Subgroups — #31912784

Order 18176.

350 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590096	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985776	C₂	1	2	yes	maximal
3	C₄×#14556038	C₂	1	1	yes	maximal, characteristic
4	#22586419	C₂	1	2	yes	maximal
5	C₇₁×#2150	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556068	C₂²	1	2	yes	—
7	C₂×#14556038	C₄	1	1	yes	characteristic
8	C₂×#14556038	C₄	1	1	yes	characteristic
9	C₂×#14556038	C₂²	1	1	yes	characteristic
10	C₂×#14556038	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇₁⋊C₈	C₄	1	1	yes	characteristic
12	C₄²×Dic₇₁	C₄	1	1	yes	characteristic
13	C₄⋊(C₇₁⋊C₁₆)	C₄	1	4	yes	complemented
14	#17985776	—	2	4	no	—
15	C₂×M₃₂×C₇₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₄⋊C₁₆×C₇₁	C₄	1	4	yes	complemented
18	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₇₁	—	2	2	no	—
21	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
23	#14556068	—	2	2	no	—
24	#14556068	—	2	2	no	—
25	C₂×C₇₁⋊C₁₆	C₂×C₄	1	4	yes	—
26	#14556038	—	2	2	no	—
27	#14556038	—	2	2	no	—
28	#14556038	—	2	1	no	—
29	#14556038	—	2	1	no	—
30	#14556038	D₄	1	2	yes	—
31	#14556038	Q₈	1	2	yes	—
32	C₄×C₇₁⋊C₈	—	2	1	no	—
33	C₄×C₇₁⋊C₈	—	2	1	no	—
34	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
37	M₃₂×C₇₁	—	2	2	no	—
38	M₃₂×C₇₁	—	2	2	no	—
39	C₂×C₁₆×C₇₁	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
42	C₂²×Dic₇₁	—	2	1	no	—
43	C₂²×Dic₇₁	—	2	1	no	—
44	C₂×C₇₁⋊C₈	—	2	1	no	—
45	C₂×C₇₁⋊C₈	—	2	1	no	—
46	C₂×C₇₁⋊C₈	—	2	1	no	—
47	C₂×C₇₁⋊C₈	—	2	1	no	—
48	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
49	C₂×C₇₁⋊C₈	M₁₆	1	2	yes	—
50	C₄×Dic₇₁	—	2	1	no	—
51	C₄×Dic₇₁	—	2	1	no	—
52	C₄×Dic₇₁	—	4	4	no	—
53	C₄×Dic₇₁	C₄⋊C₄	1	2	yes	—
54	C₄×Dic₇₁	M₁₆	1	2	yes	—
55	C₇₁⋊C₁₆	—	2	4	no	—
56	C₂²×C₄×C₇₁	M₁₆	1	2	yes	—
57	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₇₁	—	2	1	no	—
59	C₂×C₈×C₇₁	—	2	1	no	—
60	C₂×C₈×C₇₁	C₄²	1	2	yes	—
61	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
66	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
67	C₄²×C₇₁	C₄²	1	2	yes	—
68	C₄²×C₇₁	M₁₆	1	1	yes	characteristic
69	C₄²×C₇₁	M₁₆	1	1	yes	characteristic
70	C₁₆×C₇₁	—	2	4	no	—
71	C₂³×C₇₁	C₂×M₁₆	1	1	yes	characteristic
72	C₂×C₄×C₇₁	—	2	4	no	—
73	C₂×C₄×C₇₁	#97	1	2	yes	—
74	C₂×C₄×C₇₁	#97	1	2	yes	—
75	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
78	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
81	C₂×Dic₇₁	—	2	2	no	—
82	C₂×Dic₇₁	—	2	2	no	—
83	C₂×Dic₇₁	—	4	2	no	—
84	C₂×Dic₇₁	—	4	2	no	—
85	C₈×C₇₁	—	2	2	no	—
86	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
87	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
88	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
89	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
90	C₇₁⋊C₈	—	2	2	no	—
91	C₇₁⋊C₈	—	2	2	no	—
92	C₂²×C₇₁	—	2	2	no	—
93	C₂²×C₇₁	C₂×#97	1	1	yes	characteristic
94	C₂²×C₇₁	#423	1	1	yes	characteristic
95	C₂²×C₇₁	#432	1	1	yes	characteristic
96	C₄×C₇₁	—	2	2	no	—
97	C₄×C₇₁	#347	1	4	yes	complemented
98	C₄×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
99	C₄×C₇₁	C₄×M₁₆	1	1	yes	characteristic
100	C₄×C₇₁	#423	1	1	yes	characteristic
101	C₄×C₇₁	#432	1	1	yes	characteristic
102	Dic₇₁	—	4	2	no	—
103	Dic₇₁	—	4	2	no	—
104	#12658	—	71	1	no	maximal, Sylow
105	C₂×C₇₁	—	2	1	no	—
106	C₂×C₇₁	—	2	1	no	—
107	C₂×C₇₁	#1767	1	1	yes	characteristic
108	C₂×C₇₁	C₂×#347	1	2	yes	—
109	#2150	—	71	1	no	—
110	#2150	D₇₁	1	1	yes	characteristic
111	C₂×#347	—	71	2	no	—
112	C₄×#97	—	71	1	no	—
113	#1375	—	71	2	no	—
114	C₇₁	#12658	1	1	yes	characteristic, complemented, Sylow
115	C₂×M₃₂	—	71	2	no	—
116	C₂×M₃₂	C₂×D₇₁	1	2	yes	—
117	C₂×#97	—	71	1	no	—
118	C₂×#97	—	71	1	no	—
119	C₂×#97	—	71	1	no	—
120	C₂×#97	—	71	1	no	—
121	C₂×C₄×C₈	—	71	1	no	—
122	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
