Subgroups — #31913806

Order 18176.

298 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17985820	C₂	1	2	yes	maximal
2	#22586424	C₂	1	2	yes	maximal
3	#22586420	C₂	1	2	yes	maximal
4	C₇₁×#1726	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556068	C₂²	1	4	yes	—
6	#17985820	—	2	4	no	—
7	C₂×#97×C₇₁	C₄	1	1	yes	characteristic
8	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
9	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₇₁	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₇₁	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
13	#376×C₇₁	C₄	1	1	yes	characteristic
14	#14556068	—	2	4	no	—
15	#14556068	—	2	4	no	—
16	C₂×C₇₁⋊C₁₆	—	2	4	no	—
17	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
24	#97×C₇₁	—	2	2	no	—
25	#97×C₇₁	—	2	2	no	—
26	#97×C₇₁	—	2	2	no	—
27	C₄×C₈×C₇₁	—	2	2	no	—
28	C₄×C₈×C₇₁	C₈	1	4	yes	—
29	C₄×C₈×C₇₁	D₄	1	2	yes	—
30	C₄×C₈×C₇₁	Q₈	1	2	yes	—
31	#95×C₇₁	—	2	2	no	—
32	C₇₁⋊C₁₆	—	4	4	no	—
33	C₂²×C₄×C₇₁	—	2	1	no	—
34	C₂²×C₄×C₇₁	—	2	1	no	—
35	C₂²×C₄×C₇₁	M₁₆	1	2	yes	—
36	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₇₁	—	2	1	no	—
38	C₂×C₈×C₇₁	—	2	1	no	—
39	C₂×C₈×C₇₁	—	2	1	no	—
40	C₂×C₈×C₇₁	—	2	1	no	—
41	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
44	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
45	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
46	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₇₁	#41	1	2	yes	—
49	C₂×C₈×C₇₁	#41	1	2	yes	—
50	C₄²×C₇₁	—	2	1	no	—
51	C₄²×C₇₁	—	2	1	no	—
52	C₄²×C₇₁	—	2	1	no	—
53	C₄²×C₇₁	—	2	1	no	—
54	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
55	C₄²×C₇₁	C₂×C₈	1	2	yes	—
56	C₂³×C₇₁	C₂×M₁₆	1	1	yes	characteristic
57	C₂×C₄×C₇₁	—	2	2	no	—
58	C₂×C₄×C₇₁	—	2	2	no	—
59	C₂×C₄×C₇₁	—	2	2	no	—
60	C₂×C₄×C₇₁	—	4	2	no	—
61	C₂×C₄×C₇₁	—	4	2	no	—
62	C₂×C₄×C₇₁	—	4	2	no	—
63	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
65	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
66	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
68	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
69	C₈×C₇₁	—	2	2	no	—
70	C₈×C₇₁	—	2	2	no	—
71	C₈×C₇₁	C₄⋊C₈	1	4	yes	—
72	C₈×C₇₁	#124	1	2	yes	—
73	C₈×C₇₁	#125	1	2	yes	—
74	C₂²×C₇₁	—	2	2	no	—
75	C₂²×C₇₁	#423	1	1	yes	characteristic
76	C₂²×C₇₁	#431	1	1	yes	characteristic
77	C₂²×C₇₁	#432	1	1	yes	characteristic
78	C₄×C₇₁	—	4	2	no	—
79	C₄×C₇₁	—	4	2	no	—
80	C₄×C₇₁	—	4	2	no	—
81	C₄×C₇₁	#382	1	1	yes	characteristic
82	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
83	C₄×C₇₁	#431	1	1	yes	characteristic
84	C₄×C₇₁	#432	1	1	yes	characteristic
85	#12881	—	71	1	no	maximal, Sylow
86	C₂×C₇₁	—	2	1	no	—
87	C₂×C₇₁	—	2	1	no	—
88	C₂×C₇₁	#1831	1	1	yes	characteristic
89	C₂×C₇₁	#2166	1	2	yes	—
90	C₂×#364	—	71	2	no	—
91	#1726	D₇₁	1	1	yes	characteristic
92	#1377	—	71	2	no	—
93	#1375	—	71	2	no	—
94	C₇₁	#12881	1	1	yes	characteristic, complemented, Sylow
95	C₂×M₃₂	—	71	4	no	—
96	C₂×#97	Dic₇₁	1	1	yes	characteristic
97	C₂×#97	C₂×D₇₁	1	1	yes	characteristic
98	C₂×#97	C₂×D₇₁	1	1	yes	characteristic
99	C₂×C₄×C₈	Dic₇₁	1	1	yes	characteristic
100	C₂×C₄×C₈	Dic₇₁	1	1	yes	characteristic
101	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
102	#376	Dic₇₁	1	1	yes	characteristic
103	#364	—	142	4	no	—
104	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
105	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
106	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
107	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
108	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
109	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
110	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
111	M₃₂	—	142	4	no	—
112	M₃₂	—	142	4	no	—
113	C₂×C₁₆	—	142	4	no	—
114	#97	—	2	2	no	—
115	#97	—	2	2	no	—
116	#97	—	2	2	no	—
117	C₄×C₈	—	2	2	no	—
118	C₄×C₈	C₇₁⋊C₈	1	4	yes	—
119	C₄×C₈	Dic₁₄₂	1	2	yes	—
120	C₄×C₈	D₂₈₄	1	2	yes	—
121	#95	—	2	2	no	—
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	#11765849	1	2	yes	—
125	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
131	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
132	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
133	C₂×C₈	#11765849	1	2	yes	—
134	C₂×C₈	#11765849	1	2	yes	—
135	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
136	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
137	C₂×C₈	#11765865	1	2	yes	—
138	C₂×C₈	#11765865	1	2	yes	—
139	C₄²	—	2	1	no	—
140	C₄²	—	2	1	no	—
141	C₄²	—	2	1	no	—
142	C₄²	—	2	1	no	—
143	C₄²	C₄⋊Dic₇₁	1	2	yes	—
144	C₄²	C₂×C₇₁⋊C₈	1	2	yes	—
145	C₁₆	—	284	4	no	—
146	C₂³	C₂×#11765849	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
154	C₂×C₄	#14556087	1	1	yes	characteristic
155	C₂×C₄	#14556087	1	1	yes	characteristic
156	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
157	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
158	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
159	C₈	—	2	2	no	—
160	C₈	—	2	2	no	—
161	C₈	C₄⋊(C₇₁⋊C₈)	1	4	yes	—
162	C₈	#14556116	1	2	yes	—
163	C₈	#14556120	1	2	yes	—
164	C₂²	—	2	2	no	—
165	C₂²	#17986023	1	1	yes	characteristic
166	C₂²	#17986052	1	1	yes	characteristic
167	C₂²	#17986056	1	1	yes	characteristic
168	C₄	—	4	2	no	—
169	C₄	—	4	2	no	—
170	C₄	—	4	2	no	—
171	C₄	#17985865	1	1	yes	characteristic
172	C₄	C₂×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
173	C₄	#17986052	1	1	yes	characteristic
174	C₄	#17986056	1	1	yes	characteristic
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	#22588457	1	1	yes	characteristic
178	C₂	#22590156	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14556087
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1726	C₂
socle	C₂²×C₇₁	#432
prime radical	C₂³×C₇₁	C₂×M₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1726	D₇₁
hypocenter	C₇₁	#12881
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1726

Complements

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

N	H ≅ G/N	type
94. C₇₁	#12881	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17985820	2
#22586424	2
#22586420	2
C₇₁×#1726	2
#12881	71