Subgroups — C₆₇×#33580 #30712831

Order 17152.

770 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#509×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3147	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2210	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2208	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2206	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2150	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2139	C₂	1	2	yes	maximal, complemented
9	C₂×#135×C₆₇	—	2	1	no	—
10	C₂×#135×C₆₇	C₂²	1	2	yes	—
11	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
12	#532×C₆₇	—	2	1	no	—
13	C₂×C₄×D₄×C₆₇	—	2	1	no	—
14	#509×C₆₇	—	2	8	no	—
15	C₂×SD₃₂×C₆₇	C₂²	1	4	yes	—
16	C₂×D₁₆×C₆₇	C₂²	1	4	yes	—
17	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
18	#495×C₆₇	—	2	1	no	—
19	#495×C₆₇	C₂²	1	2	yes	complemented
20	#494×C₆₇	C₂²	1	1	yes	characteristic, complemented
21	#493×C₆₇	C₂²	1	1	yes	characteristic, complemented
22	#492×C₆₇	—	2	1	no	—
23	C₄×D₈×C₆₇	—	2	1	no	—
24	C₄×D₈×C₆₇	—	2	1	no	—
25	C₄×D₈×C₆₇	C₂²	1	2	yes	—
26	C₄×D₈×C₆₇	C₂²	1	2	yes	—
27	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#102×C₆₇	—	2	1	no	—
29	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	complemented
31	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	complemented
32	#358×C₆₇	C₂²	1	4	yes	complemented
33	#357×C₆₇	C₂²	1	4	yes	complemented
34	C₂×#41×C₆₇	—	2	2	no	—
35	C₂²×D₄×C₆₇	—	2	1	no	—
36	C₂³×C₄×C₆₇	—	4	1	no	—
37	#135×C₆₇	—	2	4	no	—
38	#135×C₆₇	—	2	4	no	—
39	#135×C₆₇	—	4	4	no	—
40	C₂×Q₁₆×C₆₇	—	2	1	no	—
41	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
42	C₂×SD₁₆×C₆₇	—	2	2	no	—
43	C₂×SD₁₆×C₆₇	—	2	2	no	—
44	C₂×D₈×C₆₇	—	2	1	no	—
45	C₂×D₈×C₆₇	—	2	1	no	—
46	C₂×D₈×C₆₇	—	2	2	no	—
47	C₂×D₈×C₆₇	D₄	1	4	yes	complemented
48	C₂×D₈×C₆₇	C₂³	1	2	yes	—
49	C₂×D₈×C₆₇	C₂³	1	2	yes	—
50	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₄⋊Q₈×C₆₇	—	2	1	no	—
53	#127×C₆₇	—	2	1	no	—
54	#124×C₆₇	—	2	2	no	—
55	#121×C₆₇	—	4	1	no	—
56	#121×C₆₇	—	4	1	no	—
57	C₄×D₄×C₆₇	—	2	2	no	—
58	C₄×D₄×C₆₇	—	2	2	no	—
59	C₄×D₄×C₆₇	—	4	1	no	—
60	C₄×D₄×C₆₇	—	4	1	no	—
61	C₄×D₄×C₆₇	—	4	1	no	—
62	C₄×D₄×C₆₇	—	4	1	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
64	C₂×C₂²⋊C₄×C₆₇	—	4	1	no	—
65	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
66	SD₃₂×C₆₇	—	2	8	no	—
67	D₁₆×C₆₇	—	2	8	no	—
68	M₃₂×C₆₇	—	2	4	no	—
69	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
70	#107×C₆₇	—	2	1	no	—
71	#107×C₆₇	C₂³	1	2	yes	—
72	#102×C₆₇	—	2	2	no	—
73	#102×C₆₇	—	2	2	no	—
74	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
75	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
