Subgroups — #31915564

Order 18176.

294 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590128	C₂	1	2	yes	maximal
2	#22588504	C₂	1	1	yes	maximal, characteristic
3	#22588219	C₂	1	1	yes	maximal, characteristic
4	#22586449	C₂	1	2	yes	maximal
5	C₂×#364×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556064	C₂²	1	1	yes	characteristic
7	#17986037	C₂²	1	1	yes	characteristic
8	C₂×#14556062	C₂²	1	2	yes	—
9	C₂×#14556060	—	2	1	no	—
10	#17985954	—	2	1	no	—
11	#17985865	—	2	1	no	—
12	#17985827	—	2	2	no	—
13	#17985825	—	2	2	no	—
14	#17985759	—	2	1	no	—
15	C₂×M₃₂×C₇₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	#364×C₇₁	C₄	1	4	yes	complemented
18	C₂×#11765849	—	2	1	no	—
19	#14556087	—	2	1	no	—
20	C₂×C₄⋊Dic₇₁	—	2	2	no	—
21	#14556064	—	2	1	no	—
22	#14556064	Q₈	1	2	yes	—
23	#14556062	—	2	2	no	—
24	#14556062	—	2	2	no	—
25	#14556062	D₄	1	2	yes	—
26	#14556060	—	2	1	no	—
27	#14556060	—	4	1	no	—
28	#14556060	—	4	1	no	—
29	#14556038	—	4	1	no	—
30	C₈×Dic₇₁	—	4	1	no	—
31	C₇₁⋊#95	—	4	2	no	—
32	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
35	M₃₂×C₇₁	—	2	2	no	—
36	M₃₂×C₇₁	C₂×C₄	1	4	yes	—
37	C₂×C₁₆×C₇₁	—	2	2	no	—
38	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₇₁	Q₈	1	2	yes	—
40	C₂²×Dic₇₁	—	4	2	no	—
41	#11765849	—	2	2	no	—
42	#11765849	—	4	1	no	—
43	C₂×C₇₁⋊C₈	—	4	1	no	—
44	C₄⋊Dic₇₁	—	2	2	no	—
45	C₄⋊Dic₇₁	—	4	2	no	—
46	C₄⋊Dic₇₁	—	4	2	no	—
47	C₂²⋊Dic₇₁	—	4	1	no	—
48	C₄×Dic₇₁	—	4	1	no	—
49	C₂²×C₄×C₇₁	—	2	1	no	—
50	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
52	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
53	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
54	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
55	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
60	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
61	C₄²×C₇₁	—	2	1	no	—
62	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
63	C₁₆×C₇₁	—	2	4	no	—
64	C₂³×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂×C₄×C₇₁	—	2	2	no	—
66	C₂×C₄×C₇₁	—	4	2	no	—
67	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
69	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
73	C₂×Dic₇₁	—	4	2	no	—
74	C₂×Dic₇₁	—	8	4	no	—
75	C₂×Dic₇₁	—	8	2	no	—
76	C₈×C₇₁	#107	1	2	yes	—
77	C₈×C₇₁	#107	1	2	yes	—
78	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
79	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
80	C₈×C₇₁	#125	1	1	yes	characteristic
81	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
82	C₇₁⋊C₈	—	4	2	no	—
83	C₂²×C₇₁	—	2	2	no	—
84	C₂²×C₇₁	#428	1	1	yes	characteristic
85	C₂²×C₇₁	#459	1	1	yes	characteristic
86	C₂²×C₇₁	#477	1	1	yes	characteristic
87	C₄×C₇₁	—	4	2	no	—
88	C₄×C₇₁	#389	1	1	yes	characteristic
89	C₄×C₇₁	C₂×#107	1	1	yes	characteristic
90	C₄×C₇₁	#462	1	1	yes	characteristic
91	C₄×C₇₁	#474	1	1	yes	characteristic
92	Dic₇₁	—	8	2	no	—
93	Dic₇₁	—	16	2	no	—
94	#13180	—	71	1	no	maximal, Sylow
95	C₂×C₇₁	—	2	1	no	—
96	C₂×C₇₁	—	2	1	no	—
97	C₂×C₇₁	#1947	1	1	yes	characteristic
98	C₂×C₇₁	#2212	1	2	yes	—
99	#2159	—	71	2	no	—
100	C₂×#364	D₇₁	1	1	yes	characteristic
101	#1846	—	71	1	no	—
102	#1772	—	71	1	no	—
103	#1387	—	71	2	no	—
104	C₇₁	#13180	1	1	yes	characteristic, complemented, Sylow
105	C₂×M₃₂	C₂×D₇₁	1	2	yes	—
106	C₂×#108	—	71	1	no	—
107	#427	—	71	1	no	—
108	C₂×#107	—	71	2	no	—
