Subgroups — C₆₇×#33149 #30712400

Order 17152.

892 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2857	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1987	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1947	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1921	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1852	C₂	1	2	yes	maximal
7	C₆₇×#1824	C₂	1	2	yes	maximal
8	C₆₇×#1797	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
11	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₆₇	—	2	1	no	—
14	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#107×C₆₇	C₂²	1	4	yes	—
16	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
18	#391×C₆₇	—	2	1	no	—
19	#389×C₆₇	—	2	1	no	—
20	#389×C₆₇	C₂²	1	2	yes	—
21	#384×C₆₇	C₂²	1	1	yes	characteristic
22	#380×C₆₇	—	2	1	no	—
23	#380×C₆₇	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
25	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
26	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
28	C₂×#95×C₆₇	—	2	1	no	—
29	#340×C₆₇	C₂²	1	4	yes	—
30	#340×C₆₇	C₂²	1	4	yes	—
31	#340×C₆₇	C₂²	1	4	yes	—
32	#336×C₆₇	C₂²	1	2	yes	—
33	C₂²×Q₈×C₆₇	—	2	1	no	—
34	C₂³×C₄×C₆₇	—	2	1	no	—
35	C₂³×C₄×C₆₇	—	2	1	no	—
36	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
40	C₄⋊Q₈×C₆₇	—	2	4	no	—
41	C₂²⋊Q₈×C₆₇	—	4	4	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
44	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
46	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
47	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
49	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
50	C₂×C₄⋊C₄×C₆₇	Q₈	1	4	yes	complemented
51	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
55	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
56	C₂×C₄²×C₆₇	—	2	4	no	—
57	C₂×C₄²×C₆₇	—	2	2	no	—
58	C₂×C₄²×C₆₇	—	2	2	no	—
59	C₂×C₄²×C₆₇	—	4	4	no	—
60	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
61	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
62	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
64	#107×C₆₇	—	2	8	no	—
65	C₄⋊C₈×C₆₇	—	2	4	no	—
66	C₂²⋊C₈×C₆₇	—	2	4	no	—
67	#95×C₆₇	—	2	2	no	—
68	#95×C₆₇	—	2	2	no	—
69	#95×C₆₇	—	2	2	no	—
70	#95×C₆₇	—	2	2	no	—
71	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₂×Q₈×C₆₇	—	4	2	no	—
73	C₂×Q₈×C₆₇	—	4	2	no	—
74	C₂×Q₈×C₆₇	—	4	2	no	—
75	C₂²×C₄×C₆₇	—	2	2	no	—
76	C₂²×C₄×C₆₇	—	2	2	no	—
77	C₂²×C₄×C₆₇	—	2	2	no	—
78	C₂²×C₄×C₆₇	—	2	2	no	—
79	C₂²×C₄×C₆₇	—	2	4	no	—
80	C₂²×C₄×C₆₇	—	2	4	no	—
81	C₂²×C₄×C₆₇	—	2	2	no	—
82	C₂²×C₄×C₆₇	—	2	2	no	—
83	C₂²×C₄×C₆₇	—	2	2	no	—
84	C₂²×C₄×C₆₇	—	2	2	no	—
85	C₂²×C₄×C₆₇	—	2	2	no	—
86	C₂²×C₄×C₆₇	—	2	2	no	—
87	C₂²×C₄×C₆₇	—	4	1	no	—
88	C₂²×C₄×C₆₇	—	4	2	no	—
89	C₂²×C₄×C₆₇	—	4	2	no	—
90	C₂²×C₄×C₆₇	—	4	2	no	—
91	C₂²×C₄×C₆₇	—	4	2	no	—
92	C₂²×C₄×C₆₇	Q₁₆	1	4	yes	—
93	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
99	C₂²×C₄×C₆₇	#41	1	2	yes	—
100	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
101	C₂×C₈×C₆₇	—	2	4	no	—
102	C₂×C₈×C₆₇	—	2	4	no	—
103	C₂×C₈×C₆₇	—	2	4	no	—
104	C₄⋊C₄×C₆₇	—	2	4	no	—
105	C₄⋊C₄×C₆₇	—	2	4	no	—
106	C₄⋊C₄×C₆₇	—	2	2	no	—
107	C₄⋊C₄×C₆₇	—	2	2	no	—
108	C₄⋊C₄×C₆₇	—	2	4	no	—
109	C₄⋊C₄×C₆₇	—	2	4	no	—
110	C₄⋊C₄×C₆₇	—	4	2	no	—
111	C₄⋊C₄×C₆₇	—	4	4	no	—
112	C₄⋊C₄×C₆₇	—	4	2	no	—
113	C₄⋊C₄×C₆₇	—	4	2	no	—
