Subgroups — C₂×C₆₇×#2613 #30713804

Order 17152.

7946 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#123×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#120×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2613	C₂	1	16	yes	maximal, direct product factor
5	C₂×#394×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₂×#392×C₆₇	C₂	1	4	yes	maximal, complemented
7	C₂×#381×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#379×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂⁶×C₆₇	C₂²	1	2	yes	—
10	C₂³×D₄×C₆₇	—	2	16	no	—
11	C₂³×D₄×C₆₇	—	2	8	no	—
12	C₂³×D₄×C₆₇	C₂²	1	4	yes	—
13	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×D₄×C₆₇	C₂²	1	8	yes	complemented
15	C₂³×D₄×C₆₇	C₂²	1	4	yes	complemented
16	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#123×C₆₇	—	2	8	no	—
18	C₂×#123×C₆₇	C₂²	1	8	yes	complemented
19	C₂×#120×C₆₇	—	2	16	no	—
20	C₂×#120×C₆₇	C₂²	1	16	yes	complemented
21	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	complemented
22	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
24	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	complemented
25	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	complemented
26	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	complemented
27	#394×C₆₇	C₂²	1	32	yes	complemented
28	#392×C₆₇	C₂²	1	32	yes	complemented
29	#381×C₆₇	C₂²	1	8	yes	complemented
30	#379×C₆₇	C₂²	1	16	yes	complemented
31	C₂×#95×C₆₇	C₂²	1	4	yes	complemented
32	C₂⁵×C₆₇	—	2	8	no	—
33	C₂⁵×C₆₇	—	2	8	no	—
34	C₂⁵×C₆₇	—	4	16	no	—
35	C₂⁵×C₆₇	D₄	1	16	yes	complemented
36	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
37	C₂⁵×C₆₇	C₂³	1	4	yes	—
38	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
39	C₂⁵×C₆₇	C₂³	1	8	yes	—
40	C₂²×D₄×C₆₇	—	2	128	no	—
41	C₂²×D₄×C₆₇	—	2	32	no	—
42	C₂²×D₄×C₆₇	—	2	64	no	—
43	C₂²×D₄×C₆₇	—	2	8	no	—
44	C₂²×D₄×C₆₇	—	2	16	no	—
45	C₂²×D₄×C₆₇	—	2	4	no	—
46	C₂²×D₄×C₆₇	—	2	16	no	—
47	C₂²×D₄×C₆₇	—	2	32	no	—
48	C₂²×D₄×C₆₇	—	4	16	no	—
49	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
50	C₂²×D₄×C₆₇	C₂³	1	16	yes	—
51	C₂²×D₄×C₆₇	C₂³	1	16	yes	complemented
52	C₂³×C₄×C₆₇	—	2	8	no	—
53	C₂³×C₄×C₆₇	D₄	1	8	yes	complemented
54	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
55	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
56	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
57	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
58	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
59	#123×C₆₇	—	2	32	no	—
60	#120×C₆₇	—	2	64	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
62	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	complemented
63	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
64	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
