Subgroups — C₆₇×#25867 #30705118

Order 17152.

1422 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#535×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#490×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#488×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2629	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2042	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2041	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1959	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1957	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1788	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1786	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1750	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
13	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
14	#535×C₆₇	—	2	4	no	—
15	#535×C₆₇	—	2	4	no	—
16	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic, complemented
19	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#121×C₆₇	—	2	2	no	—
22	C₂×#120×C₆₇	—	2	2	no	—
23	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
24	#490×C₆₇	—	2	4	no	—
25	#488×C₆₇	—	2	4	no	—
26	C₂×#103×C₆₇	C₂²	1	2	yes	complemented
27	C₂×#103×C₆₇	C₂²	1	2	yes	complemented
28	C₂×#102×C₆₇	C₂²	1	2	yes	—
29	C₂×#102×C₆₇	C₂²	1	2	yes	—
30	C₂×#102×C₆₇	C₂²	1	2	yes	—
31	C₂×#102×C₆₇	C₂²	1	2	yes	—
32	C₂×#102×C₆₇	C₂²	1	2	yes	complemented
33	C₂×#102×C₆₇	C₂²	1	2	yes	complemented
34	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
35	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic, complemented
36	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
37	#398×C₆₇	—	2	2	no	—
38	#396×C₆₇	—	2	2	no	—
39	#389×C₆₇	C₂²	1	2	yes	complemented
40	#384×C₆₇	C₂²	1	1	yes	characteristic, complemented
41	#380×C₆₇	C₂²	1	2	yes	—
42	#336×C₆₇	C₂²	1	2	yes	complemented
43	C₂⁵×C₆₇	—	2	1	no	—
44	C₂×#41×C₆₇	—	2	2	no	—
45	C₂×#41×C₆₇	—	2	2	no	—
46	C₂×#41×C₆₇	—	2	4	no	—
47	C₂×#41×C₆₇	—	4	2	no	—
48	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
49	C₂²×D₄×C₆₇	—	2	1	no	—
50	C₂²×D₄×C₆₇	—	2	2	no	—
51	C₂²×D₄×C₆₇	—	2	2	no	—
52	C₂²×D₄×C₆₇	—	2	1	no	—
53	C₂²×D₄×C₆₇	—	2	2	no	—
54	C₂²×D₄×C₆₇	—	2	2	no	—
55	C₂²×D₄×C₆₇	—	2	4	no	—
56	C₂²×D₄×C₆₇	—	4	2	no	—
57	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
58	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
59	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₆₇	—	2	1	no	—
61	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
63	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
64	#127×C₆₇	—	2	2	no	—
65	#127×C₆₇	—	2	2	no	—
66	#127×C₆₇	—	2	2	no	—
67	#127×C₆₇	D₄	1	4	yes	—
68	#125×C₆₇	—	2	2	no	—
69	#125×C₆₇	—	2	2	no	—
