Subgroups — C₆₇×#33215 #30712466

Order 17152.

810 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2628	C₂	1	2	yes	maximal
4	C₆₇×#1993	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1944	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1942	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1846	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1809	C₂	1	2	yes	maximal
9	C₆₇×#1808	C₂	1	2	yes	maximal
10	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
13	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
14	#500×C₆₇	—	2	4	no	—
15	C₂×#107×C₆₇	C₂²	1	2	yes	—
16	C₂×#107×C₆₇	C₂²	1	2	yes	—
17	C₂×#107×C₆₇	C₂²	1	2	yes	—
18	C₂×#106×C₆₇	C₂²	1	2	yes	—
19	#405×C₆₇	—	2	4	no	—
20	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	#400×C₆₇	—	2	2	no	—
25	#399×C₆₇	—	2	2	no	—
26	#395×C₆₇	—	2	2	no	—
27	#393×C₆₇	—	2	2	no	—
28	#389×C₆₇	C₂²	1	4	yes	—
29	#382×C₆₇	C₂²	1	2	yes	—
30	#380×C₆₇	C₂²	1	4	yes	—
31	#340×C₆₇	C₂²	1	4	yes	—
32	#340×C₆₇	C₂²	1	4	yes	—
33	C₂²×Q₈×C₆₇	—	2	2	no	—
34	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₆₇	—	2	2	no	—
36	C₂×M₁₆×C₆₇	—	2	2	no	—
37	C₂×M₁₆×C₆₇	Q₈	1	4	yes	complemented
38	C₂²×C₈×C₆₇	—	2	2	no	—
39	C₂²×C₈×C₆₇	—	2	2	no	—
40	C₂²×C₈×C₆₇	Q₈	1	4	yes	complemented
41	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
45	C₄⋊Q₈×C₆₇	—	2	4	no	—
46	C₄⋊Q₈×C₆₇	—	2	4	no	—
47	#117×C₆₇	—	2	2	no	—
48	#117×C₆₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
51	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
54	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
55	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
56	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
57	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
60	#107×C₆₇	—	2	4	no	—
61	#107×C₆₇	—	2	4	no	—
62	#107×C₆₇	—	2	4	no	—
63	#106×C₆₇	—	2	4	no	—
64	#97×C₆₇	—	2	2	no	—
65	#97×C₆₇	—	2	2	no	—
66	C₄×C₈×C₆₇	—	2	2	no	—
67	C₄×C₈×C₆₇	D₄	1	4	yes	—
68	#95×C₆₇	—	2	4	no	—
69	#95×C₆₇	—	2	2	no	—
70	#95×C₆₇	—	2	2	no	—
71	#95×C₆₇	—	2	4	no	—
72	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₂×Q₈×C₆₇	—	4	4	no	—
74	C₂×Q₈×C₆₇	—	8	4	no	—
75	C₂²×C₄×C₆₇	—	2	1	no	—
76	C₂²×C₄×C₆₇	—	2	1	no	—
77	C₂²×C₄×C₆₇	—	2	1	no	—
78	C₂²×C₄×C₆₇	—	2	1	no	—
79	C₂²×C₄×C₆₇	—	2	4	no	—
80	C₂²×C₄×C₆₇	—	2	4	no	—
81	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
87	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
88	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
89	M₁₆×C₆₇	—	2	8	no	—
90	C₂×C₈×C₆₇	—	2	1	no	—
91	C₂×C₈×C₆₇	—	2	1	no	—
92	C₂×C₈×C₆₇	—	2	1	no	—
93	C₂×C₈×C₆₇	—	2	1	no	—
94	C₂×C₈×C₆₇	—	2	8	no	—
95	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
98	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
99	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
100	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
101	C₂×C₈×C₆₇	#41	1	2	yes	—
102	C₂×C₈×C₆₇	#41	1	2	yes	—
103	C₄⋊C₄×C₆₇	—	2	2	no	—
104	C₄⋊C₄×C₆₇	—	2	2	no	—
105	C₄⋊C₄×C₆₇	—	2	4	no	—
106	C₄⋊C₄×C₆₇	—	2	4	no	—
