Subgroups — C₆₇×#33861 #30713112

Order 17152.

654 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3334	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2240	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2232	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2224	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2215	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2206	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2184	C₂	1	2	yes	maximal, complemented
9	C₄×D₁₆×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	#547×C₆₇	—	2	1	no	—
14	D₄²×C₆₇	—	2	1	no	—
15	C₂×D₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×D₁₆×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	#480×C₆₇	—	2	2	no	—
21	#474×C₆₇	—	2	1	no	—
22	#466×C₆₇	C₂²	1	2	yes	—
23	#466×C₆₇	C₂²	1	2	yes	complemented
24	#459×C₆₇	—	2	1	no	—
25	#447×C₆₇	—	2	2	no	—
26	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
28	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#107×C₆₇	C₂²	1	2	yes	complemented
30	#366×C₆₇	C₂²	1	1	yes	characteristic, complemented
31	#366×C₆₇	C₂²	1	2	yes	complemented
32	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
33	#357×C₆₇	C₂²	1	1	yes	characteristic
34	#357×C₆₇	C₂²	1	2	yes	—
35	#357×C₆₇	C₂²	1	2	yes	complemented
36	#357×C₆₇	C₂²	1	2	yes	complemented
37	#357×C₆₇	C₂²	1	1	yes	characteristic, 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	C₂×#41×C₆₇	—	4	1	no	—
42	C₂²×D₄×C₆₇	—	2	2	no	—
43	C₂²×D₄×C₆₇	—	4	1	no	—
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	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
52	C₄⋊Q₈×C₆₇	—	2	1	no	—
53	#127×C₆₇	—	2	1	no	—
54	#123×C₆₇	—	4	2	no	—
55	C₂²⋊Q₈×C₆₇	—	4	1	no	—
56	#121×C₆₇	—	2	2	no	—
57	#121×C₆₇	—	2	2	no	—
58	#121×C₆₇	—	4	1	no	—
59	#120×C₆₇	—	4	2	no	—
60	C₄×D₄×C₆₇	—	2	1	no	—
61	C₄×D₄×C₆₇	—	2	1	no	—
62	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
64	D₁₆×C₆₇	—	2	2	no	—
65	D₁₆×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	#102×C₆₇	—	2	2	no	—
77	#102×C₆₇	—	2	2	no	—
78	#102×C₆₇	—	2	1	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	#41×C₆₇	—	8	4	no	—
84	C₂×Q₈×C₆₇	—	4	1	no	—
85	C₂×D₄×C₆₇	—	2	1	no	—
86	C₂×D₄×C₆₇	—	2	1	no	—
87	C₂×D₄×C₆₇	—	2	2	no	—
88	C₂×D₄×C₆₇	—	4	4	no	—
89	C₂×D₄×C₆₇	—	4	2	no	—
90	C₂×D₄×C₆₇	—	4	2	no	—
91	C₂×D₄×C₆₇	—	4	2	no	—
92	C₂×D₄×C₆₇	—	4	1	no	—
93	C₂×D₄×C₆₇	—	4	1	no	—
94	C₂×D₄×C₆₇	—	8	2	no	—
95	C₂×D₄×C₆₇	—	8	4	no	—
96	C₂×D₄×C₆₇	D₈	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	—	4	2	no	—
98	C₂²×C₄×C₆₇	—	4	1	no	—
99	C₂²×C₄×C₆₇	—	4	1	no	—
100	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
101	D₈×C₆₇	—	2	1	no	—