70	#124×C₆₇	—	2	2	no	—
71	#124×C₆₇	—	2	2	no	—
72	#124×C₆₇	—	2	2	no	—
73	#124×C₆₇	D₄	1	4	yes	—
74	#121×C₆₇	—	2	16	no	—
75	#120×C₆₇	—	2	16	no	—
76	#117×C₆₇	—	2	2	no	—
77	#117×C₆₇	—	2	2	no	—
78	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
79	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
80	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
81	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
82	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
84	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
85	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
86	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
88	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
90	#103×C₆₇	—	2	4	no	—
91	#103×C₆₇	—	4	2	no	—
92	#102×C₆₇	—	2	4	no	—
93	#102×C₆₇	—	2	4	no	—
94	#102×C₆₇	—	2	4	no	—
95	#102×C₆₇	—	4	2	no	—
96	#102×C₆₇	—	4	2	no	—
97	#102×C₆₇	—	4	2	no	—
98	C₂²⋊C₈×C₆₇	—	2	4	no	—
99	#97×C₆₇	—	2	2	no	—
100	C₄×C₈×C₆₇	—	2	2	no	—
101	#95×C₆₇	—	2	2	no	—
102	#95×C₆₇	—	2	2	no	—
103	C₂⁴×C₆₇	—	2	4	no	—
104	C₂⁴×C₆₇	—	2	1	no	—
105	C₂⁴×C₆₇	—	2	1	no	—
106	C₂⁴×C₆₇	—	2	1	no	—
107	C₂⁴×C₆₇	—	4	2	no	—
108	C₂⁴×C₆₇	—	4	2	no	—
109	C₂⁴×C₆₇	—	8	2	no	—
110	C₂⁴×C₆₇	—	8	2	no	—
111	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
112	#41×C₆₇	—	4	8	no	—
113	#41×C₆₇	—	4	8	no	—
114	C₂×Q₈×C₆₇	—	2	1	no	—
115	C₂×Q₈×C₆₇	—	2	2	no	—
116	C₂×Q₈×C₆₇	—	2	2	no	—
117	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
118	C₂×D₄×C₆₇	—	2	4	no	—
119	C₂×D₄×C₆₇	—	2	4	no	—
120	C₂×D₄×C₆₇	—	2	8	no	—
121	C₂×D₄×C₆₇	—	2	8	no	—
122	C₂×D₄×C₆₇	—	2	1	no	—
123	C₂×D₄×C₆₇	—	2	1	no	—
124	C₂×D₄×C₆₇	—	2	1	no	—
125	C₂×D₄×C₆₇	—	2	2	no	—
126	C₂×D₄×C₆₇	—	2	2	no	—
127	C₂×D₄×C₆₇	—	2	2	no	—
128	C₂×D₄×C₆₇	—	2	2	no	—
129	C₂×D₄×C₆₇	—	2	2	no	—
130	C₂×D₄×C₆₇	—	2	2	no	—
131	C₂×D₄×C₆₇	—	4	8	no	—
132	C₂×D₄×C₆₇	—	4	8	no	—
133	C₂×D₄×C₆₇	—	4	2	no	—
134	C₂×D₄×C₆₇	—	4	2	no	—
135	C₂×D₄×C₆₇	—	4	2	no	—
136	C₂×D₄×C₆₇	—	4	2	no	—
137	C₂×D₄×C₆₇	—	8	2	no	—
138	C₂×D₄×C₆₇	—	8	2	no	—
139	C₂×D₄×C₆₇	—	8	2	no	—
140	C₂×D₄×C₆₇	—	8	2	no	—
141	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
142	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
143	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
144	C₂²×C₄×C₆₇	—	2	1	no	—
145	C₂²×C₄×C₆₇	—	2	1	no	—
146	C₂²×C₄×C₆₇	—	2	2	no	—
147	C₂²×C₄×C₆₇	—	2	4	no	—
148	C₂²×C₄×C₆₇	—	2	1	no	—
149	C₂²×C₄×C₆₇	—	2	2	no	—
150	C₂²×C₄×C₆₇	—	2	2	no	—
151	C₂²×C₄×C₆₇	—	4	2	no	—
152	C₂²×C₄×C₆₇	—	4	2	no	—
153	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
154	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
155	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₆₇	#41	1	4	yes	—