123	C₄³	—	71	1	no	—
124	C₄⋊C₁₆	—	71	4	no	—
125	C₄⋊C₁₆	Dic₇₁	1	4	yes	complemented
126	#347	—	142	4	no	—
127	C₂²×C₈	—	71	1	no	—
128	C₂²×C₈	—	71	1	no	—
129	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
131	C₂×C₄²	—	71	1	no	—
132	C₂×C₄²	—	71	1	no	—
133	C₂×C₄²	—	142	2	no	—
134	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
135	M₃₂	—	2	2	no	—
136	M₃₂	—	2	2	no	—
137	M₃₂	—	142	2	no	—
138	M₃₂	—	142	2	no	—
139	C₂×C₁₆	—	71	4	no	—
140	C₂×C₁₆	C₂×Dic₇₁	1	4	yes	—
141	#97	—	71	2	no	—
142	#97	—	71	2	no	—
143	#97	—	142	2	no	—
144	#97	—	142	2	no	—
145	#97	—	142	1	no	—
146	#97	—	142	1	no	—
147	C₄×C₈	—	142	1	no	—
148	C₄×C₈	—	142	1	no	—
149	C₄×C₈	C₄×D₇₁	1	2	yes	—
150	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
151	C₂²×C₄	—	142	1	no	—
152	C₂²×C₄	—	142	1	no	—
153	C₂²×C₄	#11765848	1	2	yes	—
154	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	71	2	no	—
158	C₂×C₈	—	71	2	no	—
159	C₂×C₈	—	142	1	no	—
160	C₂×C₈	—	142	1	no	—
161	C₂×C₈	—	142	1	no	—
162	C₂×C₈	—	142	1	no	—
163	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
164	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
165	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
166	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
167	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
168	C₂×C₈	#11765864	1	1	yes	characteristic
169	C₂×C₈	#11765866	1	1	yes	characteristic
170	C₄²	—	284	4	no	—
171	C₄²	—	71	2	no	—
172	C₄²	—	71	2	no	—
173	C₄²	—	142	1	no	—
174	C₄²	—	142	1	no	—
175	C₄²	C₄×Dic₇₁	1	2	yes	—
176	C₄²	#11765848	1	1	yes	characteristic
177	C₄²	#11765848	1	1	yes	characteristic
178	C₁₆	—	2	4	no	—
179	C₁₆	—	142	4	no	—
180	C₂³	C₂×#11765848	1	1	yes	characteristic
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	284	2	no	—
183	C₂×C₄	—	284	2	no	—
184	C₂×C₄	—	142	2	no	—
185	C₂×C₄	—	142	2	no	—
186	C₂×C₄	#14556038	1	2	yes	—
187	C₂×C₄	#14556038	1	2	yes	—
188	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
189	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
190	C₂×C₄	#14556086	1	1	yes	characteristic
191	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
192	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
193	C₂×C₄	C₂×#11765848	1	1	yes	characteristic
194	C₈	—	2	2	no	—
195	C₈	—	142	2	no	—
196	C₈	—	142	2	no	—
197	C₈	#14556090	1	1	yes	characteristic
198	C₈	D₄×Dic₇₁	1	1	yes	characteristic
199	C₈	#14556093	1	1	yes	characteristic
200	C₈	Q₈×Dic₇₁	1	1	yes	characteristic
201	C₂²	—	2	2	no	—
202	C₂²	C₂×#14556038	1	1	yes	characteristic
203	C₂²	#17986021	1	1	yes	characteristic
204	C₂²	#17986055	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	—	284	2	no	—
207	C₄	—	284	2	no	—
208	C₄	#17985776	1	4	yes	complemented
209	C₄	C₄⋊C₄×Dic₇₁	1	1	yes	characteristic
210	C₄	M₁₆×Dic₇₁	1	1	yes	characteristic
211	C₄	#17986021	1	1	yes	characteristic
212	C₄	#17986055	1	1	yes	characteristic
213	C₂	—	2	1	no	—
214	C₂	—	2	1	no	—
215	C₂	#22588195	1	1	yes	characteristic
216	C₂	C₂×#17985776	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₇₁
commutator	C₂×C₄×C₇₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#2150	C₂
socle	C₂²×C₇₁	C₂×#97
prime radical	C₂³×C₇₁	C₂×M₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#2150	D₇₁
hypocenter	C₇₁	#12658
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ #2150

Complements

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

N	H ≅ G/N	type
13. C₄⋊(C₇₁⋊C₁₆)	C₄	semidirect
17. C₄⋊C₁₆×C₇₁	C₄	semidirect
97. C₄×C₇₁	#347	semidirect
114. C₇₁	#12658	semidirect
125. C₄⋊C₁₆	Dic₇₁	semidirect
208. C₄	#17985776	semidirect

Maximal subgroups

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

subgroup	index
#22590096	2
C₂×#17985776	2
C₄×#14556038	2
#22586419	2
C₇₁×#2150	2
#12658	71