Subgroups — C₆₇×#13911 #30693162

Order 17152.

1486 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#384×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#382×C₆₇	C₂	1	2	yes	maximal
3	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1442	C₂	1	4	yes	maximal
6	C₆₇×C₄⋊#95	C₂	1	4	yes	maximal
7	C₆₇×C₄⋊#95	C₂	1	2	yes	maximal
8	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
9	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
10	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
11	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
14	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	#384×C₆₇	—	2	4	no	—
18	#382×C₆₇	—	2	8	no	—
19	C₄×C₄⋊C₄×C₆₇	—	2	4	no	—
20	C₄×C₄⋊C₄×C₆₇	C₄	1	8	yes	complemented
21	C₂×#95×C₆₇	C₂²	1	2	yes	—
22	C₂×#95×C₆₇	C₂²	1	8	yes	—
23	C₂×#95×C₆₇	C₂²	1	8	yes	—
24	C₂×#95×C₆₇	C₂²	1	2	yes	—
25	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
26	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
27	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
28	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
29	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
30	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
31	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
33	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
35	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
36	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
39	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
44	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
45	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
46	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	—
47	C₂×C₄⋊C₄×C₆₇	Q₈	1	4	yes	—
48	C₂×C₄²×C₆₇	—	2	4	no	—
49	C₂×C₄²×C₆₇	—	2	4	no	—
50	C₂×C₄²×C₆₇	—	2	4	no	—
51	C₂×C₄²×C₆₇	—	2	8	no	—
52	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	—
53	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
54	C₂×C₄²×C₆₇	D₄	1	4	yes	—
55	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
56	#95×C₆₇	—	2	4	no	—
57	#95×C₆₇	—	2	16	no	—
58	#95×C₆₇	—	4	8	no	—
59	#95×C₆₇	C₂×C₄	1	8	yes	—
60	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	—	2	2	no	—
62	C₂²×C₄×C₆₇	—	2	2	no	—
63	C₂²×C₄×C₆₇	—	2	2	no	—
64	C₂²×C₄×C₆₇	—	2	2	no	—
65	C₂²×C₄×C₆₇	—	2	4	no	—
66	C₂²×C₄×C₆₇	—	2	4	no	—
67	C₂²×C₄×C₆₇	—	2	4	no	—
68	C₂²×C₄×C₆₇	—	2	4	no	—
69	C₂²×C₄×C₆₇	—	2	4	no	—
70	C₂²×C₄×C₆₇	—	2	4	no	—
71	C₂²×C₄×C₆₇	—	2	2	no	—
72	C₂²×C₄×C₆₇	—	2	2	no	—
73	C₂²×C₄×C₆₇	—	2	4	no	—
74	C₂²×C₄×C₆₇	—	2	4	no	—
75	C₂²×C₄×C₆₇	—	2	4	no	—
76	C₂²×C₄×C₆₇	—	2	4	no	—
77	C₂²×C₄×C₆₇	—	2	4	no	—
78	C₂²×C₄×C₆₇	—	2	4	no	—
79	C₂²×C₄×C₆₇	—	2	4	no	—
80	C₂²×C₄×C₆₇	—	2	8	no	—
81	C₂²×C₄×C₆₇	—	2	8	no	—
82	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
83	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
87	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
92	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
93	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
94	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
95	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
96	C₂²×C₄×C₆₇	#41	1	4	yes	—
97	C₂²×C₄×C₆₇	#41	1	4	yes	—
98	C₂²×C₄×C₆₇	#41	1	2	yes	—
99	C₂²×C₄×C₆₇	#41	1	2	yes	—
100	C₄⋊C₄×C₆₇	—	2	16	no	—
101	C₄⋊C₄×C₆₇	—	2	8	no	—
102	C₄⋊C₄×C₆₇	—	2	8	no	—
103	C₄⋊C₄×C₆₇	—	4	8	no	—
104	C₄⋊C₄×C₆₇	—	4	8	no	—
105	C₄⋊C₄×C₆₇	—	4	8	no	—
106	C₄²×C₆₇	—	2	16	no	—
107	C₄²×C₆₇	—	4	8	no	—
108	C₄²×C₆₇	C₄⋊C₄	1	16	yes	—
109	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
110	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
