Subgroups — #31888686

Order 18176.

414 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14556062	C₂	1	1	yes	maximal, characteristic
2	#22588207	C₂	1	2	yes	maximal
3	C₂×#17985752	C₂	1	1	yes	maximal, characteristic
4	#22586180	C₂	1	2	yes	maximal
5	C₂×C₈²×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556062	C₄	1	2	yes	—
7	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
8	C₂×#14556038	—	2	2	no	—
9	C₄×C₄⋊Dic₇₁	C₂²	1	2	yes	—
10	#17985762	C₄	1	2	yes	—
11	#17985753	—	2	2	no	—
12	#17985752	C₄	1	4	yes	complemented
13	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₈²×C₇₁	C₄	1	4	yes	complemented
17	C₂²×C₇₁⋊C₈	—	2	2	no	—
18	C₂×C₄⋊Dic₇₁	C₂×C₄	1	4	yes	—
19	C₂×C₄×Dic₇₁	—	2	2	no	—
20	#14556062	—	2	4	no	—
21	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
22	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	4	yes	—
23	#14556038	—	4	4	no	—
24	C₇₁⋊#95	—	2	2	no	—
25	C₂²×C₈×C₇₁	—	2	1	no	—
26	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₇₁	—	2	1	no	—
32	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₇₁	Q₈	1	2	yes	—
37	C₂²×Dic₇₁	—	2	4	no	—
38	C₂×C₇₁⋊C₈	—	2	4	no	—
39	C₂×C₇₁⋊C₈	—	4	2	no	—
40	C₂×C₇₁⋊C₈	—	4	2	no	—
41	C₄⋊Dic₇₁	—	2	4	no	—
42	C₄⋊Dic₇₁	—	2	4	no	—
43	C₄×Dic₇₁	—	4	2	no	—
44	C₄×Dic₇₁	—	4	2	no	—
45	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
48	C₂×C₈×C₇₁	—	2	4	no	—
49	C₂×C₈×C₇₁	—	2	2	no	—
50	C₂×C₈×C₇₁	—	2	2	no	—
51	C₂×C₈×C₇₁	—	2	1	no	—
52	C₂×C₈×C₇₁	—	2	1	no	—
53	C₂×C₈×C₇₁	C₄²	1	2	yes	—
54	C₂×C₈×C₇₁	C₄²	1	2	yes	—
55	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₇₁	M₁₆	1	4	yes	—
58	C₂×C₈×C₇₁	D₈	1	2	yes	—
59	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
60	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
63	C₄²×C₇₁	#41	1	1	yes	characteristic
64	C₂³×C₇₁	C₄×D₄	1	1	yes	characteristic
65	C₂×C₄×C₇₁	—	2	2	no	—
66	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
69	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
73	C₂×C₄×C₇₁	#131	1	2	yes	—
74	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
77	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
78	C₂×Dic₇₁	—	4	4	no	—
79	C₂×Dic₇₁	—	4	4	no	—
80	C₂×Dic₇₁	—	4	4	no	—
81	C₈×C₇₁	—	2	4	no	—
82	C₈×C₇₁	—	2	2	no	—
83	C₈×C₇₁	—	2	2	no	—
84	C₈×C₇₁	#97	1	4	yes	complemented
85	C₈×C₇₁	#107	1	4	yes	complemented
86	C₇₁⋊C₈	—	4	4	no	—
87	C₂²×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₇₁	#436	1	2	yes	—
89	C₂²×C₇₁	C₄×D₈	1	1	yes	characteristic
90	C₂²×C₇₁	C₄×Q₁₆	1	1	yes	characteristic
91	C₂²×C₇₁	#443	1	2	yes	—
92	C₄×C₇₁	—	2	1	no	—
93	C₄×C₇₁	—	2	1	no	—
94	C₄×C₇₁	C₂×#97	1	1	yes	characteristic
95	C₄×C₇₁	#405	1	1	yes	characteristic
96	C₄×C₇₁	C₂×#107	1	1	yes	characteristic
97	C₄×C₇₁	#427	1	1	yes	characteristic
98	C₄×C₇₁	C₄×D₈	1	1	yes	characteristic
99	C₄×C₇₁	C₄×Q₁₆	1	1	yes	characteristic
100	C₄×C₇₁	#446	1	2	yes	—
101	Dic₇₁	—	8	4	no	—
102	#8282	—	71	1	no	maximal, Sylow
103	C₂×C₇₁	#1589	1	2	yes	—
104	C₂×C₇₁	#1591	1	2	yes	—
105	C₂×C₇₁	#1770	1	1	yes	characteristic
106	C₂×C₇₁	#1772	1	1	yes	characteristic
107	C₂×C₇₁	C₄×#107	1	1	yes	characteristic
108	C₄×#107	—	71	1	no	—
109	#1770	—	71	2	no	—
110	C₂×#335	—	71	1	no	—
111	C₂×C₈²	D₇₁	1	1	yes	characteristic
112	#1279	—	71	2	no	—
113	C₇₁	#8282	1	1	yes	characteristic, complemented, Sylow
114	C₂×#107	—	71	2	no	—
115	C₂×C₄⋊C₈	—	71	2	no	—
116	C₂×#97	—	142	2	no	—
117	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
120	C₄×C₄⋊C₄	—	71	2	no	—
121	#340	—	71	2	no	—
122	#336	—	142	2	no	—
123	#335	—	71	4	no	—
124	C₈²	Dic₇₁	1	4	yes	complemented
125	C₂²×C₈	—	2	1	no	—
126	C₂²×C₈	—	142	2	no	—
127	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
128	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
