Subgroups — #31891259

Order 18176.

322 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22588477	C₂	1	2	yes	maximal
2	#22587099	C₂	1	1	yes	maximal, characteristic
3	#22587095	C₂	1	2	yes	maximal
4	C₂×C₈×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1567	C₂	1	1	yes	maximal, characteristic
6	#17986023	C₂²	1	2	yes	—
7	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	#17985763	—	2	4	no	—
10	#17985752	—	2	2	no	—
11	#17985749	—	2	4	no	—
12	C₈×C₇₁⋊C₈	C₄	1	1	yes	characteristic
13	C₈×C₇₁⋊C₈	C₄	1	1	yes	characteristic
14	C₈×C₇₁⋊C₈	C₄	1	1	yes	characteristic
15	C₈×C₇₁⋊C₈	C₄	1	1	yes	characteristic
16	#423×C₇₁	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
18	#335×C₇₁	—	2	2	no	—
19	C₂×#11765849	—	2	2	no	—
20	C₂²×C₇₁⋊C₈	D₄	1	2	yes	—
21	C₂²×C₇₁⋊C₈	Q₈	1	2	yes	—
22	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
23	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
24	C₄×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₇₁⋊C₈	D₄	1	2	yes	—
29	C₄×C₇₁⋊C₈	Q₈	1	2	yes	—
30	C₂×M₁₆×C₇₁	—	2	2	no	—
31	C₂²×C₈×C₇₁	Q₈	1	2	yes	—
32	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₇₁	—	2	2	no	—
34	C₄⋊C₈×C₇₁	—	2	2	no	—
35	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₇₁	D₄	1	2	yes	—
38	#11765849	—	4	4	no	—
39	C₂×C₇₁⋊C₈	—	2	2	no	—
40	C₂×C₇₁⋊C₈	—	2	2	no	—
41	C₂×C₇₁⋊C₈	—	2	4	no	—
42	C₂×C₇₁⋊C₈	—	4	2	no	—
43	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
44	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
45	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
46	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
50	M₁₆×C₇₁	—	4	4	no	—
51	C₂×C₈×C₇₁	—	2	4	no	—
52	C₂×C₈×C₇₁	—	4	2	no	—
53	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
54	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
55	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
59	C₂³×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₇₁	—	2	2	no	—
61	C₂×C₄×C₇₁	—	2	2	no	—
62	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
66	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₇₁	#135	1	2	yes	—
69	C₂×C₄×C₇₁	#135	1	2	yes	—
70	C₈×C₇₁	—	2	2	no	—
71	C₈×C₇₁	—	8	2	no	—
72	C₈×C₇₁	#108	1	4	yes	—
73	C₇₁⋊C₈	—	2	2	no	—
74	C₇₁⋊C₈	—	2	2	no	—
75	C₇₁⋊C₈	—	2	2	no	—
76	C₇₁⋊C₈	—	8	2	no	—
77	C₇₁⋊C₈	#108	1	4	yes	—
78	C₂²×C₇₁	—	2	1	no	—
79	C₂²×C₇₁	—	2	1	no	—
80	C₂²×C₇₁	#384	1	1	yes	characteristic
81	C₂²×C₇₁	#499	1	2	yes	—
82	C₄×C₇₁	—	2	1	no	—
83	C₄×C₇₁	—	2	1	no	—
84	C₄×C₇₁	#427	1	2	yes	—
85	C₄×C₇₁	C₂×#108	1	2	yes	—
86	C₄×C₇₁	C₂×#108	1	2	yes	—
87	C₄×C₇₁	#495	1	1	yes	characteristic
88	C₄×C₇₁	#495	1	1	yes	characteristic
89	#8824	—	71	1	no	maximal, Sylow
90	C₂×C₇₁	—	2	2	no	—
91	C₂×C₇₁	#1846	1	1	yes	characteristic
92	C₂×C₇₁	#1848	1	1	yes	characteristic
93	C₂×C₇₁	#1848	1	1	yes	characteristic
94	#1837	—	71	2	no	—
95	#1567	—	71	1	no	—
96	#1567	D₇₁	1	1	yes	characteristic
97	#1566	—	71	2	no	—
98	C₂×C₈²	—	71	1	no	—
99	C₇₁	#8824	1	1	yes	characteristic, complemented, Sylow
100	#423	—	71	2	no	—
101	#423	C₂×D₇₁	1	2	yes	—
102	C₂×C₄×C₈	—	71	1	no	—