65	C₂×C₂²⋊C₄×C₆₇	—	2	16	no	—
66	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
67	C₂×C₂²⋊C₄×C₆₇	—	4	16	no	—
68	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
69	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
70	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	complemented
71	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	complemented
72	C₂×C₂²⋊C₄×C₆₇	C₂³	1	32	yes	complemented
73	#95×C₆₇	C₂³	1	16	yes	complemented
74	C₂⁴×C₆₇	—	2	8	no	—
75	C₂⁴×C₆₇	—	2	64	no	—
76	C₂⁴×C₆₇	—	2	8	no	—
77	C₂⁴×C₆₇	—	2	8	no	—
78	C₂⁴×C₆₇	—	2	16	no	—
79	C₂⁴×C₆₇	—	2	16	no	—
80	C₂⁴×C₆₇	—	2	16	no	—
81	C₂⁴×C₆₇	—	2	16	no	—
82	C₂⁴×C₆₇	—	2	2	no	—
83	C₂⁴×C₆₇	—	2	4	no	—
84	C₂⁴×C₆₇	—	2	8	no	—
85	C₂⁴×C₆₇	—	4	64	no	—
86	C₂⁴×C₆₇	—	4	32	no	—
87	C₂⁴×C₆₇	—	4	64	no	—
88	C₂⁴×C₆₇	—	4	64	no	—
89	C₂⁴×C₆₇	C₂×D₄	1	16	yes	—
90	C₂⁴×C₆₇	C₂×D₄	1	8	yes	—
91	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
92	C₂⁴×C₆₇	C₂×D₄	1	32	yes	complemented
93	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
94	C₂⁴×C₆₇	C₂⁴	1	8	yes	—
95	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
96	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
97	C₂×D₄×C₆₇	—	2	128	no	—
98	C₂×D₄×C₆₇	—	2	256	no	—
99	C₂×D₄×C₆₇	—	2	64	no	—
100	C₂×D₄×C₆₇	—	2	16	no	—
101	C₂×D₄×C₆₇	—	2	32	no	—
102	C₂×D₄×C₆₇	—	2	8	no	—
103	C₂×D₄×C₆₇	—	2	32	no	—
104	C₂×D₄×C₆₇	—	4	32	no	—
105	C₂×D₄×C₆₇	—	4	64	no	—
106	C₂×D₄×C₆₇	—	4	16	no	—
107	C₂²×C₄×C₆₇	—	2	32	no	—
108	C₂²×C₄×C₆₇	—	2	8	no	—
109	C₂²×C₄×C₆₇	—	2	16	no	—
110	C₂²×C₄×C₆₇	—	2	4	no	—
111	C₂²×C₄×C₆₇	—	2	4	no	—
112	C₂²×C₄×C₆₇	—	2	8	no	—
113	C₂²×C₄×C₆₇	—	2	2	no	—
114	C₂²×C₄×C₆₇	—	2	8	no	—
115	C₂²×C₄×C₆₇	—	2	16	no	—
116	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
117	C₂²×C₄×C₆₇	C₂×D₄	1	16	yes	complemented
118	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
119	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
120	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
121	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
122	C₄⋊C₄×C₆₇	—	2	16	no	—
123	C₂²⋊C₄×C₆₇	—	2	16	no	—
124	C₂²⋊C₄×C₆₇	—	2	8	no	—
125	C₂²⋊C₄×C₆₇	—	2	32	no	—
126	C₂²⋊C₄×C₆₇	—	2	16	no	—
127	C₂²⋊C₄×C₆₇	—	4	32	no	—
128	C₂³×C₆₇	—	2	32	no	—
129	C₂³×C₆₇	—	2	16	no	—
130	C₂³×C₆₇	—	2	8	no	—
131	C₂³×C₆₇	—	2	32	no	—
132	C₂³×C₆₇	—	2	64	no	—
133	C₂³×C₆₇	—	2	32	no	—
134	C₂³×C₆₇	—	2	32	no	—
135	C₂³×C₆₇	—	2	8	no	—
136	C₂³×C₆₇	—	2	8	no	—
137	C₂³×C₆₇	—	2	16	no	—
138	C₂³×C₆₇	—	2	16	no	—
139	C₂³×C₆₇	—	2	2	no	—
140	C₂³×C₆₇	—	2	4	no	—
141	C₂³×C₆₇	—	2	8	no	—
142	C₂³×C₆₇	—	4	64	no	—
143	C₂³×C₆₇	—	4	256	no	—
144	C₂³×C₆₇	—	4	64	no	—
145	C₂³×C₆₇	—	4	64	no	—
146	C₂³×C₆₇	—	4	32	no	—
147	C₂³×C₆₇	—	4	32	no	—
148	C₂³×C₆₇	—	4	4	no	—
149	C₂³×C₆₇	—	4	16	no	—
