Subgroups — #31912308

Order 18176.

586 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556068	C₂	1	2	yes	maximal
2	#22586422	C₂	1	4	yes	maximal, complemented
3	C₂×#405×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556068	—	2	2	no	—
5	C₂×#14556068	—	2	4	no	—
6	C₂×#14556068	C₄	1	8	yes	complemented
7	C₂×#14556068	C₂²	1	4	yes	—
8	C₂²×C₇₁⋊C₁₆	—	2	2	no	—
9	C₂²×M₁₆×C₇₁	C₄	1	1	yes	characteristic
10	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
11	C₂×#117×C₇₁	C₄	1	1	yes	characteristic
12	#405×C₇₁	—	2	4	no	—
13	C₂×#97×C₇₁	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
15	#14556068	—	2	8	no	—
16	#14556068	—	2	4	no	—
17	#14556068	C₂×C₄	1	8	yes	—
18	C₂×C₇₁⋊C₁₆	—	2	8	no	—
19	C₂×C₇₁⋊C₁₆	—	2	4	no	—
20	C₂³×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
21	C₂×M₁₆×C₇₁	—	2	2	no	—
22	C₂×M₁₆×C₇₁	—	2	2	no	—
23	C₂×M₁₆×C₇₁	C₈	1	4	yes	—
24	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₇₁	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₇₁	D₄	1	2	yes	—
28	C₂²×C₈×C₇₁	D₄	1	4	yes	—
29	C₂²×C₈×C₇₁	Q₈	1	2	yes	—
30	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
31	#117×C₇₁	—	2	2	no	—
32	#117×C₇₁	C₈	1	4	yes	—
33	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
34	C₂×C₂²⋊C₄×C₇₁	—	2	1	no	—
35	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
36	#97×C₇₁	—	2	4	no	—
37	C₄×C₈×C₇₁	—	2	4	no	—
38	C₇₁⋊C₁₆	—	2	8	no	—
39	C₂⁴×C₇₁	M₁₆	1	1	yes	characteristic
40	C₂²×C₄×C₇₁	—	2	2	no	—
41	C₂²×C₄×C₇₁	C₄²	1	2	yes	—
42	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₇₁	M₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₇₁	M₁₆	1	2	yes	—
48	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
49	M₁₆×C₇₁	—	2	8	no	—
50	C₂×C₈×C₇₁	—	2	2	no	—
51	C₂×C₈×C₇₁	—	2	2	no	—
52	C₂×C₈×C₇₁	C₄²	1	2	yes	—
53	C₂×C₈×C₇₁	C₂²⋊C₄	1	4	yes	—
54	C₂×C₈×C₇₁	C₂²⋊C₄	1	4	yes	—
55	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₇₁	C₄⋊C₄	1	8	yes	—
57	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
58	C₂×C₈×C₇₁	C₂×C₈	1	4	yes	—
59	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
63	C₄⋊C₄×C₇₁	—	2	4	no	—
64	C₂²⋊C₄×C₇₁	—	2	4	no	—
65	C₄²×C₇₁	—	2	2	no	—
66	C₄²×C₇₁	C₂×C₈	1	4	yes	—
67	C₂³×C₇₁	—	4	2	no	—
68	C₂³×C₇₁	#97	1	2	yes	—
69	C₂³×C₇₁	C₂²⋊C₈	1	2	yes	—
70	C₂³×C₇₁	C₄⋊C₈	1	2	yes	—
71	C₂³×C₇₁	C₂×M₁₆	1	1	yes	characteristic
72	C₂×C₄×C₇₁	—	2	4	no	—
73	C₂×C₄×C₇₁	—	4	4	no	—
74	C₂×C₄×C₇₁	#95	1	4	yes	—
75	C₂×C₄×C₇₁	C₄×C₈	1	4	yes	—
76	C₂×C₄×C₇₁	#97	1	2	yes	—
77	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
78	C₂×C₄×C₇₁	C₂²⋊C₈	1	4	yes	—
79	C₂×C₄×C₇₁	C₄⋊C₈	1	2	yes	—
80	C₂×C₄×C₇₁	C₄⋊C₈	1	4	yes	—
81	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
87	C₈×C₇₁	—	2	4	no	—
88	C₈×C₇₁	#95	1	4	yes	—
89	C₈×C₇₁	C₂×C₂²⋊C₄	1	2	yes	—
90	C₈×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
91	C₂²×C₇₁	—	2	2	no	—
92	C₂²×C₇₁	—	2	4	no	—
93	C₂²×C₇₁	—	4	2	no	—
94	C₂²×C₇₁	—	4	2	no	—
95	C₂²×C₇₁	#336	1	4	yes	—
96	C₂²×C₇₁	C₂×#97	1	1	yes	characteristic
97	C₂²×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
98	C₂²×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
99	C₄×C₇₁	—	4	4	no	—
100	C₄×C₇₁	#336	1	4	yes	—
101	C₄×C₇₁	C₂×#95	1	1	yes	characteristic
102	C₄×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
103	C₄×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
104	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
105	#12549	—	71	1	no	maximal, Sylow
106	C₂×C₇₁	—	2	1	no	—
107	C₂×C₇₁	—	2	4	no	—
108	C₂×C₇₁	—	2	1	no	—
109	C₂×C₇₁	C₂×#336	1	1	yes	characteristic
110	C₂×C₇₁	#2153	1	2	yes	—
111	C₂²×M₃₂	—	71	2	no	—
112	C₂×#405	D₇₁	1	1	yes	characteristic
113	#1376	—	71	4	no	—
114	C₇₁	#12549	1	1	yes	characteristic, complemented, Sylow
115	C₂²×M₁₆	Dic₇₁	1	1	yes	characteristic
116	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
117	C₂×#117	Dic₇₁	1	1	yes	characteristic
118	C₂×M₃₂	—	71	8	no	—
119	C₂×M₃₂	—	71	4	no	—
120	C₂×M₃₂	—	142	2	no	—
121	C₂×M₃₂	—	142	4	no	—
122	C₂²×C₁₆	—	142	2	no	—
123	#405	—	2	4	no	—
124	C₂×#97	Dic₇₁	1	2	yes	—