114	C₄⋊C₄×C₆₇	C₂×Q₈	1	4	yes	—
115	C₄⋊C₄×C₆₇	#41	1	4	yes	—
116	C₂²⋊C₄×C₆₇	—	4	4	no	—
117	C₄²×C₆₇	—	2	2	no	—
118	C₄²×C₆₇	—	2	2	no	—
119	C₄²×C₆₇	—	2	2	no	—
120	C₄²×C₆₇	—	4	4	no	—
121	C₄²×C₆₇	—	4	4	no	—
122	C₄²×C₆₇	#41	1	4	yes	—
123	C₂³×C₆₇	—	2	1	no	—
124	C₂³×C₆₇	—	2	1	no	—
125	C₂³×C₆₇	—	2	1	no	—
126	C₂³×C₆₇	—	2	1	no	—
127	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
128	C₂³×C₆₇	C₂×Q₁₆	1	2	yes	—
129	C₂³×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
130	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
131	Q₈×C₆₇	—	8	4	no	—
132	C₂×C₄×C₆₇	—	2	1	no	—
133	C₂×C₄×C₆₇	—	2	1	no	—
134	C₂×C₄×C₆₇	—	2	1	no	—
135	C₂×C₄×C₆₇	—	2	1	no	—
136	C₂×C₄×C₆₇	—	2	1	no	—
137	C₂×C₄×C₆₇	—	2	1	no	—
138	C₂×C₄×C₆₇	—	2	4	no	—
139	C₂×C₄×C₆₇	—	2	4	no	—
140	C₂×C₄×C₆₇	—	2	8	no	—
141	C₂×C₄×C₆₇	—	2	4	no	—
142	C₂×C₄×C₆₇	—	2	1	no	—
143	C₂×C₄×C₆₇	—	2	4	no	—
144	C₂×C₄×C₆₇	—	2	1	no	—
145	C₂×C₄×C₆₇	—	2	1	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	—	4	2	no	—
150	C₂×C₄×C₆₇	—	4	2	no	—
151	C₂×C₄×C₆₇	—	4	2	no	—
152	C₂×C₄×C₆₇	—	4	2	no	—
153	C₂×C₄×C₆₇	—	4	2	no	—
154	C₂×C₄×C₆₇	—	4	2	no	—
155	C₂×C₄×C₆₇	—	4	2	no	—
156	C₂×C₄×C₆₇	—	4	2	no	—
157	C₂×C₄×C₆₇	—	4	2	no	—
158	C₂×C₄×C₆₇	—	4	2	no	—
159	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
160	C₂×C₄×C₆₇	#123	1	2	yes	—
161	C₂×C₄×C₆₇	#123	1	2	yes	—
162	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
163	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
164	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
165	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
166	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
167	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
168	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
169	C₈×C₆₇	—	4	4	no	—
170	C₂²×C₆₇	—	2	2	no	—
171	C₂²×C₆₇	—	2	2	no	—
172	C₂²×C₆₇	—	2	2	no	—
173	C₂²×C₆₇	—	2	2	no	—
174	C₂²×C₆₇	—	2	2	no	—
175	C₂²×C₆₇	—	2	2	no	—
176	C₂²×C₆₇	#477	1	4	yes	—
177	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
178	C₂²×C₆₇	C₂×#123	1	1	yes	characteristic
179	C₂²×C₆₇	#525	1	1	yes	characteristic
180	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
181	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
183	C₂²×C₆₇	#575	1	1	yes	characteristic
184	C₄×C₆₇	—	2	2	no	—
185	C₄×C₆₇	—	2	2	no	—
186	C₄×C₆₇	—	2	2	no	—
187	C₄×C₆₇	—	4	4	no	—
188	C₄×C₆₇	—	4	1	no	—
189	C₄×C₆₇	—	4	4	no	—
190	C₄×C₆₇	—	8	2	no	—
191	C₄×C₆₇	—	8	2	no	—
192	C₄×C₆₇	#484	1	4	yes	—
193	C₄×C₆₇	#529	1	2	yes	—
194	C₄×C₆₇	#533	1	2	yes	—
195	#33149	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
196	C₂×C₆₇	—	2	4	no	—
197	C₂×C₆₇	#2855	1	1	yes	characteristic
198	C₂×C₆₇	C₂×#477	1	1	yes	characteristic
199	C₂×C₆₇	#3077	1	1	yes	characteristic
200	C₂×C₆₇	C₂×#484	1	1	yes	characteristic
201	C₂×C₆₇	#3094	1	1	yes	characteristic
202	C₂×C₆₇	#3104	1	1	yes	characteristic
203	C₂×C₆₇	#3109	1	1	yes	characteristic
204	#2857	C₂×C₆₇	1	1	yes	characteristic
205	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
206	#1987	C₂×C₆₇	1	2	yes	complemented
207	#1947	C₂×C₆₇	1	2	yes	complemented
208	#1921	C₂×C₆₇	1	4	yes	complemented
209	#1852	C₂×C₆₇	1	2	yes	—
210	#1824	C₂×C₆₇	1	2	yes	—
211	#1797	C₂×C₆₇	1	1	yes	characteristic
212	C₆₇	#33149	1	1	yes	characteristic, direct product factor, Sylow
