Subgroups — C₆₇×#21777 #30701028

Order 17152.

1318 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3485	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2883	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1913	C₂	1	4	yes	maximal, complemented
4	C₆₇×#1890	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1828	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1782	C₂	1	2	yes	maximal, complemented
7	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	1	yes	characteristic
8	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
9	#547×C₆₇	—	2	4	no	—
10	#546×C₆₇	—	2	4	no	—
11	#536×C₆₇	C₄	1	2	yes	complemented
12	#534×C₆₇	C₄	1	2	yes	complemented
13	#534×C₆₇	C₂²	1	2	yes	—
14	#525×C₆₇	C₄	1	2	yes	complemented
15	C₂×#123×C₆₇	C₄	1	2	yes	complemented
16	C₂×#123×C₆₇	C₂²	1	2	yes	—
17	#518×C₆₇	C₂²	1	4	yes	—
18	#518×C₆₇	C₂²	1	1	yes	characteristic
19	#410×C₆₇	—	2	4	no	—
20	C₂×#99×C₆₇	—	2	4	no	—
21	C₂×#99×C₆₇	C₂²	1	4	yes	—
22	C₂×#99×C₆₇	C₂²	1	4	yes	—
23	C₂×#99×C₆₇	C₂²	1	4	yes	complemented
24	#381×C₆₇	C₂²	1	4	yes	complemented
25	#342×C₆₇	C₂²	1	8	yes	complemented
26	H(5,𝔽₂)×C₆₇	—	4	4	no	—
27	C₂×#41×C₆₇	—	2	4	no	—
28	C₂×#41×C₆₇	—	4	4	no	—
29	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×#41×C₆₇	D₄	1	4	yes	—
31	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
32	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
33	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
34	C₂²×D₄×C₆₇	D₄	1	4	yes	—
35	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
36	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
37	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
38	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
39	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
40	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
41	C₄⋊Q₈×C₆₇	—	2	2	no	—
42	#124×C₆₇	—	2	2	no	—
43	#123×C₆₇	—	2	4	no	—
44	#123×C₆₇	—	2	4	no	—
45	#123×C₆₇	—	2	2	no	—
46	#123×C₆₇	—	2	2	no	—
47	#123×C₆₇	C₂×C₄	1	4	yes	complemented
48	#123×C₆₇	C₂×C₄	1	4	yes	complemented
49	#123×C₆₇	C₂×C₄	1	4	yes	complemented
50	C₂²⋊Q₈×C₆₇	—	2	2	no	—
51	C₂²⋊Q₈×C₆₇	—	2	4	no	—
52	#121×C₆₇	—	2	2	no	—
53	#121×C₆₇	—	2	4	no	—
54	#121×C₆₇	—	2	4	no	—
55	#120×C₆₇	—	2	2	no	—
56	#120×C₆₇	—	2	2	no	—
57	#120×C₆₇	C₂×C₄	1	4	yes	complemented
58	C₄×D₄×C₆₇	—	2	4	no	—
59	C₄×D₄×C₆₇	—	2	4	no	—
60	C₄×D₄×C₆₇	—	4	8	no	—
61	#117×C₆₇	—	2	4	no	—
62	#117×C₆₇	C₂×C₄	1	2	yes	complemented
63	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
64	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
66	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
68	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	complemented
69	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
70	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
71	#99×C₆₇	—	2	8	no	—
72	#99×C₆₇	—	2	8	no	—
73	#99×C₆₇	—	4	4	no	—
74	#95×C₆₇	—	4	4	no	—
75	C₂⁴×C₆₇	C₂²⋊C₄	1	2	yes	—
76	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
79	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