150	C₂³×C₆₇	—	4	8	no	—
151	C₂³×C₆₇	#120	1	16	yes	—
152	C₂³×C₆₇	C₂²×D₄	1	16	yes	—
153	C₂³×C₆₇	C₂²×D₄	1	8	yes	—
154	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
155	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
156	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
157	C₂³×C₆₇	H(5,𝔽₂)	1	8	yes	—
158	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
159	D₄×C₆₇	—	2	128	no	—
160	D₄×C₆₇	—	4	64	no	—
161	D₄×C₆₇	—	4	16	no	—
162	C₂×C₄×C₆₇	—	2	16	no	—
163	C₂×C₄×C₆₇	—	2	32	no	—
164	C₂×C₄×C₆₇	—	2	8	no	—
165	C₂×C₄×C₆₇	—	2	4	no	—
166	C₂×C₄×C₆₇	—	2	4	no	—
167	C₂×C₄×C₆₇	—	2	8	no	—
168	C₂×C₄×C₆₇	—	2	2	no	—
169	C₂×C₄×C₆₇	—	2	8	no	—
170	C₂×C₄×C₆₇	—	2	16	no	—
171	C₂×C₄×C₆₇	—	4	4	no	—
172	C₂×C₄×C₆₇	—	4	16	no	—
173	C₂×C₄×C₆₇	—	4	8	no	—
174	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
175	C₂²×C₆₇	—	2	32	no	—
176	C₂²×C₆₇	—	2	16	no	—
177	C₂²×C₆₇	—	2	8	no	—
178	C₂²×C₆₇	—	2	16	no	—
179	C₂²×C₆₇	—	2	16	no	—
180	C₂²×C₆₇	—	2	16	no	—
181	C₂²×C₆₇	—	4	128	no	—
182	C₂²×C₆₇	—	4	64	no	—
183	C₂²×C₆₇	—	4	16	no	—
184	C₂²×C₆₇	—	4	16	no	—
185	C₂²×C₆₇	—	4	2	no	—
186	C₂²×C₆₇	—	4	8	no	—
187	C₂²×C₆₇	—	4	4	no	—
188	C₂²×C₆₇	—	8	8	no	—
189	C₂²×C₆₇	C₂×#120	1	8	yes	—
190	C₂²×C₆₇	C₂×#120	1	4	yes	—
191	C₂²×C₆₇	#534	1	16	yes	—
192	C₂²×C₆₇	#535	1	8	yes	—
193	C₂²×C₆₇	C₂³×D₄	1	2	yes	—
194	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
195	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	4	yes	—
196	C₄×C₆₇	—	2	8	no	—
197	C₄×C₆₇	—	4	2	no	—
198	C₄×C₆₇	—	4	8	no	—
199	C₄×C₆₇	—	4	4	no	—
200	C₂×#2613	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
201	C₂×C₆₇	—	2	8	no	—
202	C₂×C₆₇	—	4	16	no	—
203	C₂×C₆₇	—	8	2	no	—
204	C₂×C₆₇	#2613	1	8	yes	direct product factor
205	C₂×C₆₇	C₂²×#120	1	1	yes	characteristic
206	C₂×C₆₇	C₂×#534	1	4	yes	—
207	C₂×C₆₇	C₂×#535	1	2	yes	—
208	C₂⁴×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
209	C₂²×#123	C₂×C₆₇	1	1	yes	characteristic, complemented
210	C₂²×#120	C₂×C₆₇	1	2	yes	complemented
211	#2613	C₂×C₆₇	1	16	yes	direct product factor
212	C₂×#394	C₂×C₆₇	1	4	yes	complemented
213	C₂×#392	C₂×C₆₇	1	4	yes	complemented
214	C₂×#381	C₂×C₆₇	1	1	yes	characteristic, complemented
215	C₂×#379	C₂×C₆₇	1	2	yes	complemented
216	C₆₇	C₂×#2613	1	1	yes	characteristic, direct product factor, Sylow
217	C₂⁶	C₂²×C₆₇	1	2	yes	—
218	C₂³×D₄	—	2	16	no	—
219	C₂³×D₄	—	2	8	no	—
220	C₂³×D₄	C₂²×C₆₇	1	4	yes	—
221	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
222	C₂³×D₄	C₂²×C₆₇	1	8	yes	complemented
223	C₂³×D₄	C₂²×C₆₇	1	4	yes	complemented
224	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
225	C₂×#123	—	2	8	no	—
226	C₂×#123	C₂²×C₆₇	1	8	yes	complemented
227	C₂×#120	—	2	16	no	—
228	C₂×#120	C₂²×C₆₇	1	16	yes	complemented