125	C₂×C₄×C₈	C₂×D₇₁	1	2	yes	—
126	C₂³×C₄	C₂×Dic₇₁	1	1	yes	characteristic
127	C₂×M₁₆	—	2	2	no	—
128	C₂×M₁₆	—	2	2	no	—
129	C₂×M₁₆	C₇₁⋊C₈	1	4	yes	—
130	C₂²×C₈	Dic₁₄₂	1	2	yes	—
131	C₂²×C₈	C₄×D₇₁	1	4	yes	—
132	C₂²×C₈	D₂₈₄	1	2	yes	—
133	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
134	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
135	C₂²×C₈	C₂²⋊D₇₁	1	4	yes	—
136	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
137	#117	—	2	2	no	—
138	#117	C₇₁⋊C₈	1	4	yes	—
139	C₂×C₄⋊C₄	—	2	1	no	—
140	C₂×C₂²⋊C₄	—	2	1	no	—
141	C₂×C₄²	C₂×Dic₇₁	1	2	yes	—
142	M₃₂	—	71	8	no	—
143	M₃₂	—	142	8	no	—
144	M₃₂	—	142	4	no	—
145	C₂×C₁₆	—	142	8	no	—
146	C₂×C₁₆	—	142	4	no	—
147	#97	—	2	4	no	—
148	C₄×C₈	—	2	4	no	—
149	C₂⁴	#11765849	1	1	yes	characteristic
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	C₄×Dic₇₁	1	2	yes	—
152	C₂²×C₄	C₂²⋊Dic₇₁	1	2	yes	—
153	C₂²×C₄	C₄⋊Dic₇₁	1	2	yes	—
154	C₂²×C₄	C₂×C₇₁⋊C₈	1	2	yes	—
155	C₂²×C₄	C₂×C₇₁⋊C₈	1	2	yes	—
156	C₂²×C₄	#11765849	1	1	yes	characteristic
157	C₂²×C₄	#11765849	1	2	yes	—
158	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
159	M₁₆	—	2	8	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
163	C₂×C₈	(C₂×D₇₁)⋊C₄	1	4	yes	—
164	C₂×C₈	(C₂×D₇₁)⋊C₄	1	4	yes	—
165	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
166	C₂×C₈	Dic₇₁⋊C₄	1	8	yes	—
167	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
168	C₂×C₈	C₂×C₇₁⋊C₈	1	4	yes	—
169	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
170	C₂×C₈	C₂×C₂²⋊D₇₁	1	2	yes	—
171	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
172	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
173	C₄⋊C₄	—	2	4	no	—
174	C₂²⋊C₄	—	2	4	no	—
175	C₄²	—	2	2	no	—
176	C₄²	C₂×C₇₁⋊C₈	1	4	yes	—
177	C₁₆	—	142	8	no	—
178	C₂³	—	4	2	no	—
179	C₂³	#14556039	1	2	yes	—
180	C₂³	C₂²⋊(C₇₁⋊C₈)	1	2	yes	—
181	C₂³	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
182	C₂³	C₂×#11765849	1	1	yes	characteristic
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	4	4	no	—
185	C₂×C₄	C₇₁⋊#95	1	4	yes	—
186	C₂×C₄	C₄×C₇₁⋊C₈	1	4	yes	—
187	C₂×C₄	#14556039	1	2	yes	—
188	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	4	yes	—
189	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	2	yes	—
190	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
191	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	4	yes	—
192	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
193	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
195	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
196	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
197	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
198	C₈	—	2	4	no	—
199	C₈	C₇₁⋊#95	1	4	yes	—
200	C₈	C₂×(C₂×D₇₁)⋊C₄	1	2	yes	—
201	C₈	C₂×Dic₇₁⋊C₄	1	2	yes	—
202	C₂²	—	2	2	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	4	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	#17985754	1	4	yes	—
207	C₂²	C₂×#14556039	1	1	yes	characteristic
208	C₂²	C₂×C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
209	C₂²	C₂×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
210	C₄	—	4	4	no	—
211	C₄	#17985754	1	4	yes	—
212	C₄	C₂×C₇₁⋊#95	1	1	yes	characteristic
213	C₄	C₂×C₄×C₇₁⋊C₈	1	1	yes	characteristic
214	C₄	C₂×C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
215	C₄	C₂×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
216	C₂	—	2	1	no	—
217	C₂	—	2	1	no	—
218	C₂	—	2	4	no	—
219	C₂	C₂×#17985754	1	1	yes	characteristic
220	C₂	#22590106	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₂₈₄
commutator	C₄×C₇₁	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#405×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₄⋊C₈
prime radical	C₂⁴×C₇₁	M₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#405	D₇₁
hypocenter	C₇₁	#12549
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #22586422	C₂	semidirect
6. C₂×#14556068	C₄	semidirect
114. C₇₁	#12549	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14556068	2
#22586422	2
C₂×#405×C₇₁	2
#12549	71