Subgroups — C₆₇×#25869 #30705120

Order 17152.

1314 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3496	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#3141	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2563	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2040	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1980	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1792	C₂	1	4	yes	maximal, complemented
7	C₄×#100×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	2	yes	—
9	#547×C₆₇	—	2	2	no	—
10	#546×C₆₇	—	2	4	no	—
11	D₄²×C₆₇	—	2	2	no	—
12	#537×C₆₇	C₂²	1	2	yes	—
13	#534×C₆₇	—	2	2	no	—
14	#532×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
16	C₂×#121×C₆₇	—	2	2	no	—
17	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
18	#489×C₆₇	—	2	4	no	—
19	#487×C₆₇	—	2	2	no	—
20	#486×C₆₇	—	2	2	no	—
21	C₂×#104×C₆₇	C₂²	1	8	yes	complemented
22	#417×C₆₇	C₂²	1	8	yes	complemented
23	C₂×#100×C₆₇	C₂²	1	2	yes	—
24	C₄×M₁₆×C₆₇	C₂²	1	2	yes	complemented
25	#393×C₆₇	—	2	2	no	—
26	#390×C₆₇	C₂²	1	2	yes	—
27	#382×C₆₇	C₂²	1	2	yes	complemented
28	#380×C₆₇	—	2	2	no	—
29	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
30	#343×C₆₇	C₂²	1	2	yes	complemented
31	H(5,𝔽₂)×C₆₇	—	2	8	no	—
32	H(5,𝔽₂)×C₆₇	—	4	4	no	—
33	C₂×#41×C₆₇	—	2	4	no	—
34	C₂×#41×C₆₇	C₂³	1	4	yes	—
35	C₂²×Q₈×C₆₇	—	2	1	no	—
36	C₂²×D₄×C₆₇	—	2	4	no	—
37	C₂²×D₄×C₆₇	—	2	2	no	—
38	C₂²×D₄×C₆₇	—	2	4	no	—
39	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	D₄	1	2	yes	—
41	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
42	C₄⋊Q₈×C₆₇	—	2	2	no	—
43	C₄⋊Q₈×C₆₇	—	2	2	no	—
44	C₄⋊Q₈×C₆₇	D₄	1	2	yes	—
45	#127×C₆₇	D₄	1	2	yes	—
46	#124×C₆₇	D₄	1	4	yes	—
47	#123×C₆₇	—	2	8	no	—
48	C₂²⋊Q₈×C₆₇	—	2	4	no	—
49	#121×C₆₇	—	2	4	no	—
50	#121×C₆₇	—	2	4	no	—
51	#121×C₆₇	—	2	4	no	—
52	#121×C₆₇	—	2	2	no	—
53	#121×C₆₇	—	2	4	no	—
54	#121×C₆₇	—	2	2	no	—
55	#120×C₆₇	—	2	8	no	—
56	C₄×D₄×C₆₇	—	2	2	no	—
57	C₄×D₄×C₆₇	—	2	2	no	—
58	C₄×D₄×C₆₇	—	2	2	no	—
59	C₄×D₄×C₆₇	—	2	2	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
63	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
66	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
67	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
68	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
69	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
70	#104×C₆₇	—	4	8	no	—
71	#103×C₆₇	—	2	8	no	—
72	#102×C₆₇	—	2	8	no	—
73	#100×C₆₇	—	2	4	no	—
74	#97×C₆₇	—	2	2	no	—
75	C₄×C₈×C₆₇	—	2	2	no	—
76	#95×C₆₇	—	2	1	no	—
77	#95×C₆₇	—	2	4	no	—
78	C₂⁴×C₆₇	—	2	4	no	—
79	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
