Subgroups — C₆₇×#33851 #30713102

Order 17152.

674 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#508×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×Q₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3285	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2249	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2216	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2215	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2211	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2189	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2185	C₂	1	2	yes	maximal, complemented
10	C₄×Q₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	D₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135×C₆₇	C₂²	1	2	yes	—
13	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
14	#548×C₆₇	—	2	2	no	—
15	#508×C₆₇	—	2	4	no	—
16	C₂×Q₃₂×C₆₇	—	2	4	no	—
17	C₂×Q₃₂×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×Q₃₂×C₆₇	C₂²	1	2	yes	—
19	C₂×Q₃₂×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	—
21	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	—
22	C₂×D₁₆×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
25	#495×C₆₇	C₂²	1	2	yes	—
26	#467×C₆₇	C₂²	1	2	yes	—
27	#466×C₆₇	C₂²	1	2	yes	—
28	#464×C₆₇	—	2	2	no	—
29	#452×C₆₇	—	2	2	no	—
30	#448×C₆₇	—	2	2	no	—
31	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
32	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
33	#366×C₆₇	C₂²	1	1	yes	characteristic
34	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
35	#358×C₆₇	C₂²	1	2	yes	—
36	#358×C₆₇	C₂²	1	2	yes	—
37	#357×C₆₇	C₂²	1	2	yes	—
38	C₂²⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
39	C₂²⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×#41×C₆₇	—	2	2	no	—
42	C₂²×Q₈×C₆₇	—	2	2	no	—
43	#135×C₆₇	—	2	4	no	—
44	#135×C₆₇	—	4	2	no	—
45	C₂×Q₁₆×C₆₇	—	2	4	no	—
46	C₂×Q₁₆×C₆₇	—	2	2	no	—
47	C₂×Q₁₆×C₆₇	—	4	2	no	—
48	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
49	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
50	C₂×SD₁₆×C₆₇	—	2	2	no	—
51	C₂×SD₁₆×C₆₇	—	2	2	no	—
52	C₂×D₈×C₆₇	C₂³	1	2	yes	—
53	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
55	#124×C₆₇	—	2	2	no	—
56	#124×C₆₇	—	4	2	no	—
57	C₂²⋊Q₈×C₆₇	—	2	2	no	—
58	C₂²⋊Q₈×C₆₇	—	4	2	no	—
59	#121×C₆₇	—	2	2	no	—
60	#121×C₆₇	—	4	2	no	—
61	C₄×Q₈×C₆₇	—	2	2	no	—
62	C₄×D₄×C₆₇	—	4	2	no	—
63	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
64	Q₃₂×C₆₇	—	2	4	no	—
65	Q₃₂×C₆₇	—	2	2	no	—
66	Q₃₂×C₆₇	D₄	1	4	yes	complemented
67	SD₃₂×C₆₇	—	2	4	no	—
68	SD₃₂×C₆₇	—	4	2	no	—
69	D₁₆×C₆₇	—	2	2	no	—
70	C₂×C₁₆×C₆₇	—	2	2	no	—
71	C₂×C₁₆×C₆₇	—	2	2	no	—
72	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
75	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
76	#107×C₆₇	C₂³	1	2	yes	—
77	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
78	#103×C₆₇	—	2	2	no	—
79	#103×C₆₇	—	2	2	no	—
80	#102×C₆₇	—	2	2	no	—
81	C₂²⋊C₈×C₆₇	D₄	1	1	yes	characteristic
82	C₂²⋊C₈×C₆₇	D₄	1	1	yes	characteristic
83	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
84	#41×C₆₇	—	4	4	no	—
85	#41×C₆₇	—	8	2	no	—
86	C₂×Q₈×C₆₇	—	2	2	no	—
87	C₂×Q₈×C₆₇	—	2	2	no	—
88	C₂×Q₈×C₆₇	—	4	4	no	—
89	C₂×Q₈×C₆₇	—	4	2	no	—
90	C₂×Q₈×C₆₇	—	8	2	no	—
