Subgroups — C₆₇×#33862 #30713113

Order 17152.

622 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3338	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2240	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2233	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2226	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2217	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2207	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2187	C₂	1	2	yes	maximal, complemented
9	C₄×SD₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	D₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#357×C₆₇	C₂	1	2	yes	maximal, complemented
12	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
13	#554×C₆₇	—	2	1	no	—
14	D₄²×C₆₇	—	2	1	no	—
15	C₂×SD₃₂×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	complemented
17	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	complemented
18	#500×C₆₇	C₂²	1	1	yes	characteristic, complemented
19	#493×C₆₇	C₂²	1	1	yes	characteristic
20	#484×C₆₇	—	2	2	no	—
21	#477×C₆₇	—	2	1	no	—
22	#467×C₆₇	C₂²	1	2	yes	complemented
23	#466×C₆₇	C₂²	1	2	yes	—
24	#459×C₆₇	—	2	1	no	—
25	#447×C₆₇	—	2	2	no	—
26	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
27	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
28	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#107×C₆₇	C₂²	1	2	yes	complemented
30	#367×C₆₇	C₂²	1	1	yes	characteristic, complemented
31	#367×C₆₇	C₂²	1	2	yes	complemented
32	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
33	#358×C₆₇	C₂²	1	1	yes	characteristic, complemented
34	#357×C₆₇	C₂²	1	1	yes	characteristic
35	#357×C₆₇	C₂²	1	2	yes	—
36	#357×C₆₇	C₂²	1	2	yes	complemented
37	#357×C₆₇	C₂²	1	2	yes	complemented
38	#357×C₆₇	C₂²	1	2	yes	complemented
39	C₂²⋊C₁₆×C₆₇	C₂²	1	2	yes	complemented
40	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
41	C₂²×D₄×C₆₇	—	2	2	no	—
42	C₂²×D₄×C₆₇	—	4	1	no	—
43	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
44	C₂×D₈×C₆₇	—	2	4	no	—
45	C₂×D₈×C₆₇	—	2	2	no	—
46	C₂×D₈×C₆₇	—	2	1	no	—
47	C₂×D₈×C₆₇	—	4	2	no	—
48	C₂×D₈×C₆₇	C₂³	1	2	yes	—
49	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
51	C₄⋊Q₈×C₆₇	—	2	1	no	—
52	#127×C₆₇	—	2	1	no	—
53	#125×C₆₇	—	4	1	no	—
54	C₂²⋊Q₈×C₆₇	—	2	2	no	—
55	C₂²⋊Q₈×C₆₇	—	4	2	no	—
56	#121×C₆₇	—	2	2	no	—
57	#121×C₆₇	—	4	1	no	—
58	#120×C₆₇	—	4	2	no	—
59	C₄×Q₈×C₆₇	—	2	1	no	—
60	C₄×D₄×C₆₇	—	2	1	no	—
61	C₄×D₄×C₆₇	—	4	1	no	—
62	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
64	SD₃₂×C₆₇	—	2	2	no	—
65	SD₃₂×C₆₇	—	4	2	no	—
66	C₂×C₁₆×C₆₇	—	4	2	no	—
67	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
68	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
69	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
70	#107×C₆₇	—	2	2	no	—
71	#107×C₆₇	—	2	1	no	—
72	#107×C₆₇	C₂³	1	1	yes	characteristic
