Subgroups — C₆₇×#13910 #30693161

Order 17152.

1774 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#389×C₆₇	C₂	1	4	yes	maximal
2	C₂×#384×C₆₇	C₂	1	2	yes	maximal
3	C₂×#384×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1444	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	8	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×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
17	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	#389×C₆₇	—	2	16	no	—
19	#384×C₆₇	—	2	8	no	—
20	#384×C₆₇	—	2	4	no	—
21	C₄×C₄⋊C₄×C₆₇	—	2	4	no	—
22	C₄×C₄⋊C₄×C₆₇	C₄	1	8	yes	complemented
23	C₂×#95×C₆₇	C₂²	1	2	yes	—
24	C₂×#95×C₆₇	C₂²	1	8	yes	—
25	C₂×#95×C₆₇	C₂²	1	2	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	4	yes	—
30	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
31	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
32	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₆₇	—	2	16	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	16	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
44	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
45	C₂×C₄⋊C₄×C₆₇	—	4	8	no	—
46	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
47	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
48	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	—
49	C₂×C₄⋊C₄×C₆₇	D₄	1	8	yes	—
50	C₂×C₄⋊C₄×C₆₇	Q₈	1	4	yes	—
51	C₂×C₄⋊C₄×C₆₇	Q₈	1	8	yes	—
52	C₂×C₄²×C₆₇	—	2	4	no	—
53	C₂×C₄²×C₆₇	—	2	4	no	—
54	C₂×C₄²×C₆₇	—	2	4	no	—
55	C₂×C₄²×C₆₇	—	2	8	no	—
56	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	—
57	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
58	C₂×C₄²×C₆₇	D₄	1	4	yes	—
59	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
60	#95×C₆₇	—	2	4	no	—
61	#95×C₆₇	—	2	16	no	—
62	#95×C₆₇	C₂×C₄	1	8	yes	—
63	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	—	2	4	no	—
65	C₂²×C₄×C₆₇	—	2	4	no	—
66	C₂²×C₄×C₆₇	—	2	2	no	—
67	C₂²×C₄×C₆₇	—	2	4	no	—
68	C₂²×C₄×C₆₇	—	2	2	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	2	no	—
75	C₂²×C₄×C₆₇	—	2	2	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	4	no	—
82	C₂²×C₄×C₆₇	—	2	8	no	—
83	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
84	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
88	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
90	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
93	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
95	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
96	C₂²×C₄×C₆₇	C₂×Q₈	1	8	yes	—
97	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
98	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
99	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
100	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
101	C₂²×C₄×C₆₇	#41	1	2	yes	—
102	C₂²×C₄×C₆₇	#41	1	2	yes	—
103	C₄⋊C₄×C₆₇	—	2	8	no	—
104	C₄⋊C₄×C₆₇	—	2	8	no	—
105	C₄⋊C₄×C₆₇	—	4	32	no	—
106	C₄⋊C₄×C₆₇	—	4	8	no	—
107	C₄⋊C₄×C₆₇	—	4	8	no	—
108	C₄⋊C₄×C₆₇	—	4	8	no	—
109	C₄⋊C₄×C₆₇	C₄⋊C₄	1	32	yes	complemented
110	C₄²×C₆₇	—	2	16	no	—
111	C₄²×C₆₇	—	4	8	no	—
112	C₄²×C₆₇	C₄⋊C₄	1	16	yes	complemented
113	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
114	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
115	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
116	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
117	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
118	C₂³×C₆₇	C₂²×Q₈	1	2	yes	—
119	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
120	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
121	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
122	C₂³×C₆₇	H(5,𝔽₂)	1	2	yes	—
123	C₂³×C₆₇	2₋¹⁺⁴	1	2	yes	—
124	C₂×C₄×C₆₇	—	2	8	no	—
125	C₂×C₄×C₆₇	—	2	4	no	—
