Subgroups — #32150873

Order 18176.

642 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22601476	C₂	1	1	yes	maximal, characteristic
2	#22597258	C₂	1	2	yes	maximal, complemented
3	#22590227	C₂	1	4	yes	maximal, complemented
4	#22590222	C₂	1	4	yes	maximal, complemented
5	#22590135	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17985807	C₂	1	2	yes	maximal, complemented
7	C₇₁×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14556150	C₂²	1	2	yes	—
9	#17986765	—	2	2	no	—
10	#17986442	C₄	1	4	yes	complemented
11	#17986437	C₄	1	4	yes	complemented
12	#17986431	C₂²	1	4	yes	—
13	#17986122	—	2	4	no	—
14	#17986117	C₂²	1	8	yes	complemented
15	C₂×#14556064	C₂²	1	2	yes	—
16	#17986017	—	2	2	no	—
17	#17985829	C₂²	1	4	yes	complemented
18	#17985807	C₂²	1	4	yes	complemented
19	#17985807	C₂²	1	4	yes	complemented
20	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
22	#405×C₇₁	C₂²	1	1	yes	characteristic
23	#346×C₇₁	C₂²	1	2	yes	complemented
24	C₄×C₁₆×C₇₁	C₂²	1	2	yes	complemented
25	C₂×#11765865	—	2	2	no	—
26	#14556150	—	2	2	no	—
27	#14556150	—	2	2	no	—
28	#14556150	C₂×C₄	1	4	yes	—
29	#14556150	C₂³	1	4	yes	—
30	C₂×Dic₂₈₄	C₂×C₄	1	2	yes	—
31	C₂×#11765854	—	2	2	no	—
32	C₂×D₅₆₈	C₂×C₄	1	2	yes	—
33	#14556136	—	4	4	no	—
34	#14556134	—	2	8	no	—
35	C₂×#11765849	—	2	2	no	—
36	M₁₆×D₇₁	—	4	4	no	—
37	Dic₅₆₈	C₂×C₄	1	4	yes	—
38	#14556073	C₂×C₄	1	8	yes	—
39	D₁₁₃₆	C₂×C₄	1	4	yes	—
40	#14556064	—	2	2	no	—
41	#14556064	C₂³	1	4	yes	—
42	#14556056	—	2	8	no	—
43	C₂×M₁₆×C₇₁	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
45	#117×C₇₁	D₄	1	1	yes	characteristic
46	M₃₂×C₇₁	C₂×C₄	1	4	yes	—
47	C₂×C₁₆×C₇₁	—	2	2	no	—
48	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
50	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
51	#97×C₇₁	D₄	1	2	yes	—
52	C₄×C₈×C₇₁	C₂³	1	2	yes	—
53	#11765865	—	2	4	no	—
54	#11765865	—	2	4	no	—
55	#11765865	—	4	2	no	—
56	#11765865	—	4	2	no	—
57	C₂×Dic₁₄₂	—	2	2	no	—
58	C₂×D₂₈₄	—	2	2	no	—
59	C₂×C₂²⋊D₇₁	—	4	2	no	—
60	C₂×C₄×D₇₁	—	4	2	no	—
61	Dic₂₈₄	—	2	2	no	—
62	Dic₂₈₄	C₂²×C₄	1	4	yes	—
63	#11765854	—	2	4	no	—
64	#11765854	—	2	4	no	—
65	D₅₆₈	—	2	2	no	—
66	D₅₆₈	C₂²×C₄	1	4	yes	—
67	#11765849	—	2	4	no	—
68	#11765849	—	4	2	no	—
69	#11765848	—	4	8	no	—
70	C₂×C₇₁⋊C₈	—	4	2	no	—
71	C₈×D₇₁	—	4	8	no	—
72	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
73	M₁₆×C₇₁	D₈	1	4	yes	—
74	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₇₁	#41	1	2	yes	—
78	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
79	C₄⋊C₄×C₇₁	—	2	1	no	—
80	C₂²⋊C₄×C₇₁	—	2	1	no	—
81	C₄²×C₇₁	C₂×D₄	1	2	yes	—
82	C₁₆×C₇₁	—	2	1	no	—
83	C₁₆×C₇₁	—	2	1	no	—
84	C₁₆×C₇₁	C₂²×C₄	1	4	yes	—
85	C₂³×C₇₁	#135	1	1	yes	characteristic
86	C₂²×D₇₁	—	4	2	no	—
87	C₂×C₄×C₇₁	—	2	2	no	—