80	#41×C₆₇	—	2	8	no	—
81	#41×C₆₇	—	2	8	no	—
82	#41×C₆₇	—	4	16	no	—
83	C₂×Q₈×C₆₇	—	4	2	no	—
84	C₂×Q₈×C₆₇	—	8	2	no	—
85	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
86	C₂×D₄×C₆₇	—	2	8	no	—
87	C₂×D₄×C₆₇	—	2	4	no	—
88	C₂×D₄×C₆₇	—	2	4	no	—
89	C₂×D₄×C₆₇	—	2	2	no	—
90	C₂×D₄×C₆₇	—	2	2	no	—
91	C₂×D₄×C₆₇	—	2	2	no	—
92	C₂×D₄×C₆₇	—	2	2	no	—
93	C₂×D₄×C₆₇	—	4	16	no	—
94	C₂×D₄×C₆₇	—	4	8	no	—
95	C₂×D₄×C₆₇	—	4	8	no	—
96	C₂×D₄×C₆₇	—	8	4	no	—
97	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
98	C₂×D₄×C₆₇	#41	1	4	yes	—
99	C₂²×C₄×C₆₇	—	2	4	no	—
100	C₂²×C₄×C₆₇	—	2	2	no	—
101	C₂²×C₄×C₆₇	—	2	4	no	—
102	C₂²×C₄×C₆₇	—	2	2	no	—
103	C₂²×C₄×C₆₇	—	2	2	no	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	—	4	4	no	—
106	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
108	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
110	M₁₆×C₆₇	—	2	4	no	—
111	M₁₆×C₆₇	—	2	4	no	—
112	C₂×C₈×C₆₇	—	2	4	no	—
113	C₄⋊C₄×C₆₇	—	2	4	no	—
114	C₄⋊C₄×C₆₇	—	2	2	no	—
115	C₄⋊C₄×C₆₇	—	2	2	no	—
116	C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₄⋊C₄×C₆₇	—	2	4	no	—
118	C₄⋊C₄×C₆₇	—	4	2	no	—
119	C₄⋊C₄×C₆₇	—	4	1	no	—
120	C₄⋊C₄×C₆₇	—	4	1	no	—
121	C₂²⋊C₄×C₆₇	—	2	8	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	—	8	4	no	—
126	C₄²×C₆₇	—	2	4	no	—
127	C₄²×C₆₇	C₂×D₄	1	2	yes	—
128	C₄²×C₆₇	C₂×D₄	1	2	yes	—
129	C₂³×C₆₇	—	2	2	no	—
130	C₂³×C₆₇	—	2	2	no	—
131	C₂³×C₆₇	—	2	4	no	—
132	C₂³×C₆₇	—	4	2	no	—
133	C₂³×C₆₇	—	4	2	no	—
134	C₂³×C₆₇	—	4	8	no	—
135	C₂³×C₆₇	—	8	4	no	—
136	C₂³×C₆₇	—	8	4	no	—
137	C₂³×C₆₇	#124	1	4	yes	—
138	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
139	Q₈×C₆₇	—	2	8	no	—
140	Q₈×C₆₇	—	8	4	no	—
141	D₄×C₆₇	—	2	8	no	—
142	D₄×C₆₇	—	2	8	no	—
143	D₄×C₆₇	—	4	16	no	—
144	D₄×C₆₇	—	4	4	no	—
145	D₄×C₆₇	—	4	4	no	—
146	D₄×C₆₇	—	4	4	no	—
147	D₄×C₆₇	—	8	2	no	—
148	D₄×C₆₇	—	8	4	no	—
149	C₂×C₄×C₆₇	—	2	2	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	2	4	no	—
152	C₂×C₄×C₆₇	—	4	1	no	—
153	C₂×C₄×C₆₇	—	4	8	no	—
154	C₂×C₄×C₆₇	—	4	8	no	—
155	C₂×C₄×C₆₇	—	4	4	no	—
156	C₂×C₄×C₆₇	—	4	2	no	—
157	C₂×C₄×C₆₇	—	4	2	no	—
158	C₂×C₄×C₆₇	—	4	4	no	—
159	C₂×C₄×C₆₇	—	4	2	no	—
160	C₂×C₄×C₆₇	—	4	2	no	—
161	C₂×C₄×C₆₇	—	4	2	no	—
162	C₂×C₄×C₆₇	—	8	2	no	—
163	C₂×C₄×C₆₇	#120	1	2	yes	—
164	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
165	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
166	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
167	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
168	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
169	C₈×C₆₇	—	4	4	no	—
170	C₂²×C₆₇	—	2	1	no	—
