Subgroups — #22597086

Order 9088.

492 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14556136	C₂	1	2	yes	maximal, complemented
2	#17986549	C₂	1	1	yes	maximal, characteristic
3	#17986078	C₂	1	2	yes	maximal, complemented
4	#17986072	C₂	1	4	yes	maximal, complemented
5	C₈×D₂₈₄	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
7	#17985967	C₂	1	2	yes	maximal, complemented
8	C₄×M₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11765865	C₂²	1	2	yes	—
10	#14556136	—	2	8	no	—
11	C₂×#11765848	C₂²	1	4	yes	—
12	C₂²×C₇₁⋊C₈	C₂²	1	2	yes	—
13	C₂×C₈×D₇₁	C₂²	1	4	yes	—
14	#14556128	C₄	1	1	yes	characteristic
15	#14556125	C₄	1	1	yes	characteristic
16	#14556116	C₄	1	2	yes	—
17	#14556102	C₄	1	2	yes	—
18	#14556102	C₄	1	2	yes	—
19	C₄×D₂₈₄	C₂²	1	2	yes	—
20	C₄×D₂₈₄	C₂²	1	2	yes	—
21	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
22	C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
23	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
24	(C₂×D₇₁)⋊C₈	C₂²	1	4	yes	—
25	(C₂×D₇₁)⋊C₈	C₂²	1	4	yes	—
26	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
27	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
29	#97×C₇₁	C₂²	1	2	yes	—
30	C₄×C₈×C₇₁	C₂²	1	2	yes	—
31	#11765865	—	2	4	no	—
32	#11765865	—	2	4	no	—
33	C₂×Dic₁₄₂	C₂×C₄	1	2	yes	—
34	C₂×D₂₈₄	C₂×C₄	1	2	yes	—
35	C₂×D₂₈₄	C₂×C₄	1	2	yes	—
36	C₂×D₂₈₄	C₂×C₄	1	2	yes	—
37	C₂×C₂²⋊D₇₁	C₂×C₄	1	4	yes	—
38	C₂×C₄×D₇₁	C₂³	1	4	yes	—
39	#11765848	—	2	8	no	—
40	C₂×C₇₁⋊C₈	—	2	4	no	—
41	C₂×C₇₁⋊C₈	C₂³	1	2	yes	—
42	C₂×C₇₁⋊C₈	C₂³	1	2	yes	—
43	C₈×D₇₁	—	2	8	no	—
44	C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
45	C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
46	(C₂×D₇₁)⋊C₄	C₂×C₄	1	4	yes	—
47	(C₂×D₇₁)⋊C₄	C₂×C₄	1	4	yes	—
48	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
51	M₁₆×C₇₁	—	2	2	no	—
52	M₁₆×C₇₁	D₄	1	4	yes	complemented
53	C₂×C₈×C₇₁	C₂³	1	2	yes	—
54	C₂×C₈×C₇₁	C₂³	1	2	yes	—
55	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₇₁	C₂³	1	1	yes	characteristic
58	C₄²×C₇₁	C₂³	1	1	yes	characteristic
59	C₂³×C₇₁	C₂²×C₄	1	1	yes	characteristic
60	C₂²×D₇₁	C₂²×C₄	1	4	yes	—
61	C₂×C₄×C₇₁	—	2	2	no	—
62	C₂×C₄×C₇₁	—	4	1	no	—
63	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₇₁	#41	1	2	yes	—
70	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
71	C₂²⋊D₇₁	—	2	8	no	—
72	C₂×Dic₇₁	C₂²×C₄	1	4	yes	—
73	D₂₈₄	—	2	4	no	—
74	D₂₈₄	—	2	4	no	—
75	D₂₈₄	—	4	2	no	—
76	C₄×D₇₁	—	2	4	no	—
77	C₄×D₇₁	—	2	4	no	—
78	Dic₁₄₂	—	2	4	no	—
79	C₈×C₇₁	—	2	2	no	—
80	C₈×C₇₁	C₂×D₄	1	4	yes	—
81	C₇₁⋊C₈	—	2	2	no	—
82	C₇₁⋊C₈	—	2	2	no	—
83	C₂²×C₇₁	—	2	1	no	—
84	C₂²×C₇₁	—	2	1	no	—
85	C₂²×C₇₁	C₄×D₄	1	2	yes	—
86	C₂²×C₇₁	C₂³×C₄	1	1	yes	characteristic
87	C₂×D₇₁	—	2	4	no	—
88	C₂×D₇₁	—	2	4	no	—
89	C₂×D₇₁	—	2	4	no	—
90	C₄×C₇₁	—	2	1	no	—
91	C₄×C₇₁	—	2	1	no	—
92	C₄×C₇₁	—	2	1	no	—
93	C₄×C₇₁	C₄×D₄	1	2	yes	—
94	C₄×C₇₁	#131	1	2	yes	complemented
95	C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
96	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
97	Dic₇₁	—	2	4	no	—
98	C₂×C₇₁	—	2	2	no	—
99	C₂×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
100	C₂×C₇₁	C₂×#131	1	1	yes	characteristic
101	C₂×C₇₁	#568	1	1	yes	characteristic
102	D₇₁	—	4	4	no	—
103	#2935	—	71	1	no	maximal, Sylow
104	C₇₁	#2935	1	1	yes	characteristic, complemented, Sylow
105	C₂×#131	—	71	2	no	—
106	#532	—	71	1	no	—