229	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	complemented
230	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
231	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
232	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	complemented
233	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	complemented
234	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	complemented
235	#394	C₂²×C₆₇	1	32	yes	complemented
236	#392	C₂²×C₆₇	1	32	yes	complemented
237	#381	C₂²×C₆₇	1	8	yes	complemented
238	#379	C₂²×C₆₇	1	16	yes	complemented
239	C₂×#95	C₂²×C₆₇	1	4	yes	complemented
240	C₂⁵	—	2	8	no	—
241	C₂⁵	—	2	8	no	—
242	C₂⁵	—	4	16	no	—
243	C₂⁵	D₄×C₆₇	1	16	yes	complemented
244	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
245	C₂⁵	C₂³×C₆₇	1	4	yes	—
246	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
247	C₂⁵	C₂³×C₆₇	1	8	yes	—
248	C₂²×D₄	—	2	128	no	—
249	C₂²×D₄	—	2	32	no	—
250	C₂²×D₄	—	2	64	no	—
251	C₂²×D₄	—	2	8	no	—
252	C₂²×D₄	—	2	16	no	—
253	C₂²×D₄	—	2	4	no	—
254	C₂²×D₄	—	2	16	no	—
255	C₂²×D₄	—	2	32	no	—
256	C₂²×D₄	—	4	16	no	—
257	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
258	C₂²×D₄	C₂³×C₆₇	1	16	yes	—
259	C₂²×D₄	C₂³×C₆₇	1	16	yes	complemented
260	C₂³×C₄	—	2	8	no	—
261	C₂³×C₄	D₄×C₆₇	1	8	yes	complemented
262	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
263	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
264	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
265	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
266	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
267	#123	—	2	32	no	—
268	#120	—	2	64	no	—
269	C₂×C₄⋊C₄	—	2	8	no	—
270	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	complemented
271	C₂×C₂²⋊C₄	—	2	8	no	—
272	C₂×C₂²⋊C₄	—	2	4	no	—
273	C₂×C₂²⋊C₄	—	2	16	no	—
274	C₂×C₂²⋊C₄	—	2	8	no	—
275	C₂×C₂²⋊C₄	—	4	16	no	—
276	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
277	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
278	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	complemented
279	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	complemented
280	C₂×C₂²⋊C₄	C₂³×C₆₇	1	32	yes	complemented
281	#95	C₂³×C₆₇	1	16	yes	complemented
282	C₂⁴	—	2	8	no	—
283	C₂⁴	—	2	64	no	—
284	C₂⁴	—	2	8	no	—
285	C₂⁴	—	2	8	no	—
286	C₂⁴	—	2	16	no	—
287	C₂⁴	—	2	16	no	—
288	C₂⁴	—	2	16	no	—
289	C₂⁴	—	2	16	no	—
290	C₂⁴	—	2	2	no	—
291	C₂⁴	—	2	4	no	—
292	C₂⁴	—	2	8	no	—
293	C₂⁴	—	4	64	no	—
294	C₂⁴	—	4	32	no	—
295	C₂⁴	—	4	64	no	—
296	C₂⁴	—	4	64	no	—
297	C₂⁴	C₂×D₄×C₆₇	1	16	yes	—
298	C₂⁴	C₂×D₄×C₆₇	1	8	yes	—
299	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
300	C₂⁴	C₂×D₄×C₆₇	1	32	yes	complemented
301	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
302	C₂⁴	C₂⁴×C₆₇	1	8	yes	—