102	D₈×C₆₇	—	2	2	no	—
103	D₈×C₆₇	—	2	1	no	—
104	D₈×C₆₇	—	2	1	no	—
105	D₈×C₆₇	—	2	2	no	—
106	D₈×C₆₇	—	4	4	no	—
107	D₈×C₆₇	—	8	1	no	—
108	D₈×C₆₇	#41	1	2	yes	complemented
109	C₂×C₈×C₆₇	—	2	2	no	—
110	C₂×C₈×C₆₇	—	2	2	no	—
111	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
112	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄×C₆₇	—	2	1	no	—
115	C₄⋊C₄×C₆₇	—	2	1	no	—
116	C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₄⋊C₄×C₆₇	—	4	2	no	—
118	C₄⋊C₄×C₆₇	—	4	1	no	—
119	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
120	C₂²⋊C₄×C₆₇	—	4	1	no	—
121	C₂²⋊C₄×C₆₇	—	4	1	no	—
122	C₂²⋊C₄×C₆₇	—	4	2	no	—
123	C₂²⋊C₄×C₆₇	—	4	2	no	—
124	C₂²⋊C₄×C₆₇	D₈	1	2	yes	—
125	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₁₆×C₆₇	—	2	1	no	—
127	C₁₆×C₆₇	—	2	1	no	—
128	C₁₆×C₆₇	—	2	2	no	—
129	C₂³×C₆₇	—	4	1	no	—
130	C₂³×C₆₇	—	4	1	no	—
131	C₂³×C₆₇	—	4	2	no	—
132	C₂³×C₆₇	—	8	4	no	—
133	C₂³×C₆₇	—	8	4	no	—
134	C₂³×C₆₇	—	16	2	no	—
135	C₂³×C₆₇	C₂×D₈	1	2	yes	—
136	Q₈×C₆₇	—	8	2	no	—
137	D₄×C₆₇	—	2	2	no	—
138	D₄×C₆₇	—	4	2	no	—
139	D₄×C₆₇	—	4	1	no	—
140	D₄×C₆₇	—	4	1	no	—
141	D₄×C₆₇	—	4	2	no	—
142	D₄×C₆₇	—	4	1	no	—
143	D₄×C₆₇	—	8	4	no	—
144	D₄×C₆₇	—	8	4	no	—
145	D₄×C₆₇	—	8	4	no	—
146	D₄×C₆₇	—	8	2	no	—
147	D₄×C₆₇	—	16	1	no	—
148	D₄×C₆₇	D₁₆	1	4	yes	complemented
149	C₂×C₄×C₆₇	—	2	2	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	4	1	no	—
152	C₂×C₄×C₆₇	—	4	1	no	—
153	C₂×C₄×C₆₇	—	4	2	no	—
154	C₂×C₄×C₆₇	—	8	4	no	—
155	C₂×C₄×C₆₇	—	8	2	no	—
156	C₂×C₄×C₆₇	—	8	2	no	—
157	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
158	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
159	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
160	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
161	C₈×C₆₇	—	2	2	no	—
162	C₈×C₆₇	—	2	1	no	—
163	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
164	C₈×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
165	C₂²×C₆₇	—	2	2	no	—
166	C₂²×C₆₇	—	2	2	no	—
167	C₂²×C₆₇	—	4	2	no	—
168	C₂²×C₆₇	—	8	1	no	—
169	C₂²×C₆₇	—	8	1	no	—
170	C₂²×C₆₇	—	8	1	no	—
171	C₂²×C₆₇	—	8	2	no	—
172	C₂²×C₆₇	—	8	2	no	—
173	C₂²×C₆₇	—	8	1	no	—
174	C₂²×C₆₇	—	8	2	no	—
175	C₂²×C₆₇	—	8	1	no	—
176	C₂²×C₆₇	—	16	8	no	—
177	C₂²×C₆₇	C₂×D₁₆	1	4	yes	—
178	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
179	C₄×C₆₇	—	2	1	no	—
180	C₄×C₆₇	—	2	2	no	—
181	C₄×C₆₇	—	8	1	no	—
182	C₄×C₆₇	—	8	1	no	—
183	C₄×C₆₇	—	8	2	no	—
184	C₄×C₆₇	C₂×D₁₆	1	2	yes	—
185	C₄×C₆₇	#546	1	1	yes	characteristic
186	C₄×C₆₇	#577	1	1	yes	characteristic
187	#33861	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
188	C₂×C₆₇	—	2	4	no	—
189	C₂×C₆₇	—	8	2	no	—
190	C₂×C₆₇	—	16	1	no	—
191	C₂×C₆₇	—	16	2	no	—