126	C₂×C₄×C₆₇	—	2	8	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	2	2	no	—
129	C₂×C₄×C₆₇	—	2	4	no	—
130	C₂×C₄×C₆₇	—	2	8	no	—
131	C₂×C₄×C₆₇	—	2	8	no	—
132	C₂×C₄×C₆₇	—	2	8	no	—
133	C₂×C₄×C₆₇	—	2	8	no	—
134	C₂×C₄×C₆₇	—	2	4	no	—
135	C₂×C₄×C₆₇	—	2	4	no	—
136	C₂×C₄×C₆₇	—	2	8	no	—
137	C₂×C₄×C₆₇	—	2	8	no	—
138	C₂×C₄×C₆₇	—	4	2	no	—
139	C₂×C₄×C₆₇	—	4	2	no	—
140	C₂×C₄×C₆₇	—	4	2	no	—
141	C₂×C₄×C₆₇	—	4	2	no	—
142	C₂×C₄×C₆₇	—	4	4	no	—
143	C₂×C₄×C₆₇	—	4	4	no	—
144	C₂×C₄×C₆₇	—	4	4	no	—
145	C₂×C₄×C₆₇	—	4	4	no	—
146	C₂×C₄×C₆₇	—	4	4	no	—
147	C₂×C₄×C₆₇	—	4	8	no	—
148	C₂×C₄×C₆₇	—	4	8	no	—
149	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
150	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	16	yes	—
151	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
152	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	8	yes	—
153	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
154	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
155	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
156	C₂×C₄×C₆₇	#121	1	4	yes	—
157	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
158	C₂×C₄×C₆₇	#125	1	4	yes	—
159	C₂×C₄×C₆₇	#127	1	4	yes	—
160	C₂×C₄×C₆₇	C₄⋊Q₈	1	4	yes	—
161	C₂×C₄×C₆₇	C₄⋊Q₈	1	8	yes	—
162	C₂×C₄×C₆₇	C₄⋊Q₈	1	4	yes	—
163	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
164	C₂²×C₆₇	C₂²×C₄⋊C₄	1	2	yes	—
165	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
166	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
167	C₂²×C₆₇	#520	1	2	yes	—
168	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
169	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
170	C₂²×C₆₇	C₂×#125	1	1	yes	characteristic
171	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
172	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
173	C₂²×C₆₇	C₂×C₄⋊Q₈	1	2	yes	—
174	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
175	C₂²×C₆₇	#535	1	1	yes	characteristic
176	C₂²×C₆₇	#536	1	1	yes	characteristic
177	C₂²×C₆₇	#544	1	2	yes	—
178	C₂²×C₆₇	D₄²	1	2	yes	—
179	C₂²×C₆₇	#547	1	2	yes	—
180	C₂²×C₆₇	D₄×Q₈	1	2	yes	—
181	C₂²×C₆₇	D₄×Q₈	1	2	yes	—
182	C₂²×C₆₇	#550	1	2	yes	—
183	C₂²×C₆₇	#554	1	2	yes	—
184	C₂²×C₆₇	#557	1	2	yes	—
185	C₂²×C₆₇	Q₈²	1	2	yes	—
186	C₄×C₆₇	—	2	4	no	—
187	C₄×C₆₇	—	2	16	no	—
188	C₄×C₆₇	—	4	4	no	—
189	C₄×C₆₇	—	4	4	no	—
190	C₄×C₆₇	—	8	4	no	—
191	C₄×C₆₇	#384	1	8	yes	complemented
192	C₄×C₆₇	#389	1	8	yes	complemented
193	#13910	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
194	C₂×C₆₇	C₂×#384	1	1	yes	characteristic
195	C₂×C₆₇	C₂×#389	1	1	yes	characteristic
196	C₂×C₆₇	#2317	1	1	yes	characteristic
197	C₂×C₆₇	D₄×C₄⋊C₄	1	2	yes	—
198	C₂×C₆₇	Q₈×C₄⋊C₄	1	2	yes	—
199	C₂×C₆₇	#2541	1	1	yes	characteristic
200	C₂×C₆₇	#2542	1	1	yes	characteristic
201	C₂×C₆₇	#2546	1	1	yes	characteristic
202	C₂×C₆₇	#2551	1	2	yes	—
203	C₂×C₆₇	#2552	1	2	yes	—
204	C₂×C₆₇	#2567	1	1	yes	characteristic
205	C₂×#389	C₂×C₆₇	1	4	yes	—
206	C₂×#384	C₂×C₆₇	1	1	yes	characteristic
207	C₂×#384	C₂×C₆₇	1	2	yes	—
208	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
209	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
210	#1444	C₂×C₆₇	1	4	yes	—
211	C₄⋊#95	C₂×C₆₇	1	2	yes	—
212	C₆₇	#13910	1	1	yes	characteristic, direct product factor, Sylow
213	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
214	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
215	C₂²×C₄⋊C₄	C₂²×C₆₇	1	8	yes	—
216	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
217	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
218	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
219	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
220	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
