Subgroups — #31008913

Order 17152.

682 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562012	C₂	1	4	yes	maximal, complemented
2	#22562008	C₂	1	4	yes	maximal, complemented
3	#22561965	C₂	1	2	yes	maximal, complemented
4	#22561820	C₂	1	2	yes	maximal
5	C₄×#14523983	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#14523983	C₂²	1	4	yes	—
8	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	8	yes	complemented
9	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	8	yes	—
10	#16863172	C₂²	1	4	yes	complemented
11	C₄×C₆₇⋊C₁₆	C₂²	1	4	yes	complemented
12	C₂³×C₈×C₆₇	C₄	1	1	yes	characteristic
13	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₆₇	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₆₇	C₄	1	1	yes	characteristic
17	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	—
18	C₂×C₄×C₈×C₆₇	C₄	1	1	yes	characteristic
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
23	#14523983	—	2	8	no	—
24	C₂×C₆₇⋊C₁₆	C₂³	1	8	yes	—
25	C₂³×C₄×C₆₇	C₈	1	2	yes	—
26	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	—	2	4	no	—
28	C₂²×C₈×C₆₇	—	2	4	no	—
29	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
36	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₆₇	C₈	1	2	yes	—
38	C₂×C₄²×C₆₇	C₈	1	2	yes	—
39	C₂×C₄²×C₆₇	C₈	1	2	yes	—
40	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₄×C₈×C₆₇	C₂³	1	2	yes	—
53	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₆₇⋊C₁₆	—	2	8	no	—
55	C₂⁴×C₆₇	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	—	2	8	no	—
57	C₂²×C₄×C₆₇	—	2	2	no	—
58	C₂²×C₄×C₆₇	—	2	2	no	—
59	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
63	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
65	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
66	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
67	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₆₇	—	2	2	no	—
70	C₂×C₈×C₆₇	—	2	2	no	—
71	C₂×C₈×C₆₇	—	2	4	no	—
72	C₂×C₈×C₆₇	—	2	4	no	—
73	C₂×C₈×C₆₇	—	2	4	no	—
74	C₂×C₈×C₆₇	—	2	4	no	—
75	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
76	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₆₇	#41	1	4	yes	—
82	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
83	C₄²×C₆₇	C₂×C₈	1	2	yes	—
84	C₄²×C₆₇	C₂×C₈	1	2	yes	—
85	C₄²×C₆₇	C₂×C₈	1	2	yes	—
86	C₄²×C₆₇	C₂×C₈	1	2	yes	—
87	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
88	C₄²×C₆₇	C₂×C₈	1	2	yes	—
89	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
90	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
92	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
93	C₂³×C₆₇	—	2	2	no	—
94	C₂³×C₆₇	—	2	4	no	—
95	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
96	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
97	C₂×C₄×C₆₇	—	2	2	no	—
98	C₂×C₄×C₆₇	—	2	4	no	—
99	C₂×C₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	—	2	2	no	—
101	C₂×C₄×C₆₇	—	2	4	no	—
102	C₂×C₄×C₆₇	—	2	4	no	—
103	C₂×C₄×C₆₇	—	2	4	no	—
104	C₂×C₄×C₆₇	—	2	4	no	—
105	C₂×C₄×C₆₇	—	2	4	no	—
106	C₂×C₄×C₆₇	M₃₂	1	8	yes	complemented
107	C₂×C₄×C₆₇	#117	1	2	yes	—
108	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
110	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
111	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
113	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
114	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
115	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
116	C₈×C₆₇	—	2	2	no	—
117	C₈×C₆₇	—	2	2	no	—
118	C₈×C₆₇	#117	1	2	yes	—
119	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
120	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
121	C₈×C₆₇	C₂×#41	1	2	yes	—
122	C₂²×C₆₇	—	2	8	no	—
123	C₂²×C₆₇	—	2	2	no	—
124	C₂²×C₆₇	—	2	2	no	—
125	C₂²×C₆₇	—	2	2	no	—
126	C₂²×C₆₇	#431	1	2	yes	—
127	C₂²×C₆₇	C₂×M₃₂	1	4	yes	—
128	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
129	C₄×C₆₇	—	2	2	no	—
130	C₄×C₆₇	—	2	2	no	—
131	C₄×C₆₇	—	2	2	no	—
132	C₄×C₆₇	#431	1	2	yes	—
133	C₄×C₆₇	C₂×M₃₂	1	4	yes	—
134	C₄×C₆₇	C₂×M₃₂	1	4	yes	—
135	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
136	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
137	#33418	—	67	1	no	maximal, Sylow
138	C₂×C₆₇	—	2	2	no	—
139	C₂×C₆₇	—	2	4	no	—
140	C₂×C₆₇	C₂×#431	1	1	yes	characteristic
141	C₂×C₆₇	C₂²×M₃₂	1	2	yes	—
142	C₂²×C₄×C₈	D₆₇	1	1	yes	characteristic
143	#2163	—	67	4	no	—
144	#2162	—	67	4	no	—
145	#2150	—	67	2	no	—
146	#2112	—	67	2	no	—
147	C₄×M₃₂	—	67	2	no	—
148	C₆₇	#33418	1	1	yes	characteristic, complemented, Sylow
149	C₂³×C₈	Dic₆₇	1	1	yes	characteristic
150	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
151	C₂²×C₄²	Dic₆₇	1	1	yes	characteristic
152	C₂×M₃₂	—	67	4	no	—
153	C₂×C₄×C₈	Dic₆₇	1	1	yes	characteristic
154	C₂×C₄×C₈	Dic₆₇	1	2	yes	—
155	C₂×C₄×C₈	Dic₆₇	1	2	yes	—
156	C₂×C₄×C₈	Dic₆₇	1	1	yes	characteristic
157	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
159	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
160	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
161	C₄⋊C₁₆	—	67	8	no	—
162	C₂²⋊C₁₆	—	67	8	no	—
163	#346	—	67	4	no	—