192	C₂×C₆₇	#3294	1	1	yes	characteristic
193	C₂×C₆₇	C₂²×D₁₆	1	1	yes	characteristic
194	C₂×C₆₇	#3416	1	1	yes	characteristic
195	#3334	C₂×C₆₇	1	1	yes	characteristic, complemented
196	D₄×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
197	#2240	C₂×C₆₇	1	1	yes	characteristic, complemented
198	#2232	C₂×C₆₇	1	2	yes	complemented
199	#2224	C₂×C₆₇	1	1	yes	characteristic, complemented
200	#2215	C₂×C₆₇	1	2	yes	complemented
201	#2206	C₂×C₆₇	1	1	yes	characteristic, complemented
202	#2184	C₂×C₆₇	1	2	yes	complemented
203	C₄×D₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
204	D₄×C₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
205	C₂×#357	C₂×C₆₇	1	2	yes	complemented
206	C₆₇	#33861	1	1	yes	characteristic, direct product factor, Sylow
207	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
208	#547	—	2	1	no	—
209	D₄²	—	2	1	no	—
210	C₂×D₁₆	C₂²×C₆₇	1	1	yes	characteristic
211	C₂×D₁₆	C₂²×C₆₇	1	2	yes	complemented
212	C₂²×C₁₆	C₂²×C₆₇	1	2	yes	complemented
213	#500	C₂²×C₆₇	1	1	yes	characteristic, complemented
214	#493	C₂²×C₆₇	1	1	yes	characteristic
215	#480	—	2	2	no	—
216	#474	—	2	1	no	—
217	#466	C₂²×C₆₇	1	2	yes	—
218	#466	C₂²×C₆₇	1	2	yes	complemented
219	#459	—	2	1	no	—
220	#447	—	2	2	no	—
221	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
222	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
223	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
224	C₂×#107	C₂²×C₆₇	1	2	yes	complemented
225	#366	C₂²×C₆₇	1	1	yes	characteristic, complemented
226	#366	C₂²×C₆₇	1	2	yes	complemented
227	C₄⋊C₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
228	#357	C₂²×C₆₇	1	1	yes	characteristic
229	#357	C₂²×C₆₇	1	2	yes	—
230	#357	C₂²×C₆₇	1	2	yes	complemented
231	#357	C₂²×C₆₇	1	2	yes	complemented
232	#357	C₂²×C₆₇	1	1	yes	characteristic, complemented
233	#357	C₂²×C₆₇	1	2	yes	complemented
234	C₂²⋊C₁₆	C₂²×C₆₇	1	2	yes	complemented
235	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
236	C₂×#41	—	4	1	no	—
237	C₂²×D₄	—	2	2	no	—
238	C₂²×D₄	—	4	1	no	—
239	C₂×D₈	—	2	4	no	—
240	C₂×D₈	—	2	2	no	—
241	C₂×D₈	—	2	1	no	—
242	C₂×D₈	—	4	2	no	—
243	C₂×D₈	C₂³×C₆₇	1	2	yes	—
244	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
245	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
246	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
247	C₄⋊Q₈	—	2	1	no	—
248	#127	—	2	1	no	—
249	#123	—	4	2	no	—
250	C₂²⋊Q₈	—	4	1	no	—
251	#121	—	2	2	no	—
252	#121	—	2	2	no	—
253	#121	—	4	1	no	—
254	#120	—	4	2	no	—
255	C₄×D₄	—	2	1	no	—
256	C₄×D₄	—	2	1	no	—
257	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
258	C₂×C₄⋊C₄	—	2	2	no	—
259	D₁₆	—	2	2	no	—
260	D₁₆	—	4	2	no	—