303	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
304	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
305	C₂×D₄	—	2	128	no	—
306	C₂×D₄	—	2	256	no	—
307	C₂×D₄	—	2	64	no	—
308	C₂×D₄	—	2	16	no	—
309	C₂×D₄	—	2	32	no	—
310	C₂×D₄	—	2	8	no	—
311	C₂×D₄	—	2	32	no	—
312	C₂×D₄	—	4	32	no	—
313	C₂×D₄	—	4	64	no	—
314	C₂×D₄	—	4	16	no	—
315	C₂²×C₄	—	2	32	no	—
316	C₂²×C₄	—	2	8	no	—
317	C₂²×C₄	—	2	16	no	—
318	C₂²×C₄	—	2	4	no	—
319	C₂²×C₄	—	2	4	no	—
320	C₂²×C₄	—	2	8	no	—
321	C₂²×C₄	—	2	2	no	—
322	C₂²×C₄	—	2	8	no	—
323	C₂²×C₄	—	2	16	no	—
324	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
325	C₂²×C₄	C₂×D₄×C₆₇	1	16	yes	complemented
326	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
327	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
328	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
329	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
330	C₄⋊C₄	—	2	16	no	—
331	C₂²⋊C₄	—	2	16	no	—
332	C₂²⋊C₄	—	2	8	no	—
333	C₂²⋊C₄	—	2	32	no	—
334	C₂²⋊C₄	—	2	16	no	—
335	C₂²⋊C₄	—	4	32	no	—
336	C₂³	—	2	32	no	—
337	C₂³	—	2	16	no	—
338	C₂³	—	2	8	no	—
339	C₂³	—	2	32	no	—
340	C₂³	—	2	64	no	—
341	C₂³	—	2	32	no	—
342	C₂³	—	2	32	no	—
343	C₂³	—	2	8	no	—
344	C₂³	—	2	8	no	—
345	C₂³	—	2	16	no	—
346	C₂³	—	2	16	no	—
347	C₂³	—	2	2	no	—
348	C₂³	—	2	4	no	—
349	C₂³	—	2	8	no	—
350	C₂³	—	4	64	no	—
351	C₂³	—	4	256	no	—
352	C₂³	—	4	64	no	—
353	C₂³	—	4	64	no	—
354	C₂³	—	4	32	no	—
355	C₂³	—	4	32	no	—
356	C₂³	—	4	4	no	—
357	C₂³	—	4	16	no	—
358	C₂³	—	4	8	no	—
359	C₂³	#120×C₆₇	1	16	yes	—
360	C₂³	C₂²×D₄×C₆₇	1	16	yes	—
361	C₂³	C₂²×D₄×C₆₇	1	8	yes	—
362	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
363	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
364	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
365	C₂³	H(5,𝔽₂)×C₆₇	1	8	yes	—
366	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
367	D₄	—	2	128	no	—
368	D₄	—	4	64	no	—
369	D₄	—	4	16	no	—
370	C₂×C₄	—	2	16	no	—
371	C₂×C₄	—	2	32	no	—
372	C₂×C₄	—	2	8	no	—
373	C₂×C₄	—	2	4	no	—
374	C₂×C₄	—	2	4	no	—
375	C₂×C₄	—	2	8	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	8	no	—
378	C₂×C₄	—	2	16	no	—
379	C₂×C₄	—	4	4	no	—
380	C₂×C₄	—	4	16	no	—
381	C₂×C₄	—	4	8	no	—
382	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
383	C₂²	—	2	32	no	—
384	C₂²	—	2	16	no	—
385	C₂²	—	2	8	no	—
386	C₂²	—	2	16	no	—
387	C₂²	—	2	16	no	—
388	C₂²	—	2	16	no	—
389	C₂²	—	4	128	no	—
390	C₂²	—	4	64	no	—
391	C₂²	—	4	16	no	—
392	C₂²	—	4	16	no	—
393	C₂²	—	4	2	no	—
394	C₂²	—	4	8	no	—
395	C₂²	—	4	4	no	—
396	C₂²	—	8	8	no	—
397	C₂²	C₂×#120×C₆₇	1	8	yes	—
398	C₂²	C₂×#120×C₆₇	1	4	yes	—
399	C₂²	#534×C₆₇	1	16	yes	—
400	C₂²	#535×C₆₇	1	8	yes	—
401	C₂²	C₂³×D₄×C₆₇	1	2	yes	—
402	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