107	#436	—	71	2	no	—
108	#435	—	71	4	no	—
109	C₈×D₄	—	71	2	no	—
110	C₂×C₄⋊C₈	—	71	1	no	—
111	#408	—	71	2	no	—
112	C₄×M₁₆	D₇₁	1	1	yes	characteristic, complemented
113	C₂×#41	—	71	2	no	—
114	#131	—	142	8	no	—
115	C₂×M₁₆	—	71	4	no	—
116	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
117	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
118	C₂²×C₈	—	71	2	no	—
119	C₂²×C₈	—	71	4	no	—
120	C₄⋊Q₈	—	71	1	no	—
121	#127	—	71	1	no	—
122	#124	—	71	2	no	—
123	#121	—	71	2	no	—
124	#121	—	71	2	no	—
125	C₄×D₄	—	71	2	no	—
126	C₄×D₄	—	71	2	no	—
127	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
128	C₄⋊C₈	—	71	1	no	—
129	C₄⋊C₈	—	71	2	no	—
130	C₄⋊C₈	—	71	1	no	—
131	C₂²⋊C₈	—	71	4	no	—
132	C₂²⋊C₈	—	71	4	no	—
133	#97	C₂×D₇₁	1	2	yes	—
134	C₄×C₈	C₂×D₇₁	1	2	yes	—
135	#41	—	142	4	no	—
136	#41	—	142	4	no	—
137	C₂×Q₈	—	71	2	no	—
138	C₂×D₄	—	71	4	no	—
139	C₂×D₄	—	71	2	no	—
140	C₂×D₄	—	71	2	no	—
141	C₂×D₄	—	71	2	no	—
142	C₂²×C₄	—	71	4	no	—
143	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
144	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
145	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
146	M₁₆	—	2	2	no	—
147	M₁₆	—	142	8	no	—
148	M₁₆	D₂₈₄	1	4	yes	complemented
149	C₂×C₈	—	71	2	no	—
150	C₂×C₈	—	71	2	no	—
151	C₂×C₈	—	142	4	no	—
152	C₂×C₈	—	142	8	no	—
153	C₂×C₈	C₂²×D₇₁	1	2	yes	—
154	C₂×C₈	C₂²×D₇₁	1	2	yes	—
155	C₄⋊C₄	—	71	2	no	—
156	C₄⋊C₄	—	71	2	no	—
157	C₂²⋊C₄	—	71	4	no	—
158	C₂²⋊C₄	—	71	4	no	—
159	C₄²	C₄×D₇₁	1	1	yes	characteristic
160	C₄²	C₄×D₇₁	1	1	yes	characteristic
161	C₄²	C₂²×D₇₁	1	1	yes	characteristic
162	C₄²	C₂²×D₇₁	1	1	yes	characteristic
163	C₂³	—	71	4	no	—
164	C₂³	C₂×C₄×D₇₁	1	1	yes	characteristic
165	Q₈	—	142	4	no	—
166	D₄	—	284	2	no	—
167	D₄	—	142	8	no	—
168	D₄	—	142	4	no	—
169	D₄	—	142	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	71	4	no	—
173	C₂×C₄	—	142	4	no	—
174	C₂×C₄	—	142	4	no	—
175	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
176	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
177	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
180	C₂×C₄	C₂×D₂₈₄	1	2	yes	—
181	C₂×C₄	#11765865	1	2	yes	—
182	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	—	142	2	no	—
185	C₈	—	142	2	no	—
186	C₈	C₂×D₂₈₄	1	4	yes	—
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	142	4	no	—
190	C₂²	—	142	4	no	—
191	C₂²	—	142	4	no	—
192	C₂²	C₄×D₂₈₄	1	2	yes	—
193	C₂²	C₂²×C₄×D₇₁	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	142	4	no	—
198	C₄	C₄×D₂₈₄	1	2	yes	—
199	C₄	#14556136	1	2	yes	complemented
200	C₄	C₂²×D₂₈₄	1	1	yes	characteristic
201	C₄	C₂×#11765865	1	1	yes	characteristic
202	C₂	—	2	2	no	—
203	C₂	—	284	4	no	—
204	C₂	C₂×C₄×D₂₈₄	1	1	yes	characteristic
205	C₂	C₂×#14556136	1	1	yes	characteristic
206	C₂	#17986765	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇₁
commutator, socle	C₂²×C₇₁	C₂³×C₄
Fitting	C₄×M₁₆×C₇₁	C₂
hypercenter	C₄×M₁₆	D₇₁
hypocenter	C₇₁	#2935
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#14556136	C₂	semidirect
3. #17986078	C₂	semidirect
4. #17986072	C₂	semidirect
5. C₈×D₂₈₄	C₂	semidirect
6. C₂×C₄⋊(C₇₁⋊C₈)	C₂	semidirect
7. #17985967	C₂	semidirect
8. C₄×M₁₆×C₇₁	C₂	semidirect
52. M₁₆×C₇₁	D₄	semidirect
94. C₄×C₇₁	#131	semidirect
104. C₇₁	#2935	semidirect
112. C₄×M₁₆	D₇₁	semidirect
148. M₁₆	D₂₈₄	semidirect
199. C₄	#14556136	semidirect