Subgroups — #32152969

Order 18176.

474 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22598879	C₂	1	2	yes	maximal
2	#22590626	C₂	1	4	yes	maximal, complemented
3	#22590622	C₂	1	2	yes	maximal, complemented
4	#22590619	C₂	1	2	yes	maximal, complemented
5	#22590128	C₂	1	2	yes	maximal
6	#22590124	C₂	1	1	yes	maximal, characteristic
7	C₂×#17985827	C₂	1	1	yes	maximal, characteristic
8	C₇₁×#2108	C₂	1	1	yes	maximal, characteristic
9	#17986627	—	2	2	no	—
10	#17986408	—	2	4	no	—
11	#17986404	C₂²	1	4	yes	—
12	#17986401	C₂²	1	4	yes	—
13	#17986041	—	2	2	no	—
14	#17986037	C₂²	1	2	yes	—
15	C₂×#14556062	C₂²	1	2	yes	—
16	#17985827	C₂²	1	1	yes	characteristic
17	#17985827	C₂²	1	2	yes	—
18	#17985827	C₂²	1	1	yes	characteristic
19	#17985827	C₂²	1	4	yes	—
20	#17985827	C₂²	1	2	yes	—
21	#17985825	C₂²	1	4	yes	—
22	#17985825	C₂²	1	2	yes	—
23	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
25	#405×C₇₁	C₂²	1	1	yes	characteristic
26	#346×C₇₁	C₂²	1	2	yes	—
27	C₄×C₁₆×C₇₁	C₂²	1	2	yes	—
28	#14556128	—	2	4	no	—
29	#14556127	—	4	2	no	—
30	#14556120	—	2	4	no	—
31	#14556119	—	4	2	no	—
32	#14556105	—	4	4	no	—
33	#14556087	—	2	2	no	—
34	C₂×C₄⋊Dic₇₁	—	2	2	no	—
35	#14556062	—	2	4	no	—
36	#14556062	—	2	2	no	—
37	#14556062	C₂³	1	2	yes	—
38	#14556062	C₂³	1	4	yes	—
39	#14556062	C₂³	1	2	yes	—
40	#14556060	—	2	4	no	—
41	#14556060	—	2	2	no	—
42	C₂×M₁₆×C₇₁	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
44	#117×C₇₁	D₄	1	1	yes	characteristic
45	M₃₂×C₇₁	Q₈	1	4	yes	complemented
46	C₂×C₁₆×C₇₁	Q₈	1	4	yes	complemented
47	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
49	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
50	#97×C₇₁	D₄	1	2	yes	—
51	C₄×C₈×C₇₁	C₂³	1	2	yes	—
52	C₂×Dic₁₄₂	—	4	4	no	—
53	C₂²×Dic₇₁	—	4	2	no	—
54	C₄⋊Dic₇₁	—	2	2	no	—
55	C₄⋊Dic₇₁	—	2	4	no	—
56	C₄⋊Dic₇₁	—	2	2	no	—
57	C₄⋊Dic₇₁	—	4	2	no	—
58	C₄⋊Dic₇₁	—	4	4	no	—
59	Dic₇₁⋊C₄	—	4	4	no	—
60	Dic₇₁⋊C₄	—	4	4	no	—
61	C₂²⋊Dic₇₁	—	4	2	no	—
62	C₄×Dic₇₁	—	4	2	no	—
63	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
64	M₁₆×C₇₁	Q₁₆	1	4	yes	—
65	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
68	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
69	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
70	C₄⋊C₄×C₇₁	D₈	1	2	yes	—
71	C₂²⋊C₄×C₇₁	D₈	1	2	yes	—
72	C₄²×C₇₁	C₂×D₄	1	2	yes	—
73	C₁₆×C₇₁	C₂×Q₈	1	2	yes	—
74	C₁₆×C₇₁	C₂×Q₈	1	2	yes	—
75	C₁₆×C₇₁	C₂×Q₈	1	4	yes	—