111	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
112	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
113	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
114	C₂³×C₆₇	C₂×#41	1	2	yes	—
115	C₂³×C₆₇	C₂×#41	1	2	yes	—
116	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
117	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
118	C₂³×C₆₇	H(5,𝔽₂)	1	2	yes	—
119	C₂³×C₆₇	2₋¹⁺⁴	1	2	yes	—
120	C₂×C₄×C₆₇	—	2	4	no	—
121	C₂×C₄×C₆₇	—	2	2	no	—
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	—	2	4	no	—
124	C₂×C₄×C₆₇	—	2	8	no	—
125	C₂×C₄×C₆₇	—	2	8	no	—
126	C₂×C₄×C₆₇	—	2	8	no	—
127	C₂×C₄×C₆₇	—	2	4	no	—
128	C₂×C₄×C₆₇	—	2	4	no	—
129	C₂×C₄×C₆₇	—	2	8	no	—
130	C₂×C₄×C₆₇	—	2	8	no	—
131	C₂×C₄×C₆₇	—	4	2	no	—
132	C₂×C₄×C₆₇	—	4	2	no	—
133	C₂×C₄×C₆₇	—	4	2	no	—
134	C₂×C₄×C₆₇	—	4	2	no	—
135	C₂×C₄×C₆₇	—	4	4	no	—
136	C₂×C₄×C₆₇	—	4	4	no	—
137	C₂×C₄×C₆₇	—	4	4	no	—
138	C₂×C₄×C₆₇	—	4	4	no	—
139	C₂×C₄×C₆₇	—	4	4	no	—
140	C₂×C₄×C₆₇	—	4	4	no	—
141	C₂×C₄×C₆₇	—	4	4	no	—
142	C₂×C₄×C₆₇	—	4	4	no	—
143	C₂×C₄×C₆₇	—	4	4	no	—
144	C₂×C₄×C₆₇	—	4	8	no	—
145	C₂×C₄×C₆₇	—	4	8	no	—
146	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
147	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
148	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
149	C₂×C₄×C₆₇	#117	1	8	yes	—
150	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
151	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
152	C₂×C₄×C₆₇	#121	1	4	yes	—
153	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
154	C₂×C₄×C₆₇	#124	1	4	yes	—
155	C₂×C₄×C₆₇	#124	1	4	yes	—
156	C₂×C₄×C₆₇	#125	1	8	yes	—
157	C₂×C₄×C₆₇	#125	1	4	yes	—
158	C₂×C₄×C₆₇	C₄⋊Q₈	1	4	yes	—
159	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
160	C₂²×C₆₇	C₂×#117	1	2	yes	—
161	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
162	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
163	C₂²×C₆₇	#520	1	2	yes	—
164	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
165	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
166	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
167	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
168	C₂²×C₆₇	C₂×#125	1	1	yes	characteristic
169	C₂²×C₆₇	C₂×#125	1	2	yes	—
170	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
171	C₂²×C₆₇	#535	1	1	yes	characteristic
172	C₂²×C₆₇	#536	1	1	yes	characteristic
173	C₂²×C₆₇	#544	1	2	yes	—
174	C₂²×C₆₇	#547	1	2	yes	—
175	C₂²×C₆₇	#550	1	2	yes	—
176	C₂²×C₆₇	#554	1	2	yes	—
177	C₂²×C₆₇	#555	1	2	yes	—
178	C₂²×C₆₇	#556	1	2	yes	—
179	C₂²×C₆₇	#556	1	2	yes	—
180	C₂²×C₆₇	#557	1	2	yes	—
181	C₂²×C₆₇	#559	1	2	yes	—
182	C₄×C₆₇	—	2	4	no	—
183	C₄×C₆₇	—	4	4	no	—
184	C₄×C₆₇	—	4	8	no	—
185	C₄×C₆₇	—	4	4	no	—
186	C₄×C₆₇	—	8	4	no	—
187	C₄×C₆₇	#382	1	8	yes	complemented
188	C₄×C₆₇	#389	1	8	yes	—
189	#13911	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
190	C₂×C₆₇	C₂×#382	1	1	yes	characteristic
191	C₂×C₆₇	C₂×#389	1	1	yes	characteristic
192	C₂×C₆₇	#2317	1	1	yes	characteristic
193	C₂×C₆₇	#2354	1	2	yes	—
194	C₂×C₆₇	#2355	1	2	yes	—
195	C₂×C₆₇	#2547	1	1	yes	characteristic
196	C₂×C₆₇	#2548	1	1	yes	characteristic
197	C₂×C₆₇	#2552	1	2	yes	—
198	C₂×C₆₇	#2563	1	1	yes	characteristic
199	C₂×C₆₇	#2568	1	1	yes	characteristic
200	C₂×C₆₇	#2571	1	2	yes	—
201	C₂×#384	C₂×C₆₇	1	1	yes	characteristic
202	C₂×#382	C₂×C₆₇	1	2	yes	—
203	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
204	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
205	#1442	C₂×C₆₇	1	4	yes	—
206	C₄⋊#95	C₂×C₆₇	1	4	yes	—
207	C₄⋊#95	C₂×C₆₇	1	2	yes	—
208	C₆₇	#13911	1	1	yes	characteristic, direct product factor, Sylow