76	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
78	C₂⁴×C₆₇	—	4	1	no	—
79	#41×C₆₇	—	4	4	no	—
80	#41×C₆₇	—	4	4	no	—
81	C₂×Q₈×C₆₇	—	2	2	no	—
82	C₂×D₄×C₆₇	—	2	4	no	—
83	C₂×D₄×C₆₇	—	2	2	no	—
84	C₂×D₄×C₆₇	—	2	2	no	—
85	C₂×D₄×C₆₇	—	4	2	no	—
86	C₂×D₄×C₆₇	—	4	1	no	—
87	C₂×D₄×C₆₇	—	4	1	no	—
88	C₂×D₄×C₆₇	—	4	1	no	—
89	C₂×D₄×C₆₇	—	4	2	no	—
90	C₂²×C₄×C₆₇	—	4	1	no	—
91	C₂²×C₄×C₆₇	—	4	2	no	—
92	C₂²×C₄×C₆₇	—	4	2	no	—
93	C₂²×C₄×C₆₇	—	4	2	no	—
94	C₂²×C₄×C₆₇	—	8	4	no	—
95	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
98	Q₁₆×C₆₇	—	2	2	no	—
99	Q₁₆×C₆₇	—	2	2	no	—
100	Q₁₆×C₆₇	—	4	2	no	—
101	SD₁₆×C₆₇	—	4	4	no	—
102	SD₁₆×C₆₇	—	4	4	no	—
103	D₈×C₆₇	—	2	4	no	—
104	D₈×C₆₇	—	2	2	no	—
105	D₈×C₆₇	—	2	2	no	—
106	D₈×C₆₇	—	2	2	no	—
107	D₈×C₆₇	—	4	2	no	—
108	D₈×C₆₇	C₂×D₄	1	4	yes	—
109	C₂×C₈×C₆₇	—	2	2	no	—
110	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
114	C₂×C₈×C₆₇	#41	1	2	yes	—
115	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
116	C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₄⋊C₄×C₆₇	—	4	1	no	—
118	C₄⋊C₄×C₆₇	—	4	1	no	—
119	C₄⋊C₄×C₆₇	—	4	1	no	—
120	C₂²⋊C₄×C₆₇	—	4	2	no	—
121	C₂²⋊C₄×C₆₇	—	4	2	no	—
122	C₂²⋊C₄×C₆₇	—	4	2	no	—
123	C₂²⋊C₄×C₆₇	—	4	2	no	—
124	C₄²×C₆₇	D₈	1	1	yes	characteristic
125	C₄²×C₆₇	D₈	1	1	yes	characteristic
126	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₁₆×C₆₇	—	2	4	no	—
129	C₂³×C₆₇	—	4	2	no	—
130	C₂³×C₆₇	—	4	4	no	—
131	C₂³×C₆₇	—	4	2	no	—
132	C₂³×C₆₇	—	8	2	no	—
133	C₂³×C₆₇	—	8	2	no	—
134	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
135	Q₈×C₆₇	—	4	2	no	—
136	Q₈×C₆₇	—	4	2	no	—
137	D₄×C₆₇	—	2	4	no	—
138	D₄×C₆₇	—	4	2	no	—
139	D₄×C₆₇	—	4	2	no	—
140	D₄×C₆₇	—	4	2	no	—
141	D₄×C₆₇	—	4	4	no	—
142	D₄×C₆₇	—	8	2	no	—
143	D₄×C₆₇	—	8	4	no	—
144	D₄×C₆₇	—	8	2	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	4	1	no	—
147	C₂×C₄×C₆₇	—	4	2	no	—
148	C₂×C₄×C₆₇	—	4	2	no	—
149	C₂×C₄×C₆₇	—	8	4	no	—
150	C₂×C₄×C₆₇	—	8	2	no	—
151	C₂×C₄×C₆₇	—	8	4	no	—
152	C₂×C₄×C₆₇	—	8	4	no	—
153	C₂×C₄×C₆₇	#121	1	2	yes	—
154	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
158	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
160	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
161	C₈×C₆₇	—	2	1	no	—
162	C₈×C₆₇	—	2	1	no	—
163	C₈×C₆₇	#121	1	2	yes	—
164	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
165	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
166	C₂²×C₆₇	—	2	1	no	—
167	C₂²×C₆₇	—	2	1	no	—
168	C₂²×C₆₇	—	4	4	no	—
169	C₂²×C₆₇	—	8	8	no	—
170	C₂²×C₆₇	—	8	2	no	—
171	C₂²×C₆₇	—	8	2	no	—
172	C₂²×C₆₇	—	8	2	no	—
173	C₂²×C₆₇	—	8	2	no	—
174	C₂²×C₆₇	—	8	2	no	—
175	C₂²×C₆₇	#459	1	2	yes	—
