Subgroups — #31913908

Order 18176.

266 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	C₂×#17985829	C₂	1	1	yes	maximal, characteristic
2	#22590124	C₂	1	1	yes	maximal, characteristic
3	#22590113	C₂	1	2	yes	maximal
4	#22588464	C₂	1	2	yes	maximal
5	C₇₁×#2108	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556064	C₂²	1	2	yes	—
7	#17986037	C₂²	1	2	yes	—
8	#17985829	—	2	1	no	—
9	#17985829	—	2	1	no	—
10	#17985827	—	2	1	no	—
11	#17985825	—	2	1	no	—
12	#17985822	—	2	4	no	—
13	#17985770	—	2	2	no	—
14	#17985755	—	2	2	no	—
15	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
17	#405×C₇₁	C₂²	1	1	yes	characteristic
18	#346×C₇₁	C₄	1	2	yes	complemented
19	C₄×C₁₆×C₇₁	C₄	1	2	yes	complemented
20	C₂×#11765849	—	2	2	no	—
21	#14556087	—	2	2	no	—
22	#14556064	—	2	2	no	—
23	#14556064	—	2	2	no	—
24	#14556062	—	2	2	no	—
25	#14556060	—	2	2	no	—
26	C₂×M₁₆×C₇₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
28	#117×C₇₁	D₄	1	1	yes	characteristic
29	M₃₂×C₇₁	D₄	1	2	yes	—
30	M₃₂×C₇₁	Q₈	1	2	yes	—
31	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
34	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
36	C₂×C₁₆×C₇₁	Q₈	1	1	yes	characteristic
37	C₂×C₁₆×C₇₁	Q₈	1	1	yes	characteristic
38	#97×C₇₁	Q₈	1	2	yes	—
39	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
40	#11765849	—	4	2	no	—
41	#11765849	—	4	2	no	—
42	C₂×C₇₁⋊C₈	—	4	2	no	—
43	C₄⋊Dic₇₁	—	4	2	no	—
44	C₂²⋊Dic₇₁	—	4	2	no	—
45	C₄×Dic₇₁	—	4	2	no	—
46	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
47	M₁₆×C₇₁	—	2	1	no	—
48	M₁₆×C₇₁	—	2	1	no	—
49	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
50	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
56	C₄⋊C₄×C₇₁	—	2	1	no	—
57	C₂²⋊C₄×C₇₁	—	2	1	no	—
58	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
59	C₁₆×C₇₁	C₄⋊C₄	1	1	yes	characteristic
60	C₁₆×C₇₁	C₄⋊C₄	1	1	yes	characteristic
61	C₁₆×C₇₁	C₄⋊C₄	1	1	yes	characteristic
62	C₁₆×C₇₁	C₄⋊C₄	1	1	yes	characteristic
63	C₁₆×C₇₁	C₄⋊C₄	1	4	yes	—
64	C₂³×C₇₁	#135	1	1	yes	characteristic
65	C₂×C₄×C₇₁	—	2	2	no	—
66	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
68	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
70	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
71	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
72	C₂×Dic₇₁	—	8	2	no	—
73	C₈×C₇₁	—	2	2	no	—
74	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
77	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
78	C₇₁⋊C₈	—	8	2	no	—
79	C₂²×C₇₁	—	4	1	no	—
80	C₂²×C₇₁	#427	1	1	yes	characteristic
81	C₂²×C₇₁	#495	1	1	yes	characteristic
82	C₂²×C₇₁	#499	1	1	yes	characteristic
83	C₄×C₇₁	—	4	2	no	—
84	C₄×C₇₁	#384	1	1	yes	characteristic
85	C₄×C₇₁	#427	1	1	yes	characteristic
86	C₄×C₇₁	#495	1	1	yes	characteristic
87	C₄×C₇₁	#499	1	1	yes	characteristic
88	Dic₇₁	—	16	2	no	—
89	#12911	—	71	1	no	maximal, Sylow
90	C₂×C₇₁	—	2	1	no	—
91	C₂×C₇₁	—	2	1	no	—
92	C₂×C₇₁	—	2	1	no	—
93	C₂×C₇₁	#1846	1	1	yes	characteristic
94	C₂×#368	—	71	1	no	—
95	#2158	—	71	1	no	—
96	#2155	—	71	2	no	—
97	#2108	D₇₁	1	1	yes	characteristic
98	#1833	—	71	2	no	—
99	C₇₁	#12911	1	1	yes	characteristic, complemented, Sylow
100	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
101	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
102	C₂×#108	—	71	2	no	—
103	#427	—	71	2	no	—
104	#405	C₂×D₇₁	1	1	yes	characteristic
105	#368	—	142	1	no	—
106	#368	—	142	1	no	—
107	#367	—	142	1	no	—
108	#366	—	142	1	no	—