80	#41×C₆₇	—	4	4	no	—
81	#41×C₆₇	—	4	8	no	—
82	#41×C₆₇	—	4	16	no	—
83	C₂×Q₈×C₆₇	—	2	2	no	—
84	C₂×Q₈×C₆₇	—	4	2	no	—
85	C₂×Q₈×C₆₇	C₂²⋊C₄	1	2	yes	—
86	C₂×D₄×C₆₇	—	2	8	no	—
87	C₂×D₄×C₆₇	—	2	4	no	—
88	C₂×D₄×C₆₇	—	2	2	no	—
89	C₂×D₄×C₆₇	—	2	2	no	—
90	C₂×D₄×C₆₇	—	2	4	no	—
91	C₂×D₄×C₆₇	—	2	4	no	—
92	C₂×D₄×C₆₇	—	4	2	no	—
93	C₂×D₄×C₆₇	—	4	4	no	—
94	C₂×D₄×C₆₇	—	4	8	no	—
95	C₂×D₄×C₆₇	C₂²⋊C₄	1	4	yes	—
96	C₂×D₄×C₆₇	C₂²⋊C₄	1	2	yes	—
97	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
98	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
99	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
100	C₂²×C₄×C₆₇	—	2	2	no	—
101	C₂²×C₄×C₆₇	—	2	2	no	—
102	C₂²×C₄×C₆₇	—	2	4	no	—
103	C₂²×C₄×C₆₇	—	2	2	no	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	—	2	2	no	—
106	C₂²×C₄×C₆₇	—	2	8	no	—
107	C₂²×C₄×C₆₇	—	4	2	no	—
108	C₂²×C₄×C₆₇	—	4	2	no	—
109	C₂²×C₄×C₆₇	—	4	4	no	—
110	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
111	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
112	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
117	C₄⋊C₄×C₆₇	—	2	4	no	—
118	C₄⋊C₄×C₆₇	—	2	4	no	—
119	C₄⋊C₄×C₆₇	—	2	4	no	—
120	C₄⋊C₄×C₆₇	—	2	2	no	—
121	C₄⋊C₄×C₆₇	—	4	4	no	—
122	C₂²⋊C₄×C₆₇	—	2	4	no	—
123	C₂²⋊C₄×C₆₇	—	2	4	no	—
124	C₂²⋊C₄×C₆₇	—	2	4	no	—
125	C₂²⋊C₄×C₆₇	—	2	2	no	—
126	C₂²⋊C₄×C₆₇	—	2	2	no	—
127	C₂²⋊C₄×C₆₇	—	2	4	no	—
128	C₂²⋊C₄×C₆₇	—	2	8	no	—
129	C₂²⋊C₄×C₆₇	—	2	8	no	—
130	C₂²⋊C₄×C₆₇	—	4	4	no	—
131	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
132	C₄²×C₆₇	—	2	2	no	—
133	C₄²×C₆₇	—	4	4	no	—
134	C₂³×C₆₇	—	2	2	no	—
135	C₂³×C₆₇	—	2	4	no	—
136	C₂³×C₆₇	—	2	1	no	—
137	C₂³×C₆₇	—	2	2	no	—
138	C₂³×C₆₇	—	4	2	no	—
139	C₂³×C₆₇	—	8	1	no	—
140	C₂³×C₆₇	—	8	1	no	—
141	C₂³×C₆₇	—	8	1	no	—
142	C₂³×C₆₇	—	8	2	no	—
143	C₂³×C₆₇	—	8	2	no	—
144	C₂³×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
145	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
146	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
147	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
148	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
149	Q₈×C₆₇	—	4	2	no	—
150	Q₈×C₆₇	—	4	4	no	—
151	D₄×C₆₇	—	2	8	no	—
152	D₄×C₆₇	—	2	8	no	—
153	D₄×C₆₇	—	2	8	no	—
154	D₄×C₆₇	—	4	2	no	—
155	D₄×C₆₇	—	4	4	no	—
156	D₄×C₆₇	—	4	4	no	—
157	D₄×C₆₇	—	4	4	no	—
158	D₄×C₆₇	—	8	4	no	—
159	C₂×C₄×C₆₇	—	2	2	no	—
160	C₂×C₄×C₆₇	—	2	4	no	—
161	C₂×C₄×C₆₇	—	2	4	no	—
162	C₂×C₄×C₆₇	—	2	1	no	—
163	C₂×C₄×C₆₇	—	2	1	no	—
164	C₂×C₄×C₆₇	—	2	1	no	—
165	C₂×C₄×C₆₇	—	2	2	no	—
166	C₂×C₄×C₆₇	—	2	2	no	—
167	C₂×C₄×C₆₇	—	2	2	no	—
168	C₂×C₄×C₆₇	—	4	4	no	—
169	C₂×C₄×C₆₇	—	4	2	no	—
170	C₂×C₄×C₆₇	—	4	4	no	—
171	C₂×C₄×C₆₇	—	4	4	no	—
172	C₂×C₄×C₆₇	—	4	4	no	—
173	C₂×C₄×C₆₇	—	4	8	no	—
174	C₂×C₄×C₆₇	—	8	2	no	—
175	C₂×C₄×C₆₇	—	8	2	no	—
176	C₂×C₄×C₆₇	—	8	8	no	—
177	C₂×C₄×C₆₇	—	16	4	no	—
178	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
179	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
180	C₂²×C₆₇	—	2	2	no	—
181	C₂²×C₆₇	—	2	4	no	—
182	C₂²×C₆₇	—	4	4	no	—
