150	C₂²×C₄×C₆₇	—	2	4	no	—
151	C₂²×C₄×C₆₇	—	2	2	no	—
152	C₂²×C₄×C₆₇	—	2	2	no	—
153	C₂²×C₄×C₆₇	—	2	4	no	—
154	C₂²×C₄×C₆₇	—	2	4	no	—
155	C₂²×C₄×C₆₇	—	2	4	no	—
156	C₂²×C₄×C₆₇	—	2	4	no	—
157	C₂²×C₄×C₆₇	—	2	4	no	—
158	C₂²×C₄×C₆₇	—	2	4	no	—
159	C₂²×C₄×C₆₇	—	2	4	no	—
160	C₂²×C₄×C₆₇	—	2	4	no	—
161	C₂²×C₄×C₆₇	—	2	4	no	—
162	C₂²×C₄×C₆₇	—	2	4	no	—
163	C₂²×C₄×C₆₇	—	2	2	no	—
164	C₂²×C₄×C₆₇	—	4	16	no	—
165	C₂²×C₄×C₆₇	—	4	8	no	—
166	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
167	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
168	C₂²×C₄×C₆₇	C₂×D₄	1	16	yes	complemented
169	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
170	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
171	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
172	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
173	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
174	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
175	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
176	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
177	C₄⋊C₄×C₆₇	—	2	8	no	—
178	C₄⋊C₄×C₆₇	—	2	8	no	—
179	C₄⋊C₄×C₆₇	—	2	8	no	—
180	C₄⋊C₄×C₆₇	—	4	8	no	—
181	C₄⋊C₄×C₆₇	—	4	8	no	—
182	C₂²⋊C₄×C₆₇	—	2	8	no	—
183	C₂²⋊C₄×C₆₇	—	2	16	no	—
184	C₂²⋊C₄×C₆₇	—	2	8	no	—
185	C₂²⋊C₄×C₆₇	—	2	8	no	—
186	C₂²⋊C₄×C₆₇	—	4	8	no	—
187	C₂²⋊C₄×C₆₇	—	4	8	no	—
188	C₂²⋊C₄×C₆₇	—	4	8	no	—
189	C₂²⋊C₄×C₆₇	—	4	8	no	—
190	C₂²⋊C₄×C₆₇	—	4	8	no	—
191	C₂³×C₆₇	—	2	8	no	—
192	C₂³×C₆₇	—	2	8	no	—
193	C₂³×C₆₇	—	2	8	no	—
194	C₂³×C₆₇	—	2	2	no	—
195	C₂³×C₆₇	—	2	4	no	—
196	C₂³×C₆₇	—	2	4	no	—
197	C₂³×C₆₇	—	2	2	no	—
198	C₂³×C₆₇	—	2	2	no	—
199	C₂³×C₆₇	—	2	4	no	—
200	C₂³×C₆₇	—	2	4	no	—
201	C₂³×C₆₇	—	2	4	no	—
202	C₂³×C₆₇	—	2	4	no	—
203	C₂³×C₆₇	—	2	4	no	—
204	C₂³×C₆₇	—	2	4	no	—
205	C₂³×C₆₇	—	2	4	no	—
206	C₂³×C₆₇	—	2	4	no	—
207	C₂³×C₆₇	—	2	2	no	—
208	C₂³×C₆₇	—	2	2	no	—
209	C₂³×C₆₇	—	2	4	no	—
210	C₂³×C₆₇	—	4	8	no	—
211	C₂³×C₆₇	—	4	8	no	—
212	C₂³×C₆₇	—	4	8	no	—
213	C₂³×C₆₇	—	4	4	no	—
214	C₂³×C₆₇	—	4	8	no	—
215	C₂³×C₆₇	—	4	4	no	—
216	C₂³×C₆₇	—	4	4	no	—
217	C₂³×C₆₇	—	4	8	no	—
218	C₂³×C₆₇	—	4	8	no	—
219	C₂³×C₆₇	—	4	8	no	—
220	C₂³×C₆₇	—	4	8	no	—
221	C₂³×C₆₇	—	4	8	no	—
222	C₂³×C₆₇	—	4	8	no	—
223	C₂³×C₆₇	—	4	8	no	—
224	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
225	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
226	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
227	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
228	C₂³×C₆₇	H(5,𝔽₂)	1	2	yes	—
229	C₂³×C₆₇	2₋¹⁺⁴	1	2	yes	—
230	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
231	D₄×C₆₇	—	4	16	no	—
232	D₄×C₆₇	—	4	16	no	—
233	D₄×C₆₇	—	4	32	no	—
234	D₄×C₆₇	—	4	32	no	—
235	C₂×C₄×C₆₇	—	2	8	no	—
236	C₂×C₄×C₆₇	—	2	8	no	—
237	C₂×C₄×C₆₇	—	2	4	no	—
238	C₂×C₄×C₆₇	—	2	4	no	—
239	C₂×C₄×C₆₇	—	2	2	no	—
240	C₂×C₄×C₆₇	—	2	2	no	—
241	C₂×C₄×C₆₇	—	2	4	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	2	2	no	—