157	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
158	C₂×C₈×C₆₇	—	2	2	no	—
159	C₂×C₈×C₆₇	—	2	2	no	—
160	C₂×C₈×C₆₇	—	2	2	no	—
161	C₂×C₈×C₆₇	—	2	2	no	—
162	C₂×C₈×C₆₇	—	2	2	no	—
163	C₂×C₈×C₆₇	—	2	2	no	—
164	C₄⋊C₄×C₆₇	—	2	4	no	—
165	C₄⋊C₄×C₆₇	—	2	2	no	—
166	C₄⋊C₄×C₆₇	—	2	2	no	—
167	C₄⋊C₄×C₆₇	—	2	2	no	—
168	C₄⋊C₄×C₆₇	—	2	2	no	—
169	C₄⋊C₄×C₆₇	—	4	4	no	—
170	C₄⋊C₄×C₆₇	—	8	2	no	—
171	C₄⋊C₄×C₆₇	—	8	2	no	—
172	C₂²⋊C₄×C₆₇	—	2	8	no	—
173	C₂²⋊C₄×C₆₇	—	2	8	no	—
174	C₂²⋊C₄×C₆₇	—	2	4	no	—
175	C₂²⋊C₄×C₆₇	—	8	2	no	—
176	C₄²×C₆₇	—	2	1	no	—
177	C₄²×C₆₇	—	2	1	no	—
178	C₄²×C₆₇	C₂×D₄	1	2	yes	—
179	C₄²×C₆₇	C₂×D₄	1	2	yes	—
180	C₂³×C₆₇	—	2	1	no	—
181	C₂³×C₆₇	—	2	1	no	—
182	C₂³×C₆₇	—	2	2	no	—
183	C₂³×C₆₇	—	2	2	no	—
184	C₂³×C₆₇	—	2	2	no	—
185	C₂³×C₆₇	—	2	2	no	—
186	C₂³×C₆₇	—	2	2	no	—
187	C₂³×C₆₇	—	4	8	no	—
188	C₂³×C₆₇	—	4	1	no	—
189	C₂³×C₆₇	—	4	1	no	—
190	C₂³×C₆₇	—	4	1	no	—
191	C₂³×C₆₇	—	4	1	no	—
192	C₂³×C₆₇	—	4	1	no	—
193	C₂³×C₆₇	—	4	1	no	—
194	C₂³×C₆₇	—	4	2	no	—
195	C₂³×C₆₇	—	4	2	no	—
196	C₂³×C₆₇	—	4	2	no	—
197	C₂³×C₆₇	—	4	2	no	—
198	C₂³×C₆₇	—	4	2	no	—
199	C₂³×C₆₇	—	4	2	no	—
200	C₂³×C₆₇	—	8	8	no	—
201	C₂³×C₆₇	—	8	8	no	—
202	C₂³×C₆₇	—	8	4	no	—
203	C₂³×C₆₇	—	8	4	no	—
204	C₂³×C₆₇	#124	1	2	yes	—
205	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
206	Q₈×C₆₇	—	2	4	no	—
207	Q₈×C₆₇	—	4	2	no	—
208	D₄×C₆₇	—	2	4	no	—
209	D₄×C₆₇	—	2	4	no	—
210	D₄×C₆₇	—	2	4	no	—
211	D₄×C₆₇	—	4	8	no	—
212	D₄×C₆₇	—	4	8	no	—
213	D₄×C₆₇	—	4	2	no	—
214	D₄×C₆₇	—	4	2	no	—
215	D₄×C₆₇	—	4	2	no	—
216	D₄×C₆₇	—	8	8	no	—
217	D₄×C₆₇	—	8	8	no	—
218	C₂×C₄×C₆₇	—	2	1	no	—
219	C₂×C₄×C₆₇	—	2	1	no	—
220	C₂×C₄×C₆₇	—	2	2	no	—
221	C₂×C₄×C₆₇	—	2	2	no	—
222	C₂×C₄×C₆₇	—	2	4	no	—
223	C₂×C₄×C₆₇	—	2	4	no	—
224	C₂×C₄×C₆₇	—	2	2	no	—
225	C₂×C₄×C₆₇	—	2	2	no	—
226	C₂×C₄×C₆₇	—	2	2	no	—
227	C₂×C₄×C₆₇	—	2	2	no	—
228	C₂×C₄×C₆₇	—	4	8	no	—
229	C₂×C₄×C₆₇	—	4	1	no	—
230	C₂×C₄×C₆₇	—	4	1	no	—
231	C₂×C₄×C₆₇	—	4	2	no	—
232	C₂×C₄×C₆₇	—	4	2	no	—
233	C₂×C₄×C₆₇	—	4	2	no	—
234	C₂×C₄×C₆₇	—	4	2	no	—
235	C₂×C₄×C₆₇	—	4	2	no	—
236	C₂×C₄×C₆₇	—	4	2	no	—
237	C₂×C₄×C₆₇	#120	1	4	yes	—
238	C₂×C₄×C₆₇	#124	1	2	yes	—
239	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
240	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
242	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
243	C₈×C₆₇	—	4	2	no	—
244	C₈×C₆₇	—	4	2	no	—
245	C₂²×C₆₇	—	2	4	no	—
246	C₂²×C₆₇	—	2	2	no	—
247	C₂²×C₆₇	—	4	1	no	—
248	C₂²×C₆₇	—	4	1	no	—
249	C₂²×C₆₇	—	4	2	no	—
250	C₂²×C₆₇	—	4	2	no	—
251	C₂²×C₆₇	—	4	2	no	—
252	C₂²×C₆₇	—	4	2	no	—
253	C₂²×C₆₇	—	4	4	no	—
254	C₂²×C₆₇	—	4	4	no	—
255	C₂²×C₆₇	—	4	4	no	—
256	C₂²×C₆₇	—	8	16	no	—
257	C₂²×C₆₇	—	8	2	no	—
258	C₂²×C₆₇	—	8	2	no	—
259	C₂²×C₆₇	—	8	2	no	—
260	C₂²×C₆₇	—	8	1	no	—