171	C₂²×C₆₇	—	2	1	no	—
172	C₂²×C₆₇	—	2	4	no	—
173	C₂²×C₆₇	—	4	4	no	—
174	C₂²×C₆₇	—	4	2	no	—
175	C₂²×C₆₇	—	4	2	no	—
176	C₂²×C₆₇	—	4	4	no	—
177	C₂²×C₆₇	—	4	4	no	—
178	C₂²×C₆₇	—	4	4	no	—
179	C₂²×C₆₇	—	8	2	no	—
180	C₂²×C₆₇	—	16	8	no	—
181	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
182	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
183	C₂²×C₆₇	#535	1	1	yes	characteristic
184	C₄×C₆₇	—	2	1	no	—
185	C₄×C₆₇	—	2	2	no	—
186	C₄×C₆₇	—	4	4	no	—
187	C₄×C₆₇	—	4	2	no	—
188	C₄×C₆₇	—	8	2	no	—
189	C₄×C₆₇	—	8	2	no	—
190	C₄×C₆₇	#455	1	2	yes	—
191	C₄×C₆₇	#546	1	4	yes	—
192	#25869	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
193	C₂×C₆₇	—	2	1	no	—
194	C₂×C₆₇	—	2	1	no	—
195	C₂×C₆₇	—	4	4	no	—
196	C₂×C₆₇	—	8	4	no	—
197	C₂×C₆₇	#2435	1	1	yes	characteristic
198	C₂×C₆₇	C₂×#455	1	1	yes	characteristic
199	C₂×C₆₇	#3021	1	1	yes	characteristic
200	#3496	C₂×C₆₇	1	1	yes	characteristic
201	#3141	C₂×C₆₇	1	2	yes	complemented
202	#2563	C₂×C₆₇	1	1	yes	characteristic, complemented
203	#2040	C₂×C₆₇	1	4	yes	complemented
204	#1980	C₂×C₆₇	1	2	yes	complemented
205	#1792	C₂×C₆₇	1	4	yes	complemented
206	C₄×#100	C₂×C₆₇	1	1	yes	characteristic, complemented
207	C₆₇	#25869	1	1	yes	characteristic, direct product factor, Sylow
208	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	2	yes	—
209	#547	—	2	2	no	—
210	#546	—	2	4	no	—
211	D₄²	—	2	2	no	—
212	#537	C₂²×C₆₇	1	2	yes	—
213	#534	—	2	2	no	—
214	#532	C₂²×C₆₇	1	2	yes	—
215	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
216	C₂×#121	—	2	2	no	—
217	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
218	#489	—	2	4	no	—
219	#487	—	2	2	no	—
220	#486	—	2	2	no	—
221	C₂×#104	C₂²×C₆₇	1	8	yes	complemented
222	#417	C₂²×C₆₇	1	8	yes	complemented
223	C₂×#100	C₂²×C₆₇	1	2	yes	—
224	C₄×M₁₆	C₂²×C₆₇	1	2	yes	complemented
225	#393	—	2	2	no	—
226	#390	C₂²×C₆₇	1	2	yes	—
227	#382	C₂²×C₆₇	1	2	yes	complemented
228	#380	—	2	2	no	—
229	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
230	#343	C₂²×C₆₇	1	2	yes	complemented
231	H(5,𝔽₂)	—	2	8	no	—
232	H(5,𝔽₂)	—	4	4	no	—
233	C₂×#41	—	2	4	no	—
234	C₂×#41	C₂³×C₆₇	1	4	yes	—
235	C₂²×Q₈	—	2	1	no	—
236	C₂²×D₄	—	2	4	no	—
237	C₂²×D₄	—	2	2	no	—
238	C₂²×D₄	—	2	4	no	—
239	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
240	C₂³×C₄	D₄×C₆₇	1	2	yes	—
241	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
242	C₄⋊Q₈	—	2	2	no	—
243	C₄⋊Q₈	—	2	2	no	—
244	C₄⋊Q₈	D₄×C₆₇	1	2	yes	—
245	#127	D₄×C₆₇	1	2	yes	—
246	#124	D₄×C₆₇	1	4	yes	—
247	#123	—	2	8	no	—
248	C₂²⋊Q₈	—	2	4	no	—
249	#121	—	2	4	no	—
250	#121	—	2	4	no	—
251	#121	—	2	4	no	—
