Subgroups — #32150642

Order 18176.

690 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986415	C₂	1	2	yes	maximal, complemented
2	#22597053	C₂	1	1	yes	maximal, characteristic
3	#22590173	C₂	1	4	yes	maximal, complemented
4	C₁₆×D₂₈₄	C₂	1	4	yes	maximal, complemented
5	#22590096	C₂	1	1	yes	maximal, characteristic, complemented
6	#22589953	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14556134	C₂²	1	2	yes	—
9	#17986549	C₄	1	1	yes	characteristic
10	#17986415	—	2	8	no	—
11	C₂×#14556068	C₂²	1	2	yes	—
12	C₂×#14556067	C₂²	1	4	yes	—
13	C₂×C₁₆×D₇₁	C₂²	1	4	yes	—
14	#17986078	C₄	1	4	yes	—
15	C₈×D₂₈₄	C₂²	1	4	yes	—
16	#17986023	C₄	1	1	yes	characteristic
17	#17985965	C₄	1	2	yes	—
18	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	4	yes	—
19	(C₂×D₇₁)⋊C₁₆	C₂²	1	8	yes	—
20	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
23	C₄×C₁₆×C₇₁	C₂²	1	2	yes	—
24	C₄×C₁₆×C₇₁	C₂²	1	2	yes	—
25	C₂×#11765865	C₂×C₄	1	2	yes	—
26	#14556134	—	2	4	no	—
27	#14556134	—	2	4	no	—
28	C₂×#11765849	C₂×C₄	1	2	yes	—
29	C₂×#11765848	C₂×C₄	1	4	yes	—
30	C₂×C₈×D₇₁	C₂³	1	4	yes	—
31	#14556128	C₈	1	1	yes	characteristic
32	#14556125	C₈	1	1	yes	characteristic
33	#14556116	C₈	1	2	yes	—
34	#14556102	C₈	1	4	yes	—
35	C₄×D₂₈₄	C₂×C₄	1	4	yes	—
36	#14556068	—	2	4	no	—
37	#14556067	—	2	8	no	—
38	C₂×C₇₁⋊C₁₆	C₂³	1	4	yes	—
39	C₁₆×D₇₁	—	2	8	no	—
40	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	4	yes	—
41	(C₂×D₇₁)⋊C₈	C₂×C₄	1	8	yes	—
42	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₁₆×C₇₁	—	2	2	no	—
46	C₂×C₁₆×C₇₁	D₄	1	4	yes	complemented
47	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
48	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
50	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₇₁	C₂³	1	2	yes	—
52	#11765865	—	2	4	no	—
53	#11765865	—	2	4	no	—
54	C₂×Dic₁₄₂	C₂×C₈	1	2	yes	—
55	C₂×D₂₈₄	C₂×C₈	1	2	yes	—
56	C₂×D₂₈₄	C₂×C₈	1	4	yes	—
57	C₂×C₂²⋊D₇₁	C₂×C₈	1	4	yes	—
58	C₂×C₄×D₇₁	C₂²×C₄	1	4	yes	—
59	#11765849	—	2	4	no	—
60	#11765848	—	2	8	no	—
61	C₂×C₇₁⋊C₈	C₂²×C₄	1	4	yes	—
62	C₈×D₇₁	—	2	4	no	—
63	C₈×D₇₁	—	2	4	no	—
64	C₄⋊Dic₇₁	C₂×C₈	1	4	yes	—
65	(C₂×D₇₁)⋊C₄	C₂×C₈	1	8	yes	—
66	C₇₁⋊C₁₆	—	2	4	no	—
67	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
68	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₇₁	—	2	2	no	—
70	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₇₁	#41	1	2	yes	—
74	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
75	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
76	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
77	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
78	C₁₆×C₇₁	—	2	2	no	—
79	C₁₆×C₇₁	C₂×D₄	1	2	yes	—
80	C₁₆×C₇₁	C₂×D₄	1	2	yes	—
81	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