221	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
222	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
223	#389	—	2	16	no	—
224	#384	—	2	8	no	—
225	#384	—	2	4	no	—
226	C₄×C₄⋊C₄	—	2	4	no	—
227	C₄×C₄⋊C₄	C₄×C₆₇	1	8	yes	complemented
228	C₂×#95	C₂²×C₆₇	1	8	yes	—
229	C₂×#95	C₂²×C₆₇	1	2	yes	—
230	C₂×#95	C₂²×C₆₇	1	2	yes	—
231	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
232	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
233	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
234	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
235	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
236	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
237	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
238	C₂×C₄⋊C₄	—	2	2	no	—
239	C₂×C₄⋊C₄	—	2	2	no	—
240	C₂×C₄⋊C₄	—	2	2	no	—
241	C₂×C₄⋊C₄	—	2	2	no	—
242	C₂×C₄⋊C₄	—	2	16	no	—
243	C₂×C₄⋊C₄	—	2	4	no	—
244	C₂×C₄⋊C₄	—	2	4	no	—
245	C₂×C₄⋊C₄	—	2	4	no	—
246	C₂×C₄⋊C₄	—	2	4	no	—
247	C₂×C₄⋊C₄	—	2	16	no	—
248	C₂×C₄⋊C₄	—	2	8	no	—
249	C₂×C₄⋊C₄	—	2	4	no	—
250	C₂×C₄⋊C₄	—	4	8	no	—
251	C₂×C₄⋊C₄	—	4	4	no	—
252	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	8	yes	—
253	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	—
254	C₂×C₄⋊C₄	D₄×C₆₇	1	8	yes	—
255	C₂×C₄⋊C₄	Q₈×C₆₇	1	4	yes	—
256	C₂×C₄⋊C₄	Q₈×C₆₇	1	8	yes	—
257	C₂×C₄²	—	2	4	no	—
258	C₂×C₄²	—	2	4	no	—
259	C₂×C₄²	—	2	4	no	—
260	C₂×C₄²	—	2	8	no	—
261	C₂×C₄²	C₂×C₄×C₆₇	1	8	yes	—
262	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
263	C₂×C₄²	D₄×C₆₇	1	4	yes	—
264	C₂×C₄²	Q₈×C₆₇	1	4	yes	—
265	#95	—	2	4	no	—
266	#95	—	2	16	no	—
267	#95	C₂×C₄×C₆₇	1	8	yes	—
268	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
269	C₂²×C₄	—	2	4	no	—
270	C₂²×C₄	—	2	4	no	—
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	—	2	4	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	2	4	no	—
275	C₂²×C₄	—	2	4	no	—
276	C₂²×C₄	—	2	2	no	—
277	C₂²×C₄	—	2	2	no	—
278	C₂²×C₄	—	2	4	no	—
279	C₂²×C₄	—	2	2	no	—
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	—	2	4	no	—
282	C₂²×C₄	—	2	4	no	—
283	C₂²×C₄	—	2	8	no	—
284	C₂²×C₄	—	2	4	no	—
285	C₂²×C₄	—	2	4	no	—
286	C₂²×C₄	—	2	4	no	—
287	C₂²×C₄	—	2	8	no	—
288	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
289	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
290	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
291	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
292	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
293	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
294	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
295	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
296	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
297	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
298	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
299	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
300	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
301	C₂²×C₄	C₂×Q₈×C₆₇	1	8	yes	—
302	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
303	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
304	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
305	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
306	C₂²×C₄	#41×C₆₇	1	2	yes	—
307	C₂²×C₄	#41×C₆₇	1	2	yes	—
308	C₄⋊C₄	—	2	8	no	—
309	C₄⋊C₄	—	2	8	no	—
310	C₄⋊C₄	—	4	32	no	—
311	C₄⋊C₄	—	4	8	no	—
312	C₄⋊C₄	—	4	8	no	—
313	C₄⋊C₄	—	4	8	no	—
314	C₄⋊C₄	C₄⋊C₄×C₆₇	1	32	yes	complemented
315	C₄²	—	2	16	no	—
316	C₄²	—	4	8	no	—
317	C₄²	C₄⋊C₄×C₆₇	1	16	yes	complemented
318	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
319	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
320	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