252	#121	—	2	2	no	—
253	#121	—	2	4	no	—
254	#121	—	2	2	no	—
255	#120	—	2	8	no	—
256	C₄×D₄	—	2	2	no	—
257	C₄×D₄	—	2	2	no	—
258	C₄×D₄	—	2	2	no	—
259	C₄×D₄	—	2	2	no	—
260	C₂×C₄⋊C₄	—	2	2	no	—
261	C₂×C₄⋊C₄	—	2	1	no	—
262	C₂×C₄⋊C₄	—	2	1	no	—
263	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
264	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
265	C₂×C₂²⋊C₄	—	2	4	no	—
266	C₂×C₂²⋊C₄	—	2	4	no	—
267	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
268	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
269	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
270	#104	—	4	8	no	—
271	#103	—	2	8	no	—
272	#102	—	2	8	no	—
273	#100	—	2	4	no	—
274	#97	—	2	2	no	—
275	C₄×C₈	—	2	2	no	—
276	#95	—	2	1	no	—
277	#95	—	2	4	no	—
278	C₂⁴	—	2	4	no	—
279	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
280	#41	—	2	8	no	—
281	#41	—	2	8	no	—
282	#41	—	4	16	no	—
283	C₂×Q₈	—	4	2	no	—
284	C₂×Q₈	—	8	2	no	—
285	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
286	C₂×D₄	—	2	8	no	—
287	C₂×D₄	—	2	4	no	—
288	C₂×D₄	—	2	4	no	—
289	C₂×D₄	—	2	2	no	—
290	C₂×D₄	—	2	2	no	—
291	C₂×D₄	—	2	2	no	—
292	C₂×D₄	—	2	2	no	—
293	C₂×D₄	—	4	16	no	—
294	C₂×D₄	—	4	8	no	—
295	C₂×D₄	—	4	8	no	—
296	C₂×D₄	—	8	4	no	—
297	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
298	C₂×D₄	#41×C₆₇	1	4	yes	—
299	C₂²×C₄	—	2	4	no	—
300	C₂²×C₄	—	2	2	no	—
301	C₂²×C₄	—	2	4	no	—
302	C₂²×C₄	—	2	2	no	—
303	C₂²×C₄	—	2	2	no	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	4	4	no	—
306	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
307	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
308	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
309	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
310	M₁₆	—	2	4	no	—
311	M₁₆	—	2	4	no	—
312	C₂×C₈	—	2	4	no	—
313	C₄⋊C₄	—	2	4	no	—
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	2	4	no	—
318	C₄⋊C₄	—	4	2	no	—
319	C₄⋊C₄	—	4	1	no	—
320	C₄⋊C₄	—	4	1	no	—
321	C₂²⋊C₄	—	2	8	no	—
322	C₂²⋊C₄	—	2	4	no	—
323	C₂²⋊C₄	—	2	4	no	—
324	C₂²⋊C₄	—	2	4	no	—
325	C₂²⋊C₄	—	8	4	no	—
326	C₄²	—	2	4	no	—
327	C₄²	C₂×D₄×C₆₇	1	2	yes	—
328	C₄²	C₂×D₄×C₆₇	1	2	yes	—
329	C₂³	—	2	2	no	—
330	C₂³	—	2	2	no	—
331	C₂³	—	2	4	no	—
332	C₂³	—	4	2	no	—
333	C₂³	—	4	2	no	—
334	C₂³	—	4	8	no	—
335	C₂³	—	8	4	no	—
336	C₂³	—	8	4	no	—
337	C₂³	#124×C₆₇	1	4	yes	—
338	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
339	Q₈	—	2	8	no	—
340	Q₈	—	8	4	no	—
341	D₄	—	2	8	no	—
342	D₄	—	2	8	no	—
343	D₄	—	4	16	no	—
344	D₄	—	4	4	no	—
345	D₄	—	4	4	no	—
346	D₄	—	4	4	no	—
347	D₄	—	8	2	no	—
348	D₄	—	8	4	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	4	no	—