129	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
130	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
131	C₂×C₄⋊C₄	—	71	4	no	—
132	C₂×C₄²	—	142	2	no	—
133	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
134	#107	—	142	4	no	—
135	C₄⋊C₈	—	71	4	no	—
136	C₄⋊C₈	—	142	2	no	—
137	#97	—	284	4	no	—
138	C₄×C₈	—	2	1	no	—
139	C₄×C₈	Dic₁₄₂	1	2	yes	—
140	C₄×C₈	C₄×D₇₁	1	2	yes	—
141	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
142	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
143	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
144	#95	—	142	2	no	—
145	C₂²×C₄	—	142	4	no	—
146	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
147	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
148	C₂²×C₄	#11765865	1	1	yes	characteristic
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	284	2	no	—
155	C₂×C₈	—	284	2	no	—
156	C₂×C₈	—	142	4	no	—
157	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
158	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
159	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
160	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
161	C₂×C₈	#11765848	1	4	yes	—
162	C₂×C₈	D₅₆₈	1	2	yes	—
163	C₂×C₈	Dic₂₈₄	1	2	yes	—
164	C₄⋊C₄	—	142	4	no	—
165	C₄⋊C₄	—	142	4	no	—
166	C₄²	—	284	2	no	—
167	C₄²	—	284	2	no	—
168	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
169	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
170	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
171	C₄²	#11765865	1	1	yes	characteristic
172	C₂³	C₄×D₂₈₄	1	1	yes	characteristic
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	284	4	no	—
175	C₂×C₄	—	284	4	no	—
176	C₂×C₄	—	284	4	no	—
177	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
179	C₂×C₄	#14556087	1	1	yes	characteristic
180	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
181	C₂×C₄	C₄×Dic₁₄₂	1	1	yes	characteristic
182	C₂×C₄	C₄×Dic₁₄₂	1	1	yes	characteristic
183	C₂×C₄	C₂×#11765848	1	2	yes	—
184	C₂×C₄	#14556134	1	2	yes	—
185	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
186	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
187	C₂×C₄	#14556150	1	1	yes	characteristic
188	C₂×C₄	#14556150	1	1	yes	characteristic
189	C₈	—	2	4	no	—
190	C₈	—	2	2	no	—
191	C₈	—	2	2	no	—
192	C₈	—	284	4	no	—
193	C₈	#14556038	1	4	yes	complemented
194	C₈	#14556062	1	4	yes	complemented
195	C₂²	C₄×C₄⋊Dic₇₁	1	1	yes	characteristic
196	C₂²	#17986078	1	2	yes	—
197	C₂²	C₄×D₅₆₈	1	1	yes	characteristic
198	C₂²	C₄×Dic₂₈₄	1	1	yes	characteristic
199	C₂²	#17986117	1	2	yes	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	568	4	no	—
203	C₄	C₂×#14556038	1	1	yes	characteristic
204	C₄	#17985954	1	1	yes	characteristic
205	C₄	C₂×#14556062	1	1	yes	characteristic
206	C₄	#17986037	1	1	yes	characteristic
207	C₄	C₄×D₅₆₈	1	1	yes	characteristic
208	C₄	C₄×Dic₂₈₄	1	1	yes	characteristic
209	C₄	#17986129	1	2	yes	—
210	C₂	#22587226	1	2	yes	—
211	C₂	#22587238	1	2	yes	—
212	C₂	#22588207	1	1	yes	characteristic
213	C₂	#22588219	1	1	yes	characteristic
214	C₂	C₄×#14556062	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₂×C₈²×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₄×D₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₂×C₈²	D₇₁
hypocenter	C₇₁	#8282
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
12. #17985752	C₄	semidirect
16. C₈²×C₇₁	C₄	semidirect
84. C₈×C₇₁	#97	semidirect
85. C₈×C₇₁	#107	semidirect
113. C₇₁	#8282	semidirect
124. C₈²	Dic₇₁	semidirect
193. C₈	#14556038	semidirect
194. C₈	#14556062	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14556062	2
#22588207	2
C₂×#17985752	2
#22586180	2
C₂×C₈²×C₇₁	2
#8282	71