88	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
90	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
91	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
92	C₂²⋊D₇₁	—	4	4	no	—
93	C₂²⋊D₇₁	—	4	4	no	—
94	C₂×Dic₇₁	—	4	2	no	—
95	D₂₈₄	—	2	4	no	—
96	D₂₈₄	—	4	2	no	—
97	C₄×D₇₁	—	4	4	no	—
98	C₄×D₇₁	—	4	4	no	—
99	Dic₁₄₂	—	2	4	no	—
100	Dic₁₄₂	—	4	2	no	—
101	C₈×C₇₁	C₄×D₄	1	2	yes	—
102	C₈×C₇₁	C₂×D₈	1	4	yes	—
103	C₈×C₇₁	C₂³×C₄	1	1	yes	characteristic
104	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
105	C₇₁⋊C₈	—	4	4	no	—
106	C₂²×C₇₁	—	4	1	no	—
107	C₂²×C₇₁	C₄×D₈	1	2	yes	—
108	C₂²×C₇₁	C₂×#135	1	1	yes	characteristic
109	C₂×D₇₁	—	4	4	no	—
110	C₂×D₇₁	—	16	2	no	—
111	C₄×C₇₁	—	4	2	no	—
112	C₄×C₇₁	C₄×D₈	1	2	yes	—
113	C₄×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
114	C₄×C₇₁	C₂²×D₈	1	1	yes	characteristic
115	Dic₇₁	—	4	4	no	—
116	#33464	—	71	1	no	maximal, Sylow
117	C₂×C₇₁	—	2	1	no	—
118	C₂×C₇₁	—	2	2	no	—
119	C₂×C₇₁	C₂×C₄×D₈	1	1	yes	characteristic
120	D₇₁	—	8	4	no	—
121	#3414	—	71	1	no	—
122	#2957	—	71	2	no	—
123	#2179	—	71	4	no	—
124	#2178	—	71	4	no	—
125	#2161	—	71	1	no	—
126	C₂×#359	—	71	2	no	—
127	#2108	D₇₁	1	1	yes	characteristic, complemented
128	C₇₁	#33464	1	1	yes	characteristic, complemented, Sylow
129	C₂×#135	—	71	2	no	—
130	#568	—	142	2	no	—
131	#510	—	71	4	no	—
132	#509	—	71	4	no	—
133	#508	—	71	4	no	—
134	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
135	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
136	#444	—	142	4	no	—
137	#443	—	71	8	no	—
138	C₂×#108	—	71	2	no	—
139	#421	—	142	2	no	—
140	#405	C₂×D₇₁	1	1	yes	characteristic
141	#368	—	71	4	no	—
142	#359	—	71	4	no	—
143	#359	—	71	4	no	—
144	#346	C₂×D₇₁	1	2	yes	complemented
145	C₄×C₁₆	C₂×D₇₁	1	2	yes	complemented
146	C₂×#41	—	142	2	no	—
147	#135	—	71	4	no	—
148	#135	—	71	4	no	—
149	#135	—	142	2	no	—
150	#135	—	142	2	no	—
151	C₂×Q₁₆	—	71	2	no	—
152	C₂×SD₁₆	—	142	2	no	—
153	C₂×D₈	—	71	2	no	—
154	#131	—	284	4	no	—
155	#131	—	142	8	no	—
156	C₂×M₁₆	—	284	4	no	—
157	C₂×M₁₆	—	142	2	no	—
158	C₂×M₁₆	D₂₈₄	1	1	yes	characteristic
159	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
160	#117	D₂₈₄	1	1	yes	characteristic
161	Q₃₂	—	71	4	no	—
162	SD₃₂	—	71	8	no	—
163	D₁₆	—	71	4	no	—
164	M₃₂	C₄×D₇₁	1	4	yes	—
165	C₂×C₁₆	—	2	2	no	—
166	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
167	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
168	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
169	#108	—	71	4	no	—
170	#108	—	142	2	no	—
171	#104	—	142	8	no	—
172	#97	D₂₈₄	1	2	yes	—
173	C₄×C₈	C₂²×D₇₁	1	2	yes	—
174	#41	—	284	2	no	—
175	#41	—	284	2	no	—
176	#41	—	142	4	no	—
177	#41	—	142	4	no	—
178	C₂×Q₈	—	142	2	no	—
179	C₂×D₄	—	284	2	no	—
180	C₂×D₄	—	142	2	no	—