176	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
177	C₄×C₆₇	—	2	1	no	—
178	C₄×C₆₇	—	2	1	no	—
179	C₄×C₆₇	—	2	1	no	—
180	C₄×C₆₇	—	8	2	no	—
181	C₄×C₆₇	—	8	2	no	—
182	C₄×C₆₇	#459	1	2	yes	—
183	C₄×C₆₇	#509	1	2	yes	complemented
184	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
185	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
186	#33580	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
187	C₂×C₆₇	—	2	2	no	—
188	C₂×C₆₇	—	8	4	no	—
189	C₂×C₆₇	—	16	2	no	—
190	C₂×C₆₇	C₂×#459	1	1	yes	characteristic
191	C₂×C₆₇	C₂×#509	1	1	yes	characteristic
192	C₂×C₆₇	#3414	1	1	yes	characteristic
193	C₂×#509	C₂×C₆₇	1	2	yes	complemented
194	#3147	C₂×C₆₇	1	1	yes	characteristic, complemented
195	C₂×C₄×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
196	#2210	C₂×C₆₇	1	4	yes	complemented
197	#2208	C₂×C₆₇	1	2	yes	complemented
198	#2206	C₂×C₆₇	1	2	yes	complemented
199	#2150	C₂×C₆₇	1	1	yes	characteristic, complemented
200	#2139	C₂×C₆₇	1	2	yes	complemented
201	C₆₇	#33580	1	1	yes	characteristic, direct product factor, Sylow
202	C₂×#135	—	2	1	no	—
203	C₂×#135	C₂²×C₆₇	1	2	yes	—
204	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
205	#532	—	2	1	no	—
206	C₂×C₄×D₄	—	2	1	no	—
207	#509	—	2	8	no	—
208	C₂×SD₃₂	C₂²×C₆₇	1	4	yes	—
209	C₂×D₁₆	C₂²×C₆₇	1	4	yes	—
210	C₂×M₃₂	C₂²×C₆₇	1	2	yes	—
211	#495	—	2	1	no	—
212	#495	C₂²×C₆₇	1	2	yes	complemented
213	#494	C₂²×C₆₇	1	1	yes	characteristic, complemented
214	#493	C₂²×C₆₇	1	1	yes	characteristic, complemented
215	#492	—	2	1	no	—
216	C₄×D₈	—	2	1	no	—
217	C₄×D₈	—	2	1	no	—
218	C₄×D₈	C₂²×C₆₇	1	2	yes	—
219	C₄×D₈	C₂²×C₆₇	1	2	yes	—
220	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
221	C₂×#102	—	2	1	no	—
222	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
223	C₄⋊C₁₆	C₂²×C₆₇	1	2	yes	complemented
224	C₄⋊C₁₆	C₂²×C₆₇	1	2	yes	complemented
225	#358	C₂²×C₆₇	1	4	yes	complemented
226	#357	C₂²×C₆₇	1	4	yes	complemented
227	C₂×#41	—	2	2	no	—
228	C₂²×D₄	—	2	1	no	—
229	C₂³×C₄	—	4	1	no	—
230	#135	—	2	4	no	—
231	#135	—	2	4	no	—
232	#135	—	4	4	no	—
233	C₂×Q₁₆	—	2	1	no	—
234	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
235	C₂×SD₁₆	—	2	2	no	—
236	C₂×SD₁₆	—	2	2	no	—
237	C₂×D₈	—	2	1	no	—
238	C₂×D₈	—	2	1	no	—
239	C₂×D₈	—	2	2	no	—
240	C₂×D₈	D₄×C₆₇	1	4	yes	complemented
241	C₂×D₈	C₂³×C₆₇	1	2	yes	—
242	C₂×D₈	C₂³×C₆₇	1	2	yes	—
243	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
244	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
245	C₄⋊Q₈	—	2	1	no	—
246	#127	—	2	1	no	—
247	#124	—	2	2	no	—
248	#121	—	4	1	no	—
249	#121	—	4	1	no	—
250	C₄×D₄	—	2	2	no	—
251	C₄×D₄	—	2	2	no	—
252	C₄×D₄	—	4	1	no	—
253	C₄×D₄	—	4	1	no	—
254	C₄×D₄	—	4	1	no	—