91	C₂×D₄×C₆₇	—	2	2	no	—
92	C₂×D₄×C₆₇	—	4	2	no	—
93	C₂×D₄×C₆₇	—	4	2	no	—
94	C₂×D₄×C₆₇	D₈	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	—	4	2	no	—
96	C₂²×C₄×C₆₇	—	4	2	no	—
97	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
99	Q₁₆×C₆₇	—	2	2	no	—
100	Q₁₆×C₆₇	—	2	2	no	—
101	Q₁₆×C₆₇	—	2	2	no	—
102	Q₁₆×C₆₇	—	4	4	no	—
103	Q₁₆×C₆₇	C₂×D₄	1	4	yes	—
104	SD₁₆×C₆₇	—	4	4	no	—
105	SD₁₆×C₆₇	—	8	2	no	—
106	D₈×C₆₇	—	2	2	no	—
107	D₈×C₆₇	—	2	2	no	—
108	C₂×C₈×C₆₇	—	2	1	no	—
109	C₂×C₈×C₆₇	—	2	1	no	—
110	C₂×C₈×C₆₇	—	2	1	no	—
111	C₂×C₈×C₆₇	—	2	1	no	—
112	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
116	C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₄⋊C₄×C₆₇	—	4	2	no	—
118	C₄⋊C₄×C₆₇	—	4	2	no	—
119	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
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	C₂²⋊C₄×C₆₇	D₈	1	1	yes	characteristic
125	C₂²⋊C₄×C₆₇	D₈	1	1	yes	characteristic
126	C₄²×C₆₇	—	4	2	no	—
127	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₁₆×C₆₇	—	2	1	no	—
129	C₁₆×C₆₇	—	2	1	no	—
130	C₁₆×C₆₇	—	2	1	no	—
131	C₁₆×C₆₇	C₂×D₄	1	2	yes	—
132	C₂³×C₆₇	—	4	2	no	—
133	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
134	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
135	Q₈×C₆₇	—	2	4	no	—
136	Q₈×C₆₇	—	4	2	no	—
137	Q₈×C₆₇	—	4	2	no	—
138	Q₈×C₆₇	—	4	2	no	—
139	Q₈×C₆₇	—	8	4	no	—
140	D₄×C₆₇	—	2	2	no	—
141	D₄×C₆₇	—	4	2	no	—
142	D₄×C₆₇	—	4	2	no	—
143	D₄×C₆₇	—	8	4	no	—
144	D₄×C₆₇	—	8	4	no	—
145	C₂×C₄×C₆₇	—	2	1	no	—
146	C₂×C₄×C₆₇	—	2	1	no	—
147	C₂×C₄×C₆₇	—	2	1	no	—
148	C₂×C₄×C₆₇	—	2	1	no	—
149	C₂×C₄×C₆₇	—	4	2	no	—
150	C₂×C₄×C₆₇	—	4	2	no	—
151	C₂×C₄×C₆₇	—	4	2	no	—
152	C₂×C₄×C₆₇	—	8	4	no	—
153	C₂×C₄×C₆₇	—	8	4	no	—
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	C₂²×D₄	1	1	yes	characteristic
159	C₈×C₆₇	—	2	1	no	—
160	C₈×C₆₇	—	2	1	no	—
161	C₈×C₆₇	—	2	1	no	—
162	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
163	C₈×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
164	C₂²×C₆₇	—	2	1	no	—
165	C₂²×C₆₇	—	2	1	no	—
166	C₂²×C₆₇	—	2	1	no	—
167	C₂²×C₆₇	—	2	1	no	—
168	C₂²×C₆₇	—	2	1	no	—
169	C₂²×C₆₇	—	8	2	no	—
170	C₂²×C₆₇	—	8	2	no	—
171	C₂²×C₆₇	—	8	2	no	—
172	C₂²×C₆₇	#508	1	2	yes	complemented
173	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
174	C₄×C₆₇	—	2	1	no	—
175	C₄×C₆₇	—	2	1	no	—
176	C₄×C₆₇	—	2	1	no	—
177	C₄×C₆₇	—	2	1	no	—
178	C₄×C₆₇	—	4	4	no	—
179	C₄×C₆₇	—	8	2	no	—
180	C₄×C₆₇	—	8	2	no	—
181	C₄×C₆₇	D₄²	1	1	yes	characteristic
182	C₄×C₆₇	#576	1	1	yes	characteristic
183	#33851	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
184	C₂×C₆₇	—	2	2	no	—
185	C₂×C₆₇	—	4	1	no	—
186	C₂×C₆₇	—	16	2	no	—
187	C₂×C₆₇	D₄×D₈	1	1	yes	characteristic
188	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
189	C₂×C₆₇	#3417	1	1	yes	characteristic
190	C₂×#508	C₂×C₆₇	1	1	yes	characteristic, complemented
191	C₂²×Q₃₂	C₂×C₆₇	1	1	yes	characteristic, complemented
192	#3285	C₂×C₆₇	1	2	yes	complemented
193	#2249	C₂×C₆₇	1	1	yes	characteristic, complemented
194	#2216	C₂×C₆₇	1	1	yes	characteristic, complemented
195	#2215	C₂×C₆₇	1	1	yes	characteristic, complemented
196	#2211	C₂×C₆₇	1	2	yes	complemented
197	#2189	C₂×C₆₇	1	2	yes	complemented
198	#2185	C₂×C₆₇	1	2	yes	complemented
