Subgroups — #31915569

Order 18176.

310 conjugacy classes of subgroups.

The subgroup lattice has 214 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	#22588500	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	—	2	1	no	—
9	C₂×#14556062	C₂²	1	2	yes	—
10	#17985954	—	2	1	no	—
11	#17985869	—	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	C₂×C₄⋊Dic₇₁	—	4	2	no	—
22	#14556064	—	2	1	no	—
23	#14556064	Q₈	1	2	yes	—
24	#14556062	—	2	2	no	—
25	#14556062	—	2	2	no	—
26	#14556062	—	4	1	no	—
27	#14556062	—	4	1	no	—
28	#14556062	D₄	1	1	yes	characteristic
29	#14556062	D₄	1	1	yes	characteristic
30	#14556060	—	2	1	no	—
31	#14556038	—	4	1	no	—
32	C₈×Dic₇₁	—	4	1	no	—
33	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
36	M₃₂×C₇₁	—	2	2	no	—
37	M₃₂×C₇₁	C₂×C₄	1	4	yes	—
38	C₂×C₁₆×C₇₁	—	2	2	no	—
39	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₇₁	Q₈	1	2	yes	—
41	C₂²×Dic₇₁	—	4	2	no	—
42	#11765849	—	2	2	no	—
43	#11765849	—	4	1	no	—
44	C₂×C₇₁⋊C₈	—	4	1	no	—
45	C₄⋊Dic₇₁	—	2	1	no	—
46	C₄⋊Dic₇₁	—	2	1	no	—
47	C₄⋊Dic₇₁	—	4	2	no	—
48	C₄⋊Dic₇₁	—	4	2	no	—
49	C₄⋊Dic₇₁	—	8	4	no	—
50	C₂²⋊Dic₇₁	—	4	1	no	—
51	C₄×Dic₇₁	—	4	1	no	—
52	C₂²×C₄×C₇₁	—	2	1	no	—
53	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
55	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
56	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
57	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
58	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
63	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
64	C₄²×C₇₁	—	2	1	no	—
65	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
66	C₁₆×C₇₁	—	2	4	no	—
67	C₂³×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₂×C₄×C₇₁	—	2	2	no	—
69	C₂×C₄×C₇₁	—	2	2	no	—
70	C₂×C₄×C₇₁	—	2	2	no	—
71	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
73	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
77	C₂×Dic₇₁	—	4	1	no	—
78	C₂×Dic₇₁	—	4	1	no	—
79	C₂×Dic₇₁	—	8	4	no	—
80	C₂×Dic₇₁	—	8	2	no	—
81	C₈×C₇₁	#107	1	2	yes	—
82	C₈×C₇₁	#107	1	2	yes	—
83	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
84	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
85	C₈×C₇₁	#125	1	1	yes	characteristic
86	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
87	C₇₁⋊C₈	—	4	2	no	—
88	C₂²×C₇₁	—	2	1	no	—
89	C₂²×C₇₁	—	2	1	no	—
90	C₂²×C₇₁	#428	1	1	yes	characteristic
91	C₂²×C₇₁	#459	1	1	yes	characteristic
92	C₂²×C₇₁	#477	1	1	yes	characteristic
93	C₄×C₇₁	—	2	4	no	—
94	C₄×C₇₁	#389	1	1	yes	characteristic
95	C₄×C₇₁	C₂×#107	1	1	yes	characteristic
96	C₄×C₇₁	#462	1	1	yes	characteristic
97	C₄×C₇₁	#474	1	1	yes	characteristic
98	Dic₇₁	—	8	1	no	—
99	Dic₇₁	—	8	1	no	—
100	Dic₇₁	—	16	2	no	—
101	#13181	—	71	1	no	maximal, Sylow
102	C₂×C₇₁	—	2	1	no	—
103	C₂×C₇₁	—	2	1	no	—
104	C₂×C₇₁	#1947	1	1	yes	characteristic
105	C₂×C₇₁	#2213	1	1	yes	characteristic
106	C₂×C₇₁	#2214	1	1	yes	characteristic
107	#2159	—	71	2	no	—
108	C₂×#364	D₇₁	1	1	yes	characteristic
109	#1845	—	71	1	no	—
110	#1772	—	71	1	no	—
111	#1387	—	71	2	no	—
112	C₇₁	#13181	1	1	yes	characteristic, complemented, Sylow
113	C₂×M₃₂	C₂×D₇₁	1	2	yes	—
114	C₂×#108	—	71	1	no	—
115	#427	—	71	1	no	—
116	C₂×#107	—	71	2	no	—
117	C₂×#107	—	142	1	no	—
118	#405	—	142	1	no	—