255	C₄×D₄	—	4	1	no	—
256	C₂×C₄⋊C₄	—	2	1	no	—
257	C₂×C₂²⋊C₄	—	4	1	no	—
258	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
259	SD₃₂	—	2	8	no	—
260	D₁₆	—	2	8	no	—
261	M₃₂	—	2	4	no	—
262	C₂×C₁₆	C₂³×C₆₇	1	4	yes	—
263	#107	—	2	1	no	—
264	#107	C₂³×C₆₇	1	2	yes	—
265	#102	—	2	2	no	—
266	#102	—	2	2	no	—
267	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
268	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
269	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
270	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
271	C₂⁴	—	4	1	no	—
272	#41	—	4	4	no	—
273	#41	—	4	4	no	—
274	C₂×Q₈	—	2	2	no	—
275	C₂×D₄	—	2	4	no	—
276	C₂×D₄	—	2	2	no	—
277	C₂×D₄	—	2	2	no	—
278	C₂×D₄	—	4	2	no	—
279	C₂×D₄	—	4	1	no	—
280	C₂×D₄	—	4	1	no	—
281	C₂×D₄	—	4	1	no	—
282	C₂×D₄	—	4	2	no	—
283	C₂²×C₄	—	4	1	no	—
284	C₂²×C₄	—	4	2	no	—
285	C₂²×C₄	—	4	2	no	—
286	C₂²×C₄	—	4	2	no	—
287	C₂²×C₄	—	8	4	no	—
288	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
289	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
290	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
291	Q₁₆	—	2	2	no	—
292	Q₁₆	—	2	2	no	—
293	Q₁₆	—	4	2	no	—
294	SD₁₆	—	4	4	no	—
295	SD₁₆	—	4	4	no	—
296	D₈	—	2	4	no	—
297	D₈	—	2	2	no	—
298	D₈	—	2	2	no	—
299	D₈	—	2	2	no	—
300	D₈	—	4	2	no	—
301	D₈	C₂×D₄×C₆₇	1	4	yes	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
304	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
305	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
306	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
307	C₂×C₈	#41×C₆₇	1	2	yes	—
308	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	4	1	no	—
311	C₄⋊C₄	—	4	1	no	—
312	C₄⋊C₄	—	4	1	no	—
313	C₂²⋊C₄	—	4	2	no	—
314	C₂²⋊C₄	—	4	2	no	—
315	C₂²⋊C₄	—	4	2	no	—
316	C₂²⋊C₄	—	4	2	no	—
317	C₄²	D₈×C₆₇	1	1	yes	characteristic
318	C₄²	D₈×C₆₇	1	1	yes	characteristic
319	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
320	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
321	C₁₆	—	2	4	no	—
322	C₂³	—	4	4	no	—
323	C₂³	—	4	2	no	—
324	C₂³	—	4	2	no	—
325	C₂³	—	8	2	no	—
326	C₂³	—	8	2	no	—
327	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
328	Q₈	—	4	2	no	—
329	Q₈	—	4	2	no	—
330	D₄	—	2	4	no	—
331	D₄	—	4	4	no	—
332	D₄	—	4	2	no	—
333	D₄	—	4	2	no	—
334	D₄	—	4	2	no	—
335	D₄	—	8	2	no	—
336	D₄	—	8	4	no	—
337	D₄	—	8	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	4	1	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	4	2	no	—
342	C₂×C₄	—	8	4	no	—
343	C₂×C₄	—	8	2	no	—
344	C₂×C₄	—	8	4	no	—
345	C₂×C₄	—	8	4	no	—
346	C₂×C₄	#121×C₆₇	1	2	yes	—
347	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