244	C₂×C₄×C₆₇	—	2	4	no	—
245	C₂×C₄×C₆₇	—	2	4	no	—
246	C₂×C₄×C₆₇	—	2	4	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	4	no	—
251	C₂×C₄×C₆₇	—	2	4	no	—
252	C₂×C₄×C₆₇	—	2	4	no	—
253	C₂×C₄×C₆₇	—	2	4	no	—
254	C₂×C₄×C₆₇	—	2	2	no	—
255	C₂×C₄×C₆₇	—	4	16	no	—
256	C₂×C₄×C₆₇	—	4	4	no	—
257	C₂×C₄×C₆₇	—	4	4	no	—
258	C₂×C₄×C₆₇	—	4	8	no	—
259	C₂×C₄×C₆₇	—	4	4	no	—
260	C₂×C₄×C₆₇	—	4	8	no	—
261	C₂×C₄×C₆₇	—	4	8	no	—
262	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
263	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
264	C₂²×C₆₇	—	2	4	no	—
265	C₂²×C₆₇	—	2	4	no	—
266	C₂²×C₆₇	—	2	4	no	—
267	C₂²×C₆₇	—	4	4	no	—
268	C₂²×C₆₇	—	4	4	no	—
269	C₂²×C₆₇	—	4	16	no	—
270	C₂²×C₆₇	—	4	4	no	—
271	C₂²×C₆₇	—	4	2	no	—
272	C₂²×C₆₇	—	4	4	no	—
273	C₂²×C₆₇	—	4	2	no	—
274	C₂²×C₆₇	—	4	2	no	—
275	C₂²×C₆₇	—	4	4	no	—
276	C₂²×C₆₇	—	4	4	no	—
277	C₂²×C₆₇	—	4	4	no	—
278	C₂²×C₆₇	—	4	4	no	—
279	C₂²×C₆₇	—	4	4	no	—
280	C₂²×C₆₇	—	4	4	no	—
281	C₂²×C₆₇	—	4	4	no	—
282	C₂²×C₆₇	—	8	16	no	—
283	C₂²×C₆₇	—	8	8	no	—
284	C₂²×C₆₇	#534	1	4	yes	—
285	C₂²×C₆₇	#535	1	4	yes	—
286	C₂²×C₆₇	#537	1	4	yes	—
287	C₂²×C₆₇	#560	1	8	yes	—
288	C₂²×C₆₇	#562	1	8	yes	—
289	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
290	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
291	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
292	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
293	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
294	C₄×C₆₇	—	2	4	no	—
295	C₄×C₆₇	—	2	4	no	—
296	C₄×C₆₇	—	4	2	no	—
297	C₄×C₆₇	—	4	2	no	—
298	C₄×C₆₇	—	4	4	no	—
299	C₄×C₆₇	—	4	2	no	—
300	C₄×C₆₇	—	4	4	no	—
301	C₄×C₆₇	—	4	4	no	—
302	C₂×#2813	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
303	C₂×C₆₇	—	4	4	no	—
304	C₂×C₆₇	—	8	4	no	—
305	C₂×C₆₇	—	8	2	no	—
306	C₂×C₆₇	#2813	1	8	yes	direct product factor
307	C₂×C₆₇	C₂×#534	1	1	yes	characteristic
308	C₂×C₆₇	C₂×#535	1	1	yes	characteristic
309	C₂×C₆₇	C₂×#537	1	1	yes	characteristic
310	C₂×C₆₇	C₂×#560	1	2	yes	—
311	C₂×C₆₇	C₂×#562	1	2	yes	—
312	C₂²×#123	C₂×C₆₇	1	1	yes	characteristic, complemented
313	C₂²×#121	C₂×C₆₇	1	1	yes	characteristic, complemented
314	#2813	C₂×C₆₇	1	16	yes	direct product factor
315	C₂×#400	C₂×C₆₇	1	1	yes	characteristic, complemented
316	C₂×#399	C₂×C₆₇	1	1	yes	characteristic, complemented
317	C₂×#396	C₂×C₆₇	1	2	yes	complemented
318	C₂×#394	C₂×C₆₇	1	2	yes	complemented
319	C₂×#394	C₂×C₆₇	1	2	yes	complemented
320	C₂×#394	C₂×C₆₇	1	1	yes	characteristic, complemented
321	C₂×#392	C₂×C₆₇	1	1	yes	characteristic, complemented
322	C₂×#392	C₂×C₆₇	1	2	yes	complemented
323	C₂×#380	C₂×C₆₇	1	1	yes	characteristic, complemented
324	C₆₇	C₂×#2813	1	1	yes	characteristic, direct product factor, Sylow
325	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
326	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
327	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