82	C₂²×D₇₁	C₂²×C₈	1	4	yes	—
83	C₂×C₄×C₇₁	—	2	4	no	—
84	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
86	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₇₁	#131	1	2	yes	—
89	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
90	C₂²⋊D₇₁	—	2	8	no	—
91	C₂×Dic₇₁	C₂²×C₈	1	4	yes	—
92	D₂₈₄	—	2	4	no	—
93	D₂₈₄	—	2	8	no	—
94	C₄×D₇₁	—	2	4	no	—
95	C₄×D₇₁	—	2	4	no	—
96	Dic₁₄₂	—	2	4	no	—
97	C₈×C₇₁	—	2	2	no	—
98	C₈×C₇₁	C₄×D₄	1	2	yes	—
99	C₈×C₇₁	C₂²×D₄	1	1	yes	characteristic
100	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
101	C₇₁⋊C₈	—	2	4	no	—
102	C₂²×C₇₁	—	2	2	no	—
103	C₂²×C₇₁	C₈×D₄	1	2	yes	—
104	C₂²×C₇₁	C₂³×C₈	1	1	yes	characteristic
105	C₂×D₇₁	—	2	8	no	—
106	C₂×D₇₁	—	2	4	no	—
107	C₄×C₇₁	—	2	2	no	—
108	C₄×C₇₁	C₈×D₄	1	2	yes	—
109	C₄×C₇₁	#504	1	4	yes	complemented
110	C₄×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
111	C₄×C₇₁	C₂×#131	1	1	yes	characteristic
112	Dic₇₁	—	2	4	no	—
113	#33434	—	71	1	no	maximal, Sylow
114	C₂×C₇₁	—	2	2	no	—
115	C₂×C₇₁	C₂×C₈×D₄	1	1	yes	characteristic
116	C₂×C₇₁	C₂×#504	1	2	yes	—
117	D₇₁	—	4	4	no	—
118	C₂×#504	—	71	2	no	—
119	#2931	—	71	1	no	—
120	#2169	—	71	4	no	—
121	D₄×C₁₆	—	71	4	no	—
122	#2150	—	71	1	no	—
123	#2113	—	71	2	no	—
124	C₂×C₄×C₁₆	D₇₁	1	1	yes	characteristic, complemented
125	C₇₁	#33434	1	1	yes	characteristic, complemented, Sylow
126	C₂×#131	—	71	2	no	—
127	#532	—	71	1	no	—
128	#504	—	142	8	no	—
129	C₂×M₃₂	—	71	2	no	—
130	C₂×M₃₂	—	71	4	no	—
131	C₂²×C₁₆	—	71	4	no	—
132	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
133	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
134	#436	—	71	4	no	—
135	C₈×D₄	—	71	4	no	—
136	#423	—	71	1	no	—
137	#408	—	71	2	no	—
138	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
139	C₄⋊C₁₆	—	71	4	no	—
140	C₂²⋊C₁₆	—	71	8	no	—
141	C₄×C₁₆	C₂×D₇₁	1	2	yes	—
142	C₄×C₁₆	C₂×D₇₁	1	2	yes	—
143	C₂×#41	—	71	2	no	—
144	#131	—	142	4	no	—
145	#131	—	142	4	no	—
146	C₂×M₁₆	—	71	2	no	—
147	C₂×M₁₆	—	71	4	no	—
148	C₂²×C₈	—	71	4	no	—
149	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
150	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
151	C₄⋊Q₈	—	71	1	no	—
152	#127	—	71	1	no	—
153	#124	—	71	2	no	—
154	#121	—	71	4	no	—
155	C₄×D₄	—	71	4	no	—
156	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
157	M₃₂	—	142	4	no	—
158	M₃₂	—	142	8	no	—
159	C₂×C₁₆	—	2	2	no	—
160	C₂×C₁₆	—	71	4	no	—
161	C₂×C₁₆	—	142	8	no	—
162	C₂×C₁₆	D₂₈₄	1	4	yes	complemented
163	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
164	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
165	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
166	C₄⋊C₈	—	71	4	no	—
167	C₂²⋊C₈	—	71	8	no	—
168	C₄×C₈	C₄×D₇₁	1	2	yes	—
169	C₄×C₈	C₂²×D₇₁	1	2	yes	—
