Subgroups — #31913885

Order 18176.

374 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17985824	C₂	1	4	yes	maximal
2	C₂×C₄×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#22588500	C₂	1	1	yes	maximal, characteristic
4	C₇₁×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14556062	—	2	1	no	—
7	C₂×#14556062	C₂²	1	2	yes	—
8	#17985869	—	2	1	no	—
9	#17985824	—	2	8	no	—
10	C₄×C₇₁⋊C₁₆	C₄	1	4	yes	complemented
11	C₂×#107×C₇₁	—	2	1	no	—
12	C₂×#107×C₇₁	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	#384×C₇₁	—	2	1	no	—
15	C₂×C₄⋊Dic₇₁	—	2	2	no	—
16	C₂×C₄⋊Dic₇₁	—	4	2	no	—
17	C₂×C₇₁⋊C₁₆	C₂×C₄	1	4	yes	—
18	C₂×C₇₁⋊C₁₆	D₄	1	4	yes	—
19	C₂×C₇₁⋊C₁₆	Q₈	1	4	yes	—
20	#14556062	—	2	4	no	—
21	#14556062	—	4	2	no	—
22	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
25	C₂×C₄⋊C₄×C₇₁	—	4	2	no	—
26	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
27	#107×C₇₁	—	2	4	no	—
28	#107×C₇₁	—	4	2	no	—
29	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₇₁	Q₈	1	2	yes	—
31	C₂²×Dic₇₁	—	4	2	no	—
32	C₄⋊Dic₇₁	—	4	2	no	—
33	C₄⋊Dic₇₁	—	4	2	no	—
34	C₄⋊Dic₇₁	—	8	4	no	—
35	C₇₁⋊C₁₆	C₄⋊C₄	1	8	yes	complemented
36	C₂²×C₄×C₇₁	—	4	2	no	—
37	C₂²×C₄×C₇₁	D₈	1	2	yes	—
38	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
40	C₂×C₈×C₇₁	D₈	1	2	yes	—
41	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
42	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
44	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
45	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
46	C₄⋊C₄×C₇₁	—	4	2	no	—
47	C₄⋊C₄×C₇₁	—	4	2	no	—
48	C₄⋊C₄×C₇₁	—	8	4	no	—
49	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
50	C₄²×C₇₁	Q₁₆	1	2	yes	—
51	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₇₁	—	8	2	no	—
53	C₂×C₄×C₇₁	—	8	4	no	—
54	C₂×C₄×C₇₁	#107	1	4	yes	—
55	C₂×C₄×C₇₁	D₁₆	1	4	yes	—
56	C₂×C₄×C₇₁	Q₃₂	1	4	yes	—
57	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
59	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
62	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×Dic₇₁	—	8	2	no	—
64	C₂×Dic₇₁	—	8	4	no	—
65	C₈×C₇₁	#107	1	4	yes	—
66	C₈×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
67	C₈×C₇₁	C₄⋊Q₈	1	2	yes	—
68	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
69	C₂²×C₇₁	#493	1	1	yes	characteristic
70	C₂²×C₇₁	#500	1	1	yes	characteristic
71	C₂²×C₇₁	C₂×D₁₆	1	2	yes	—
72	C₂²×C₇₁	C₂×Q₃₂	1	2	yes	—
73	C₄×C₇₁	—	16	2	no	—
74	C₄×C₇₁	#366	1	8	yes	complemented
75	C₄×C₇₁	#384	1	1	yes	characteristic
76	C₄×C₇₁	C₂×#107	1	1	yes	characteristic
77	C₄×C₇₁	#494	1	1	yes	characteristic
78	C₄×C₇₁	#500	1	1	yes	characteristic
79	Dic₇₁	—	16	2	no	—
80	#12905	—	71	1	no	maximal, Sylow
81	C₂×C₇₁	#1845	1	1	yes	characteristic
82	C₂×C₇₁	C₂×#366	1	2	yes	—
83	C₂×C₇₁	#2246	1	1	yes	characteristic
84	C₂×C₇₁	#2247	1	1	yes	characteristic
85	C₂×C₇₁	#2252	1	2	yes	—
86	C₂×#366	—	71	4	no	—
87	C₂×C₄×C₁₆	—	71	1	no	—
88	#1845	—	71	1	no	—
89	#1845	D₇₁	1	1	yes	characteristic
90	C₇₁	#12905	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₁₆	—	71	2	no	—
92	C₂×#107	—	2	1	no	—
93	C₂×#107	—	71	2	no	—
94	C₂×#107	—	142	1	no	—
95	C₂×#107	C₂×D₇₁	1	2	yes	—