181	C₂²×C₄	—	284	2	no	—
182	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
183	Q₁₆	—	71	4	no	—
184	Q₁₆	—	142	2	no	—
185	SD₁₆	—	142	4	no	—
186	SD₁₆	—	142	4	no	—
187	D₈	—	71	4	no	—
188	D₈	—	142	2	no	—
189	M₁₆	—	284	2	no	—
190	M₁₆	—	284	8	no	—
191	M₁₆	—	142	4	no	—
192	M₁₆	D₅₆₈	1	4	yes	—
193	C₂×C₈	—	284	2	no	—
194	C₂×C₈	—	284	8	no	—
195	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
196	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
197	C₂×C₈	C₂×D₂₈₄	1	2	yes	—
198	C₂×C₈	#11765865	1	2	yes	—
199	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
200	C₄⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	2	1	no	—
202	C₄²	C₂×D₂₈₄	1	2	yes	—
203	C₁₆	—	2	1	no	—
204	C₁₆	—	2	1	no	—
205	C₁₆	C₂×C₄×D₇₁	1	4	yes	—
206	C₂³	—	284	2	no	—
207	C₂³	#14556150	1	1	yes	characteristic
208	Q₈	—	284	2	no	—
209	Q₈	—	142	4	no	—
210	D₄	—	284	4	no	—
211	D₄	—	284	4	no	—
212	D₄	—	284	2	no	—
213	D₄	—	142	4	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	284	2	no	—
216	C₂×C₄	—	284	4	no	—
217	C₂×C₄	—	284	4	no	—
218	C₂×C₄	C₄×D₂₈₄	1	2	yes	—
219	C₂×C₄	C₂×D₅₆₈	1	2	yes	—
220	C₂×C₄	#14556150	1	1	yes	characteristic
221	C₂×C₄	C₂²×D₂₈₄	1	1	yes	characteristic
222	C₈	—	284	4	no	—
223	C₈	C₄×D₂₈₄	1	2	yes	—
224	C₈	C₂×D₅₆₈	1	4	yes	—
225	C₈	C₂²×C₄×D₇₁	1	1	yes	characteristic
226	C₈	C₂×#11765865	1	1	yes	characteristic
227	C₂²	—	4	1	no	—
228	C₂²	—	284	4	no	—
229	C₂²	—	1136	2	no	—
230	C₂²	C₄×D₅₆₈	1	2	yes	—
231	C₂²	C₂×#14556150	1	1	yes	characteristic
232	C₄	—	4	2	no	—
233	C₄	—	284	4	no	—
234	C₄	C₄×D₅₆₈	1	2	yes	—
235	C₄	C₂×C₄×D₂₈₄	1	1	yes	characteristic
236	C₄	C₂²×D₅₆₈	1	1	yes	characteristic
237	C₂	—	2	1	no	—
238	C₂	—	2	2	no	—
239	C₂	—	568	4	no	—
240	C₂	C₂×C₄×D₅₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×D₅₆₈
commutator	C₈×C₇₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₇₁×#2108	C₂
socle	C₂×C₇₁	C₂×C₄×D₈
prime radical	#22601476	C₂
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	#2108	D₇₁
hypocenter	C₇₁	#33464
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #22597258	C₂	semidirect
3. #22590227	C₂	semidirect
4. #22590222	C₂	semidirect
5. #22590135	C₂	semidirect
6. C₂×#17985807	C₂	semidirect
7. C₇₁×#2108	C₂	semidirect
10. #17986442	C₄	semidirect
11. #17986437	C₄	semidirect
14. #17986117	C₂²	semidirect
17. #17985829	C₂²	semidirect
18. #17985807	C₂²	semidirect
19. #17985807	C₂²	semidirect
23. #346×C₇₁	C₂²	semidirect
24. C₄×C₁₆×C₇₁	C₂²	semidirect
127. #2108	D₇₁	semidirect
128. C₇₁	#33464	semidirect
144. #346	C₂×D₇₁	semidirect
145. C₄×C₁₆	C₂×D₇₁	semidirect

Maximal subgroups

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

subgroup	index
#22601476	2
#22597258	2
#22590227	2
#22590222	2
#22590135	2
C₂×#17985807	2
C₇₁×#2108	2
#33464	71