328	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
329	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
330	C₂×#123	—	2	8	no	—
331	C₂×#123	C₂²×C₆₇	1	8	yes	complemented
332	C₂×#121	—	2	8	no	—
333	C₂×#121	C₂²×C₆₇	1	8	yes	complemented
334	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
335	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
336	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
337	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
338	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
339	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
340	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
341	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
342	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
343	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
344	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
345	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
346	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
347	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
348	#400	C₂²×C₆₇	1	8	yes	complemented
349	#399	C₂²×C₆₇	1	8	yes	complemented
350	#396	C₂²×C₆₇	1	16	yes	complemented
351	#394	C₂²×C₆₇	1	8	yes	complemented
352	#394	C₂²×C₆₇	1	16	yes	complemented
353	#394	C₂²×C₆₇	1	16	yes	complemented
354	#392	C₂²×C₆₇	1	8	yes	complemented
355	#392	C₂²×C₆₇	1	16	yes	complemented
356	#380	C₂²×C₆₇	1	8	yes	complemented
357	C₂×#95	C₂²×C₆₇	1	2	yes	—
358	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
359	C₂×#95	C₂²×C₆₇	1	2	yes	—
360	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
361	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
362	C₂⁵	C₂³×C₆₇	1	2	yes	—
363	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
364	C₂²×D₄	—	2	8	no	—
365	C₂²×D₄	—	2	4	no	—
366	C₂²×D₄	—	2	4	no	—
367	C₂²×D₄	—	4	4	no	—
368	C₂²×D₄	—	4	4	no	—
369	C₂²×D₄	—	4	4	no	—
370	C₂²×D₄	—	4	4	no	—
371	C₂²×D₄	—	4	4	no	—
372	C₂²×D₄	—	4	4	no	—
373	C₂²×D₄	—	4	4	no	—
374	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
375	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
376	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
377	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
378	C₂³×C₄	—	4	4	no	—
379	C₂³×C₄	D₄×C₆₇	1	8	yes	complemented
380	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
381	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
382	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
383	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
384	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
385	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
386	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
387	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
388	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
389	#123	—	2	32	no	—
390	#121	—	2	32	no	—
391	C₂×C₄⋊C₄	—	2	4	no	—