73	#107×C₆₇	C₂³	1	2	yes	—
74	#107×C₆₇	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
76	#103×C₆₇	—	2	2	no	—
77	#103×C₆₇	—	2	1	no	—
78	#102×C₆₇	—	2	2	no	—
79	#102×C₆₇	—	2	1	no	—
80	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
81	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
82	C₂⁴×C₆₇	—	4	2	no	—
83	C₂×Q₈×C₆₇	—	2	1	no	—
84	C₂×Q₈×C₆₇	—	4	1	no	—
85	C₂×D₄×C₆₇	—	2	1	no	—
86	C₂×D₄×C₆₇	—	2	2	no	—
87	C₂×D₄×C₆₇	—	4	4	no	—
88	C₂×D₄×C₆₇	—	4	2	no	—
89	C₂×D₄×C₆₇	—	4	2	no	—
90	C₂×D₄×C₆₇	—	4	1	no	—
91	C₂×D₄×C₆₇	—	4	1	no	—
92	C₂×D₄×C₆₇	—	8	2	no	—
93	C₂×D₄×C₆₇	—	8	4	no	—
94	C₂×D₄×C₆₇	D₈	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	—	4	2	no	—
96	C₂²×C₄×C₆₇	—	4	1	no	—
97	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
98	Q₁₆×C₆₇	—	2	1	no	—
99	Q₁₆×C₆₇	—	2	1	no	—
100	D₈×C₆₇	—	2	2	no	—
101	D₈×C₆₇	—	2	1	no	—
102	D₈×C₆₇	—	2	2	no	—
103	D₈×C₆₇	—	4	4	no	—
104	D₈×C₆₇	—	8	1	no	—
105	D₈×C₆₇	#41	1	2	yes	—
106	C₂×C₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	2	2	no	—
108	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
111	C₄⋊C₄×C₆₇	—	2	1	no	—
112	C₄⋊C₄×C₆₇	—	2	1	no	—
113	C₄⋊C₄×C₆₇	—	2	2	no	—
114	C₄⋊C₄×C₆₇	—	4	2	no	—
115	C₄⋊C₄×C₆₇	—	4	1	no	—
116	C₄⋊C₄×C₆₇	—	4	1	no	—
117	C₄⋊C₄×C₆₇	—	4	2	no	—
118	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
119	C₂²⋊C₄×C₆₇	—	4	2	no	—
120	C₂²⋊C₄×C₆₇	—	4	1	no	—
121	C₂²⋊C₄×C₆₇	—	4	2	no	—
122	C₂²⋊C₄×C₆₇	D₈	1	2	yes	—
123	C₄²×C₆₇	—	4	1	no	—
124	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₁₆×C₆₇	—	2	1	no	—
126	C₁₆×C₆₇	—	2	1	no	—
127	C₁₆×C₆₇	—	2	2	no	—
128	C₂³×C₆₇	—	4	1	no	—
129	C₂³×C₆₇	—	4	2	no	—
130	C₂³×C₆₇	—	8	4	no	—
131	C₂³×C₆₇	—	8	4	no	—
132	C₂³×C₆₇	—	16	2	no	—
133	C₂³×C₆₇	C₂×D₈	1	2	yes	—
134	Q₈×C₆₇	—	4	1	no	—
135	Q₈×C₆₇	—	4	1	no	—
136	Q₈×C₆₇	—	16	1	no	—
137	D₄×C₆₇	—	2	2	no	—
138	D₄×C₆₇	—	4	2	no	—
139	D₄×C₆₇	—	4	1	no	—
140	D₄×C₆₇	—	4	2	no	—
141	D₄×C₆₇	—	8	4	no	—
142	D₄×C₆₇	—	8	4	no	—
143	D₄×C₆₇	—	8	2	no	—
144	D₄×C₆₇	—	16	1	no	—
145	D₄×C₆₇	SD₃₂	1	4	yes	complemented
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	2	2	no	—
148	C₂×C₄×C₆₇	—	4	1	no	—
149	C₂×C₄×C₆₇	—	4	1	no	—
150	C₂×C₄×C₆₇	—	4	1	no	—
151	C₂×C₄×C₆₇	—	4	2	no	—
152	C₂×C₄×C₆₇	—	8	2	no	—
153	C₂×C₄×C₆₇	—	16	2	no	—
154	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
156	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
158	C₈×C₆₇	—	2	2	no	—
159	C₈×C₆₇	—	2	1	no	—
160	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
161	C₈×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂²×C₆₇	—	2	2	no	—
163	C₂²×C₆₇	—	2	2	no	—
164	C₂²×C₆₇	—	4	2	no	—
165	C₂²×C₆₇	—	8	1	no	—
166	C₂²×C₆₇	—	8	1	no	—
167	C₂²×C₆₇	—	8	2	no	—
168	C₂²×C₆₇	—	8	2	no	—
169	C₂²×C₆₇	—	8	2	no	—
170	C₂²×C₆₇	—	16	8	no	—
171	C₂²×C₆₇	C₂×SD₃₂	1	4	yes	—
172	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