107	C₄⋊C₄×C₆₇	—	2	4	no	—
108	C₄⋊C₄×C₆₇	—	2	4	no	—
109	C₄⋊C₄×C₆₇	—	4	4	no	—
110	C₄⋊C₄×C₆₇	—	8	4	no	—
111	C₂²⋊C₄×C₆₇	—	2	2	no	—
112	C₂²⋊C₄×C₆₇	—	2	2	no	—
113	C₂²⋊C₄×C₆₇	—	8	4	no	—
114	C₄²×C₆₇	—	2	1	no	—
115	C₄²×C₆₇	—	2	1	no	—
116	C₄²×C₆₇	C₂×D₄	1	2	yes	—
117	C₄²×C₆₇	#41	1	2	yes	—
118	C₂³×C₆₇	—	2	1	no	—
119	C₂³×C₆₇	—	2	1	no	—
120	C₂³×C₆₇	—	2	2	no	—
121	C₂³×C₆₇	—	2	2	no	—
122	C₂³×C₆₇	C₄⋊Q₈	1	2	yes	—
123	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
124	Q₈×C₆₇	—	8	8	no	—
125	C₂×C₄×C₆₇	—	2	1	no	—
126	C₂×C₄×C₆₇	—	2	1	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	2	2	no	—
129	C₂×C₄×C₆₇	—	2	2	no	—
130	C₂×C₄×C₆₇	—	2	2	no	—
131	C₂×C₄×C₆₇	—	2	2	no	—
132	C₂×C₄×C₆₇	—	2	2	no	—
133	C₂×C₄×C₆₇	—	2	1	no	—
134	C₂×C₄×C₆₇	—	2	1	no	—
135	C₂×C₄×C₆₇	—	2	2	no	—
136	C₂×C₄×C₆₇	—	2	2	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	#121	1	2	yes	—
144	C₂×C₄×C₆₇	#121	1	2	yes	—
145	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
146	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
148	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
149	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
151	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
152	C₈×C₆₇	—	2	2	no	—
153	C₈×C₆₇	—	2	2	no	—
154	C₈×C₆₇	C₂²⋊Q₈	1	4	yes	—
155	C₈×C₆₇	C₂²⋊Q₈	1	4	yes	—
156	C₂²×C₆₇	—	2	2	no	—
157	C₂²×C₆₇	—	2	4	no	—
158	C₂²×C₆₇	—	4	2	no	—
159	C₂²×C₆₇	—	4	1	no	—
160	C₂²×C₆₇	—	4	1	no	—
161	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
162	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
163	C₂²×C₆₇	#532	1	1	yes	characteristic
164	C₂²×C₆₇	#576	1	2	yes	—
165	C₂²×C₆₇	#578	1	2	yes	—
166	C₄×C₆₇	—	2	2	no	—
167	C₄×C₆₇	—	2	2	no	—
168	C₄×C₆₇	—	2	2	no	—
169	C₄×C₆₇	—	4	1	no	—
170	C₄×C₆₇	—	4	1	no	—
171	C₄×C₆₇	—	8	4	no	—
172	C₄×C₆₇	—	8	4	no	—
173	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
174	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
175	C₄×C₆₇	#533	1	1	yes	characteristic
176	C₄×C₆₇	#533	1	1	yes	characteristic
177	#33215	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
178	C₂×C₆₇	—	4	2	no	—
179	C₂×C₆₇	#2857	1	1	yes	characteristic
180	C₂×C₆₇	#3058	1	1	yes	characteristic
181	C₂×C₆₇	#3059	1	1	yes	characteristic
182	C₂×C₆₇	#3164	1	2	yes	—
183	C₂×C₆₇	#3175	1	2	yes	—
184	C₂×#500	C₂×C₆₇	1	1	yes	characteristic, complemented
185	C₂×#405	C₂×C₆₇	1	1	yes	characteristic
186	#2628	C₂×C₆₇	1	2	yes	—
187	#1993	C₂×C₆₇	1	2	yes	complemented
188	#1944	C₂×C₆₇	1	2	yes	complemented
189	#1942	C₂×C₆₇	1	2	yes	complemented
190	#1846	C₂×C₆₇	1	1	yes	characteristic, complemented
191	#1809	C₂×C₆₇	1	2	yes	—
192	#1808	C₂×C₆₇	1	2	yes	—
193	C₆₇	#33215	1	1	yes	characteristic, direct product factor, Sylow
194	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
195	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
196	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
197	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
198	#500	—	2	4	no	—
199	C₂×#107	C₂²×C₆₇	1	2	yes	—