392	C₂×C₄⋊C₄	—	2	4	no	—
393	C₂×C₄⋊C₄	—	2	4	no	—
394	C₂×C₄⋊C₄	—	4	4	no	—
395	C₂×C₄⋊C₄	—	4	4	no	—
396	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
397	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
398	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
399	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
400	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
401	C₂×C₂²⋊C₄	—	2	4	no	—
402	C₂×C₂²⋊C₄	—	2	8	no	—
403	C₂×C₂²⋊C₄	—	2	4	no	—
404	C₂×C₂²⋊C₄	—	2	4	no	—
405	C₂×C₂²⋊C₄	—	4	4	no	—
406	C₂×C₂²⋊C₄	—	4	4	no	—
407	C₂×C₂²⋊C₄	—	4	4	no	—
408	C₂×C₂²⋊C₄	—	4	4	no	—
409	C₂×C₂²⋊C₄	—	4	4	no	—
410	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
411	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
412	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
413	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
414	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
415	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
416	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
417	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
418	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
419	#95	C₂³×C₆₇	1	4	yes	—
420	#95	C₂³×C₆₇	1	4	yes	—
421	#95	C₂³×C₆₇	1	8	yes	—
422	#95	C₂³×C₆₇	1	8	yes	—
423	C₂⁴	—	2	2	no	—
424	C₂⁴	—	2	4	no	—
425	C₂⁴	—	2	4	no	—
426	C₂⁴	—	2	2	no	—
427	C₂⁴	—	2	2	no	—
428	C₂⁴	—	2	4	no	—
429	C₂⁴	—	2	4	no	—
430	C₂⁴	—	2	4	no	—
431	C₂⁴	—	2	4	no	—
432	C₂⁴	—	2	4	no	—
433	C₂⁴	—	2	4	no	—
434	C₂⁴	—	2	4	no	—
435	C₂⁴	—	2	4	no	—
436	C₂⁴	—	2	2	no	—
437	C₂⁴	—	2	2	no	—
438	C₂⁴	—	2	4	no	—
439	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
440	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
441	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
442	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
443	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
444	C₂×D₄	—	2	16	no	—
445	C₂×D₄	—	2	8	no	—
446	C₂×D₄	—	2	8	no	—
447	C₂×D₄	—	4	8	no	—
448	C₂×D₄	—	4	8	no	—
449	C₂×D₄	—	4	8	no	—
450	C₂×D₄	—	4	8	no	—
451	C₂×D₄	—	4	8	no	—
452	C₂×D₄	—	4	8	no	—
453	C₂×D₄	—	4	8	no	—
454	C₂×D₄	—	4	16	no	—
455	C₂×D₄	—	4	16	no	—
456	C₂×D₄	—	4	32	no	—
457	C₂×D₄	—	4	32	no	—
458	C₂²×C₄	—	2	4	no	—
459	C₂²×C₄	—	2	4	no	—
460	C₂²×C₄	—	2	2	no	—
461	C₂²×C₄	—	2	2	no	—
462	C₂²×C₄	—	2	4	no	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	4	no	—
466	C₂²×C₄	—	2	4	no	—
467	C₂²×C₄	—	2	4	no	—
468	C₂²×C₄	—	2	4	no	—
469	C₂²×C₄	—	2	4	no	—
470	C₂²×C₄	—	2	4	no	—
471	C₂²×C₄	—	2	4	no	—
472	C₂²×C₄	—	2	4	no	—
473	C₂²×C₄	—	2	4	no	—
474	C₂²×C₄	—	2	4	no	—
475	C₂²×C₄	—	2	2	no	—
476	C₂²×C₄	—	4	16	no	—
477	C₂²×C₄	—	4	8	no	—
478	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
479	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
480	C₂²×C₄	C₂×D₄×C₆₇	1	16	yes	complemented
481	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