183	C₂²×C₆₇	—	4	2	no	—
184	C₂²×C₆₇	—	4	1	no	—
185	C₂²×C₆₇	—	4	2	no	—
186	C₂²×C₆₇	—	4	4	no	—
187	C₂²×C₆₇	—	4	2	no	—
188	C₂²×C₆₇	—	4	1	no	—
189	C₂²×C₆₇	—	8	4	no	—
190	C₂²×C₆₇	—	8	2	no	—
191	C₂²×C₆₇	—	8	2	no	—
192	C₂²×C₆₇	—	16	2	no	—
193	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
194	C₂²×C₆₇	#518	1	2	yes	—
195	C₂²×C₆₇	#534	1	4	yes	—
196	C₄×C₆₇	—	2	2	no	—
197	C₄×C₆₇	—	2	4	no	—
198	C₄×C₆₇	—	4	4	no	—
199	C₄×C₆₇	—	4	1	no	—
200	C₄×C₆₇	—	4	1	no	—
201	C₄×C₆₇	—	4	1	no	—
202	C₄×C₆₇	—	8	8	no	—
203	#21777	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
204	C₂×C₆₇	—	2	1	no	—
205	C₂×C₆₇	—	2	4	no	—
206	C₂×C₆₇	—	2	2	no	—
207	C₂×C₆₇	—	4	2	no	—
208	C₂×C₆₇	—	4	4	no	—
209	C₂×C₆₇	—	8	1	no	—
210	C₂×C₆₇	#2308	1	1	yes	characteristic
211	#3485	C₂×C₆₇	1	1	yes	characteristic
212	#2883	C₂×C₆₇	1	2	yes	complemented
213	#1913	C₂×C₆₇	1	4	yes	complemented
214	#1890	C₂×C₆₇	1	4	yes	complemented
215	#1828	C₂×C₆₇	1	2	yes	complemented
216	#1782	C₂×C₆₇	1	2	yes	complemented
217	C₆₇	#21777	1	1	yes	characteristic, direct product factor, Sylow
218	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	1	yes	characteristic
219	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
220	#547	—	2	4	no	—
221	#546	—	2	4	no	—
222	#536	C₄×C₆₇	1	2	yes	complemented
223	#534	C₄×C₆₇	1	2	yes	complemented
224	#534	C₂²×C₆₇	1	2	yes	—
225	#525	C₄×C₆₇	1	2	yes	complemented
226	C₂×#123	C₄×C₆₇	1	2	yes	complemented
227	C₂×#123	C₂²×C₆₇	1	2	yes	—
228	#518	C₂²×C₆₇	1	4	yes	—
229	#518	C₂²×C₆₇	1	1	yes	characteristic
230	#410	—	2	4	no	—
231	C₂×#99	—	2	4	no	—
232	C₂×#99	C₂²×C₆₇	1	4	yes	—
233	C₂×#99	C₂²×C₆₇	1	4	yes	—
234	C₂×#99	C₂²×C₆₇	1	4	yes	complemented
235	#381	C₂²×C₆₇	1	4	yes	complemented
236	#342	C₂²×C₆₇	1	8	yes	complemented
237	H(5,𝔽₂)	—	4	4	no	—
238	C₂×#41	—	2	4	no	—
239	C₂×#41	—	4	4	no	—
240	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
241	C₂×#41	D₄×C₆₇	1	4	yes	—
242	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic, complemented
243	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
244	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic, complemented
245	C₂²×D₄	D₄×C₆₇	1	4	yes	—
246	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
247	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
248	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
249	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic, complemented
250	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic, complemented
251	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
252	C₄⋊Q₈	—	2	2	no	—
253	#124	—	2	2	no	—
254	#123	—	2	4	no	—
255	#123	—	2	4	no	—
256	#123	—	2	2	no	—
257	#123	—	2	2	no	—
258	#123	C₂×C₄×C₆₇	1	4	yes	complemented
259	#123	C₂×C₄×C₆₇	1	4	yes	complemented
260	#123	C₂×C₄×C₆₇	1	4	yes	complemented
261	C₂²⋊Q₈	—	2	2	no	—
262	C₂²⋊Q₈	—	2	4	no	—
263	#121	—	2	2	no	—
264	#121	—	2	4	no	—
265	#121	—	2	4	no	—
266	#120	—	2	2	no	—
267	#120	—	2	2	no	—
268	#120	C₂×C₄×C₆₇	1	4	yes	complemented
269	C₄×D₄	—	2	4	no	—
270	C₄×D₄	—	2	4	no	—
271	C₄×D₄	—	4	8	no	—