261	C₂²×C₆₇	—	8	1	no	—
262	C₂²×C₆₇	—	8	1	no	—
263	C₂²×C₆₇	—	8	1	no	—
264	C₂²×C₆₇	—	8	1	no	—
265	C₂²×C₆₇	—	8	1	no	—
266	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
267	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
268	C₂²×C₆₇	#535	1	1	yes	characteristic
269	C₂²×C₆₇	#576	1	2	yes	—
270	C₂²×C₆₇	#577	1	2	yes	—
271	C₄×C₆₇	—	2	2	no	—
272	C₄×C₆₇	—	2	4	no	—
273	C₄×C₆₇	—	4	2	no	—
274	C₄×C₆₇	—	4	2	no	—
275	C₄×C₆₇	—	8	2	no	—
276	C₄×C₆₇	—	8	2	no	—
277	C₄×C₆₇	#546	1	2	yes	—
278	C₄×C₆₇	#546	1	2	yes	—
279	#25867	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
280	C₂×C₆₇	—	4	2	no	—
281	C₂×C₆₇	—	8	2	no	—
282	C₂×C₆₇	—	8	2	no	—
283	C₂×C₆₇	—	8	2	no	—
284	C₂×C₆₇	#2435	1	1	yes	characteristic
285	C₂×C₆₇	#3012	1	2	yes	—
286	C₂×C₆₇	#3136	1	2	yes	—
287	C₂×C₆₇	#3220	1	1	yes	characteristic
288	C₂×C₆₇	#3222	1	1	yes	characteristic
289	C₂×#535	C₂×C₆₇	1	1	yes	characteristic
290	C₂×#490	C₂×C₆₇	1	1	yes	characteristic, complemented
291	C₂×#488	C₂×C₆₇	1	1	yes	characteristic, complemented
292	#2629	C₂×C₆₇	1	1	yes	characteristic, complemented
293	#2042	C₂×C₆₇	1	2	yes	complemented
294	#2041	C₂×C₆₇	1	2	yes	complemented
295	#1959	C₂×C₆₇	1	1	yes	characteristic, complemented
296	#1957	C₂×C₆₇	1	1	yes	characteristic, complemented
297	#1788	C₂×C₆₇	1	2	yes	complemented
298	#1786	C₂×C₆₇	1	2	yes	complemented
299	#1750	C₂×C₆₇	1	1	yes	characteristic, complemented
300	C₆₇	#25867	1	1	yes	characteristic, direct product factor, Sylow
301	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
302	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
303	#535	—	2	4	no	—
304	#535	—	2	4	no	—
305	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
306	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
307	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic, complemented
308	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
309	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
310	C₂×#121	—	2	2	no	—
311	C₂×#120	—	2	2	no	—
312	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
313	#490	—	2	4	no	—
314	#488	—	2	4	no	—
315	C₂×#103	C₂²×C₆₇	1	2	yes	complemented
316	C₂×#103	C₂²×C₆₇	1	2	yes	complemented
317	C₂×#102	C₂²×C₆₇	1	2	yes	—
318	C₂×#102	C₂²×C₆₇	1	2	yes	—
319	C₂×#102	C₂²×C₆₇	1	2	yes	—
320	C₂×#102	C₂²×C₆₇	1	2	yes	—
321	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
322	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
323	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
324	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic, complemented
325	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
326	#398	—	2	2	no	—
327	#396	—	2	2	no	—
328	#389	C₂²×C₆₇	1	2	yes	complemented