119	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
120	#384	—	142	1	no	—
121	#367	—	142	2	no	—
122	#366	—	142	2	no	—
123	#364	Dic₇₁	1	4	yes	complemented
124	#339	—	142	1	no	—
125	C₂×M₁₆	—	142	1	no	—
126	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
128	#117	—	142	1	no	—
129	C₂×C₄⋊C₄	—	284	2	no	—
130	C₂×C₄⋊C₄	—	142	2	no	—
131	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
132	M₃₂	—	2	2	no	—
133	M₃₂	C₂×Dic₇₁	1	4	yes	—
134	C₂×C₁₆	—	2	2	no	—
135	#108	—	71	2	no	—
136	#108	—	142	1	no	—
137	#107	—	284	1	no	—
138	#107	—	284	1	no	—
139	#107	—	71	1	no	—
140	#107	—	71	1	no	—
141	#107	—	142	2	no	—
142	#107	—	142	2	no	—
143	#106	—	142	1	no	—
144	#97	—	284	1	no	—
145	C₄×C₈	—	284	1	no	—
146	C₄×C₈	Dic₁₄₂	1	2	yes	—
147	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	—	284	2	no	—
150	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
151	M₁₆	—	284	1	no	—
152	M₁₆	—	142	2	no	—
153	C₂×C₈	—	284	1	no	—
154	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
155	C₂×C₈	D₅₆₈	1	1	yes	characteristic
156	C₂×C₈	D₅₆₈	1	1	yes	characteristic
157	C₂×C₈	Dic₂₈₄	1	2	yes	—
158	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
159	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
160	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
161	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
162	C₂×C₈	#11765864	1	1	yes	characteristic
163	C₂×C₈	#11765866	1	1	yes	characteristic
164	C₄⋊C₄	—	284	2	no	—
165	C₄⋊C₄	—	284	2	no	—
166	C₄⋊C₄	—	568	4	no	—
167	C₄⋊C₄	—	142	1	no	—
168	C₄⋊C₄	—	142	1	no	—
169	C₂²⋊C₄	—	284	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	284	1	no	—
172	C₄²	C₄⋊Dic₇₁	1	2	yes	—
173	C₁₆	—	2	4	no	—
174	C₂³	#14556156	1	1	yes	characteristic
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	284	1	no	—
179	C₂×C₄	—	284	1	no	—
180	C₂×C₄	—	568	4	no	—
181	C₂×C₄	—	568	2	no	—
182	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
183	C₂×C₄	#14556103	1	1	yes	characteristic
184	C₂×C₄	#14556108	1	1	yes	characteristic
185	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
186	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
187	C₂×C₄	#14556161	1	1	yes	characteristic
188	C₈	—	284	2	no	—
189	C₈	#14556062	1	2	yes	—
190	C₈	#14556062	1	2	yes	—
191	C₈	D₄×Dic₇₁	1	1	yes	characteristic
192	C₈	Q₈×Dic₇₁	1	1	yes	characteristic
193	C₈	#14556119	1	1	yes	characteristic
194	C₈	#14556127	1	1	yes	characteristic
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	#17986041	1	1	yes	characteristic
198	C₂²	#17986195	1	1	yes	characteristic
199	C₂²	#17986295	1	1	yes	characteristic
200	C₄	—	2	4	no	—
201	C₄	—	568	1	no	—
202	C₄	—	568	1	no	—
203	C₄	—	1136	2	no	—
204	C₄	#17985898	1	1	yes	characteristic
205	C₄	C₂×#14556062	1	1	yes	characteristic
206	C₄	#17986210	1	1	yes	characteristic
207	C₄	#17986280	1	1	yes	characteristic
208	C₂	—	2	1	no	—
209	C₂	—	2	1	no	—
210	C₂	#22589099	1	1	yes	characteristic
211	C₂	#22590417	1	1	yes	characteristic
212	C₂	#22590422	1	1	yes	characteristic

	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	#13181
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
112. C₇₁	#13181	semidirect
123. #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
#22588500	2
#22588219	2
#22586449	2
C₂×#364×C₇₁	2
#13181	71