272	#117	—	2	4	no	—
273	#117	C₂×C₄×C₆₇	1	2	yes	complemented
274	C₂×C₄⋊C₄	—	2	2	no	—
275	C₂×C₄⋊C₄	—	2	2	no	—
276	C₂×C₂²⋊C₄	—	2	2	no	—
277	C₂×C₂²⋊C₄	—	2	2	no	—
278	C₂×C₂²⋊C₄	—	2	4	no	—
279	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	complemented
280	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
281	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
282	#99	—	2	8	no	—
283	#99	—	2	8	no	—
284	#99	—	4	4	no	—
285	#95	—	4	4	no	—
286	C₂⁴	C₂²⋊C₄×C₆₇	1	2	yes	—
287	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
288	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
289	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
290	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
291	#41	—	4	4	no	—
292	#41	—	4	8	no	—
293	#41	—	4	16	no	—
294	C₂×Q₈	—	2	2	no	—
295	C₂×Q₈	—	4	2	no	—
296	C₂×Q₈	C₂²⋊C₄×C₆₇	1	2	yes	—
297	C₂×D₄	—	2	8	no	—
298	C₂×D₄	—	2	4	no	—
299	C₂×D₄	—	2	2	no	—
300	C₂×D₄	—	2	2	no	—
301	C₂×D₄	—	2	4	no	—
302	C₂×D₄	—	2	4	no	—
303	C₂×D₄	—	4	2	no	—
304	C₂×D₄	—	4	4	no	—
305	C₂×D₄	—	4	8	no	—
306	C₂×D₄	C₂²⋊C₄×C₆₇	1	4	yes	—
307	C₂×D₄	C₂²⋊C₄×C₆₇	1	2	yes	—
308	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
309	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
310	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	4	no	—
314	C₂²×C₄	—	2	2	no	—
315	C₂²×C₄	—	2	2	no	—
316	C₂²×C₄	—	2	2	no	—
317	C₂²×C₄	—	2	8	no	—
318	C₂²×C₄	—	4	2	no	—
319	C₂²×C₄	—	4	2	no	—
320	C₂²×C₄	—	4	4	no	—
321	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
322	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
323	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
324	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
325	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
326	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
327	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
328	C₄⋊C₄	—	2	4	no	—
329	C₄⋊C₄	—	2	4	no	—
330	C₄⋊C₄	—	2	4	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	4	4	no	—
333	C₂²⋊C₄	—	2	4	no	—
334	C₂²⋊C₄	—	2	4	no	—
335	C₂²⋊C₄	—	2	4	no	—
336	C₂²⋊C₄	—	2	2	no	—
337	C₂²⋊C₄	—	2	2	no	—
338	C₂²⋊C₄	—	2	4	no	—
339	C₂²⋊C₄	—	2	8	no	—
340	C₂²⋊C₄	—	2	8	no	—
341	C₂²⋊C₄	—	4	4	no	—
342	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
343	C₄²	—	2	2	no	—
344	C₄²	—	4	4	no	—
345	C₂³	—	2	2	no	—
346	C₂³	—	2	4	no	—
347	C₂³	—	2	1	no	—
348	C₂³	—	2	2	no	—
349	C₂³	—	4	2	no	—
350	C₂³	—	8	1	no	—
351	C₂³	—	8	1	no	—
352	C₂³	—	8	1	no	—
353	C₂³	—	8	2	no	—
354	C₂³	—	8	2	no	—
355	C₂³	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
356	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
357	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
358	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
359	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
360	Q₈	—	4	2	no	—
361	Q₈	—	4	4	no	—
362	D₄	—	2	8	no	—
363	D₄	—	2	8	no	—
364	D₄	—	2	8	no	—
365	D₄	—	4	2	no	—
366	D₄	—	4	4	no	—
367	D₄	—	4	4	no	—
368	D₄	—	4	4	no	—