261	C₂×C₁₆	—	4	2	no	—
262	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
263	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
264	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
265	#107	—	2	2	no	—
266	#107	—	2	1	no	—
267	#107	C₂³×C₆₇	1	1	yes	characteristic
268	#107	C₂³×C₆₇	1	2	yes	—
269	#107	C₂³×C₆₇	1	1	yes	characteristic
270	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
271	#102	—	2	1	no	—
272	#102	—	2	1	no	—
273	#102	—	2	2	no	—
274	#102	—	2	2	no	—
275	C₂²⋊C₈	D₄×C₆₇	1	2	yes	—
276	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
277	C₂⁴	—	4	2	no	—
278	#41	—	8	4	no	—
279	C₂×Q₈	—	4	1	no	—
280	C₂×D₄	—	2	1	no	—
281	C₂×D₄	—	2	1	no	—
282	C₂×D₄	—	2	2	no	—
283	C₂×D₄	—	4	4	no	—
284	C₂×D₄	—	4	2	no	—
285	C₂×D₄	—	4	2	no	—
286	C₂×D₄	—	4	2	no	—
287	C₂×D₄	—	4	1	no	—
288	C₂×D₄	—	4	1	no	—
289	C₂×D₄	—	8	2	no	—
290	C₂×D₄	—	8	4	no	—
291	C₂×D₄	D₈×C₆₇	1	1	yes	characteristic
292	C₂²×C₄	—	4	2	no	—
293	C₂²×C₄	—	4	1	no	—
294	C₂²×C₄	—	4	1	no	—
295	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
296	D₈	—	2	1	no	—
297	D₈	—	2	2	no	—
298	D₈	—	2	1	no	—
299	D₈	—	2	1	no	—
300	D₈	—	2	2	no	—
301	D₈	—	4	4	no	—
302	D₈	—	8	1	no	—
303	D₈	#41×C₆₇	1	2	yes	complemented
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
307	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
308	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	4	2	no	—
313	C₄⋊C₄	—	4	1	no	—
314	C₄⋊C₄	D₈×C₆₇	1	1	yes	characteristic
315	C₂²⋊C₄	—	4	1	no	—
316	C₂²⋊C₄	—	4	1	no	—
317	C₂²⋊C₄	—	4	2	no	—
318	C₂²⋊C₄	—	4	2	no	—
319	C₂²⋊C₄	D₈×C₆₇	1	2	yes	—
320	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
321	C₁₆	—	2	1	no	—
322	C₁₆	—	2	1	no	—
323	C₁₆	—	2	2	no	—
324	C₂³	—	4	1	no	—
325	C₂³	—	4	1	no	—
326	C₂³	—	4	2	no	—
327	C₂³	—	8	4	no	—
328	C₂³	—	8	4	no	—
329	C₂³	—	16	2	no	—
330	C₂³	C₂×D₈×C₆₇	1	2	yes	—
331	Q₈	—	8	2	no	—
332	D₄	—	2	2	no	—
333	D₄	—	4	2	no	—
334	D₄	—	4	1	no	—
335	D₄	—	4	1	no	—
336	D₄	—	4	2	no	—
337	D₄	—	4	1	no	—
338	D₄	—	8	4	no	—
339	D₄	—	8	4	no	—
340	D₄	—	8	4	no	—
341	D₄	—	8	2	no	—
342	D₄	—	16	1	no	—
343	D₄	D₁₆×C₆₇	1	4	yes	complemented
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	8	4	no	—
350	C₂×C₄	—	8	2	no	—
351	C₂×C₄	—	8	2	no	—
352	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
353	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
354	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
355	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
356	C₈	—	2	2	no	—
357	C₈	—	2	1	no	—
358	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
359	C₈	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
360	C₂²	—	2	2	no	—
361	C₂²	—	2	2	no	—