109	#365	—	142	4	no	—
110	#346	Dic₇₁	1	2	yes	complemented
111	C₄×C₁₆	Dic₇₁	1	2	yes	complemented
112	#344	—	142	2	no	—
113	H(3,ℤ/4)	—	142	2	no	—
114	C₂×M₁₆	—	142	2	no	—
115	C₂×M₁₆	D₂₈₄	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
117	#117	—	142	2	no	—
118	#117	D₂₈₄	1	1	yes	characteristic
119	M₃₂	Dic₁₄₂	1	2	yes	—
120	M₃₂	D₂₈₄	1	2	yes	—
121	C₂×C₁₆	Dic₁₄₂	1	1	yes	characteristic
122	C₂×C₁₆	Dic₁₄₂	1	1	yes	characteristic
123	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
124	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
125	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
126	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
127	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
128	#108	—	142	2	no	—
129	#108	—	142	2	no	—
130	#107	—	142	2	no	—
131	#106	—	142	2	no	—
132	#97	Dic₁₄₂	1	2	yes	—
133	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
134	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
135	M₁₆	—	2	1	no	—
136	M₁₆	—	2	1	no	—
137	M₁₆	—	284	2	no	—
138	M₁₆	—	284	2	no	—
139	C₂×C₈	—	284	2	no	—
140	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
141	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
142	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
143	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
144	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
145	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
146	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
147	C₄⋊C₄	—	2	1	no	—
148	C₄⋊C₄	—	284	2	no	—
149	C₂²⋊C₄	—	2	1	no	—
150	C₂²⋊C₄	—	284	2	no	—
151	C₄²	—	284	2	no	—
152	C₄²	C₄⋊Dic₇₁	1	2	yes	—
153	C₁₆	C₄⋊Dic₇₁	1	1	yes	characteristic
154	C₁₆	C₄⋊Dic₇₁	1	1	yes	characteristic
155	C₁₆	C₄⋊Dic₇₁	1	1	yes	characteristic
156	C₁₆	C₄⋊Dic₇₁	1	1	yes	characteristic
157	C₁₆	C₄⋊Dic₇₁	1	4	yes	—
158	C₂³	#14556150	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	568	2	no	—
161	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
162	C₂×C₄	#14556125	1	1	yes	characteristic
163	C₂×C₄	#14556128	1	1	yes	characteristic
164	C₂×C₄	#14556150	1	1	yes	characteristic
165	C₂×C₄	#14556150	1	1	yes	characteristic
166	C₂×C₄	#14556150	1	1	yes	characteristic
167	C₈	—	2	2	no	—
168	C₈	—	568	2	no	—
169	C₈	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
170	C₈	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
171	C₈	#14556128	1	1	yes	characteristic
172	C₈	#14556128	1	1	yes	characteristic
173	C₂²	—	4	1	no	—
174	C₂²	#17986037	1	1	yes	characteristic
175	C₂²	#17986383	1	1	yes	characteristic
176	C₂²	#17986401	1	1	yes	characteristic
177	C₄	—	4	2	no	—
178	C₄	—	1136	2	no	—
179	C₄	#17985869	1	1	yes	characteristic
180	C₄	#17986037	1	1	yes	characteristic
181	C₄	#17986383	1	1	yes	characteristic
182	C₄	#17986401	1	1	yes	characteristic
183	C₂	—	2	1	no	—
184	C₂	—	2	1	no	—
185	C₂	—	2	1	no	—
186	C₂	#22588504	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#17986037
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#2108	C₂
socle	C₂×C₇₁	#1846
prime radical	C₂³×C₇₁	#135
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#2108	D₇₁
hypocenter	C₇₁	#12911
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
18. #346×C₇₁	C₄	semidirect
19. C₄×C₁₆×C₇₁	C₄	semidirect
99. C₇₁	#12911	semidirect
110. #346	Dic₇₁	semidirect
111. C₄×C₁₆	Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17985829	2
#22590124	2
#22590113	2
#22588464	2
C₇₁×#2108	2
#12911	71