164	C₄×C₁₆	—	67	4	no	—
165	C₂³×C₄	C₆₇⋊C₈	1	2	yes	—
166	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
167	C₂²×C₈	—	2	4	no	—
168	C₂²×C₈	—	2	4	no	—
169	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
170	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
171	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
172	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
173	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
174	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
175	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
176	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
177	C₂×C₄²	C₆₇⋊C₈	1	2	yes	—
178	C₂×C₄²	C₆₇⋊C₈	1	2	yes	—
179	C₂×C₄²	C₆₇⋊C₈	1	2	yes	—
180	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
181	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
182	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
183	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
184	M₃₂	—	134	8	no	—
185	C₂×C₁₆	—	67	8	no	—
186	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
187	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
188	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
189	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
190	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
191	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
192	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
194	C₄×C₈	C₂²×D₆₇	1	2	yes	—
195	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
196	C₂⁴	C₂×C₆₇⋊C₈	1	1	yes	characteristic
197	C₂²×C₄	—	2	8	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
202	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
203	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
204	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
205	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
206	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
207	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
208	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
209	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	#11761612	1	4	yes	—
217	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
218	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
219	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
220	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
221	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
222	C₂×C₈	#11761628	1	4	yes	—
223	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
224	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
225	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
226	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
227	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
228	C₄²	C₂×C₆₇⋊C₈	1	1	yes	characteristic
229	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
230	C₄²	C₂×C₆₇⋊C₈	1	1	yes	characteristic
231	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
232	C₄²	C₂²×Dic₆₇	1	2	yes	—
233	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
234	C₁₆	—	134	8	no	—
235	C₂³	—	2	4	no	—
236	C₂³	—	2	2	no	—
237	C₂³	C₂²×C₆₇⋊C₈	1	2	yes	—
238	C₂³	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	#14523983	1	8	yes	complemented
249	C₂×C₄	#14524002	1	2	yes	—
250	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
251	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
252	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
253	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
254	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
255	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
256	C₂×C₄	C₂×#11761612	1	2	yes	—
257	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	2	2	no	—
260	C₈	#14524002	1	2	yes	—
261	C₈	C₂×#11761612	1	2	yes	—
262	C₈	C₂×#11761612	1	2	yes	—
263	C₈	C₂×#11761628	1	2	yes	—
264	C₂²	—	2	8	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	#16863449	1	2	yes	—
269	C₂²	C₂×#14523983	1	4	yes	—
270	C₂²	C₂³×C₆₇⋊C₈	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	#16863449	1	2	yes	—
275	C₄	C₂×#14523983	1	4	yes	—
276	C₄	C₂×#14523983	1	4	yes	—
277	C₄	C₂×#14524002	1	1	yes	characteristic
278	C₄	C₂²×#11761612	1	1	yes	characteristic
279	C₂	—	2	4	no	—
280	C₂	—	2	2	no	—
281	C₂	C₂×#16863449	1	1	yes	characteristic
282	C₂	C₂²×#14523983	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂²×C₄×C₈×C₆₇	C₂
prime radical	C₂⁴×C₆₇	C₂×C₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂²×C₄×C₈	D₆₇
hypocenter	C₆₇	#33418
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. #22562012	C₂	semidirect
2. #22562008	C₂	semidirect
3. #22561965	C₂	semidirect
5. C₄×#14523983	C₂	semidirect
8. C₄⋊(C₆₇⋊C₁₆)	C₂²	semidirect
10. #16863172	C₂²	semidirect
11. C₄×C₆₇⋊C₁₆	C₂²	semidirect
106. C₂×C₄×C₆₇	M₃₂	semidirect
148. C₆₇	#33418	semidirect
248. C₂×C₄	#14523983	semidirect

Maximal subgroups

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

subgroup	index
#22562012	2
#22562008	2
#22561965	2
#22561820	2
C₄×#14523983	2
C₂²×C₄×C₈×C₆₇	2
#33418	67