362	C₂²	—	4	2	no	—
363	C₂²	—	8	1	no	—
364	C₂²	—	8	1	no	—
365	C₂²	—	8	2	no	—
366	C₂²	—	8	1	no	—
367	C₂²	—	8	1	no	—
368	C₂²	—	8	2	no	—
369	C₂²	—	8	2	no	—
370	C₂²	—	8	1	no	—
371	C₂²	—	16	8	no	—
372	C₂²	C₂×D₁₆×C₆₇	1	4	yes	—
373	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
374	C₄	—	2	2	no	—
375	C₄	—	2	1	no	—
376	C₄	—	8	1	no	—
377	C₄	—	8	1	no	—
378	C₄	—	8	2	no	—
379	C₄	C₂×D₁₆×C₆₇	1	2	yes	—
380	C₄	#546×C₆₇	1	1	yes	characteristic
381	C₄	#577×C₆₇	1	1	yes	characteristic
382	C₂	—	2	4	no	—
383	C₂	—	8	2	no	—
384	C₂	—	16	2	no	—
385	C₂	—	16	1	no	—
386	C₂	C₆₇×#3294	1	1	yes	characteristic
387	C₂	C₂²×D₁₆×C₆₇	1	1	yes	characteristic
388	C₂	C₆₇×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
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×#3334	C₂	semidirect
2. D₄×D₈×C₆₇	C₂	semidirect
3. C₆₇×#2240	C₂	semidirect
4. C₆₇×#2232	C₂	semidirect
5. C₆₇×#2224	C₂	semidirect
6. C₆₇×#2215	C₂	semidirect
7. C₆₇×#2206	C₂	semidirect
8. C₆₇×#2184	C₂	semidirect
9. C₄×D₁₆×C₆₇	C₂	semidirect
10. D₄×C₁₆×C₆₇	C₂	semidirect
11. C₂×#357×C₆₇	C₂	semidirect
16. C₂×D₁₆×C₆₇	C₂²	semidirect
17. C₂²×C₁₆×C₆₇	C₂²	semidirect
18. #500×C₆₇	C₂²	semidirect
23. #466×C₆₇	C₂²	semidirect
27. C₄×D₈×C₆₇	C₂²	semidirect
29. C₂×#107×C₆₇	C₂²	semidirect
30. #366×C₆₇	C₂²	semidirect
31. #366×C₆₇	C₂²	semidirect
32. C₄⋊C₁₆×C₆₇	C₂²	semidirect
35. #357×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
108. D₈×C₆₇	#41	semidirect
148. D₄×C₆₇	D₁₆	semidirect
187. #33861	C₆₇	direct (N × H)
195. #3334	C₂×C₆₇	semidirect
196. D₄×D₈	C₂×C₆₇	semidirect
197. #2240	C₂×C₆₇	semidirect
198. #2232	C₂×C₆₇	semidirect
199. #2224	C₂×C₆₇	semidirect
200. #2215	C₂×C₆₇	semidirect
201. #2206	C₂×C₆₇	semidirect
202. #2184	C₂×C₆₇	semidirect
203. C₄×D₁₆	C₂×C₆₇	semidirect
204. D₄×C₁₆	C₂×C₆₇	semidirect
205. C₂×#357	C₂×C₆₇	semidirect
206. C₆₇	#33861	direct (N × H)
211. C₂×D₁₆	C₂²×C₆₇	semidirect
212. C₂²×C₁₆	C₂²×C₆₇	semidirect
213. #500	C₂²×C₆₇	semidirect
218. #466	C₂²×C₆₇	semidirect
222. C₄×D₈	C₂²×C₆₇	semidirect
224. C₂×#107	C₂²×C₆₇	semidirect
225. #366	C₂²×C₆₇	semidirect
226. #366	C₂²×C₆₇	semidirect
227. C₄⋊C₁₆	C₂²×C₆₇	semidirect
230. #357	C₂²×C₆₇	semidirect
231. #357	C₂²×C₆₇	semidirect
232. #357	C₂²×C₆₇	semidirect
233. #357	C₂²×C₆₇	semidirect
234. C₂²⋊C₁₆	C₂²×C₆₇	semidirect
235. C₄×C₁₆	C₂²×C₆₇	semidirect
303. D₈	#41×C₆₇	semidirect
343. D₄	D₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3334	2
D₄×D₈×C₆₇	2
C₆₇×#2240	2
C₆₇×#2232	2
C₆₇×#2224	2
C₆₇×#2215	2
C₆₇×#2206	2
C₆₇×#2184	2
C₄×D₁₆×C₆₇	2
D₄×C₁₆×C₆₇	2
C₂×#357×C₆₇	2
#33861	67

Subgroups — C67×#33861 #30713112

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33861 #30713112