Subgroups — #31912814

Order 18176.

334 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22589939	C₂	1	2	yes	maximal
2	C₂×C₄×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#22588219	C₂	1	1	yes	maximal, characteristic
4	#22586430	C₂	1	2	yes	maximal
5	C₂×#364×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×#14556064	C₄	1	1	yes	characteristic
9	#17986037	C₄	1	1	yes	characteristic
10	#17985954	C₂²	1	2	yes	—
11	#17985774	—	2	2	no	—
12	C₄×C₇₁⋊C₁₆	C₄	1	4	yes	complemented
13	C₁₆×Dic₇₁	—	2	2	no	—
14	#17985759	C₄	1	2	yes	—
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	C₂×#11765849	C₂×C₄	1	2	yes	—
19	#14556087	C₂×C₄	1	2	yes	—
20	C₂×C₇₁⋊C₁₆	—	2	1	no	—
21	C₂×C₇₁⋊C₁₆	—	2	1	no	—
22	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
25	C₂×C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
26	C₂×C₇₁⋊C₁₆	Q₈	1	2	yes	—
27	#14556064	C₈	1	4	yes	—
28	#14556062	C₈	1	2	yes	—
29	#14556060	C₈	1	2	yes	—
30	#14556038	—	2	2	no	—
31	C₈×Dic₇₁	—	2	2	no	—
32	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
35	M₃₂×C₇₁	—	2	2	no	—
36	M₃₂×C₇₁	C₂×C₄	1	4	yes	—
37	C₂×C₁₆×C₇₁	—	2	2	no	—
38	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
40	#11765849	—	2	2	no	—
41	#11765849	C₂×C₈	1	4	yes	—
42	C₂×C₇₁⋊C₈	—	2	2	no	—
43	C₄⋊Dic₇₁	C₂×C₈	1	4	yes	—
44	C₂²⋊Dic₇₁	—	2	2	no	—
45	C₄×Dic₇₁	—	2	2	no	—
46	C₇₁⋊C₁₆	—	2	2	no	—
47	C₇₁⋊C₁₆	C₄⋊C₄	1	2	yes	—
48	C₇₁⋊C₁₆	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₇₁	—	2	1	no	—
50	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₁	C₄²	1	2	yes	—
52	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
53	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
54	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
55	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
60	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
61	C₄²×C₇₁	—	2	1	no	—
62	C₄²×C₇₁	C₄²	1	2	yes	—
63	C₁₆×C₇₁	—	2	4	no	—
64	C₂³×C₇₁	#131	1	1	yes	characteristic
65	C₂×C₄×C₇₁	—	2	4	no	—
66	C₂×C₄×C₇₁	—	2	2	no	—
67	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
73	C₂×Dic₇₁	—	2	4	no	—
74	C₈×C₇₁	C₄×C₈	1	2	yes	—
75	C₈×C₇₁	C₄×C₈	1	2	yes	—
76	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₈×C₇₁	#117	1	1	yes	characteristic
78	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
79	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
80	C₇₁⋊C₈	—	2	4	no	—
81	C₂²×C₇₁	—	2	2	no	—
82	C₂²×C₇₁	#405	1	1	yes	characteristic
83	C₂²×C₇₁	C₈×D₄	1	1	yes	characteristic
84	C₂²×C₇₁	C₈×Q₈	1	1	yes	characteristic
85	C₄×C₇₁	—	2	4	no	—
86	C₄×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
87	C₄×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
88	C₄×C₇₁	C₈×D₄	1	1	yes	characteristic
89	C₄×C₇₁	C₈×Q₈	1	1	yes	characteristic
90	Dic₇₁	—	4	4	no	—
91	#12664	—	71	1	no	maximal, Sylow
92	C₂×C₇₁	—	2	1	no	—
93	C₂×C₇₁	—	2	1	no	—
94	C₂×C₇₁	C₈×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₇₁	#2170	1	2	yes	—
96	C₂×#364	D₇₁	1	1	yes	characteristic
97	#2108	—	71	2	no	—
98	C₂×C₄×C₁₆	—	71	1	no	—
99	#1772	—	71	1	no	—
100	#1379	—	71	2	no	—
101	C₇₁	#12664	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₃₂	C₂×D₇₁	1	2	yes	—
103	C₂²×C₁₆	—	71	1	no	—
104	C₂²×C₁₆	—	71	1	no	—
105	C₂×#108	—	71	1	no	—
106	#427	—	71	1	no	—
107	#405	—	71	2	no	—