348	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
349	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
350	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
351	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
352	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
353	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
354	C₈	—	2	1	no	—
355	C₈	—	2	1	no	—
356	C₈	#121×C₆₇	1	2	yes	—
357	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
358	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
359	C₂²	—	2	1	no	—
360	C₂²	—	2	1	no	—
361	C₂²	—	4	4	no	—
362	C₂²	—	8	8	no	—
363	C₂²	—	8	2	no	—
364	C₂²	—	8	2	no	—
365	C₂²	—	8	2	no	—
366	C₂²	—	8	2	no	—
367	C₂²	—	8	2	no	—
368	C₂²	#459×C₆₇	1	2	yes	—
369	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
370	C₄	—	2	1	no	—
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	8	2	no	—
374	C₄	—	8	2	no	—
375	C₄	#459×C₆₇	1	2	yes	—
376	C₄	#509×C₆₇	1	2	yes	complemented
377	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
378	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
379	C₂	—	2	2	no	—
380	C₂	—	8	4	no	—
381	C₂	—	16	2	no	—
382	C₂	C₂×#459×C₆₇	1	1	yes	characteristic
383	C₂	C₂×#509×C₆₇	1	1	yes	characteristic
384	C₂	C₆₇×#3414	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 ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#509×C₆₇	C₂	semidirect
2. C₆₇×#3147	C₂	semidirect
3. C₂×C₄×D₈×C₆₇	C₂	semidirect
4. C₆₇×#2210	C₂	semidirect
5. C₆₇×#2208	C₂	semidirect
6. C₆₇×#2206	C₂	semidirect
7. C₆₇×#2150	C₂	semidirect
8. C₆₇×#2139	C₂	semidirect
19. #495×C₆₇	C₂²	semidirect
20. #494×C₆₇	C₂²	semidirect
21. #493×C₆₇	C₂²	semidirect
30. C₄⋊C₁₆×C₆₇	C₂²	semidirect
31. C₄⋊C₁₆×C₆₇	C₂²	semidirect
32. #358×C₆₇	C₂²	semidirect
33. #357×C₆₇	C₂²	semidirect
47. C₂×D₈×C₆₇	D₄	semidirect
183. C₄×C₆₇	#509	semidirect
186. #33580	C₆₇	direct (N × H)
193. C₂×#509	C₂×C₆₇	semidirect
194. #3147	C₂×C₆₇	semidirect
195. C₂×C₄×D₈	C₂×C₆₇	semidirect
196. #2210	C₂×C₆₇	semidirect
197. #2208	C₂×C₆₇	semidirect
198. #2206	C₂×C₆₇	semidirect
199. #2150	C₂×C₆₇	semidirect
200. #2139	C₂×C₆₇	semidirect
201. C₆₇	#33580	direct (N × H)
212. #495	C₂²×C₆₇	semidirect
213. #494	C₂²×C₆₇	semidirect
214. #493	C₂²×C₆₇	semidirect
223. C₄⋊C₁₆	C₂²×C₆₇	semidirect
224. C₄⋊C₁₆	C₂²×C₆₇	semidirect
225. #358	C₂²×C₆₇	semidirect
226. #357	C₂²×C₆₇	semidirect
240. C₂×D₈	D₄×C₆₇	semidirect
376. C₄	#509×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#509×C₆₇	2
C₆₇×#3147	2
C₂×C₄×D₈×C₆₇	2
C₆₇×#2210	2
C₆₇×#2208	2
C₆₇×#2206	2
C₆₇×#2150	2
C₆₇×#2139	2
#33580	67

Subgroups — C67×#33580 #30712831

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33580 #30712831