213	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
214	C₂×C₂²⋊Q₈	—	2	1	no	—
215	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
216	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
217	C₂²×C₄²	—	2	1	no	—
218	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
219	C₂×#107	C₂²×C₆₇	1	4	yes	—
220	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
221	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
222	#391	—	2	1	no	—
223	#389	—	2	1	no	—
224	#389	C₂²×C₆₇	1	2	yes	—
225	#384	C₂²×C₆₇	1	1	yes	characteristic
226	#380	—	2	1	no	—
227	#380	C₂²×C₆₇	1	2	yes	—
228	C₄×C₄⋊C₄	—	2	1	no	—
229	C₄×C₄⋊C₄	—	2	1	no	—
230	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
231	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
232	C₂×#95	—	2	1	no	—
233	#340	C₂²×C₆₇	1	4	yes	—
234	#340	C₂²×C₆₇	1	4	yes	—
235	#340	C₂²×C₆₇	1	4	yes	—
236	#336	C₂²×C₆₇	1	2	yes	—
237	C₂²×Q₈	—	2	1	no	—
238	C₂³×C₄	—	2	1	no	—
239	C₂³×C₄	—	2	1	no	—
240	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
241	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
242	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
243	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
244	C₄⋊Q₈	—	2	4	no	—
245	C₂²⋊Q₈	—	4	4	no	—
246	C₂×C₄⋊C₄	—	2	1	no	—
247	C₂×C₄⋊C₄	—	2	1	no	—
248	C₂×C₄⋊C₄	—	2	2	no	—
249	C₂×C₄⋊C₄	—	2	4	no	—
250	C₂×C₄⋊C₄	—	2	1	no	—
251	C₂×C₄⋊C₄	—	2	2	no	—
252	C₂×C₄⋊C₄	—	2	1	no	—
253	C₂×C₄⋊C₄	—	2	1	no	—
254	C₂×C₄⋊C₄	Q₈×C₆₇	1	4	yes	complemented
255	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
256	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
257	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
258	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
259	C₂×C₂²⋊C₄	—	2	2	no	—
260	C₂×C₄²	—	2	4	no	—
261	C₂×C₄²	—	2	2	no	—
262	C₂×C₄²	—	2	2	no	—
263	C₂×C₄²	—	4	4	no	—
264	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
265	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
266	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
267	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
268	#107	—	2	8	no	—
269	C₄⋊C₈	—	2	4	no	—
270	C₂²⋊C₈	—	2	4	no	—
271	#95	—	2	2	no	—
272	#95	—	2	2	no	—
273	#95	—	2	2	no	—
274	#95	—	2	2	no	—
275	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
276	C₂×Q₈	—	4	2	no	—
277	C₂×Q₈	—	4	2	no	—
278	C₂×Q₈	—	4	2	no	—
279	C₂²×C₄	—	2	2	no	—
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	2	2	no	—
283	C₂²×C₄	—	2	4	no	—
284	C₂²×C₄	—	2	4	no	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	2	2	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	—	2	2	no	—
291	C₂²×C₄	—	4	1	no	—
292	C₂²×C₄	—	4	2	no	—
293	C₂²×C₄	—	4	2	no	—
294	C₂²×C₄	—	4	2	no	—
295	C₂²×C₄	—	4	2	no	—
296	C₂²×C₄	Q₁₆×C₆₇	1	4	yes	—
297	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
298	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
299	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
300	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
301	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
302	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
303	C₂²×C₄	#41×C₆₇	1	2	yes	—
304	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
305	C₂×C₈	—	2	4	no	—
306	C₂×C₈	—	2	4	no	—
307	C₂×C₈	—	2	4	no	—