369	D₄	—	8	4	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	4	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	2	1	no	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	4	4	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	4	no	—
382	C₂×C₄	—	4	4	no	—
383	C₂×C₄	—	4	4	no	—
384	C₂×C₄	—	4	8	no	—
385	C₂×C₄	—	8	2	no	—
386	C₂×C₄	—	8	2	no	—
387	C₂×C₄	—	8	8	no	—
388	C₂×C₄	—	16	4	no	—
389	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
390	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
391	C₂²	—	2	2	no	—
392	C₂²	—	2	4	no	—
393	C₂²	—	4	4	no	—
394	C₂²	—	4	2	no	—
395	C₂²	—	4	1	no	—
396	C₂²	—	4	2	no	—
397	C₂²	—	4	4	no	—
398	C₂²	—	4	2	no	—
399	C₂²	—	4	1	no	—
400	C₂²	—	8	4	no	—
401	C₂²	—	8	2	no	—
402	C₂²	—	8	2	no	—
403	C₂²	—	16	2	no	—
404	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
405	C₂²	#518×C₆₇	1	2	yes	—
406	C₂²	#534×C₆₇	1	4	yes	—
407	C₄	—	2	2	no	—
408	C₄	—	2	4	no	—
409	C₄	—	4	4	no	—
410	C₄	—	4	1	no	—
411	C₄	—	4	1	no	—
412	C₄	—	4	1	no	—
413	C₄	—	8	8	no	—
414	C₂	—	2	1	no	—
415	C₂	—	2	4	no	—
416	C₂	—	2	2	no	—
417	C₂	—	4	2	no	—
418	C₂	—	4	4	no	—
419	C₂	—	8	1	no	—
420	C₂	C₆₇×#2308	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₇	#2308
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂⁴	C₂⁴×C₆₇
prime radical	C₆₇×#3485	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₇ ◁ C₂⁴×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2883	C₂	semidirect
3. C₆₇×#1913	C₂	semidirect
4. C₆₇×#1890	C₂	semidirect
5. C₆₇×#1828	C₂	semidirect
6. C₆₇×#1782	C₂	semidirect
11. #536×C₆₇	C₄	semidirect
12. #534×C₆₇	C₄	semidirect
14. #525×C₆₇	C₄	semidirect
15. C₂×#123×C₆₇	C₄	semidirect
23. C₂×#99×C₆₇	C₂²	semidirect
24. #381×C₆₇	C₂²	semidirect
25. #342×C₆₇	C₂²	semidirect
31. C₂²×Q₈×C₆₇	C₂×C₄	semidirect
33. C₂²×D₄×C₆₇	C₂×C₄	semidirect
38. C₂³×C₄×C₆₇	C₂×C₄	semidirect
39. C₂³×C₄×C₆₇	C₂×C₄	semidirect
47. #123×C₆₇	C₂×C₄	semidirect
48. #123×C₆₇	C₂×C₄	semidirect
49. #123×C₆₇	C₂×C₄	semidirect
57. #120×C₆₇	C₂×C₄	semidirect
62. #117×C₆₇	C₂×C₄	semidirect
68. C₂×C₂²⋊C₄×C₆₇	C₂×C₄	semidirect
203. #21777	C₆₇	direct (N × H)
212. #2883	C₂×C₆₇	semidirect
213. #1913	C₂×C₆₇	semidirect
214. #1890	C₂×C₆₇	semidirect
215. #1828	C₂×C₆₇	semidirect
216. #1782	C₂×C₆₇	semidirect
217. C₆₇	#21777	direct (N × H)
222. #536	C₄×C₆₇	semidirect
223. #534	C₄×C₆₇	semidirect
225. #525	C₄×C₆₇	semidirect
226. C₂×#123	C₄×C₆₇	semidirect
234. C₂×#99	C₂²×C₆₇	semidirect
235. #381	C₂²×C₆₇	semidirect
236. #342	C₂²×C₆₇	semidirect
242. C₂²×Q₈	C₂×C₄×C₆₇	semidirect
244. C₂²×D₄	C₂×C₄×C₆₇	semidirect
249. C₂³×C₄	C₂×C₄×C₆₇	semidirect
250. C₂³×C₄	C₂×C₄×C₆₇	semidirect
258. #123	C₂×C₄×C₆₇	semidirect
259. #123	C₂×C₄×C₆₇	semidirect
260. #123	C₂×C₄×C₆₇	semidirect
268. #120	C₂×C₄×C₆₇	semidirect
273. #117	C₂×C₄×C₆₇	semidirect
279. C₂×C₂²⋊C₄	C₂×C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3485	2
C₆₇×#2883	2
C₆₇×#1913	2
C₆₇×#1890	2
C₆₇×#1828	2
C₆₇×#1782	2
#21777	67

Subgroups — C67×#21777 #30701028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21777 #30701028