200	C₂×#107	C₂²×C₆₇	1	2	yes	—
201	C₂×#107	C₂²×C₆₇	1	2	yes	—
202	C₂×#106	C₂²×C₆₇	1	2	yes	—
203	#405	—	2	4	no	—
204	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
205	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
206	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
207	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
208	#400	—	2	2	no	—
209	#399	—	2	2	no	—
210	#395	—	2	2	no	—
211	#393	—	2	2	no	—
212	#389	C₂²×C₆₇	1	4	yes	—
213	#382	C₂²×C₆₇	1	2	yes	—
214	#380	C₂²×C₆₇	1	4	yes	—
215	#340	C₂²×C₆₇	1	4	yes	—
216	#340	C₂²×C₆₇	1	4	yes	—
217	C₂²×Q₈	—	2	2	no	—
218	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
219	C₂×M₁₆	—	2	2	no	—
220	C₂×M₁₆	—	2	2	no	—
221	C₂×M₁₆	Q₈×C₆₇	1	4	yes	complemented
222	C₂²×C₈	—	2	2	no	—
223	C₂²×C₈	—	2	2	no	—
224	C₂²×C₈	Q₈×C₆₇	1	4	yes	complemented
225	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
226	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
227	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
228	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
229	C₄⋊Q₈	—	2	4	no	—
230	C₄⋊Q₈	—	2	4	no	—
231	#117	—	2	2	no	—
232	#117	—	2	2	no	—
233	C₂×C₄⋊C₄	—	2	2	no	—
234	C₂×C₄⋊C₄	—	2	4	no	—
235	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
236	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
237	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
238	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
239	C₂×C₂²⋊C₄	—	2	4	no	—
240	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
241	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
242	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
243	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
244	#107	—	2	4	no	—
245	#107	—	2	4	no	—
246	#107	—	2	4	no	—
247	#106	—	2	4	no	—
248	#97	—	2	2	no	—
249	#97	—	2	2	no	—
250	C₄×C₈	—	2	2	no	—
251	C₄×C₈	D₄×C₆₇	1	4	yes	—
252	#95	—	2	4	no	—
253	#95	—	2	2	no	—
254	#95	—	2	2	no	—
255	#95	—	2	4	no	—
256	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
257	C₂×Q₈	—	4	4	no	—
258	C₂×Q₈	—	8	4	no	—
259	C₂²×C₄	—	2	1	no	—
260	C₂²×C₄	—	2	1	no	—
261	C₂²×C₄	—	2	1	no	—
262	C₂²×C₄	—	2	1	no	—
263	C₂²×C₄	—	2	4	no	—
264	C₂²×C₄	—	2	4	no	—
265	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
266	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
270	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
271	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
272	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
273	M₁₆	—	2	8	no	—
274	C₂×C₈	—	2	1	no	—
275	C₂×C₈	—	2	1	no	—
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	2	8	no	—
279	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
280	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
281	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
282	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
283	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
284	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
285	C₂×C₈	#41×C₆₇	1	2	yes	—