308	C₄⋊C₄	—	2	4	no	—
309	C₄⋊C₄	—	2	4	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	2	4	no	—
313	C₄⋊C₄	—	2	4	no	—
314	C₄⋊C₄	—	4	2	no	—
315	C₄⋊C₄	—	4	4	no	—
316	C₄⋊C₄	—	4	2	no	—
317	C₄⋊C₄	—	4	2	no	—
318	C₄⋊C₄	C₂×Q₈×C₆₇	1	4	yes	—
319	C₄⋊C₄	#41×C₆₇	1	4	yes	—
320	C₂²⋊C₄	—	4	4	no	—
321	C₄²	—	2	2	no	—
322	C₄²	—	2	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	—	4	4	no	—
325	C₄²	—	4	4	no	—
326	C₄²	#41×C₆₇	1	4	yes	—
327	C₂³	—	2	1	no	—
328	C₂³	—	2	1	no	—
329	C₂³	—	2	1	no	—
330	C₂³	—	2	1	no	—
331	C₂³	C₂²⋊Q₈×C₆₇	1	2	yes	—
332	C₂³	C₂×Q₁₆×C₆₇	1	2	yes	—
333	C₂³	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
334	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
335	Q₈	—	8	4	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	8	no	—
343	C₂×C₄	—	2	4	no	—
344	C₂×C₄	—	2	4	no	—
345	C₂×C₄	—	2	4	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	2	4	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	4	4	no	—
351	C₂×C₄	—	4	8	no	—
352	C₂×C₄	—	4	8	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	4	2	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
364	C₂×C₄	#123×C₆₇	1	2	yes	—
365	C₂×C₄	#123×C₆₇	1	2	yes	—
366	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
367	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
368	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
369	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
370	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
371	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
372	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
373	C₈	—	4	4	no	—
374	C₂²	—	2	2	no	—
375	C₂²	—	2	2	no	—
376	C₂²	—	2	2	no	—
377	C₂²	—	2	2	no	—
378	C₂²	—	2	2	no	—
379	C₂²	—	2	2	no	—
380	C₂²	#477×C₆₇	1	4	yes	—
381	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
382	C₂²	C₂×#123×C₆₇	1	1	yes	characteristic
383	C₂²	#525×C₆₇	1	1	yes	characteristic
384	C₂²	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
385	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
386	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
387	C₂²	#575×C₆₇	1	1	yes	characteristic
388	C₄	—	2	2	no	—
389	C₄	—	2	2	no	—
390	C₄	—	2	2	no	—
391	C₄	—	4	4	no	—
392	C₄	—	4	1	no	—
393	C₄	—	4	4	no	—
394	C₄	—	8	2	no	—
395	C₄	—	8	2	no	—
396	C₄	#484×C₆₇	1	4	yes	—
397	C₄	#529×C₆₇	1	2	yes	—
398	C₄	#533×C₆₇	1	2	yes	—
399	C₂	—	2	4	no	—
400	C₂	C₆₇×#2855	1	1	yes	characteristic
401	C₂	C₂×#477×C₆₇	1	1	yes	characteristic
402	C₂	C₆₇×#3077	1	1	yes	characteristic
403	C₂	C₂×#484×C₆₇	1	1	yes	characteristic
404	C₂	C₆₇×#3094	1	1	yes	characteristic
405	C₂	C₆₇×#3104	1	1	yes	characteristic
406	C₂	C₆₇×#3109	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1987	C₂	semidirect
4. C₆₇×#1947	C₂	semidirect
5. C₆₇×#1921	C₂	semidirect
50. C₂×C₄⋊C₄×C₆₇	Q₈	semidirect
195. #33149	C₆₇	direct (N × H)
206. #1987	C₂×C₆₇	semidirect
207. #1947	C₂×C₆₇	semidirect
208. #1921	C₂×C₆₇	semidirect
212. C₆₇	#33149	direct (N × H)
254. C₂×C₄⋊C₄	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2857	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₆₇×#1987	2
C₆₇×#1947	2
C₆₇×#1921	2
C₆₇×#1852	2
C₆₇×#1824	2
C₆₇×#1797	2
#33149	67

Subgroups — C67×#33149 #30712400

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33149 #30712400