103	C₂×C₄×C₈	—	71	1	no	—
104	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
105	#341	—	142	4	no	—
106	#335	—	2	2	no	—
107	#335	—	142	2	no	—
108	#334	—	142	4	no	—
109	C₈²	—	71	1	no	—
110	C₈²	—	71	1	no	—
111	C₈²	—	71	1	no	—
112	C₈²	—	71	1	no	—
113	C₂×M₁₆	—	2	2	no	—
114	C₂×M₁₆	—	142	2	no	—
115	C₂²×C₈	—	71	2	no	—
116	C₂²×C₈	—	71	2	no	—
117	C₂²×C₈	Dic₁₄₂	1	2	yes	—
118	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	142	2	no	—
122	C₄⋊C₈	—	142	2	no	—
123	C₄×C₈	—	71	1	no	—
124	C₄×C₈	—	71	1	no	—
125	C₄×C₈	—	71	1	no	—
126	C₄×C₈	—	71	1	no	—
127	C₄×C₈	—	71	2	no	—
128	C₄×C₈	—	71	2	no	—
129	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
130	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
131	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
132	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
133	C₂²×C₄	Q₈×D₇₁	1	1	yes	characteristic
134	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
135	M₁₆	—	4	4	no	—
136	M₁₆	—	284	4	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	284	2	no	—
140	C₂×C₈	—	71	2	no	—
141	C₂×C₈	—	71	2	no	—
142	C₂×C₈	—	71	2	no	—
143	C₂×C₈	—	71	2	no	—
144	C₂×C₈	—	142	4	no	—
145	C₂×C₈	—	142	2	no	—
146	C₂×C₈	—	142	2	no	—
147	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
148	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
149	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
150	C₄²	D₄×D₇₁	1	1	yes	characteristic
151	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
152	C₄²	Q₈×D₇₁	1	1	yes	characteristic
153	C₂³	#14556127	1	1	yes	characteristic
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
157	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
158	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
159	C₂×C₄	#14556124	1	1	yes	characteristic
160	C₂×C₄	#14556126	1	1	yes	characteristic
161	C₂×C₄	#14556127	1	1	yes	characteristic
162	C₂×C₄	#14556147	1	2	yes	—
163	C₂×C₄	#14556151	1	2	yes	—
164	C₈	—	2	2	no	—
165	C₈	—	8	2	no	—
166	C₈	—	568	2	no	—
167	C₈	—	71	4	no	—
168	C₈	—	142	2	no	—
169	C₈	—	142	2	no	—
170	C₈	—	142	2	no	—
171	C₈	#14556063	1	4	yes	—
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	#17985868	1	1	yes	characteristic
175	C₂²	#17986400	1	2	yes	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	#17986036	1	2	yes	—
179	C₄	#108×D₇₁	1	2	yes	—
180	C₄	C₂×#14556063	1	2	yes	—
181	C₄	#17986382	1	1	yes	characteristic
182	C₄	#17986384	1	1	yes	characteristic
183	C₂	—	2	2	no	—
184	C₂	#22588503	1	1	yes	characteristic
185	C₂	#22588510	1	1	yes	characteristic
186	C₂	#22588511	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556124
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1567	C₂
socle	C₂²×C₇₁	#384
prime radical	C₂³×C₇₁	C₄⋊Q₈
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1567	D₇₁
hypocenter	C₇₁	#8824
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
99. C₇₁	#8824	semidirect

Maximal subgroups

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

subgroup	index
#22588477	2
#22587099	2
#22587095	2
C₂×C₈×C₇₁⋊C₈	2
C₇₁×#1567	2
#8824	71