173	C₄×C₆₇	—	2	1	no	—
174	C₄×C₆₇	—	2	2	no	—
175	C₄×C₆₇	—	8	1	no	—
176	C₄×C₆₇	—	8	1	no	—
177	C₄×C₆₇	—	8	2	no	—
178	C₄×C₆₇	—	8	1	no	—
179	C₄×C₆₇	C₂×SD₃₂	1	2	yes	—
180	C₄×C₆₇	#546	1	1	yes	characteristic
181	C₄×C₆₇	#577	1	1	yes	characteristic
182	#33862	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
183	C₂×C₆₇	—	2	4	no	—
184	C₂×C₆₇	—	8	2	no	—
185	C₂×C₆₇	—	16	2	no	—
186	C₂×C₆₇	#3294	1	1	yes	characteristic
187	C₂×C₆₇	C₂²×SD₃₂	1	1	yes	characteristic
188	C₂×C₆₇	#3415	1	1	yes	characteristic
189	#3338	C₂×C₆₇	1	1	yes	characteristic, complemented
190	D₄×D₈	C₂×C₆₇	1	1	yes	characteristic
191	#2240	C₂×C₆₇	1	1	yes	characteristic, complemented
192	#2233	C₂×C₆₇	1	2	yes	complemented
193	#2226	C₂×C₆₇	1	1	yes	characteristic, complemented
194	#2217	C₂×C₆₇	1	2	yes	complemented
195	#2207	C₂×C₆₇	1	1	yes	characteristic, complemented
196	#2187	C₂×C₆₇	1	2	yes	complemented
197	C₄×SD₃₂	C₂×C₆₇	1	1	yes	characteristic, complemented
198	D₄×C₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
199	C₂×#357	C₂×C₆₇	1	2	yes	complemented
200	C₆₇	#33862	1	1	yes	characteristic, direct product factor, Sylow
201	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
202	#554	—	2	1	no	—
203	D₄²	—	2	1	no	—
204	C₂×SD₃₂	C₂²×C₆₇	1	1	yes	characteristic
205	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	complemented
206	C₂²×C₁₆	C₂²×C₆₇	1	2	yes	complemented
207	#500	C₂²×C₆₇	1	1	yes	characteristic, complemented
208	#493	C₂²×C₆₇	1	1	yes	characteristic
209	#484	—	2	2	no	—
210	#477	—	2	1	no	—
211	#467	C₂²×C₆₇	1	2	yes	complemented
212	#466	C₂²×C₆₇	1	2	yes	—
213	#459	—	2	1	no	—
214	#447	—	2	2	no	—
215	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
216	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
217	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
218	C₂×#107	C₂²×C₆₇	1	2	yes	complemented
219	#367	C₂²×C₆₇	1	1	yes	characteristic, complemented
220	#367	C₂²×C₆₇	1	2	yes	complemented
221	C₄⋊C₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
222	#358	C₂²×C₆₇	1	1	yes	characteristic, complemented
223	#357	C₂²×C₆₇	1	1	yes	characteristic
224	#357	C₂²×C₆₇	1	2	yes	—
225	#357	C₂²×C₆₇	1	2	yes	complemented
226	#357	C₂²×C₆₇	1	2	yes	complemented
227	#357	C₂²×C₆₇	1	2	yes	complemented
228	C₂²⋊C₁₆	C₂²×C₆₇	1	2	yes	complemented
229	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
230	C₂²×D₄	—	2	2	no	—
231	C₂²×D₄	—	4	1	no	—
232	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
233	C₂×D₈	—	2	4	no	—
234	C₂×D₈	—	2	2	no	—
235	C₂×D₈	—	2	1	no	—
236	C₂×D₈	—	4	2	no	—
237	C₂×D₈	C₂³×C₆₇	1	2	yes	—
238	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
239	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
240	C₄⋊Q₈	—	2	1	no	—
241	#127	—	2	1	no	—
242	#125	—	4	1	no	—
243	C₂²⋊Q₈	—	2	2	no	—
244	C₂²⋊Q₈	—	4	2	no	—
245	#121	—	2	2	no	—