170	#41	—	142	4	no	—
171	#41	—	142	4	no	—
172	C₂×Q₈	—	71	2	no	—
173	C₂×D₄	—	71	4	no	—
174	C₂×D₄	—	71	4	no	—
175	C₂×D₄	—	71	2	no	—
176	C₂²×C₄	—	71	4	no	—
177	C₂²×C₄	C₈×D₇₁	1	2	yes	—
178	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
179	M₁₆	—	142	4	no	—
180	M₁₆	—	142	8	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	71	4	no	—
183	C₂×C₈	—	142	4	no	—
184	C₂×C₈	—	142	4	no	—
185	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
186	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
187	C₂×C₈	C₂×D₂₈₄	1	2	yes	—
188	C₂×C₈	#11765865	1	2	yes	—
189	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
190	C₄⋊C₄	—	71	4	no	—
191	C₂²⋊C₄	—	71	8	no	—
192	C₄²	C₈×D₇₁	1	1	yes	characteristic
193	C₄²	C₈×D₇₁	1	1	yes	characteristic
194	C₄²	C₂×C₄×D₇₁	1	2	yes	—
195	C₁₆	—	2	2	no	—
196	C₁₆	—	142	4	no	—
197	C₁₆	C₂×D₂₈₄	1	2	yes	—
198	C₁₆	C₂×D₂₈₄	1	2	yes	—
199	C₂³	—	71	4	no	—
200	C₂³	C₂×C₈×D₇₁	1	1	yes	characteristic
201	Q₈	—	142	4	no	—
202	D₄	—	142	8	no	—
203	D₄	—	142	4	no	—
204	D₄	—	142	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	71	4	no	—
207	C₂×C₄	—	142	4	no	—
208	C₂×C₄	—	142	4	no	—
209	C₂×C₄	C₄×D₂₈₄	1	2	yes	—
210	C₂×C₄	C₂×C₈×D₇₁	1	2	yes	—
211	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
212	C₂×C₄	C₂×C₈×D₇₁	1	2	yes	—
213	C₂×C₄	#14556134	1	2	yes	—
214	C₂×C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	142	4	no	—
217	C₈	C₄×D₂₈₄	1	2	yes	—
218	C₈	C₂²×D₂₈₄	1	1	yes	characteristic
219	C₈	C₂×#11765865	1	1	yes	characteristic
220	C₂²	—	2	2	no	—
221	C₂²	—	142	4	no	—
222	C₂²	—	142	8	no	—
223	C₂²	C₈×D₂₈₄	1	2	yes	—
224	C₂²	C₂²×C₈×D₇₁	1	1	yes	characteristic
225	C₄	—	2	2	no	—
226	C₄	—	142	4	no	—
227	C₄	C₈×D₂₈₄	1	2	yes	—
228	C₄	#17986415	1	4	yes	complemented
229	C₄	C₂×C₄×D₂₈₄	1	1	yes	characteristic
230	C₄	C₂×#14556134	1	1	yes	characteristic
231	C₂	—	2	2	no	—
232	C₂	—	284	4	no	—
233	C₂	C₂×C₈×D₂₈₄	1	1	yes	characteristic
234	C₂	C₂×#17986415	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₇₁
commutator, socle	C₂²×C₇₁	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₂×C₄×C₁₆×C₇₁	C₂
prime radical	#17986549	C₄
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₇₁
hypocenter	C₇₁	#33434
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#17986415	C₂	semidirect
3. #22590173	C₂	semidirect
4. C₁₆×D₂₈₄	C₂	semidirect
5. #22590096	C₂	semidirect
6. #22589953	C₂	semidirect
7. C₂×C₄×C₁₆×C₇₁	C₂	semidirect
46. C₂×C₁₆×C₇₁	D₄	semidirect
109. C₄×C₇₁	#504	semidirect
124. C₂×C₄×C₁₆	D₇₁	semidirect
125. C₇₁	#33434	semidirect
162. C₂×C₁₆	D₂₈₄	semidirect
228. C₄	#17986415	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986415	2
#22597053	2
#22590173	2
C₁₆×D₂₈₄	2
#22590096	2
#22589953	2
C₂×C₄×C₁₆×C₇₁	2
#33434	71