209	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
210	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
211	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
212	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
213	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
214	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
215	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
216	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
217	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
218	#384	—	2	4	no	—
219	#382	—	2	8	no	—
220	C₄×C₄⋊C₄	—	2	4	no	—
221	C₄×C₄⋊C₄	C₄×C₆₇	1	8	yes	complemented
222	C₂×#95	C₂²×C₆₇	1	8	yes	—
223	C₂×#95	C₂²×C₆₇	1	2	yes	—
224	C₂×#95	C₂²×C₆₇	1	8	yes	—
225	C₂×#95	C₂²×C₆₇	1	2	yes	—
226	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
227	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
228	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
229	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
230	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
231	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
232	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
233	C₂×C₄⋊C₄	—	2	2	no	—
234	C₂×C₄⋊C₄	—	2	2	no	—
235	C₂×C₄⋊C₄	—	2	4	no	—
236	C₂×C₄⋊C₄	—	2	4	no	—
237	C₂×C₄⋊C₄	—	2	2	no	—
238	C₂×C₄⋊C₄	—	2	2	no	—
239	C₂×C₄⋊C₄	—	2	4	no	—
240	C₂×C₄⋊C₄	—	2	4	no	—
241	C₂×C₄⋊C₄	—	2	4	no	—
242	C₂×C₄⋊C₄	—	2	4	no	—
243	C₂×C₄⋊C₄	—	2	4	no	—
244	C₂×C₄⋊C₄	—	2	8	no	—
245	C₂×C₄⋊C₄	—	4	4	no	—
246	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	8	yes	—
247	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	—
248	C₂×C₄⋊C₄	Q₈×C₆₇	1	4	yes	—
249	C₂×C₄²	—	2	4	no	—
250	C₂×C₄²	—	2	4	no	—
251	C₂×C₄²	—	2	4	no	—
252	C₂×C₄²	—	2	8	no	—
253	C₂×C₄²	C₂×C₄×C₆₇	1	8	yes	—
254	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
255	C₂×C₄²	D₄×C₆₇	1	4	yes	—
256	C₂×C₄²	Q₈×C₆₇	1	4	yes	—
257	#95	—	2	4	no	—
258	#95	—	2	16	no	—
259	#95	—	4	8	no	—
260	#95	C₂×C₄×C₆₇	1	8	yes	—
261	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	4	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	4	no	—
268	C₂²×C₄	—	2	4	no	—
269	C₂²×C₄	—	2	4	no	—
270	C₂²×C₄	—	2	4	no	—
271	C₂²×C₄	—	2	4	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	2	4	no	—
275	C₂²×C₄	—	2	4	no	—
276	C₂²×C₄	—	2	4	no	—
277	C₂²×C₄	—	2	4	no	—
278	C₂²×C₄	—	2	4	no	—
279	C₂²×C₄	—	2	4	no	—
280	C₂²×C₄	—	2	4	no	—
281	C₂²×C₄	—	2	8	no	—
282	C₂²×C₄	—	2	8	no	—
283	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
284	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
285	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
286	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
287	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
288	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
289	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
290	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
291	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
292	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
293	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
294	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
295	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
296	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
297	C₂²×C₄	#41×C₆₇	1	4	yes	—
298	C₂²×C₄	#41×C₆₇	1	4	yes	—
299	C₂²×C₄	#41×C₆₇	1	2	yes	—
300	C₂²×C₄	#41×C₆₇	1	2	yes	—
301	C₄⋊C₄	—	2	16	no	—
302	C₄⋊C₄	—	2	8	no	—
303	C₄⋊C₄	—	2	8	no	—
304	C₄⋊C₄	—	4	8	no	—
305	C₄⋊C₄	—	4	8	no	—
306	C₄⋊C₄	—	4	8	no	—
307	C₄²	—	2	16	no	—
308	C₄²	—	4	8	no	—
309	C₄²	C₄⋊C₄×C₆₇	1	16	yes	—