246	#121	—	4	1	no	—
247	#120	—	4	2	no	—
248	C₄×Q₈	—	2	1	no	—
249	C₄×D₄	—	2	1	no	—
250	C₄×D₄	—	4	1	no	—
251	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
252	C₂×C₄⋊C₄	—	2	2	no	—
253	SD₃₂	—	2	2	no	—
254	SD₃₂	—	4	2	no	—
255	C₂×C₁₆	—	4	2	no	—
256	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
257	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
258	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
259	#107	—	2	2	no	—
260	#107	—	2	1	no	—
261	#107	C₂³×C₆₇	1	1	yes	characteristic
262	#107	C₂³×C₆₇	1	2	yes	—
263	#107	C₂³×C₆₇	1	1	yes	characteristic
264	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
265	#103	—	2	1	no	—
266	#103	—	2	2	no	—
267	#102	—	2	1	no	—
268	#102	—	2	2	no	—
269	C₂²⋊C₈	D₄×C₆₇	1	2	yes	—
270	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
271	C₂⁴	—	4	2	no	—
272	C₂×Q₈	—	2	1	no	—
273	C₂×Q₈	—	4	1	no	—
274	C₂×D₄	—	2	1	no	—
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	—	4	4	no	—
277	C₂×D₄	—	4	2	no	—
278	C₂×D₄	—	4	2	no	—
279	C₂×D₄	—	4	1	no	—
280	C₂×D₄	—	4	1	no	—
281	C₂×D₄	—	8	2	no	—
282	C₂×D₄	—	8	4	no	—
283	C₂×D₄	D₈×C₆₇	1	1	yes	characteristic
284	C₂²×C₄	—	4	2	no	—
285	C₂²×C₄	—	4	1	no	—
286	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
287	Q₁₆	—	2	1	no	—
288	Q₁₆	—	2	1	no	—
289	D₈	—	2	2	no	—
290	D₈	—	2	1	no	—
291	D₈	—	2	2	no	—
292	D₈	—	4	4	no	—
293	D₈	—	8	1	no	—
294	D₈	#41×C₆₇	1	2	yes	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
298	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
299	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	2	1	no	—
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	—	4	2	no	—
304	C₄⋊C₄	—	4	1	no	—
305	C₄⋊C₄	—	4	1	no	—
306	C₄⋊C₄	—	4	2	no	—
307	C₄⋊C₄	D₈×C₆₇	1	1	yes	characteristic
308	C₂²⋊C₄	—	4	2	no	—
309	C₂²⋊C₄	—	4	1	no	—
310	C₂²⋊C₄	—	4	2	no	—
311	C₂²⋊C₄	D₈×C₆₇	1	2	yes	—
312	C₄²	—	4	1	no	—
313	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
314	C₁₆	—	2	1	no	—
315	C₁₆	—	2	1	no	—
316	C₁₆	—	2	2	no	—
317	C₂³	—	4	1	no	—
318	C₂³	—	4	2	no	—
319	C₂³	—	8	4	no	—
320	C₂³	—	8	4	no	—
321	C₂³	—	16	2	no	—
322	C₂³	C₂×D₈×C₆₇	1	2	yes	—
323	Q₈	—	4	1	no	—
324	Q₈	—	4	1	no	—
325	Q₈	—	16	1	no	—
326	D₄	—	2	2	no	—
327	D₄	—	4	2	no	—
328	D₄	—	4	1	no	—
329	D₄	—	4	2	no	—
330	D₄	—	8	4	no	—
331	D₄	—	8	4	no	—
332	D₄	—	8	2	no	—
333	D₄	—	16	1	no	—
334	D₄	SD₃₂×C₆₇	1	4	yes	complemented
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	4	1	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	8	2	no	—
342	C₂×C₄	—	16	2	no	—
343	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
344	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
345	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
346	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
347	C₈	—	2	2	no	—
348	C₈	—	2	1	no	—
349	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
