Subgroups — #22597286

Order 9088.

356 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986552	C₂	1	1	yes	maximal, characteristic
2	#17986128	C₂	1	2	yes	maximal, complemented
3	#17986128	C₂	1	2	yes	maximal, complemented
4	#17986126	C₂	1	2	yes	maximal, complemented
5	#17986123	C₂	1	2	yes	maximal, complemented
6	#17986027	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
8	#17985955	C₂	1	1	yes	maximal, characteristic
9	#17985954	C₂	1	1	yes	maximal, characteristic
10	M₁₆×Dic₇₁	C₂	1	1	yes	maximal, characteristic
11	#424×C₇₁	C₂	1	1	yes	maximal, characteristic
12	C₂×#11765849	C₂²	1	1	yes	characteristic
13	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
14	#14556127	C₄	1	1	yes	characteristic
15	#14556127	C₄	1	1	yes	characteristic
16	#14556119	C₄	1	1	yes	characteristic
17	#14556119	C₄	1	1	yes	characteristic
18	#14556105	C₄	1	2	yes	—
19	#14556105	C₄	1	2	yes	—
20	C₄×Dic₁₄₂	C₂²	1	2	yes	—
21	C₄×Dic₁₄₂	C₂²	1	2	yes	—
22	#14556087	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₇₁	C₂²	1	1	yes	characteristic
24	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
25	Dic₇₁⋊C₈	C₂²	1	2	yes	—
26	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
27	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
28	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
29	Dic₇₁⋊C₈	C₂²	1	2	yes	—
30	#14556039	C₂²	1	2	yes	—
31	#14556038	C₂²	1	2	yes	—
32	#14556038	C₂²	1	2	yes	—
33	C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
34	C₈×Dic₇₁	C₂²	1	2	yes	—
35	C₈×Dic₇₁	C₂²	1	2	yes	—
36	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
38	#117×C₇₁	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
40	C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
41	C₂×Dic₁₄₂	C₂×C₄	1	2	yes	—
42	C₂×Dic₁₄₂	C₂×C₄	1	2	yes	—
43	C₂²×Dic₇₁	C₂×C₄	1	1	yes	characteristic
44	C₂²×Dic₇₁	C₂×C₄	1	1	yes	characteristic
45	#11765849	Q₈	1	4	yes	complemented
46	C₂×C₇₁⋊C₈	—	2	2	no	—
47	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
48	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
49	C₂×C₇₁⋊C₈	C₂³	1	2	yes	—
50	C₄⋊Dic₇₁	C₂×C₄	1	1	yes	characteristic
51	C₄⋊Dic₇₁	C₂×C₄	1	1	yes	characteristic
52	C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
53	C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
54	Dic₇₁⋊C₄	C₂×C₄	1	2	yes	—
55	Dic₇₁⋊C₄	C₂×C₄	1	2	yes	—
56	Dic₇₁⋊C₄	C₂×C₄	1	2	yes	—
57	Dic₇₁⋊C₄	C₂×C₄	1	2	yes	—
58	C₂²⋊Dic₇₁	C₂×C₄	1	2	yes	—
59	C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
60	C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
62	C₄×Dic₇₁	C₂³	1	2	yes	—
63	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
65	M₁₆×C₇₁	—	2	2	no	—
66	C₂×C₈×C₇₁	—	2	2	no	—
67	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₇₁	C₂³	1	2	yes	—
70	C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
71	C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
72	C₂²⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
73	C₄²×C₇₁	C₂³	1	2	yes	—
74	C₂³×C₇₁	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₇₁	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₇₁	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₇₁	C₂×Q₈	1	2	yes	—
79	C₂×C₄×C₇₁	#41	1	2	yes	—
80	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
81	C₂×Dic₇₁	—	2	2	no	—
82	C₂×Dic₇₁	—	4	1	no	—
83	C₂×Dic₇₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₇₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₇₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₇₁	C₂²×C₄	1	2	yes	—
87	C₂×Dic₇₁	C₂²×C₄	1	2	yes	—
88	C₂×Dic₇₁	C₂²×C₄	1	1	yes	characteristic
89	Dic₁₄₂	—	2	4	no	—
90	Dic₁₄₂	—	4	2	no	—
91	C₈×C₇₁	—	2	1	no	—
92	C₈×C₇₁	—	2	1	no	—
93	C₈×C₇₁	—	2	2	no	—
94	C₇₁⋊C₈	—	2	1	no	—
95	C₇₁⋊C₈	—	2	1	no	—
96	C₇₁⋊C₈	C₂×Q₈	1	4	yes	—
97	C₂²×C₇₁	—	2	1	no	—
98	C₂²×C₇₁	—	2	1	no	—
99	C₂²×C₇₁	C₄×Q₈	1	2	yes	—
100	C₂²×C₇₁	C₂³×C₄	1	1	yes	characteristic
101	C₄×C₇₁	—	2	2	no	—
102	C₄×C₇₁	—	2	2	no	—
103	C₄×C₇₁	C₄×Q₈	1	2	yes	—