310	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
311	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
312	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
313	C₂³	C₂²×Q₈×C₆₇	1	1	yes	characteristic
314	C₂³	C₂²×Q₈×C₆₇	1	1	yes	characteristic
315	C₂³	C₂×#41×C₆₇	1	2	yes	—
316	C₂³	C₂×#41×C₆₇	1	2	yes	—
317	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
318	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
319	C₂³	H(5,𝔽₂)×C₆₇	1	2	yes	—
320	C₂³	2₋¹⁺⁴×C₆₇	1	2	yes	—
321	C₂×C₄	—	2	4	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	4	no	—
325	C₂×C₄	—	2	8	no	—
326	C₂×C₄	—	2	8	no	—
327	C₂×C₄	—	2	8	no	—
328	C₂×C₄	—	2	4	no	—
329	C₂×C₄	—	2	4	no	—
330	C₂×C₄	—	2	8	no	—
331	C₂×C₄	—	2	8	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	4	no	—
337	C₂×C₄	—	4	4	no	—
338	C₂×C₄	—	4	4	no	—
339	C₂×C₄	—	4	4	no	—
340	C₂×C₄	—	4	4	no	—
341	C₂×C₄	—	4	4	no	—
342	C₂×C₄	—	4	4	no	—
343	C₂×C₄	—	4	4	no	—
344	C₂×C₄	—	4	4	no	—
345	C₂×C₄	—	4	8	no	—
346	C₂×C₄	—	4	8	no	—
347	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
348	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
349	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
350	C₂×C₄	#117×C₆₇	1	8	yes	—
351	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
352	C₂×C₄	C₄×Q₈×C₆₇	1	4	yes	—
353	C₂×C₄	#121×C₆₇	1	4	yes	—
354	C₂×C₄	C₂²⋊Q₈×C₆₇	1	4	yes	—
355	C₂×C₄	#124×C₆₇	1	4	yes	—
356	C₂×C₄	#124×C₆₇	1	4	yes	—
357	C₂×C₄	#125×C₆₇	1	8	yes	—
358	C₂×C₄	#125×C₆₇	1	4	yes	—
359	C₂×C₄	C₄⋊Q₈×C₆₇	1	4	yes	—
360	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
361	C₂²	C₂×#117×C₆₇	1	2	yes	—
362	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
363	C₂²	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
364	C₂²	#520×C₆₇	1	2	yes	—
365	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
366	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
367	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
368	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
369	C₂²	C₂×#125×C₆₇	1	2	yes	—
370	C₂²	C₂×#125×C₆₇	1	1	yes	characteristic
371	C₂²	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
372	C₂²	#535×C₆₇	1	1	yes	characteristic
373	C₂²	#536×C₆₇	1	1	yes	characteristic
374	C₂²	#544×C₆₇	1	2	yes	—
375	C₂²	#547×C₆₇	1	2	yes	—
376	C₂²	#550×C₆₇	1	2	yes	—
377	C₂²	#554×C₆₇	1	2	yes	—
378	C₂²	#555×C₆₇	1	2	yes	—
379	C₂²	#556×C₆₇	1	2	yes	—
380	C₂²	#556×C₆₇	1	2	yes	—
381	C₂²	#557×C₆₇	1	2	yes	—
382	C₂²	#559×C₆₇	1	2	yes	—
383	C₄	—	2	4	no	—
384	C₄	—	4	4	no	—
385	C₄	—	4	4	no	—
386	C₄	—	4	8	no	—
387	C₄	—	8	4	no	—
388	C₄	#382×C₆₇	1	8	yes	complemented
389	C₄	#389×C₆₇	1	8	yes	—
390	C₂	C₂×#382×C₆₇	1	1	yes	characteristic
391	C₂	C₂×#389×C₆₇	1	1	yes	characteristic
392	C₂	C₆₇×#2317	1	1	yes	characteristic
393	C₂	C₆₇×#2354	1	2	yes	—
394	C₂	C₆₇×#2355	1	2	yes	—
395	C₂	C₆₇×#2547	1	1	yes	characteristic
396	C₂	C₆₇×#2548	1	1	yes	characteristic
397	C₂	C₆₇×#2552	1	2	yes	—
398	C₂	C₆₇×#2563	1	1	yes	characteristic
399	C₂	C₆₇×#2568	1	1	yes	characteristic
400	C₂	C₆₇×#2571	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₆₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂⁴	C₂⁴×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
20. C₄×C₄⋊C₄×C₆₇	C₄	semidirect
187. C₄×C₆₇	#382	semidirect
189. #13911	C₆₇	direct (N × H)
208. C₆₇	#13911	direct (N × H)
221. C₄×C₄⋊C₄	C₄×C₆₇	semidirect
388. C₄	#382×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#384×C₆₇	2
C₂×#382×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₆₇×#1442	2
C₆₇×C₄⋊#95	2
#13911	67

Subgroups — C67×#13911 #30693162

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#13911 #30693162