350	C₈	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
351	C₂²	—	2	2	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	4	2	no	—
354	C₂²	—	8	1	no	—
355	C₂²	—	8	1	no	—
356	C₂²	—	8	2	no	—
357	C₂²	—	8	2	no	—
358	C₂²	—	8	2	no	—
359	C₂²	—	16	8	no	—
360	C₂²	C₂×SD₃₂×C₆₇	1	4	yes	—
361	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
362	C₄	—	2	2	no	—
363	C₄	—	2	1	no	—
364	C₄	—	8	1	no	—
365	C₄	—	8	1	no	—
366	C₄	—	8	1	no	—
367	C₄	—	8	2	no	—
368	C₄	C₂×SD₃₂×C₆₇	1	2	yes	—
369	C₄	#546×C₆₇	1	1	yes	characteristic
370	C₄	#577×C₆₇	1	1	yes	characteristic
371	C₂	—	2	4	no	—
372	C₂	—	8	2	no	—
373	C₂	—	16	2	no	—
374	C₂	C₆₇×#3294	1	1	yes	characteristic
375	C₂	C₂²×SD₃₂×C₆₇	1	1	yes	characteristic
376	C₂	C₆₇×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	D₄×D₈×C₆₇	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄×D₄×C₆₇ ◁ C₈×D₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3338	C₂	semidirect
3. C₆₇×#2240	C₂	semidirect
4. C₆₇×#2233	C₂	semidirect
5. C₆₇×#2226	C₂	semidirect
6. C₆₇×#2217	C₂	semidirect
7. C₆₇×#2207	C₂	semidirect
8. C₆₇×#2187	C₂	semidirect
9. C₄×SD₃₂×C₆₇	C₂	semidirect
10. D₄×C₁₆×C₆₇	C₂	semidirect
11. C₂×#357×C₆₇	C₂	semidirect
16. C₂×SD₃₂×C₆₇	C₂²	semidirect
17. C₂²×C₁₆×C₆₇	C₂²	semidirect
18. #500×C₆₇	C₂²	semidirect
22. #467×C₆₇	C₂²	semidirect
26. C₄×Q₁₆×C₆₇	C₂²	semidirect
29. C₂×#107×C₆₇	C₂²	semidirect
30. #367×C₆₇	C₂²	semidirect
31. #367×C₆₇	C₂²	semidirect
32. C₄⋊C₁₆×C₆₇	C₂²	semidirect
33. #358×C₆₇	C₂²	semidirect
36. #357×C₆₇	C₂²	semidirect
37. #357×C₆₇	C₂²	semidirect
38. #357×C₆₇	C₂²	semidirect
39. C₂²⋊C₁₆×C₆₇	C₂²	semidirect
40. C₄×C₁₆×C₆₇	C₂²	semidirect
145. D₄×C₆₇	SD₃₂	semidirect
182. #33862	C₆₇	direct (N × H)
189. #3338	C₂×C₆₇	semidirect
191. #2240	C₂×C₆₇	semidirect
192. #2233	C₂×C₆₇	semidirect
193. #2226	C₂×C₆₇	semidirect
194. #2217	C₂×C₆₇	semidirect
195. #2207	C₂×C₆₇	semidirect
196. #2187	C₂×C₆₇	semidirect
197. C₄×SD₃₂	C₂×C₆₇	semidirect
198. D₄×C₁₆	C₂×C₆₇	semidirect
199. C₂×#357	C₂×C₆₇	semidirect
200. C₆₇	#33862	direct (N × H)
205. C₂×SD₃₂	C₂²×C₆₇	semidirect
206. C₂²×C₁₆	C₂²×C₆₇	semidirect
207. #500	C₂²×C₆₇	semidirect
211. #467	C₂²×C₆₇	semidirect
215. C₄×Q₁₆	C₂²×C₆₇	semidirect
218. C₂×#107	C₂²×C₆₇	semidirect
219. #367	C₂²×C₆₇	semidirect
220. #367	C₂²×C₆₇	semidirect
221. C₄⋊C₁₆	C₂²×C₆₇	semidirect
222. #358	C₂²×C₆₇	semidirect
225. #357	C₂²×C₆₇	semidirect
226. #357	C₂²×C₆₇	semidirect
227. #357	C₂²×C₆₇	semidirect
228. C₂²⋊C₁₆	C₂²×C₆₇	semidirect
229. C₄×C₁₆	C₂²×C₆₇	semidirect
334. D₄	SD₃₂×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3338	2
D₄×D₈×C₆₇	2
C₆₇×#2240	2
C₆₇×#2233	2
C₆₇×#2226	2
C₆₇×#2217	2
C₆₇×#2207	2
C₆₇×#2187	2
C₄×SD₃₂×C₆₇	2
D₄×C₁₆×C₆₇	2
C₂×#357×C₆₇	2
#33862	67

Subgroups — C67×#33862 #30713113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33862 #30713113