286	C₂×C₈	#41×C₆₇	1	2	yes	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	4	no	—
290	C₄⋊C₄	—	2	4	no	—
291	C₄⋊C₄	—	2	4	no	—
292	C₄⋊C₄	—	2	4	no	—
293	C₄⋊C₄	—	4	4	no	—
294	C₄⋊C₄	—	8	4	no	—
295	C₂²⋊C₄	—	2	2	no	—
296	C₂²⋊C₄	—	2	2	no	—
297	C₂²⋊C₄	—	8	4	no	—
298	C₄²	—	2	1	no	—
299	C₄²	—	2	1	no	—
300	C₄²	C₂×D₄×C₆₇	1	2	yes	—
301	C₄²	#41×C₆₇	1	2	yes	—
302	C₂³	—	2	1	no	—
303	C₂³	—	2	1	no	—
304	C₂³	—	2	2	no	—
305	C₂³	—	2	2	no	—
306	C₂³	C₄⋊Q₈×C₆₇	1	2	yes	—
307	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
308	Q₈	—	8	8	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	4	4	no	—
322	C₂×C₄	—	4	4	no	—
323	C₂×C₄	—	4	4	no	—
324	C₂×C₄	—	4	4	no	—
325	C₂×C₄	—	4	4	no	—
326	C₂×C₄	—	4	4	no	—
327	C₂×C₄	#121×C₆₇	1	2	yes	—
328	C₂×C₄	#121×C₆₇	1	2	yes	—
329	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
330	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
331	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
332	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
333	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
334	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
335	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
336	C₈	—	2	2	no	—
337	C₈	—	2	2	no	—
338	C₈	C₂²⋊Q₈×C₆₇	1	4	yes	—
339	C₈	C₂²⋊Q₈×C₆₇	1	4	yes	—
340	C₂²	—	2	2	no	—
341	C₂²	—	2	4	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	4	2	no	—
345	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
346	C₂²	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
347	C₂²	#532×C₆₇	1	1	yes	characteristic
348	C₂²	#576×C₆₇	1	2	yes	—
349	C₂²	#578×C₆₇	1	2	yes	—
350	C₄	—	2	2	no	—
351	C₄	—	2	2	no	—
352	C₄	—	2	2	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	8	4	no	—
356	C₄	—	8	4	no	—
357	C₄	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
358	C₄	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
359	C₄	#533×C₆₇	1	1	yes	characteristic
360	C₄	#533×C₆₇	1	1	yes	characteristic
361	C₂	—	4	2	no	—
362	C₂	C₆₇×#2857	1	1	yes	characteristic
363	C₂	C₆₇×#3058	1	1	yes	characteristic
364	C₂	C₆₇×#3059	1	1	yes	characteristic
365	C₂	C₆₇×#3164	1	2	yes	—
366	C₂	C₆₇×#3175	1	2	yes	—

	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 ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#500×C₆₇	C₂	semidirect
4. C₆₇×#1993	C₂	semidirect
5. C₆₇×#1944	C₂	semidirect
6. C₆₇×#1942	C₂	semidirect
7. C₆₇×#1846	C₂	semidirect
37. C₂×M₁₆×C₆₇	Q₈	semidirect
40. C₂²×C₈×C₆₇	Q₈	semidirect
177. #33215	C₆₇	direct (N × H)
184. C₂×#500	C₂×C₆₇	semidirect
187. #1993	C₂×C₆₇	semidirect
188. #1944	C₂×C₆₇	semidirect
189. #1942	C₂×C₆₇	semidirect
190. #1846	C₂×C₆₇	semidirect
193. C₆₇	#33215	direct (N × H)
221. C₂×M₁₆	Q₈×C₆₇	semidirect
224. C₂²×C₈	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#500×C₆₇	2
C₂×#405×C₆₇	2
C₆₇×#2628	2
C₆₇×#1993	2
C₆₇×#1944	2
C₆₇×#1942	2
C₆₇×#1846	2
C₆₇×#1809	2
C₆₇×#1808	2
#33215	67

Subgroups — C67×#33215 #30712466

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33215 #30712466