Subgroups — #31912788

Order 18176.

346 conjugacy classes of subgroups.

The subgroup lattice has 210 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	1	yes	maximal, characteristic
2	C₂×#17985776	C₂	1	2	yes	maximal
3	C₄×#14556038	C₂	1	1	yes	maximal, characteristic
4	#22586423	C₂	1	2	yes	maximal
5	C₂×#364×C₇₁	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	#17985820	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	#364×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	C₂×C₄	1	4	yes	—
25	C₂×C₇₁⋊C₁₆	—	2	2	no	—
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	2	no	—
33	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
36	M₃₂×C₇₁	—	2	2	no	—
37	M₃₂×C₇₁	C₂×C₄	1	4	yes	—
38	C₂×C₁₆×C₇₁	—	2	2	no	—
39	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
41	C₂²×Dic₇₁	—	2	1	no	—
42	C₂²×Dic₇₁	M₁₆	1	2	yes	—
43	C₂×C₇₁⋊C₈	—	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	C₄⋊C₄	1	2	yes	—
48	C₂×C₇₁⋊C₈	M₁₆	1	2	yes	—
49	C₄×Dic₇₁	—	2	1	no	—
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	C₇₁⋊C₁₆	—	2	4	no	—
55	C₂²×C₄×C₇₁	—	2	1	no	—
56	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₇₁	C₄²	1	2	yes	—
58	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
59	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
60	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
65	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
66	C₄²×C₇₁	—	2	1	no	—
67	C₄²×C₇₁	C₄²	1	2	yes	—
68	C₁₆×C₇₁	—	2	4	no	—
69	C₂³×C₇₁	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₇₁	—	2	4	no	—
71	C₂×C₄×C₇₁	—	2	2	no	—
72	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
75	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
78	C₂×Dic₇₁	—	2	2	no	—
79	C₂×Dic₇₁	—	2	2	no	—
80	C₂×Dic₇₁	—	4	2	no	—
81	C₂×Dic₇₁	—	4	2	no	—
82	C₈×C₇₁	#97	1	4	yes	—
83	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
84	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
85	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
86	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
87	C₇₁⋊C₈	—	2	2	no	—
88	C₇₁⋊C₈	—	2	2	no	—
89	C₂²×C₇₁	—	2	2	no	—
90	C₂²×C₇₁	C₄×M₁₆	1	1	yes	characteristic
91	C₂²×C₇₁	#423	1	1	yes	characteristic
92	C₂²×C₇₁	#432	1	1	yes	characteristic
93	C₄×C₇₁	—	2	4	no	—
94	C₄×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
95	C₄×C₇₁	C₂×#97	1	1	yes	characteristic
96	C₄×C₇₁	#423	1	1	yes	characteristic
97	C₄×C₇₁	#432	1	1	yes	characteristic
98	Dic₇₁	—	4	2	no	—
99	Dic₇₁	—	4	2	no	—
100	#12659	—	71	1	no	maximal, Sylow
101	C₂×C₇₁	—	2	1	no	—
102	C₂×C₇₁	—	2	1	no	—
103	C₂×C₇₁	#1767	1	1	yes	characteristic
104	C₂×C₇₁	#2165	1	2	yes	—
105	C₂×#364	—	71	1	no	—
106	C₂×#364	D₇₁	1	1	yes	characteristic
107	C₂×#347	—	71	2	no	—
108	C₄×#97	—	71	1	no	—
109	#1377	—	71	2	no	—
110	C₇₁	#12659	1	1	yes	characteristic, complemented, Sylow
111	C₂×M₃₂	—	71	2	no	—
112	C₂×M₃₂	C₂×D₇₁	1	2	yes	—
113	C₂×#97	—	71	1	no	—
114	C₂×#97	—	71	1	no	—
115	C₂×#97	—	71	1	no	—
116	C₂×#97	—	71	1	no	—
117	C₂×C₄×C₈	—	71	1	no	—
118	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
119	C₄³	—	71	1	no	—
120	#364	—	71	4	no	—
121	#364	Dic₇₁	1	4	yes	complemented
122	#347	—	142	4	no	—
123	C₂²×C₈	—	71	1	no	—
124	C₂²×C₈	—	71	1	no	—
