Subgroups — #31912849

Order 18176.

310 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590124	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985774	C₂	1	1	yes	maximal, characteristic
3	#22588222	C₂	1	2	yes	maximal
4	#22586437	C₂	1	2	yes	maximal
5	C₂×#368×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
7	#17986037	C₂²	1	2	yes	—
8	#17985957	—	2	2	no	—
9	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
10	#17985827	C₄	1	2	yes	complemented
11	#17985825	C₄	1	2	yes	complemented
12	#17985774	C₄	1	4	yes	complemented
13	#17985760	—	2	2	no	—
14	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×#108×C₇₁	C₂²	1	2	yes	—
16	#368×C₇₁	C₄	1	4	yes	complemented
17	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
18	#14556087	—	2	2	no	—
19	C₂×C₄×Dic₇₁	D₄	1	1	yes	characteristic
20	C₂×C₇₁⋊C₁₆	—	2	1	no	—
21	C₂×C₇₁⋊C₁₆	—	2	1	no	—
22	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
23	#14556062	C₂×C₄	1	4	yes	—
24	#14556060	—	2	2	no	—
25	#14556038	—	4	2	no	—
26	C₈×Dic₇₁	—	4	2	no	—
27	C₈×Dic₇₁	C₂×C₄	1	2	yes	—
28	C₈×Dic₇₁	Q₈	1	2	yes	—
29	C₂×M₁₆×C₇₁	—	2	2	no	—
30	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
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	C₂×C₄	1	2	yes	—
35	#108×C₇₁	—	2	2	no	—
36	#108×C₇₁	C₂×C₄	1	4	yes	—
37	C₂²×Dic₇₁	—	2	1	no	—
38	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
39	C₂×C₇₁⋊C₈	D₈	1	2	yes	—
40	C₂×C₇₁⋊C₈	Q₁₆	1	2	yes	—
41	C₄⋊Dic₇₁	—	2	4	no	—
42	C₂²⋊Dic₇₁	—	4	2	no	—
43	C₄×Dic₇₁	—	2	1	no	—
44	C₄×Dic₇₁	—	4	2	no	—
45	C₄×Dic₇₁	C₄⋊C₄	1	2	yes	—
46	C₇₁⋊C₁₆	—	2	2	no	—
47	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
48	M₁₆×C₇₁	—	2	4	no	—
49	M₁₆×C₇₁	—	4	2	no	—
50	C₂×C₈×C₇₁	—	4	2	no	—
51	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
56	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
57	C₁₆×C₇₁	C₄²	1	2	yes	—
58	C₁₆×C₇₁	C₄²	1	2	yes	—
59	C₂³×C₇₁	#135	1	1	yes	characteristic
60	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
62	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
65	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
66	C₂×Dic₇₁	—	2	2	no	—
67	C₂×Dic₇₁	—	4	2	no	—
68	C₂×Dic₇₁	—	4	4	no	—
69	C₈×C₇₁	—	4	4	no	—
70	C₈×C₇₁	C₂×C₄²	1	1	yes	characteristic
71	C₈×C₇₁	#117	1	1	yes	characteristic
72	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
73	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
74	C₇₁⋊C₈	#107	1	4	yes	—
75	C₂²×C₇₁	—	2	2	no	—
76	C₂²×C₇₁	#427	1	1	yes	characteristic
77	C₂²×C₇₁	C₄×D₈	1	1	yes	characteristic
78	C₂²×C₇₁	C₄×Q₁₆	1	1	yes	characteristic
79	C₄×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
80	C₄×C₇₁	C₂×#107	1	1	yes	characteristic
81	C₄×C₇₁	C₄×D₈	1	1	yes	characteristic
82	C₄×C₇₁	C₄×Q₁₆	1	1	yes	characteristic
83	Dic₇₁	—	4	2	no	—
84	Dic₇₁	—	8	4	no	—
85	#12671	—	71	1	no	maximal, Sylow
86	C₂×C₇₁	—	2	1	no	—
87	C₂×C₇₁	—	2	1	no	—
88	C₂×C₇₁	C₄×#107	1	1	yes	characteristic
89	C₂×C₇₁	#2179	1	2	yes	—
90	C₂×#368	D₇₁	1	1	yes	characteristic
91	#2158	—	71	1	no	—
92	C₂×#346	—	71	1	no	—
93	#1772	—	71	2	no	—
94	#1382	—	71	2	no	—
95	C₇₁	#12671	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₁₆	—	71	1	no	—
97	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
98	C₂×#108	C₂×D₇₁	1	2	yes	—
99	#427	—	71	2	no	—