199	C₄×Q₃₂	C₂×C₆₇	1	1	yes	characteristic, complemented
200	D₄×C₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
201	C₆₇	#33851	1	1	yes	characteristic, direct product factor, Sylow
202	C₂×#135	C₂²×C₆₇	1	2	yes	—
203	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
204	#548	—	2	2	no	—
205	#508	—	2	4	no	—
206	C₂×Q₃₂	—	2	4	no	—
207	C₂×Q₃₂	C₂²×C₆₇	1	2	yes	—
208	C₂×Q₃₂	C₂²×C₆₇	1	1	yes	characteristic
209	C₂×Q₃₂	C₂²×C₆₇	1	1	yes	characteristic
210	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	—
211	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	—
212	C₂×D₁₆	C₂²×C₆₇	1	1	yes	characteristic
213	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
214	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
215	#495	C₂²×C₆₇	1	2	yes	—
216	#467	C₂²×C₆₇	1	2	yes	—
217	#466	C₂²×C₆₇	1	2	yes	—
218	#464	—	2	2	no	—
219	#452	—	2	2	no	—
220	#448	—	2	2	no	—
221	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
222	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
223	#366	C₂²×C₆₇	1	1	yes	characteristic
224	C₄⋊C₁₆	C₂²×C₆₇	1	1	yes	characteristic
225	#358	C₂²×C₆₇	1	2	yes	—
226	#358	C₂²×C₆₇	1	2	yes	—
227	#357	C₂²×C₆₇	1	2	yes	—
228	C₂²⋊C₁₆	C₂²×C₆₇	1	1	yes	characteristic
229	C₂²⋊C₁₆	C₂²×C₆₇	1	1	yes	characteristic
230	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
231	C₂×#41	—	2	2	no	—
232	C₂²×Q₈	—	2	2	no	—
233	#135	—	2	4	no	—
234	#135	—	4	2	no	—
235	C₂×Q₁₆	—	2	4	no	—
236	C₂×Q₁₆	—	2	2	no	—
237	C₂×Q₁₆	—	4	2	no	—
238	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
239	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
240	C₂×SD₁₆	—	2	2	no	—
241	C₂×SD₁₆	—	2	2	no	—
242	C₂×D₈	C₂³×C₆₇	1	2	yes	—
243	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
244	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
245	#124	—	2	2	no	—
246	#124	—	4	2	no	—
247	C₂²⋊Q₈	—	2	2	no	—
248	C₂²⋊Q₈	—	4	2	no	—
249	#121	—	2	2	no	—
250	#121	—	4	2	no	—
251	C₄×Q₈	—	2	2	no	—
252	C₄×D₄	—	4	2	no	—
253	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
254	Q₃₂	—	2	4	no	—
255	Q₃₂	—	2	2	no	—
256	Q₃₂	D₄×C₆₇	1	4	yes	complemented
257	SD₃₂	—	2	4	no	—
258	SD₃₂	—	4	2	no	—
259	D₁₆	—	2	2	no	—
260	C₂×C₁₆	—	2	2	no	—
261	C₂×C₁₆	—	2	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	1	yes	characteristic
265	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
266	#107	C₂³×C₆₇	1	2	yes	—
267	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
268	#103	—	2	2	no	—
269	#103	—	2	2	no	—
270	#102	—	2	2	no	—
271	C₂²⋊C₈	D₄×C₆₇	1	1	yes	characteristic
272	C₂²⋊C₈	D₄×C₆₇	1	1	yes	characteristic
273	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
274	#41	—	4	4	no	—
275	#41	—	8	2	no	—
276	C₂×Q₈	—	2	2	no	—
277	C₂×Q₈	—	2	2	no	—
278	C₂×Q₈	—	4	4	no	—
279	C₂×Q₈	—	4	2	no	—
280	C₂×Q₈	—	8	2	no	—
281	C₂×D₄	—	2	2	no	—
282	C₂×D₄	—	4	2	no	—
283	C₂×D₄	—	4	2	no	—
284	C₂×D₄	D₈×C₆₇	1	1	yes	characteristic
285	C₂²×C₄	—	4	2	no	—
286	C₂²×C₄	—	4	2	no	—
287	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
288	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
289	Q₁₆	—	2	2	no	—
290	Q₁₆	—	2	2	no	—
291	Q₁₆	—	2	2	no	—
292	Q₁₆	—	4	4	no	—
293	Q₁₆	C₂×D₄×C₆₇	1	4	yes	—
294	SD₁₆	—	4	4	no	—
295	SD₁₆	—	8	2	no	—
296	D₈	—	2	2	no	—
297	D₈	—	2	2	no	—