108	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
109	#364	Dic₇₁	1	4	yes	complemented
110	#346	—	142	2	no	—
111	C₄×C₁₆	—	71	4	no	—
112	C₄×C₁₆	—	142	2	no	—
113	#339	—	71	2	no	—
114	C₂×M₁₆	—	71	2	no	—
115	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
117	#117	—	71	2	no	—
118	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
119	M₃₂	—	2	2	no	—
120	M₃₂	C₂×Dic₇₁	1	4	yes	—
121	C₂×C₁₆	—	2	2	no	—
122	C₂×C₁₆	—	71	2	no	—
123	C₂×C₁₆	—	71	1	no	—
124	C₂×C₁₆	—	71	1	no	—
125	C₂×C₁₆	—	71	2	no	—
126	C₂×C₁₆	—	71	2	no	—
127	C₂×C₁₆	—	142	1	no	—
128	C₂×C₁₆	—	142	1	no	—
129	#108	—	71	4	no	—
130	#107	—	71	2	no	—
131	#106	—	71	2	no	—
132	#97	—	142	2	no	—
133	C₄×C₈	—	142	2	no	—
134	C₄×C₈	C₄×D₇₁	1	2	yes	—
135	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
138	M₁₆	—	71	4	no	—
139	M₁₆	—	142	2	no	—
140	C₂×C₈	—	142	2	no	—
141	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
142	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
143	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
144	C₂×C₈	C₈×D₇₁	1	2	yes	—
145	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
146	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
147	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
148	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
149	C₂×C₈	#11765864	1	1	yes	characteristic
150	C₂×C₈	#11765866	1	1	yes	characteristic
151	C₄⋊C₄	—	71	4	no	—
152	C₂²⋊C₄	—	142	2	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	142	2	no	—
155	C₄²	C₄×Dic₇₁	1	2	yes	—
156	C₁₆	—	2	4	no	—
157	C₁₆	—	71	2	no	—
158	C₁₆	—	71	2	no	—
159	C₁₆	—	142	2	no	—
160	C₂³	#14556136	1	1	yes	characteristic
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	142	4	no	—
164	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
165	C₂×C₄	#14556090	1	1	yes	characteristic
166	C₂×C₄	#14556093	1	1	yes	characteristic
167	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
168	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
169	C₂×C₄	#14556136	1	1	yes	characteristic
170	C₈	—	142	4	no	—
171	C₈	C₈×Dic₇₁	1	2	yes	—
172	C₈	C₈×Dic₇₁	1	2	yes	—
173	C₈	C₄⋊C₄×D₇₁	1	1	yes	characteristic
174	C₈	#14556086	1	1	yes	characteristic
175	C₈	D₄×Dic₇₁	1	1	yes	characteristic
176	C₈	Q₈×Dic₇₁	1	1	yes	characteristic
177	C₂²	—	2	2	no	—
178	C₂²	#17985957	1	1	yes	characteristic
179	C₂²	#17986064	1	1	yes	characteristic
180	C₂²	#17986123	1	1	yes	characteristic
181	C₄	—	2	4	no	—
182	C₄	—	284	4	no	—
183	C₄	C₄⋊C₄×Dic₇₁	1	1	yes	characteristic
184	C₄	C₂×C₈×Dic₇₁	1	1	yes	characteristic
185	C₄	#17986064	1	1	yes	characteristic
186	C₄	#17986123	1	1	yes	characteristic
187	C₂	—	2	1	no	—
188	C₂	—	2	1	no	—
189	C₂	#22588205	1	1	yes	characteristic
190	C₂	#22590176	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₇₁
commutator	C₄×C₇₁	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#364×C₇₁	C₂
socle	C₂²×C₇₁	C₈×D₄
prime radical	C₂³×C₇₁	#131
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#364	D₇₁
hypocenter	C₇₁	#12664
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
12. C₄×C₇₁⋊C₁₆	C₄	semidirect
17. #364×C₇₁	C₄	semidirect
101. C₇₁	#12664	semidirect
109. #364	Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
#22589939	2
C₂×C₄×C₇₁⋊C₁₆	2
#22588219	2
#22586430	2
C₂×#364×C₇₁	2
#12664	71