76	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₇₁	C₄⋊Q₈	1	2	yes	—
78	C₂×C₄×C₇₁	C₂×D₈	1	4	yes	—
79	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
81	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
82	C₂×Dic₇₁	—	4	2	no	—
83	C₂×Dic₇₁	—	4	2	no	—
84	C₂×Dic₇₁	—	4	4	no	—
85	C₂×Dic₇₁	—	16	2	no	—
86	Dic₁₄₂	—	16	4	no	—
87	C₈×C₇₁	C₄⋊Q₈	1	2	yes	—
88	C₈×C₇₁	C₂×Q₁₆	1	4	yes	—
89	C₈×C₇₁	C₂²×Q₈	1	1	yes	characteristic
90	C₈×C₇₁	C₂²×Q₈	1	1	yes	characteristic
91	C₂²×C₇₁	—	2	2	no	—
92	C₂²×C₇₁	#500	1	2	yes	—
93	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
94	C₄×C₇₁	—	2	4	no	—
95	C₄×C₇₁	#500	1	2	yes	—
96	C₄×C₇₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
97	C₄×C₇₁	C₂²×Q₁₆	1	1	yes	characteristic
98	Dic₇₁	—	8	2	no	—
99	Dic₇₁	—	8	4	no	—
100	Dic₇₁	—	8	2	no	—
101	#33730	—	71	1	no	maximal, Sylow
102	C₂×C₇₁	—	2	2	no	—
103	C₂×C₇₁	C₂×#500	1	1	yes	characteristic
104	C₂×C₇₁	#3416	1	2	yes	—
105	#3164	—	71	2	no	—
106	#2255	—	71	4	no	—
107	#2254	—	71	2	no	—
108	#2253	—	71	2	no	—
109	#2159	—	71	2	no	—
110	#2158	—	71	1	no	—
111	C₂×#367	—	71	1	no	—
112	#2108	D₇₁	1	1	yes	characteristic
113	C₇₁	#33730	1	1	yes	characteristic, complemented, Sylow
114	#544	—	142	2	no	—
115	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
116	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
117	#501	—	142	4	no	—
118	#500	—	71	4	no	—
119	#499	—	71	4	no	—
120	#428	—	142	2	no	—
121	#427	—	71	2	no	—
122	C₂×#107	—	71	2	no	—
123	#405	C₂×D₇₁	1	1	yes	characteristic
124	#367	—	71	2	no	—
125	#367	—	71	2	no	—
126	#367	—	71	1	no	—
127	#367	—	71	1	no	—
128	#367	—	71	4	no	—
129	#366	—	71	2	no	—
130	#366	—	71	4	no	—
131	#346	C₂×D₇₁	1	2	yes	—
132	C₄×C₁₆	C₂×D₇₁	1	2	yes	—
133	C₂×M₁₆	D₂₈₄	1	1	yes	characteristic
134	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
135	C₄⋊Q₈	—	284	2	no	—
136	C₄⋊Q₈	—	142	4	no	—
137	#125	—	284	2	no	—
138	#125	—	142	4	no	—
139	C₂²⋊Q₈	—	284	4	no	—
140	#117	—	142	2	no	—
141	#117	D₂₈₄	1	1	yes	characteristic
142	C₂×C₄⋊C₄	—	142	2	no	—
143	M₃₂	Dic₁₄₂	1	4	yes	complemented
144	C₂×C₁₆	Dic₁₄₂	1	4	yes	complemented
145	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
146	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
147	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
148	#107	—	71	2	no	—
149	#107	—	71	4	no	—
150	#107	—	71	2	no	—
151	#107	—	142	2	no	—
152	#107	—	142	4	no	—