352	C₂×C₄	—	4	1	no	—
353	C₂×C₄	—	4	8	no	—
354	C₂×C₄	—	4	4	no	—
355	C₂×C₄	—	4	8	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	4	4	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	8	2	no	—
363	C₂×C₄	#120×C₆₇	1	2	yes	—
364	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
365	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
366	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
367	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
368	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
369	C₈	—	4	4	no	—
370	C₂²	—	2	1	no	—
371	C₂²	—	2	1	no	—
372	C₂²	—	2	4	no	—
373	C₂²	—	4	4	no	—
374	C₂²	—	4	2	no	—
375	C₂²	—	4	2	no	—
376	C₂²	—	4	4	no	—
377	C₂²	—	4	4	no	—
378	C₂²	—	4	4	no	—
379	C₂²	—	8	2	no	—
380	C₂²	—	16	8	no	—
381	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
382	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
383	C₂²	#535×C₆₇	1	1	yes	characteristic
384	C₄	—	2	1	no	—
385	C₄	—	2	2	no	—
386	C₄	—	4	2	no	—
387	C₄	—	4	4	no	—
388	C₄	—	8	2	no	—
389	C₄	—	8	2	no	—
390	C₄	#455×C₆₇	1	2	yes	—
391	C₄	#546×C₆₇	1	4	yes	—
392	C₂	—	2	1	no	—
393	C₂	—	2	1	no	—
394	C₂	—	4	4	no	—
395	C₂	—	8	4	no	—
396	C₂	C₆₇×#2435	1	1	yes	characteristic
397	C₂	C₂×#455×C₆₇	1	1	yes	characteristic
398	C₂	C₆₇×#3021	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#3496	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#3141	C₂	semidirect
3. C₆₇×#2563	C₂	semidirect
4. C₆₇×#2040	C₂	semidirect
5. C₆₇×#1980	C₂	semidirect
6. C₆₇×#1792	C₂	semidirect
7. C₄×#100×C₆₇	C₂	semidirect
15. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
21. C₂×#104×C₆₇	C₂²	semidirect
22. #417×C₆₇	C₂²	semidirect
24. C₄×M₁₆×C₆₇	C₂²	semidirect
27. #382×C₆₇	C₂²	semidirect
29. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
30. #343×C₆₇	C₂²	semidirect
192. #25869	C₆₇	direct (N × H)
201. #3141	C₂×C₆₇	semidirect
202. #2563	C₂×C₆₇	semidirect
203. #2040	C₂×C₆₇	semidirect
204. #1980	C₂×C₆₇	semidirect
205. #1792	C₂×C₆₇	semidirect
206. C₄×#100	C₂×C₆₇	semidirect
207. C₆₇	#25869	direct (N × H)
215. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
221. C₂×#104	C₂²×C₆₇	semidirect
222. #417	C₂²×C₆₇	semidirect
224. C₄×M₁₆	C₂²×C₆₇	semidirect
227. #382	C₂²×C₆₇	semidirect
229. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
230. #343	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3496	2
C₆₇×#3141	2
C₆₇×#2563	2
C₆₇×#2040	2
C₆₇×#1980	2
C₆₇×#1792	2
C₄×#100×C₆₇	2
#25869	67

Subgroups — C67×#25869 #30705120

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25869 #30705120