329	#384	C₂²×C₆₇	1	1	yes	characteristic, complemented
330	#380	C₂²×C₆₇	1	2	yes	—
331	#336	C₂²×C₆₇	1	2	yes	complemented
332	C₂⁵	—	2	1	no	—
333	C₂×#41	—	2	2	no	—
334	C₂×#41	—	2	2	no	—
335	C₂×#41	—	2	4	no	—
336	C₂×#41	—	4	2	no	—
337	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
338	C₂²×D₄	—	2	1	no	—
339	C₂²×D₄	—	2	2	no	—
340	C₂²×D₄	—	2	2	no	—
341	C₂²×D₄	—	2	1	no	—
342	C₂²×D₄	—	2	2	no	—
343	C₂²×D₄	—	2	2	no	—
344	C₂²×D₄	—	2	4	no	—
345	C₂²×D₄	—	4	2	no	—
346	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
347	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
348	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
349	C₂³×C₄	—	2	1	no	—
350	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
351	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
352	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
353	#127	—	2	2	no	—
354	#127	—	2	2	no	—
355	#127	—	2	2	no	—
356	#127	D₄×C₆₇	1	4	yes	—
357	#125	—	2	2	no	—
358	#125	—	2	2	no	—
359	#124	—	2	2	no	—
360	#124	—	2	2	no	—
361	#124	—	2	2	no	—
362	#124	D₄×C₆₇	1	4	yes	—
363	#121	—	2	16	no	—
364	#120	—	2	16	no	—
365	#117	—	2	2	no	—
366	#117	—	2	2	no	—
367	C₂×C₄⋊C₄	—	2	2	no	—
368	C₂×C₄⋊C₄	—	2	2	no	—
369	C₂×C₄⋊C₄	—	2	2	no	—
370	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
371	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
372	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
373	C₂×C₂²⋊C₄	—	2	2	no	—
374	C₂×C₂²⋊C₄	—	2	2	no	—
375	C₂×C₂²⋊C₄	—	2	2	no	—
376	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
377	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
378	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
379	#103	—	2	4	no	—
380	#103	—	4	2	no	—
381	#102	—	2	4	no	—
382	#102	—	2	4	no	—
383	#102	—	2	4	no	—
384	#102	—	4	2	no	—
385	#102	—	4	2	no	—
386	#102	—	4	2	no	—
387	C₂²⋊C₈	—	2	4	no	—
388	#97	—	2	2	no	—
389	C₄×C₈	—	2	2	no	—
390	#95	—	2	2	no	—
391	#95	—	2	2	no	—
392	C₂⁴	—	2	4	no	—
393	C₂⁴	—	2	1	no	—
394	C₂⁴	—	2	1	no	—
395	C₂⁴	—	2	1	no	—
396	C₂⁴	—	4	2	no	—
397	C₂⁴	—	4	2	no	—
398	C₂⁴	—	8	2	no	—
399	C₂⁴	—	8	2	no	—
400	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
401	#41	—	4	8	no	—
402	#41	—	4	8	no	—
403	C₂×Q₈	—	2	1	no	—
404	C₂×Q₈	—	2	2	no	—
405	C₂×Q₈	—	2	2	no	—
406	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
407	C₂×D₄	—	2	4	no	—
408	C₂×D₄	—	2	4	no	—
409	C₂×D₄	—	2	8	no	—
410	C₂×D₄	—	2	8	no	—
411	C₂×D₄	—	2	1	no	—
412	C₂×D₄	—	2	1	no	—
413	C₂×D₄	—	2	1	no	—
414	C₂×D₄	—	2	2	no	—
415	C₂×D₄	—	2	2	no	—
416	C₂×D₄	—	2	2	no	—
417	C₂×D₄	—	2	2	no	—
418	C₂×D₄	—	2	2	no	—
419	C₂×D₄	—	2	2	no	—
420	C₂×D₄	—	4	8	no	—
421	C₂×D₄	—	4	8	no	—
422	C₂×D₄	—	4	2	no	—
423	C₂×D₄	—	4	2	no	—
424	C₂×D₄	—	4	2	no	—