153	#106	—	142	2	no	—
154	#106	—	142	4	no	—
155	#97	D₂₈₄	1	2	yes	—
156	C₄×C₈	C₂²×D₇₁	1	2	yes	—
157	C₂×Q₈	—	284	4	no	—
158	C₂²×C₄	—	284	2	no	—
159	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
160	M₁₆	Dic₂₈₄	1	4	yes	—
161	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
162	C₂×C₈	C₂×D₂₈₄	1	2	yes	—
163	C₂×C₈	C₂×Dic₁₄₂	1	2	yes	—
164	C₂×C₈	C₂×Dic₁₄₂	1	2	yes	—
165	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
166	C₄⋊C₄	—	284	4	no	—
167	C₄⋊C₄	—	284	4	no	—
168	C₄⋊C₄	—	284	2	no	—
169	C₄⋊C₄	—	284	4	no	—
170	C₄⋊C₄	—	142	2	no	—
171	C₄⋊C₄	—	142	4	no	—
172	C₄⋊C₄	—	142	2	no	—
173	C₄⋊C₄	D₅₆₈	1	2	yes	—
174	C₂²⋊C₄	—	284	2	no	—
175	C₂²⋊C₄	D₅₆₈	1	2	yes	—
176	C₄²	—	284	2	no	—
177	C₄²	C₂×D₂₈₄	1	2	yes	—
178	C₁₆	C₂×Dic₁₄₂	1	2	yes	—
179	C₁₆	C₂×Dic₁₄₂	1	2	yes	—
180	C₁₆	C₂×Dic₁₄₂	1	4	yes	—
181	C₂³	C₂×D₅₆₈	1	1	yes	characteristic
182	Q₈	—	1136	4	no	—
183	C₂×C₄	—	284	2	no	—
184	C₂×C₄	—	284	2	no	—
185	C₂×C₄	—	284	4	no	—
186	C₂×C₄	—	1136	2	no	—
187	C₂×C₄	#14556128	1	2	yes	—
188	C₂×C₄	C₂×D₅₆₈	1	4	yes	—
189	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
190	C₂×C₄	C₂×Dic₂₈₄	1	2	yes	—
191	C₂×C₄	C₂²×D₂₈₄	1	1	yes	characteristic
192	C₈	#14556128	1	2	yes	—
193	C₈	C₂×Dic₂₈₄	1	4	yes	—
194	C₈	C₂²×Dic₁₄₂	1	1	yes	characteristic
195	C₈	C₂²×Dic₁₄₂	1	1	yes	characteristic
196	C₂²	—	2	2	no	—
197	C₂²	#17986404	1	2	yes	—
198	C₂²	C₂²×D₅₆₈	1	1	yes	characteristic
199	C₄	—	2	4	no	—
200	C₄	—	568	4	no	—
201	C₄	—	568	2	no	—
202	C₄	—	568	2	no	—
203	C₄	#17986404	1	2	yes	—
204	C₄	C₂×#14556128	1	1	yes	characteristic
205	C₄	C₂²×Dic₂₈₄	1	1	yes	characteristic
206	C₂	—	2	2	no	—
207	C₂	C₂×#17986404	1	1	yes	characteristic
208	C₂	#22601488	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₅₆₈
commutator, cosocle	C₂×C₈×C₇₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₇₁×#2108	C₂
socle	C₂²×C₇₁	C₂²×D₈
prime radical	C₂³×C₇₁	C₂×D₈
hypercenter	#2108	D₇₁
hypocenter	C₇₁	#33730
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #22590626	C₂	semidirect
3. #22590622	C₂	semidirect
4. #22590619	C₂	semidirect
45. M₃₂×C₇₁	Q₈	semidirect
46. C₂×C₁₆×C₇₁	Q₈	semidirect
113. C₇₁	#33730	semidirect
143. M₃₂	Dic₁₄₂	semidirect
144. C₂×C₁₆	Dic₁₄₂	semidirect

Maximal subgroups

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

subgroup	index
#22598879	2
#22590626	2
#22590622	2
#22590619	2
#22590128	2
#22590124	2
C₂×#17985827	2
C₇₁×#2108	2
#33730	71