403	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	4	yes	—
404	C₄	—	2	8	no	—
405	C₄	—	4	2	no	—
406	C₄	—	4	8	no	—
407	C₄	—	4	4	no	—
408	C₂	—	2	8	no	—
409	C₂	—	4	16	no	—
410	C₂	—	8	2	no	—
411	C₂	C₆₇×#2613	1	8	yes	direct product factor
412	C₂	C₂²×#120×C₆₇	1	1	yes	characteristic
413	C₂	C₂×#534×C₆₇	1	4	yes	—
414	C₂	C₂×#535×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. C₂⁴×D₄×C₆₇	C₂	semidirect
2. C₂²×#123×C₆₇	C₂	semidirect
3. C₂²×#120×C₆₇	C₂	semidirect
4. C₆₇×#2613	C₂	direct (N × H)
5. C₂×#394×C₆₇	C₂	semidirect
6. C₂×#392×C₆₇	C₂	semidirect
7. C₂×#381×C₆₇	C₂	semidirect
8. C₂×#379×C₆₇	C₂	semidirect
14. C₂³×D₄×C₆₇	C₂²	semidirect
15. C₂³×D₄×C₆₇	C₂²	semidirect
18. C₂×#123×C₆₇	C₂²	semidirect
20. C₂×#120×C₆₇	C₂²	semidirect
21. C₂²×C₄⋊C₄×C₆₇	C₂²	semidirect
24. C₂²×C₂²⋊C₄×C₆₇	C₂²	semidirect
25. C₂²×C₂²⋊C₄×C₆₇	C₂²	semidirect
26. C₂²×C₂²⋊C₄×C₆₇	C₂²	semidirect
27. #394×C₆₇	C₂²	semidirect
28. #392×C₆₇	C₂²	semidirect
29. #381×C₆₇	C₂²	semidirect
30. #379×C₆₇	C₂²	semidirect
31. C₂×#95×C₆₇	C₂²	semidirect
35. C₂⁵×C₆₇	D₄	semidirect
51. C₂²×D₄×C₆₇	C₂³	semidirect
53. C₂³×C₄×C₆₇	D₄	semidirect
62. C₂×C₄⋊C₄×C₆₇	C₂³	semidirect
70. C₂×C₂²⋊C₄×C₆₇	C₂³	semidirect
71. C₂×C₂²⋊C₄×C₆₇	C₂³	semidirect
72. C₂×C₂²⋊C₄×C₆₇	C₂³	semidirect
73. #95×C₆₇	C₂³	semidirect
92. C₂⁴×C₆₇	C₂×D₄	semidirect
117. C₂²×C₄×C₆₇	C₂×D₄	semidirect
200. C₂×#2613	C₆₇	direct (N × H)
204. C₂×C₆₇	#2613	direct (N × H)
208. C₂⁴×D₄	C₂×C₆₇	semidirect
209. C₂²×#123	C₂×C₆₇	semidirect
210. C₂²×#120	C₂×C₆₇	semidirect
211. #2613	C₂×C₆₇	direct (N × H)
212. C₂×#394	C₂×C₆₇	semidirect
213. C₂×#392	C₂×C₆₇	semidirect
214. C₂×#381	C₂×C₆₇	semidirect
215. C₂×#379	C₂×C₆₇	semidirect
216. C₆₇	C₂×#2613	direct (N × H)
222. C₂³×D₄	C₂²×C₆₇	semidirect
223. C₂³×D₄	C₂²×C₆₇	semidirect
226. C₂×#123	C₂²×C₆₇	semidirect
228. C₂×#120	C₂²×C₆₇	semidirect
229. C₂²×C₄⋊C₄	C₂²×C₆₇	semidirect
232. C₂²×C₂²⋊C₄	C₂²×C₆₇	semidirect
233. C₂²×C₂²⋊C₄	C₂²×C₆₇	semidirect
234. C₂²×C₂²⋊C₄	C₂²×C₆₇	semidirect
235. #394	C₂²×C₆₇	semidirect
236. #392	C₂²×C₆₇	semidirect
237. #381	C₂²×C₆₇	semidirect
238. #379	C₂²×C₆₇	semidirect
239. C₂×#95	C₂²×C₆₇	semidirect
243. C₂⁵	D₄×C₆₇	semidirect
259. C₂²×D₄	C₂³×C₆₇	semidirect
261. C₂³×C₄	D₄×C₆₇	semidirect
270. C₂×C₄⋊C₄	C₂³×C₆₇	semidirect
278. C₂×C₂²⋊C₄	C₂³×C₆₇	semidirect
279. C₂×C₂²⋊C₄	C₂³×C₆₇	semidirect
280. C₂×C₂²⋊C₄	C₂³×C₆₇	semidirect
281. #95	C₂³×C₆₇	semidirect
300. C₂⁴	C₂×D₄×C₆₇	semidirect
325. C₂²×C₄	C₂×D₄×C₆₇	semidirect
411. C₂	C₆₇×#2613	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₄×C₆₇	2
C₂²×#123×C₆₇	2
C₂²×#120×C₆₇	2
C₆₇×#2613	2
C₂×#394×C₆₇	2
C₂×#392×C₆₇	2
C₂×#381×C₆₇	2
C₂×#379×C₆₇	2
C₂×#2613	67

Subgroups — C2×C67×#2613 #30713804

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2613 #30713804