Subgroups — #31915512

Order 18176.

274 conjugacy classes of subgroups.

The subgroup lattice has 192 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	C₂×#17985825	C₂	1	1	yes	maximal, characteristic
3	#22589125	C₂	1	2	yes	maximal
4	#22586449	C₂	1	2	yes	maximal
5	C₇₁×#2108	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556064	C₂²	1	2	yes	—
7	#17986042	—	2	2	no	—
8	C₂×#14556062	C₂²	1	2	yes	—
9	#17985829	C₄	1	4	yes	complemented
10	#17985825	—	2	1	no	—
11	#17985825	—	2	1	no	—
12	#17985767	—	2	2	no	—
13	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
15	#405×C₇₁	C₂²	1	1	yes	characteristic
16	#346×C₇₁	—	2	1	no	—
17	C₄×C₁₆×C₇₁	—	2	1	no	—
18	C₂×#11765849	—	2	2	no	—
19	C₂×C₄⋊Dic₇₁	—	2	2	no	—
20	#14556064	—	2	2	no	—
21	#14556064	C₂×C₄	1	4	yes	—
22	#14556062	—	2	2	no	—
23	#14556062	—	2	2	no	—
24	#14556061	—	4	2	no	—
25	#14556059	—	4	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	—	2	1	no	—
30	M₃₂×C₇₁	Q₈	1	2	yes	—
31	C₂×C₁₆×C₇₁	—	2	1	no	—
32	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
33	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₇₁	Q₈	1	2	yes	—
36	#97×C₇₁	—	2	1	no	—
37	C₄×C₈×C₇₁	—	2	1	no	—
38	C₂²×Dic₇₁	—	4	2	no	—
39	#11765849	—	2	4	no	—
40	#11765849	—	4	2	no	—
41	C₂×C₇₁⋊C₈	—	4	2	no	—
42	C₄⋊Dic₇₁	—	4	2	no	—
43	C₄⋊Dic₇₁	—	4	2	no	—
44	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
45	M₁₆×C₇₁	—	2	1	no	—
46	M₁₆×C₇₁	Q₁₆	1	2	yes	—
47	C₂×C₈×C₇₁	—	2	1	no	—
48	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
53	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
54	C₄⋊C₄×C₇₁	D₈	1	1	yes	characteristic
55	C₄⋊C₄×C₇₁	D₈	1	1	yes	characteristic
56	C₂²⋊C₄×C₇₁	—	2	1	no	—
57	C₄²×C₇₁	—	2	1	no	—
58	C₁₆×C₇₁	—	2	2	no	—
59	C₁₆×C₇₁	C₄⋊C₄	1	2	yes	—
60	C₁₆×C₇₁	C₄⋊C₄	1	2	yes	—
61	C₂³×C₇₁	#135	1	1	yes	characteristic
62	C₂×C₄×C₇₁	—	2	1	no	—
63	C₂×C₄×C₇₁	—	2	1	no	—
64	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
65	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
66	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
70	C₂×Dic₇₁	—	8	2	no	—
71	C₂×Dic₇₁	—	16	2	no	—
72	C₈×C₇₁	—	2	2	no	—
73	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
75	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
76	C₈×C₇₁	#125	1	1	yes	characteristic
77	C₇₁⋊C₈	—	4	4	no	—
78	C₂²×C₇₁	—	2	1	no	—
79	C₂²×C₇₁	—	2	1	no	—
80	C₂²×C₇₁	C₄×Q₁₆	1	1	yes	characteristic
81	C₂²×C₇₁	#466	1	1	yes	characteristic
82	C₂²×C₇₁	#499	1	1	yes	characteristic
83	C₄×C₇₁	—	4	1	no	—
84	C₄×C₇₁	—	4	1	no	—
85	C₄×C₇₁	#389	1	1	yes	characteristic
86	C₄×C₇₁	C₄×D₈	1	1	yes	characteristic
87	C₄×C₇₁	#467	1	1	yes	characteristic
88	C₄×C₇₁	#500	1	1	yes	characteristic
89	Dic₇₁	—	16	2	no	—
90	#13171	—	71	1	no	maximal, Sylow
91	C₂×C₇₁	—	2	1	no	—
92	C₂×C₇₁	—	2	1	no	—
93	C₂×C₇₁	#1942	1	1	yes	characteristic
94	C₂×C₇₁	#2221	1	1	yes	characteristic
95	C₂×C₇₁	#2222	1	1	yes	characteristic
96	C₂×#368	—	71	1	no	—
97	C₂×#366	—	71	1	no	—
98	#2108	D₇₁	1	1	yes	characteristic
99	#1952	—	71	2	no	—
100	#1387	—	71	2	no	—
101	C₇₁	#13171	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
103	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
104	C₂×#108	—	71	2	no	—
