Finite Group Databasebeta

Subgroups — #31917958

Order 18176.

194 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17985832	C₂	1	1	yes	maximal, characteristic
2	#22590121	C₂	1	1	yes	maximal, characteristic
3	#366×D₇₁	C₂	1	1	yes	maximal, characteristic
4	#22586541	C₂	1	1	yes	maximal, characteristic, complemented
5	#22586540	C₂	1	1	yes	maximal, characteristic, complemented
6	#22586539	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#1424	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₆×D₇₁	C₂²	1	1	yes	characteristic
9	#17986036	—	2	1	no	—
10	#107×D₇₁	—	2	1	no	—
11	#17985832	C₄	1	4	yes	complemented
12	C₂×C₇₁⋊C₃₂	C₂²	1	1	yes	characteristic
13	#17985826	—	2	1	no	—
14	#17985825	C₂²	1	1	yes	characteristic
15	#17985825	C₂²	1	1	yes	characteristic
16	#17985824	—	2	1	no	—
17	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
18	#366×C₇₁	C₂²	1	1	yes	characteristic
19	#366×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₇₁	D₄	1	1	yes	characteristic
21	#14556086	—	4	1	no	—
22	C₄⋊C₄×D₇₁	—	4	1	no	—
23	C₂×C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
24	C₁₆×D₇₁	C₂×C₄	1	2	yes	—
25	C₁₆×D₇₁	Q₈	1	2	yes	—
26	#14556062	—	2	1	no	—
27	#14556062	—	2	1	no	—
28	#14556061	—	4	1	no	—
29	#14556059	—	4	1	no	—
30	C₇₁⋊C₃₂	C₂×C₄	1	2	yes	complemented
31	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
32	#107×C₇₁	—	2	1	no	—
33	#107×C₇₁	—	2	1	no	—
34	C₃₂×C₇₁	C₂×C₄	1	2	yes	complemented
35	C₂×C₄×D₇₁	—	8	1	no	—
36	C₂×C₄×D₇₁	D₈	1	1	yes	characteristic
37	C₂×C₇₁⋊C₈	D₈	1	1	yes	characteristic
38	C₈×D₇₁	C₄⋊C₄	1	2	yes	—
39	C₈×D₇₁	Q₁₆	1	2	yes	—
40	C₄⋊Dic₇₁	—	4	1	no	—
41	C₄⋊Dic₇₁	—	4	1	no	—
42	Dic₇₁⋊C₄	—	8	1	no	—
43	(C₂×D₇₁)⋊C₄	—	8	1	no	—
44	C₄×Dic₇₁	—	8	1	no	—
45	C₇₁⋊C₁₆	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
47	C₄⋊C₄×C₇₁	—	4	1	no	—
48	C₄⋊C₄×C₇₁	—	4	1	no	—
49	C₁₆×C₇₁	C₂²×C₄	1	1	yes	characteristic
50	C₁₆×C₇₁	C₂×Q₈	1	1	yes	characteristic
51	C₂²×D₇₁	D₁₆	1	1	yes	characteristic
52	C₂×C₄×C₇₁	—	8	1	no	—
53	C₂×C₄×C₇₁	—	8	1	no	—
54	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
55	C₂×Dic₇₁	—	8	1	no	—
56	C₂×Dic₇₁	—	8	1	no	—
57	C₂×Dic₇₁	D₁₆	1	1	yes	characteristic
58	C₄×D₇₁	—	16	2	no	—
59	C₄×D₇₁	#107	1	2	yes	—
60	C₄×D₇₁	Q₃₂	1	2	yes	—
61	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₈×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
63	C₇₁⋊C₈	#107	1	2	yes	—
64	C₂²×C₇₁	C₂×D₁₆	1	1	yes	characteristic
65	C₂×D₇₁	—	2	1	no	—
66	C₂×D₇₁	—	2	1	no	—
67	C₂×D₇₁	#366	1	2	yes	—
68	C₄×C₇₁	—	16	1	no	—
69	C₄×C₇₁	—	16	1	no	—
70	C₄×C₇₁	C₂×#107	1	1	yes	characteristic
71	C₄×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
72	Dic₇₁	—	16	1	no	—
73	Dic₇₁	—	16	1	no	—
74	Dic₇₁	#366	1	2	yes	—
75	#13634	—	71	1	no	maximal, Sylow
76	C₂×C₇₁	C₂×#366	1	1	yes	characteristic
77	C₂×C₇₁	#2263	1	1	yes	characteristic
78	C₂×C₇₁	#2264	1	1	yes	characteristic
79	D₇₁	—	2	2	no	—
80	C₂×M₆₄	—	71	1	no	—
81	#2158	—	71	1	no	—
82	C₂×#366	—	71	1	no	—