109	C₂×#106	—	142	1	no	—
110	#405	—	142	1	no	—
111	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
112	#382	—	142	1	no	—
113	#367	—	142	2	no	—
114	#366	—	142	2	no	—
115	#364	Dic₇₁	1	4	yes	complemented
116	#339	—	142	1	no	—
117	C₂×M₁₆	—	142	1	no	—
118	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
120	#117	—	142	1	no	—
121	C₂×C₄⋊C₄	—	142	2	no	—
122	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
123	M₃₂	—	2	2	no	—
124	M₃₂	C₂×Dic₇₁	1	4	yes	—
125	C₂×C₁₆	—	2	2	no	—
126	#108	—	71	2	no	—
127	#108	—	142	1	no	—
128	#107	—	71	2	no	—
129	#107	—	142	2	no	—
130	#107	—	142	2	no	—
131	#106	—	284	1	no	—
132	#106	—	284	1	no	—
133	#106	—	142	1	no	—
134	#97	—	284	1	no	—
135	C₄×C₈	—	284	1	no	—
136	C₄×C₈	Dic₁₄₂	1	2	yes	—
137	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
138	#95	—	284	2	no	—
139	C₂²×C₄	—	2	1	no	—
140	C₂²×C₄	—	284	2	no	—
141	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
142	M₁₆	—	284	1	no	—
143	M₁₆	—	142	2	no	—
144	C₂×C₈	—	284	1	no	—
145	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
146	C₂×C₈	D₅₆₈	1	1	yes	characteristic
147	C₂×C₈	D₅₆₈	1	1	yes	characteristic
148	C₂×C₈	Dic₂₈₄	1	2	yes	—
149	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
150	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
151	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
152	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
153	C₂×C₈	#11765864	1	1	yes	characteristic
154	C₂×C₈	#11765866	1	1	yes	characteristic
155	C₄⋊C₄	—	284	2	no	—
156	C₄⋊C₄	—	284	2	no	—
157	C₄⋊C₄	—	142	2	no	—
158	C₂²⋊C₄	—	284	1	no	—
159	C₄²	—	2	1	no	—
160	C₄²	—	284	1	no	—
161	C₄²	C₄⋊Dic₇₁	1	2	yes	—
162	C₁₆	—	2	4	no	—
163	C₂³	#14556156	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	284	2	no	—
167	C₂×C₄	—	568	4	no	—
168	C₂×C₄	—	568	2	no	—
169	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
170	C₂×C₄	#14556103	1	1	yes	characteristic
171	C₂×C₄	#14556108	1	1	yes	characteristic
172	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
173	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
174	C₂×C₄	#14556161	1	1	yes	characteristic
175	C₈	—	284	2	no	—
176	C₈	#14556062	1	2	yes	—
177	C₈	#14556062	1	2	yes	—
178	C₈	D₄×Dic₇₁	1	1	yes	characteristic
179	C₈	Q₈×Dic₇₁	1	1	yes	characteristic
180	C₈	#14556119	1	1	yes	characteristic
181	C₈	#14556127	1	1	yes	characteristic
182	C₂²	—	2	2	no	—
183	C₂²	#17986041	1	1	yes	characteristic
184	C₂²	#17986195	1	1	yes	characteristic
185	C₂²	#17986295	1	1	yes	characteristic
186	C₄	—	4	2	no	—
187	C₄	—	568	2	no	—
188	C₄	—	1136	2	no	—
189	C₄	#17985898	1	1	yes	characteristic
190	C₄	C₂×#14556062	1	1	yes	characteristic
191	C₄	#17986210	1	1	yes	characteristic
192	C₄	#17986280	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	#22589099	1	1	yes	characteristic
196	C₂	#22590410	1	2	yes	—

	subgroup	G/H
center	C₂²	#17986195
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#364×C₇₁	C₂
socle	C₂²×C₇₁	#459
prime radical	C₂³×C₇₁	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#364	D₇₁
hypocenter	C₇₁	#13180
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
17. #364×C₇₁	C₄	semidirect
104. C₇₁	#13180	semidirect
115. #364	Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
#22590128	2
#22588504	2
#22588219	2
#22586449	2
C₂×#364×C₇₁	2
#13180	71