104	C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
105	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
106	Dic₇₁	—	2	2	no	—
107	Dic₇₁	—	2	1	no	—
108	Dic₇₁	—	2	1	no	—
109	Dic₇₁	—	2	2	no	—
110	Dic₇₁	—	2	1	no	—
111	Dic₇₁	#131	1	2	yes	—
112	C₂×C₇₁	—	2	2	no	—
113	C₂×C₇₁	C₂×C₄×Q₈	1	1	yes	characteristic
114	C₂×C₇₁	C₂×#131	1	1	yes	characteristic
115	C₂×C₇₁	#568	1	1	yes	characteristic
116	#2962	—	71	1	no	maximal, Sylow
117	C₇₁	#2962	1	1	yes	characteristic, complemented, Sylow
118	#533	—	71	1	no	—
119	#446	—	71	2	no	—
120	#446	—	71	2	no	—
121	#446	—	71	2	no	—
122	C₈×Q₈	—	71	2	no	—
123	#424	—	71	1	no	—
124	#424	D₇₁	1	1	yes	characteristic
125	C₂×C₄⋊C₈	—	71	1	no	—
126	#405	—	71	1	no	—
127	#405	—	71	1	no	—
128	C₄×M₁₆	—	71	1	no	—
129	C₂×M₁₆	—	71	1	no	—
130	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
131	C₂²×C₈	—	71	1	no	—
132	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
133	C₄⋊Q₈	—	71	1	no	—
134	C₄⋊Q₈	—	71	1	no	—
135	#125	—	71	1	no	—
136	#125	—	71	1	no	—
137	C₂²⋊Q₈	—	71	2	no	—
138	C₂²⋊Q₈	—	71	2	no	—
139	C₄×Q₈	—	71	2	no	—
140	C₄×Q₈	—	71	2	no	—
141	#117	—	71	1	no	—
142	#117	C₂×D₇₁	1	1	yes	characteristic
143	C₂×C₄²	—	71	1	no	—
144	C₄⋊C₈	—	71	1	no	—
145	C₄⋊C₈	—	71	1	no	—
146	C₄⋊C₈	—	71	1	no	—
147	C₄⋊C₈	—	71	1	no	—
148	C₄⋊C₈	—	71	2	no	—
149	C₄⋊C₈	—	71	2	no	—
150	C₄⋊C₈	C₂×D₇₁	1	2	yes	—
151	C₄⋊C₈	C₂×D₇₁	1	2	yes	—
152	#97	—	71	2	no	—
153	#97	—	71	2	no	—
154	#97	—	71	2	no	—
155	C₄×C₈	—	71	2	no	—
156	C₄×C₈	—	71	2	no	—
157	C₄×C₈	—	71	2	no	—
158	C₂×Q₈	—	71	2	no	—
159	C₂×Q₈	—	71	2	no	—
160	C₂²×C₄	—	71	1	no	—
161	C₂²×C₄	—	71	1	no	—
162	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
163	M₁₆	—	2	2	no	—
164	M₁₆	—	71	4	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	71	2	no	—
167	C₂×C₈	—	71	1	no	—
168	C₂×C₈	—	71	1	no	—
169	C₂×C₈	—	142	2	no	—
170	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₇₁	1	2	yes	—
172	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
173	C₄⋊C₄	—	71	2	no	—
174	C₄⋊C₄	—	71	2	no	—
175	C₄⋊C₄	—	71	2	no	—
176	C₄⋊C₄	—	71	2	no	—
177	C₄⋊C₄	—	71	1	no	—
178	C₄⋊C₄	—	71	1	no	—
179	C₄⋊C₄	—	71	2	no	—
180	C₄⋊C₄	—	71	2	no	—
181	C₄⋊C₄	C₄×D₇₁	1	1	yes	characteristic
182	C₄⋊C₄	C₄×D₇₁	1	1	yes	characteristic
183	C₂²⋊C₄	—	71	2	no	—
184	C₂²⋊C₄	C₄×D₇₁	1	2	yes	—
185	C₄²	—	71	1	no	—
186	C₄²	—	71	1	no	—
187	C₄²	—	71	1	no	—
188	C₄²	—	71	2	no	—
189	C₄²	—	71	1	no	—
190	C₄²	C₂²×D₇₁	1	2	yes	—
191	C₂³	C₂×C₄×D₇₁	1	1	yes	characteristic
192	Q₈	—	284	2	no	—
193	Q₈	—	142	4	no	—
194	C₂×C₄	—	284	1	no	—
195	C₂×C₄	—	71	1	no	—
196	C₂×C₄	—	71	1	no	—
197	C₂×C₄	—	71	1	no	—
198	C₂×C₄	—	71	1	no	—
199	C₂×C₄	—	71	2	no	—
200	C₂×C₄	—	71	2	no	—
201	C₂×C₄	—	142	2	no	—
202	C₂×C₄	C₂×C₄×D₇₁	1	2	yes	—
203	C₂×C₄	C₂×C₄×D₇₁	1	2	yes	—
204	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
205	C₂×C₄	Q₈×D₇₁	1	2	yes	—
206	C₂×C₄	#11765864	1	2	yes	—
207	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
208	C₈	—	2	1	no	—
209	C₈	—	2	1	no	—
210	C₈	—	2	2	no	—
211	C₈	—	71	4	no	—
212	C₈	—	142	1	no	—
213	C₈	—	142	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	#14556093	1	2	yes	—
217	C₂²	C₂²×C₄×D₇₁	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	2	no	—
220	C₄	—	71	2	no	—
221	C₄	—	142	1	no	—
222	C₄	—	142	2	no	—
223	C₄	—	142	1	no	—
224	C₄	—	142	1	no	—
225	C₄	—	142	2	no	—
226	C₄	#14556093	1	2	yes	—
227	C₄	C₂×Q₈×D₇₁	1	1	yes	characteristic
228	C₄	C₂×#11765864	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	C₂×#14556093	1	1	yes	characteristic
231	C₂	#131×D₇₁	1	1	yes	characteristic
232	C₂	#17986764	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇₁
commutator, socle	C₂²×C₇₁	C₂³×C₄
Fitting	#424×C₇₁	C₂
hypercenter	#424	D₇₁
hypocenter	C₇₁	#2962
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
2. #17986128	C₂	semidirect
3. #17986128	C₂	semidirect
4. #17986126	C₂	semidirect
5. #17986123	C₂	semidirect
45. #11765849	Q₈	semidirect
117. C₇₁	#2962	semidirect