321	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
322	C₂³	C₂²×Q₈×C₆₇	1	1	yes	characteristic
323	C₂³	C₂²×Q₈×C₆₇	1	2	yes	—
324	C₂³	C₂²×Q₈×C₆₇	1	1	yes	characteristic
325	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
326	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
327	C₂³	H(5,𝔽₂)×C₆₇	1	2	yes	—
328	C₂³	2₋¹⁺⁴×C₆₇	1	2	yes	—
329	C₂×C₄	—	2	8	no	—
330	C₂×C₄	—	2	4	no	—
331	C₂×C₄	—	2	8	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	4	no	—
335	C₂×C₄	—	2	8	no	—
336	C₂×C₄	—	2	8	no	—
337	C₂×C₄	—	2	8	no	—
338	C₂×C₄	—	2	8	no	—
339	C₂×C₄	—	2	4	no	—
340	C₂×C₄	—	2	4	no	—
341	C₂×C₄	—	2	8	no	—
342	C₂×C₄	—	2	8	no	—
343	C₂×C₄	—	4	2	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	4	no	—
348	C₂×C₄	—	4	4	no	—
349	C₂×C₄	—	4	4	no	—
350	C₂×C₄	—	4	4	no	—
351	C₂×C₄	—	4	4	no	—
352	C₂×C₄	—	4	8	no	—
353	C₂×C₄	—	4	8	no	—
354	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
355	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	16	yes	—
356	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
357	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	8	yes	—
358	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
359	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
360	C₂×C₄	C₄×Q₈×C₆₇	1	4	yes	—
361	C₂×C₄	#121×C₆₇	1	4	yes	—
362	C₂×C₄	C₂²⋊Q₈×C₆₇	1	4	yes	—
363	C₂×C₄	#125×C₆₇	1	4	yes	—
364	C₂×C₄	#127×C₆₇	1	4	yes	—
365	C₂×C₄	C₄⋊Q₈×C₆₇	1	4	yes	—
366	C₂×C₄	C₄⋊Q₈×C₆₇	1	8	yes	—
367	C₂×C₄	C₄⋊Q₈×C₆₇	1	4	yes	—
368	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
369	C₂²	C₂²×C₄⋊C₄×C₆₇	1	2	yes	—
370	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
371	C₂²	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
372	C₂²	#520×C₆₇	1	2	yes	—
373	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
374	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
375	C₂²	C₂×#125×C₆₇	1	1	yes	characteristic
376	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
377	C₂²	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
378	C₂²	C₂×C₄⋊Q₈×C₆₇	1	2	yes	—
379	C₂²	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
380	C₂²	#535×C₆₇	1	1	yes	characteristic
381	C₂²	#536×C₆₇	1	1	yes	characteristic
382	C₂²	#544×C₆₇	1	2	yes	—
383	C₂²	D₄²×C₆₇	1	2	yes	—
384	C₂²	#547×C₆₇	1	2	yes	—
385	C₂²	D₄×Q₈×C₆₇	1	2	yes	—
386	C₂²	D₄×Q₈×C₆₇	1	2	yes	—
387	C₂²	#550×C₆₇	1	2	yes	—
388	C₂²	#554×C₆₇	1	2	yes	—
389	C₂²	#557×C₆₇	1	2	yes	—
390	C₂²	Q₈²×C₆₇	1	2	yes	—
391	C₄	—	2	4	no	—
392	C₄	—	2	16	no	—
393	C₄	—	4	4	no	—
394	C₄	—	4	4	no	—
395	C₄	—	8	4	no	—
396	C₄	#384×C₆₇	1	8	yes	complemented
397	C₄	#389×C₆₇	1	8	yes	complemented
398	C₂	C₂×#384×C₆₇	1	1	yes	characteristic
399	C₂	C₂×#389×C₆₇	1	1	yes	characteristic
400	C₂	C₆₇×#2317	1	1	yes	characteristic
401	C₂	D₄×C₄⋊C₄×C₆₇	1	2	yes	—
402	C₂	Q₈×C₄⋊C₄×C₆₇	1	2	yes	—
403	C₂	C₆₇×#2541	1	1	yes	characteristic
404	C₂	C₆₇×#2542	1	1	yes	characteristic
405	C₂	C₆₇×#2546	1	1	yes	characteristic
406	C₂	C₆₇×#2551	1	2	yes	—
407	C₂	C₆₇×#2552	1	2	yes	—
408	C₂	C₆₇×#2567	1	1	yes	characteristic

	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
22. C₄×C₄⋊C₄×C₆₇	C₄	semidirect
109. C₄⋊C₄×C₆₇	C₄⋊C₄	semidirect
112. C₄²×C₆₇	C₄⋊C₄	semidirect
191. C₄×C₆₇	#384	semidirect
192. C₄×C₆₇	#389	semidirect
193. #13910	C₆₇	direct (N × H)
212. C₆₇	#13910	direct (N × H)
227. C₄×C₄⋊C₄	C₄×C₆₇	semidirect
314. C₄⋊C₄	C₄⋊C₄×C₆₇	semidirect
317. C₄²	C₄⋊C₄×C₆₇	semidirect
396. C₄	#384×C₆₇	semidirect
397. C₄	#389×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#389×C₆₇	2
C₂×#384×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₆₇×#1444	2
C₆₇×C₄⋊#95	2
#13910	67

Subgroups — C67×#13910 #30693161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#13910 #30693161