482	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
483	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
484	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
485	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
486	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
487	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
488	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
489	C₄⋊C₄	—	2	8	no	—
490	C₄⋊C₄	—	2	8	no	—
491	C₄⋊C₄	—	2	8	no	—
492	C₄⋊C₄	—	4	8	no	—
493	C₄⋊C₄	—	4	8	no	—
494	C₂²⋊C₄	—	2	8	no	—
495	C₂²⋊C₄	—	2	16	no	—
496	C₂²⋊C₄	—	2	8	no	—
497	C₂²⋊C₄	—	2	8	no	—
498	C₂²⋊C₄	—	4	8	no	—
499	C₂²⋊C₄	—	4	8	no	—
500	C₂²⋊C₄	—	4	8	no	—
501	C₂²⋊C₄	—	4	8	no	—
502	C₂²⋊C₄	—	4	8	no	—
503	C₂³	—	2	8	no	—
504	C₂³	—	2	8	no	—
505	C₂³	—	2	8	no	—
506	C₂³	—	2	2	no	—
507	C₂³	—	2	4	no	—
508	C₂³	—	2	4	no	—
509	C₂³	—	2	2	no	—
510	C₂³	—	2	2	no	—
511	C₂³	—	2	4	no	—
512	C₂³	—	2	4	no	—
513	C₂³	—	2	4	no	—
514	C₂³	—	2	4	no	—
515	C₂³	—	2	4	no	—
516	C₂³	—	2	2	no	—
517	C₂³	—	2	4	no	—
518	C₂³	—	2	4	no	—
519	C₂³	—	2	4	no	—
520	C₂³	—	2	2	no	—
521	C₂³	—	2	4	no	—
522	C₂³	—	4	8	no	—
523	C₂³	—	4	8	no	—
524	C₂³	—	4	8	no	—
525	C₂³	—	4	4	no	—
526	C₂³	—	4	8	no	—
527	C₂³	—	4	4	no	—
528	C₂³	—	4	4	no	—
529	C₂³	—	4	8	no	—
530	C₂³	—	4	8	no	—
531	C₂³	—	4	8	no	—
532	C₂³	—	4	8	no	—
533	C₂³	—	4	8	no	—
534	C₂³	—	4	8	no	—
535	C₂³	—	4	8	no	—
536	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
537	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
538	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
539	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
540	C₂³	H(5,𝔽₂)×C₆₇	1	2	yes	—
541	C₂³	2₋¹⁺⁴×C₆₇	1	2	yes	—
542	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
543	D₄	—	4	16	no	—
544	D₄	—	4	16	no	—
545	D₄	—	4	32	no	—
546	D₄	—	4	32	no	—
547	C₂×C₄	—	2	8	no	—
548	C₂×C₄	—	2	8	no	—
549	C₂×C₄	—	2	4	no	—
550	C₂×C₄	—	2	4	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	4	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	4	no	—
556	C₂×C₄	—	2	2	no	—
557	C₂×C₄	—	2	4	no	—
558	C₂×C₄	—	2	4	no	—
559	C₂×C₄	—	2	4	no	—
560	C₂×C₄	—	2	4	no	—
561	C₂×C₄	—	2	4	no	—
562	C₂×C₄	—	2	4	no	—
563	C₂×C₄	—	2	4	no	—
564	C₂×C₄	—	2	4	no	—
565	C₂×C₄	—	2	4	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	4	16	no	—
568	C₂×C₄	—	4	4	no	—
569	C₂×C₄	—	4	8	no	—
570	C₂×C₄	—	4	4	no	—
571	C₂×C₄	—	4	4	no	—
572	C₂×C₄	—	4	8	no	—
573	C₂×C₄	—	4	8	no	—
574	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
575	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
576	C₂²	—	2	4	no	—
577	C₂²	—	2	4	no	—
578	C₂²	—	2	4	no	—
579	C₂²	—	4	4	no	—
580	C₂²	—	4	4	no	—
581	C₂²	—	4	16	no	—
582	C₂²	—	4	4	no	—
583	C₂²	—	4	4	no	—
584	C₂²	—	4	4	no	—
585	C₂²	—	4	4	no	—
586	C₂²	—	4	4	no	—
587	C₂²	—	4	4	no	—
588	C₂²	—	4	4	no	—
589	C₂²	—	4	2	no	—
590	C₂²	—	4	4	no	—
591	C₂²	—	4	2	no	—
592	C₂²	—	4	2	no	—
593	C₂²	—	4	4	no	—
594	C₂²	—	8	16	no	—