105	#428	—	142	2	no	—
106	C₂×#107	—	71	2	no	—
107	#405	C₂×D₇₁	1	1	yes	characteristic
108	#368	—	71	4	no	—
109	#366	—	142	1	no	—
110	#366	—	142	1	no	—
111	#346	—	2	1	no	—
112	C₄×C₁₆	—	2	1	no	—
113	#343	—	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	D₂₈₄	1	1	yes	characteristic
118	C₂×C₄⋊C₄	—	142	2	no	—
119	M₃₂	—	2	1	no	—
120	M₃₂	Dic₁₄₂	1	2	yes	—
121	C₂×C₁₆	—	2	1	no	—
122	C₂×C₁₆	Dic₁₄₂	1	2	yes	—
123	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
124	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
125	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
126	#108	—	71	4	no	—
127	#108	—	142	2	no	—
128	#107	—	284	2	no	—
129	#107	—	142	2	no	—
130	#107	—	142	2	no	—
131	#106	—	284	2	no	—
132	#97	—	2	1	no	—
133	C₄×C₈	—	2	1	no	—
134	C₂²×C₄	—	284	2	no	—
135	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
136	M₁₆	—	2	1	no	—
137	M₁₆	—	284	2	no	—
138	M₁₆	—	142	4	no	—
139	M₁₆	Dic₂₈₄	1	2	yes	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	284	2	no	—
142	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
143	C₂×C₈	C₂×C₂²⋊D₇₁	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	#11765865	1	1	yes	characteristic
147	C₂×C₈	#11765865	1	1	yes	characteristic
148	C₄⋊C₄	—	284	2	no	—
149	C₄⋊C₄	—	284	2	no	—
150	C₄⋊C₄	D₅₆₈	1	1	yes	characteristic
151	C₄⋊C₄	D₅₆₈	1	1	yes	characteristic
152	C₂²⋊C₄	—	2	1	no	—
153	C₄²	—	2	1	no	—
154	C₁₆	—	2	2	no	—
155	C₁₆	Dic₇₁⋊C₄	1	2	yes	—
156	C₁₆	Dic₇₁⋊C₄	1	2	yes	—
157	C₂³	#14556150	1	1	yes	characteristic
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	568	2	no	—
161	C₂×C₄	—	1136	2	no	—
162	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
163	C₂×C₄	#14556102	1	1	yes	characteristic
164	C₂×C₄	#14556128	1	1	yes	characteristic
165	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
166	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
167	C₂×C₄	#14556150	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	—	284	4	no	—
170	C₈	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
171	C₈	C₄×Dic₁₄₂	1	1	yes	characteristic
172	C₈	#14556107	1	1	yes	characteristic
173	C₈	#14556120	1	1	yes	characteristic
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	C₄×Dic₂₈₄	1	1	yes	characteristic
177	C₂²	#17986235	1	1	yes	characteristic
178	C₂²	#17986401	1	1	yes	characteristic
179	C₄	—	4	1	no	—
180	C₄	—	4	1	no	—
181	C₄	—	1136	2	no	—
182	C₄	#17985897	1	1	yes	characteristic
183	C₄	C₄×D₅₆₈	1	1	yes	characteristic
184	C₄	#17986240	1	1	yes	characteristic
185	C₄	#17986404	1	1	yes	characteristic
186	C₂	—	2	1	no	—
187	C₂	—	2	1	no	—
188	C₂	#22589073	1	1	yes	characteristic
189	C₂	#22590460	1	1	yes	characteristic
190	C₂	#22590465	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17986235
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#2108	C₂
socle	C₂²×C₇₁	#466
prime radical	C₂³×C₇₁	#135
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#2108	D₇₁
hypocenter	C₇₁	#13171
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
9. #17985829	C₄	semidirect
101. C₇₁	#13171	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
C₂×#17985825	2
#22589125	2
#22586449	2
C₇₁×#2108	2
#13171	71