83	#1424	—	71	1	no	—
84	#1424	D₇₁	1	1	yes	characteristic, complemented
85	#1423	—	71	1	no	—
86	#1423	—	71	1	no	—
87	C₇₁	#13634	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₁₆	—	71	1	no	—
89	#427	—	142	1	no	—
90	C₂×#107	—	142	1	no	—
91	M₆₄	—	71	4	no	—
92	C₂×C₃₂	—	71	1	no	—
93	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
94	#367	—	142	1	no	—
95	#366	—	71	1	no	—
96	#366	—	71	1	no	—
97	#366	—	142	1	no	—
98	#366	C₂×D₇₁	1	1	yes	characteristic
99	#366	C₂×D₇₁	1	1	yes	characteristic
100	C₂²×C₈	—	71	1	no	—
101	#117	—	284	1	no	—
102	C₂×C₄⋊C₄	—	284	1	no	—
103	C₂×C₁₆	—	71	1	no	—
104	C₂×C₁₆	—	71	2	no	—
105	C₂×C₁₆	—	71	2	no	—
106	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
107	#107	—	2	1	no	—
108	#107	—	2	1	no	—
109	#107	—	284	1	no	—
110	#107	—	142	1	no	—
111	#107	—	142	1	no	—
112	#106	—	284	1	no	—
113	C₃₂	—	71	2	no	—
114	C₃₂	C₄×D₇₁	1	2	yes	complemented
115	C₂²×C₄	—	568	1	no	—
116	C₂²×C₄	—	71	1	no	—
117	C₂×C₈	—	71	1	no	—
118	C₂×C₈	—	71	2	no	—
119	C₂×C₈	—	71	2	no	—
120	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
121	C₄⋊C₄	—	4	1	no	—
122	C₄⋊C₄	—	4	1	no	—
123	C₄⋊C₄	—	284	1	no	—
124	C₄⋊C₄	—	284	1	no	—
125	C₄⋊C₄	—	568	1	no	—
126	C₂²⋊C₄	—	568	1	no	—
127	C₄²	—	568	1	no	—
128	C₁₆	—	71	2	no	—
129	C₁₆	C₂×C₄×D₇₁	1	1	yes	characteristic
130	C₁₆	Q₈×D₇₁	1	1	yes	characteristic
131	C₂³	—	71	1	no	—
132	C₂×C₄	—	8	1	no	—
133	C₂×C₄	—	8	1	no	—
134	C₂×C₄	—	568	1	no	—
135	C₂×C₄	—	568	1	no	—
136	C₂×C₄	—	71	1	no	—
137	C₂×C₄	—	71	2	no	—
138	C₂×C₄	—	71	2	no	—
139	C₂×C₄	—	1136	2	no	—
140	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
141	C₈	—	71	2	no	—
142	C₈	C₄⋊C₄×D₇₁	1	1	yes	characteristic
143	C₈	Q₁₆×D₇₁	1	1	yes	characteristic
144	C₂²	—	71	2	no	—
145	C₂²	—	142	1	no	—
146	C₂²	—	142	1	no	—
147	C₂²	D₁₆×D₇₁	1	1	yes	characteristic
148	C₄	—	16	1	no	—
149	C₄	—	16	1	no	—
150	C₄	—	71	2	no	—
151	C₄	—	1136	1	no	—
152	C₄	—	1136	1	no	—
153	C₄	#107×D₇₁	1	1	yes	characteristic
154	C₄	Q₃₂×D₇₁	1	1	yes	characteristic
155	C₂	—	142	2	no	—
156	C₂	#366×D₇₁	1	1	yes	characteristic
157	C₂	#22590654	1	1	yes	characteristic
158	C₂	#22590656	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₇₁
commutator	C₁₆×C₇₁	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₇₁
Fitting	C₇₁×#1424	C₂
socle	C₂²×C₇₁	C₂×D₁₆
prime radical	C₂²×D₇₁	D₁₆
cosocle	C₂×C₁₆×C₇₁	C₂³
hypercenter	#1424	D₇₁
hypocenter	C₇₁	#13634
LCS	G ▷ C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1424

Complements

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

N	H ≅ G/N	type
4. #22586541	C₂	semidirect
5. #22586540	C₂	semidirect
6. #22586539	C₂	semidirect
7. C₇₁×#1424	C₂	semidirect
11. #17985832	C₄	semidirect
30. C₇₁⋊C₃₂	C₂×C₄	semidirect
34. C₃₂×C₇₁	C₂×C₄	semidirect
84. #1424	D₇₁	semidirect
87. C₇₁	#13634	semidirect
114. C₃₂	C₄×D₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17985832	2
#22590121	2
#366×D₇₁	2
#22586541	2
#22586540	2
#22586539	2
C₇₁×#1424	2
#13634	71