595	C₂²	—	8	8	no	—
596	C₂²	#534×C₆₇	1	4	yes	—
597	C₂²	#535×C₆₇	1	4	yes	—
598	C₂²	#537×C₆₇	1	4	yes	—
599	C₂²	#560×C₆₇	1	8	yes	—
600	C₂²	#562×C₆₇	1	8	yes	—
601	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
602	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
603	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
604	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
605	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
606	C₄	—	2	4	no	—
607	C₄	—	2	4	no	—
608	C₄	—	4	4	no	—
609	C₄	—	4	4	no	—
610	C₄	—	4	2	no	—
611	C₄	—	4	4	no	—
612	C₄	—	4	2	no	—
613	C₄	—	4	2	no	—
614	C₂	—	4	4	no	—
615	C₂	—	8	2	no	—
616	C₂	—	8	4	no	—
617	C₂	C₆₇×#2813	1	8	yes	direct product factor
618	C₂	C₂×#534×C₆₇	1	1	yes	characteristic
619	C₂	C₂×#535×C₆₇	1	1	yes	characteristic
620	C₂	C₂×#537×C₆₇	1	1	yes	characteristic
621	C₂	C₂×#560×C₆₇	1	2	yes	—
622	C₂	C₂×#562×C₆₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂²×#123×C₆₇	C₂	semidirect
2. C₂²×#121×C₆₇	C₂	semidirect
3. C₆₇×#2813	C₂	direct (N × H)
4. C₂×#400×C₆₇	C₂	semidirect
5. C₂×#399×C₆₇	C₂	semidirect
6. C₂×#396×C₆₇	C₂	semidirect
7. C₂×#394×C₆₇	C₂	semidirect
8. C₂×#394×C₆₇	C₂	semidirect
9. C₂×#394×C₆₇	C₂	semidirect
10. C₂×#392×C₆₇	C₂	semidirect
11. C₂×#392×C₆₇	C₂	semidirect
12. C₂×#380×C₆₇	C₂	semidirect
19. C₂×#123×C₆₇	C₂²	semidirect
21. C₂×#121×C₆₇	C₂²	semidirect
36. #400×C₆₇	C₂²	semidirect
37. #399×C₆₇	C₂²	semidirect
38. #396×C₆₇	C₂²	semidirect
39. #394×C₆₇	C₂²	semidirect
40. #394×C₆₇	C₂²	semidirect
41. #394×C₆₇	C₂²	semidirect
42. #392×C₆₇	C₂²	semidirect
43. #392×C₆₇	C₂²	semidirect
44. #380×C₆₇	C₂²	semidirect
67. C₂³×C₄×C₆₇	D₄	semidirect
168. C₂²×C₄×C₆₇	C₂×D₄	semidirect
302. C₂×#2813	C₆₇	direct (N × H)
306. C₂×C₆₇	#2813	direct (N × H)
312. C₂²×#123	C₂×C₆₇	semidirect
313. C₂²×#121	C₂×C₆₇	semidirect
314. #2813	C₂×C₆₇	direct (N × H)
315. C₂×#400	C₂×C₆₇	semidirect
316. C₂×#399	C₂×C₆₇	semidirect
317. C₂×#396	C₂×C₆₇	semidirect
318. C₂×#394	C₂×C₆₇	semidirect
319. C₂×#394	C₂×C₆₇	semidirect
320. C₂×#394	C₂×C₆₇	semidirect
321. C₂×#392	C₂×C₆₇	semidirect
322. C₂×#392	C₂×C₆₇	semidirect
323. C₂×#380	C₂×C₆₇	semidirect
324. C₆₇	C₂×#2813	direct (N × H)
331. C₂×#123	C₂²×C₆₇	semidirect
333. C₂×#121	C₂²×C₆₇	semidirect
348. #400	C₂²×C₆₇	semidirect
349. #399	C₂²×C₆₇	semidirect
350. #396	C₂²×C₆₇	semidirect
351. #394	C₂²×C₆₇	semidirect
352. #394	C₂²×C₆₇	semidirect
353. #394	C₂²×C₆₇	semidirect
354. #392	C₂²×C₆₇	semidirect
355. #392	C₂²×C₆₇	semidirect
356. #380	C₂²×C₆₇	semidirect
379. C₂³×C₄	D₄×C₆₇	semidirect
480. C₂²×C₄	C₂×D₄×C₆₇	semidirect
617. C₂	C₆₇×#2813	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×#123×C₆₇	2
C₂²×#121×C₆₇	2
C₆₇×#2813	2
C₂×#400×C₆₇	2
C₂×#399×C₆₇	2
C₂×#396×C₆₇	2
C₂×#394×C₆₇	2
C₂×#392×C₆₇	2
C₂×#380×C₆₇	2
C₂×#2813	67

Subgroups — C2×C67×#2813 #30715901

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2813 #30715901