425	C₂×D₄	—	4	2	no	—
426	C₂×D₄	—	8	2	no	—
427	C₂×D₄	—	8	2	no	—
428	C₂×D₄	—	8	2	no	—
429	C₂×D₄	—	8	2	no	—
430	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
431	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
432	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
433	C₂²×C₄	—	2	1	no	—
434	C₂²×C₄	—	2	1	no	—
435	C₂²×C₄	—	2	2	no	—
436	C₂²×C₄	—	2	4	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	2	no	—
439	C₂²×C₄	—	2	2	no	—
440	C₂²×C₄	—	4	2	no	—
441	C₂²×C₄	—	4	2	no	—
442	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
443	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
444	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
445	C₂²×C₄	#41×C₆₇	1	4	yes	—
446	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
447	C₂×C₈	—	2	2	no	—
448	C₂×C₈	—	2	2	no	—
449	C₂×C₈	—	2	2	no	—
450	C₂×C₈	—	2	2	no	—
451	C₂×C₈	—	2	2	no	—
452	C₂×C₈	—	2	2	no	—
453	C₄⋊C₄	—	2	4	no	—
454	C₄⋊C₄	—	2	2	no	—
455	C₄⋊C₄	—	2	2	no	—
456	C₄⋊C₄	—	2	2	no	—
457	C₄⋊C₄	—	2	2	no	—
458	C₄⋊C₄	—	4	4	no	—
459	C₄⋊C₄	—	8	2	no	—
460	C₄⋊C₄	—	8	2	no	—
461	C₂²⋊C₄	—	2	8	no	—
462	C₂²⋊C₄	—	2	8	no	—
463	C₂²⋊C₄	—	2	4	no	—
464	C₂²⋊C₄	—	8	2	no	—
465	C₄²	—	2	1	no	—
466	C₄²	—	2	1	no	—
467	C₄²	C₂×D₄×C₆₇	1	2	yes	—
468	C₄²	C₂×D₄×C₆₇	1	2	yes	—
469	C₂³	—	2	1	no	—
470	C₂³	—	2	1	no	—
471	C₂³	—	2	2	no	—
472	C₂³	—	2	2	no	—
473	C₂³	—	2	2	no	—
474	C₂³	—	2	2	no	—
475	C₂³	—	2	2	no	—
476	C₂³	—	4	8	no	—
477	C₂³	—	4	1	no	—
478	C₂³	—	4	1	no	—
479	C₂³	—	4	1	no	—
480	C₂³	—	4	1	no	—
481	C₂³	—	4	1	no	—
482	C₂³	—	4	1	no	—
483	C₂³	—	4	2	no	—
484	C₂³	—	4	2	no	—
485	C₂³	—	4	2	no	—
486	C₂³	—	4	2	no	—
487	C₂³	—	4	2	no	—
488	C₂³	—	4	2	no	—
489	C₂³	—	8	8	no	—
490	C₂³	—	8	8	no	—
491	C₂³	—	8	4	no	—
492	C₂³	—	8	4	no	—
493	C₂³	#124×C₆₇	1	2	yes	—
494	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
495	Q₈	—	2	4	no	—
496	Q₈	—	4	2	no	—
497	D₄	—	2	4	no	—
498	D₄	—	2	4	no	—
499	D₄	—	2	4	no	—
500	D₄	—	4	8	no	—
501	D₄	—	4	8	no	—
502	D₄	—	4	2	no	—
503	D₄	—	4	2	no	—
504	D₄	—	4	2	no	—
505	D₄	—	8	8	no	—
506	D₄	—	8	8	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	2	4	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	4	8	no	—
518	C₂×C₄	—	4	1	no	—
519	C₂×C₄	—	4	1	no	—
520	C₂×C₄	—	4	2	no	—
521	C₂×C₄	—	4	2	no	—
522	C₂×C₄	—	4	2	no	—
523	C₂×C₄	—	4	2	no	—
524	C₂×C₄	—	4	2	no	—
525	C₂×C₄	—	4	2	no	—
526	C₂×C₄	#120×C₆₇	1	4	yes	—
527	C₂×C₄	#124×C₆₇	1	2	yes	—
528	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
529	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
530	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
531	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
532	C₈	—	4	2	no	—
533	C₈	—	4	2	no	—
534	C₂²	—	2	4	no	—