125	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
127	C₂×C₄²	—	71	1	no	—
128	C₂×C₄²	—	71	1	no	—
129	C₂×C₄²	—	142	2	no	—
130	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
131	M₃₂	—	2	2	no	—
132	M₃₂	—	71	4	no	—
133	M₃₂	—	142	2	no	—
134	M₃₂	C₂×Dic₇₁	1	4	yes	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	—	142	2	no	—
137	#97	—	71	2	no	—
138	#97	—	71	2	no	—
139	#97	—	142	2	no	—
140	#97	—	142	2	no	—
141	#97	—	142	1	no	—
142	#97	—	142	1	no	—
143	C₄×C₈	—	142	2	no	—
144	C₄×C₈	C₄×D₇₁	1	2	yes	—
145	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
146	C₂²×C₄	—	2	1	no	—
147	C₂²×C₄	—	71	2	no	—
148	C₂²×C₄	—	142	1	no	—
149	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
150	C₂×C₈	—	71	2	no	—
151	C₂×C₈	—	71	2	no	—
152	C₂×C₈	—	142	1	no	—
153	C₂×C₈	—	142	1	no	—
154	C₂×C₈	—	142	1	no	—
155	C₂×C₈	—	142	1	no	—
156	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
157	C₂×C₈	#11765848	1	2	yes	—
158	C₂×C₈	#11765848	1	2	yes	—
159	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
160	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
161	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
162	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
163	C₂×C₈	#11765864	1	1	yes	characteristic
164	C₂×C₈	#11765866	1	1	yes	characteristic
165	C₄²	—	2	1	no	—
166	C₄²	—	284	4	no	—
167	C₄²	—	71	2	no	—
168	C₄²	—	142	1	no	—
169	C₄²	—	142	1	no	—
170	C₄²	—	142	1	no	—
171	C₄²	C₄×Dic₇₁	1	2	yes	—
172	C₁₆	—	2	4	no	—
173	C₁₆	—	142	4	no	—
174	C₂³	M₁₆×D₇₁	1	1	yes	characteristic
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	284	2	no	—
178	C₂×C₄	—	284	2	no	—
179	C₂×C₄	—	142	2	no	—
180	C₂×C₄	—	142	2	no	—
181	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
182	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
183	C₂×C₄	#14556086	1	1	yes	characteristic
184	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
185	C₂×C₄	C₂×#11765848	1	1	yes	characteristic
186	C₂×C₄	C₂×#11765848	1	1	yes	characteristic
187	C₈	—	142	2	no	—
188	C₈	—	142	2	no	—
189	C₈	#14556038	1	4	yes	—
190	C₈	#14556090	1	1	yes	characteristic
191	C₈	D₄×Dic₇₁	1	1	yes	characteristic
192	C₈	#14556093	1	1	yes	characteristic
193	C₈	Q₈×Dic₇₁	1	1	yes	characteristic
194	C₂²	—	2	2	no	—
195	C₂²	M₁₆×Dic₇₁	1	1	yes	characteristic
196	C₂²	#17986021	1	1	yes	characteristic
197	C₂²	#17986055	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	—	284	2	no	—
200	C₄	—	284	2	no	—
201	C₄	C₄⋊C₄×Dic₇₁	1	1	yes	characteristic
202	C₄	C₂×#14556038	1	1	yes	characteristic
203	C₄	#17986021	1	1	yes	characteristic
204	C₄	#17986055	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	#22588195	1	1	yes	characteristic
208	C₂	#22590151	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14556086
commutator	C₂×C₄×C₇₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#364×C₇₁	C₂
socle	C₂²×C₇₁	#432
prime radical	C₂³×C₇₁	C₂×M₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#364	D₇₁
hypocenter	C₇₁	#12659
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
13. #17985820	C₄	semidirect
17. #364×C₇₁	C₄	semidirect
110. C₇₁	#12659	semidirect
121. #364	Dic₇₁	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
C₂×#17985776	2
C₄×#14556038	2
#22586423	2
C₂×#364×C₇₁	2
#12659	71