100	#405	—	142	2	no	—
101	C₂×C₄×C₈	—	71	1	no	—
102	#368	Dic₇₁	1	4	yes	complemented
103	#367	—	71	2	no	—
104	#366	—	71	2	no	—
105	#346	—	71	4	no	—
106	#339	—	142	2	no	—
107	C₂×M₁₆	—	2	2	no	—
108	C₂²×C₈	—	71	1	no	—
109	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
110	#117	—	142	2	no	—
111	C₂×C₄²	—	71	1	no	—
112	C₂×C₁₆	—	71	2	no	—
113	C₂×C₁₆	—	142	1	no	—
114	C₂×C₁₆	—	142	1	no	—
115	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
116	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
117	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
118	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
119	#108	—	2	2	no	—
120	#108	C₂×Dic₇₁	1	4	yes	—
121	#107	—	71	4	no	—
122	#106	—	142	2	no	—
123	#97	—	284	2	no	—
124	C₄×C₈	—	284	2	no	—
125	C₄×C₈	—	71	2	no	—
126	C₄×C₈	—	71	2	no	—
127	C₂²×C₄	—	142	1	no	—
128	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
129	M₁₆	—	2	4	no	—
130	M₁₆	—	4	2	no	—
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	—	71	2	no	—
133	C₂×C₈	—	71	2	no	—
134	C₂×C₈	—	71	2	no	—
135	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
136	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
137	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
138	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
139	C₂×C₈	#11765864	1	1	yes	characteristic
140	C₂×C₈	#11765866	1	1	yes	characteristic
141	C₄⋊C₄	—	142	4	no	—
142	C₂²⋊C₄	—	284	2	no	—
143	C₄²	—	284	2	no	—
144	C₄²	—	71	2	no	—
145	C₄²	—	142	1	no	—
146	C₁₆	—	142	2	no	—
147	C₁₆	C₄×Dic₇₁	1	2	yes	—
148	C₁₆	C₄×Dic₇₁	1	2	yes	—
149	C₂³	#14556151	1	1	yes	characteristic
150	C₂×C₄	—	284	2	no	—
151	C₂×C₄	—	284	4	no	—
152	C₂×C₄	—	142	2	no	—
153	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
154	C₂×C₄	#14556090	1	1	yes	characteristic
155	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
156	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
157	C₂×C₄	Q₁₆×D₇₁	1	1	yes	characteristic
158	C₂×C₄	#14556147	1	1	yes	characteristic
159	C₈	—	4	4	no	—
160	C₈	—	71	4	no	—
161	C₈	C₂×C₄×Dic₇₁	1	1	yes	characteristic
162	C₈	#14556086	1	1	yes	characteristic
163	C₈	#14556093	1	1	yes	characteristic
164	C₈	Q₈×Dic₇₁	1	1	yes	characteristic
165	C₂²	—	2	2	no	—
166	C₂²	#17986036	1	1	yes	characteristic
167	C₂²	#17986081	1	1	yes	characteristic
168	C₂²	Q₁₆×Dic₇₁	1	1	yes	characteristic
169	C₄	—	284	2	no	—
170	C₄	—	568	4	no	—
171	C₄	C₄⋊C₄×Dic₇₁	1	1	yes	characteristic
172	C₄	#107×D₇₁	1	1	yes	characteristic
173	C₄	D₈×Dic₇₁	1	1	yes	characteristic
174	C₄	#17986093	1	1	yes	characteristic
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	#107×Dic₇₁	1	1	yes	characteristic
178	C₂	#22590225	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₇₁
commutator	C₈×C₇₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#368×C₇₁	C₂
socle	C₂²×C₇₁	C₄×D₈
prime radical	C₂³×C₇₁	#135
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#368	D₇₁
hypocenter	C₇₁	#12671
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
10. #17985827	C₄	semidirect
11. #17985825	C₄	semidirect
12. #17985774	C₄	semidirect
16. #368×C₇₁	C₄	semidirect
95. C₇₁	#12671	semidirect
102. #368	Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
#22590124	2
C₂×#17985774	2
#22588222	2
#22586437	2
C₂×#368×C₇₁	2
#12671	71