96	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
97	#384	—	2	1	no	—
98	#384	—	142	1	no	—
99	#366	—	142	8	no	—
100	C₄×C₁₆	—	71	4	no	—
101	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
103	C₂×C₄⋊C₄	—	2	2	no	—
104	C₂×C₄⋊C₄	—	4	2	no	—
105	C₂×C₄⋊C₄	—	284	2	no	—
106	C₂×C₄⋊C₄	—	142	2	no	—
107	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
108	C₂×C₁₆	—	71	4	no	—
109	C₂×C₁₆	—	71	4	no	—
110	C₂×C₁₆	—	71	4	no	—
111	#107	—	2	4	no	—
112	#107	—	4	2	no	—
113	#107	—	284	2	no	—
114	#107	—	142	4	no	—
115	C₄×C₈	Dic₁₄₂	1	2	yes	—
116	C₄×C₈	C₄×D₇₁	1	2	yes	—
117	C₂²×C₄	—	4	2	no	—
118	C₂²×C₄	—	284	2	no	—
119	C₂²×C₄	#11765850	1	2	yes	—
120	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
121	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
122	C₂×C₈	#11765850	1	2	yes	—
123	C₂×C₈	#11765855	1	2	yes	—
124	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
125	C₂×C₈	D₄×D₇₁	1	2	yes	—
126	C₂×C₈	Q₈×D₇₁	1	2	yes	—
127	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
128	C₄⋊C₄	—	4	2	no	—
129	C₄⋊C₄	—	4	2	no	—
130	C₄⋊C₄	—	8	4	no	—
131	C₄⋊C₄	—	284	2	no	—
132	C₄⋊C₄	—	284	2	no	—
133	C₄⋊C₄	—	568	4	no	—
134	C₄²	Dic₇₁⋊C₄	1	2	yes	—
135	C₄²	#11765855	1	2	yes	—
136	C₁₆	—	71	8	no	—
137	C₂³	C₂×#11765850	1	1	yes	characteristic
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	8	4	no	—
140	C₂×C₄	—	568	2	no	—
141	C₂×C₄	—	568	4	no	—
142	C₂×C₄	#14556061	1	4	yes	—
143	C₂×C₄	#14556069	1	4	yes	—
144	C₂×C₄	#14556074	1	4	yes	—
145	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
146	C₂×C₄	#14556124	1	1	yes	characteristic
147	C₂×C₄	#14556126	1	1	yes	characteristic
148	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
149	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
150	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
151	C₈	#14556061	1	4	yes	—
152	C₈	C₄⋊C₄×D₇₁	1	2	yes	—
153	C₈	#14556127	1	2	yes	—
154	C₂²	C₂×#14556061	1	1	yes	characteristic
155	C₂²	#17986375	1	1	yes	characteristic
156	C₂²	#17986402	1	1	yes	characteristic
157	C₂²	C₂×#14556069	1	2	yes	—
158	C₂²	C₂×#14556074	1	2	yes	—
159	C₄	—	16	2	no	—
160	C₄	—	1136	2	no	—
161	C₄	#17985824	1	8	yes	complemented
162	C₄	#17985868	1	1	yes	characteristic
163	C₄	C₂×#14556061	1	1	yes	characteristic
164	C₄	#17986378	1	1	yes	characteristic
165	C₄	#17986402	1	1	yes	characteristic
166	C₂	#22588499	1	1	yes	characteristic
167	C₂	C₂×#17985824	1	2	yes	—
168	C₂	#22590589	1	1	yes	characteristic
169	C₂	#22590592	1	1	yes	characteristic
170	C₂	#22590613	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11765850
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1845	C₂
socle, prime radical	C₂³×C₇₁	C₂×D₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1845	D₇₁
hypocenter	C₇₁	#12905
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

Complements

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

N	H ≅ G/N	type
10. C₄×C₇₁⋊C₁₆	C₄	semidirect
35. C₇₁⋊C₁₆	C₄⋊C₄	semidirect
74. C₄×C₇₁	#366	semidirect
90. C₇₁	#12905	semidirect
161. C₄	#17985824	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17985824	2
C₂×C₄×C₇₁⋊C₁₆	2
#22588500	2
C₇₁×#1845	2
#12905	71