298	C₂×C₈	—	2	1	no	—
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	2	1	no	—
301	C₂×C₈	—	2	1	no	—
302	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
303	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
304	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
305	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	4	2	no	—
308	C₄⋊C₄	—	4	2	no	—
309	C₄⋊C₄	D₈×C₆₇	1	1	yes	characteristic
310	C₂²⋊C₄	—	4	2	no	—
311	C₂²⋊C₄	—	4	2	no	—
312	C₂²⋊C₄	—	4	2	no	—
313	C₂²⋊C₄	—	4	2	no	—
314	C₂²⋊C₄	D₈×C₆₇	1	1	yes	characteristic
315	C₂²⋊C₄	D₈×C₆₇	1	1	yes	characteristic
316	C₄²	—	4	2	no	—
317	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
318	C₁₆	—	2	1	no	—
319	C₁₆	—	2	1	no	—
320	C₁₆	—	2	1	no	—
321	C₁₆	C₂×D₄×C₆₇	1	2	yes	—
322	C₂³	—	4	2	no	—
323	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
324	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
325	Q₈	—	2	4	no	—
326	Q₈	—	4	2	no	—
327	Q₈	—	4	2	no	—
328	Q₈	—	4	2	no	—
329	Q₈	—	8	4	no	—
330	D₄	—	2	2	no	—
331	D₄	—	4	2	no	—
332	D₄	—	4	2	no	—
333	D₄	—	8	4	no	—
334	D₄	—	8	4	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	4	2	no	—
342	C₂×C₄	—	8	4	no	—
343	C₂×C₄	—	8	4	no	—
344	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
345	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
346	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
347	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
348	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
349	C₈	—	2	1	no	—
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
353	C₈	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	8	2	no	—
360	C₂²	—	8	2	no	—
361	C₂²	—	8	2	no	—
362	C₂²	#508×C₆₇	1	2	yes	complemented
363	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	4	4	no	—
369	C₄	—	8	2	no	—
370	C₄	—	8	2	no	—
371	C₄	D₄²×C₆₇	1	1	yes	characteristic
372	C₄	#576×C₆₇	1	1	yes	characteristic
373	C₂	—	2	2	no	—
374	C₂	—	4	1	no	—
375	C₂	—	16	2	no	—
376	C₂	D₄×D₈×C₆₇	1	1	yes	characteristic
377	C₂	C₂×#508×C₆₇	1	1	yes	characteristic
378	C₂	C₆₇×#3417	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. C₂×#508×C₆₇	C₂	semidirect
2. C₂²×Q₃₂×C₆₇	C₂	semidirect
3. C₆₇×#3285	C₂	semidirect
4. C₆₇×#2249	C₂	semidirect
5. C₆₇×#2216	C₂	semidirect
6. C₆₇×#2215	C₂	semidirect
7. C₆₇×#2211	C₂	semidirect
8. C₆₇×#2189	C₂	semidirect
9. C₆₇×#2185	C₂	semidirect
10. C₄×Q₃₂×C₆₇	C₂	semidirect
11. D₄×C₁₆×C₆₇	C₂	semidirect
66. Q₃₂×C₆₇	D₄	semidirect
172. C₂²×C₆₇	#508	semidirect
183. #33851	C₆₇	direct (N × H)
190. C₂×#508	C₂×C₆₇	semidirect
191. C₂²×Q₃₂	C₂×C₆₇	semidirect
192. #3285	C₂×C₆₇	semidirect
193. #2249	C₂×C₆₇	semidirect
194. #2216	C₂×C₆₇	semidirect
195. #2215	C₂×C₆₇	semidirect
196. #2211	C₂×C₆₇	semidirect
197. #2189	C₂×C₆₇	semidirect
198. #2185	C₂×C₆₇	semidirect
199. C₄×Q₃₂	C₂×C₆₇	semidirect
200. D₄×C₁₆	C₂×C₆₇	semidirect
201. C₆₇	#33851	direct (N × H)
256. Q₃₂	D₄×C₆₇	semidirect
362. C₂²	#508×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#508×C₆₇	2
C₂²×Q₃₂×C₆₇	2
C₆₇×#3285	2
C₆₇×#2249	2
C₆₇×#2216	2
C₆₇×#2215	2
C₆₇×#2211	2
C₆₇×#2189	2
C₆₇×#2185	2
C₄×Q₃₂×C₆₇	2
D₄×C₁₆×C₆₇	2
#33851	67

Subgroups — C67×#33851 #30713102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33851 #30713102