535	C₂²	—	2	2	no	—
536	C₂²	—	4	2	no	—
537	C₂²	—	4	1	no	—
538	C₂²	—	4	1	no	—
539	C₂²	—	4	4	no	—
540	C₂²	—	4	4	no	—
541	C₂²	—	4	4	no	—
542	C₂²	—	4	2	no	—
543	C₂²	—	4	2	no	—
544	C₂²	—	4	2	no	—
545	C₂²	—	8	16	no	—
546	C₂²	—	8	2	no	—
547	C₂²	—	8	2	no	—
548	C₂²	—	8	2	no	—
549	C₂²	—	8	1	no	—
550	C₂²	—	8	1	no	—
551	C₂²	—	8	1	no	—
552	C₂²	—	8	1	no	—
553	C₂²	—	8	1	no	—
554	C₂²	—	8	1	no	—
555	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
556	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
557	C₂²	#535×C₆₇	1	1	yes	characteristic
558	C₂²	#576×C₆₇	1	2	yes	—
559	C₂²	#577×C₆₇	1	2	yes	—
560	C₄	—	2	4	no	—
561	C₄	—	2	2	no	—
562	C₄	—	4	2	no	—
563	C₄	—	4	2	no	—
564	C₄	—	8	2	no	—
565	C₄	—	8	2	no	—
566	C₄	#546×C₆₇	1	2	yes	—
567	C₄	#546×C₆₇	1	2	yes	—
568	C₂	—	4	2	no	—
569	C₂	—	8	2	no	—
570	C₂	—	8	2	no	—
571	C₂	—	8	2	no	—
572	C₂	C₆₇×#2435	1	1	yes	characteristic
573	C₂	C₆₇×#3012	1	2	yes	—
574	C₂	C₆₇×#3136	1	2	yes	—
575	C₂	C₆₇×#3220	1	1	yes	characteristic
576	C₂	C₆₇×#3222	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#535×C₆₇	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#490×C₆₇	C₂	semidirect
3. C₂×#488×C₆₇	C₂	semidirect
4. C₆₇×#2629	C₂	semidirect
5. C₆₇×#2042	C₂	semidirect
6. C₆₇×#2041	C₂	semidirect
7. C₆₇×#1959	C₂	semidirect
8. C₆₇×#1957	C₂	semidirect
9. C₆₇×#1788	C₂	semidirect
10. C₆₇×#1786	C₂	semidirect
11. C₆₇×#1750	C₂	semidirect
18. C₂×#125×C₆₇	C₂²	semidirect
26. C₂×#103×C₆₇	C₂²	semidirect
27. C₂×#103×C₆₇	C₂²	semidirect
32. C₂×#102×C₆₇	C₂²	semidirect
33. C₂×#102×C₆₇	C₂²	semidirect
35. C₂×#97×C₆₇	C₂²	semidirect
36. C₂×C₄×C₈×C₆₇	C₂²	semidirect
39. #389×C₆₇	C₂²	semidirect
40. #384×C₆₇	C₂²	semidirect
42. #336×C₆₇	C₂²	semidirect
279. #25867	C₆₇	direct (N × H)
290. C₂×#490	C₂×C₆₇	semidirect
291. C₂×#488	C₂×C₆₇	semidirect
292. #2629	C₂×C₆₇	semidirect
293. #2042	C₂×C₆₇	semidirect
294. #2041	C₂×C₆₇	semidirect
295. #1959	C₂×C₆₇	semidirect
296. #1957	C₂×C₆₇	semidirect
297. #1788	C₂×C₆₇	semidirect
298. #1786	C₂×C₆₇	semidirect
299. #1750	C₂×C₆₇	semidirect
300. C₆₇	#25867	direct (N × H)
307. C₂×#125	C₂²×C₆₇	semidirect
315. C₂×#103	C₂²×C₆₇	semidirect
316. C₂×#103	C₂²×C₆₇	semidirect
321. C₂×#102	C₂²×C₆₇	semidirect
322. C₂×#102	C₂²×C₆₇	semidirect
324. C₂×#97	C₂²×C₆₇	semidirect
325. C₂×C₄×C₈	C₂²×C₆₇	semidirect
328. #389	C₂²×C₆₇	semidirect
329. #384	C₂²×C₆₇	semidirect
331. #336	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#535×C₆₇	2
C₂×#490×C₆₇	2
C₂×#488×C₆₇	2
C₆₇×#2629	2
C₆₇×#2042	2
C₆₇×#2041	2
C₆₇×#1959	2
C₆₇×#1957	2
C₆₇×#1788	2
C₆₇×#1786	2
C₆₇×#1750	2
#25867	67

Subgroups — C67×#25867 #30705118

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25867 #30705118