Subgroups — #32150815

Order 18176.

718 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₁₃₆	C₂	1	1	yes	maximal, characteristic
2	#22597193	C₂	1	2	yes	maximal, complemented
3	#22590217	C₂	1	4	yes	maximal, complemented
4	C₄×D₁₁₃₆	C₂	1	4	yes	maximal, complemented
5	#22590124	C₂	1	1	yes	maximal, characteristic, complemented
6	#22590057	C₂	1	2	yes	maximal, complemented
7	C₇₁×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₅₆₈	C₂²	1	2	yes	—
9	#17986473	—	2	2	no	—
10	C₂×D₁₁₃₆	C₄	1	8	yes	complemented
11	C₂×D₁₁₃₆	C₂²	1	2	yes	—
12	C₂×D₁₁₃₆	C₂²	1	2	yes	—
13	#17986112	—	2	4	no	—
14	C₄×D₅₆₈	C₂²	1	8	yes	complemented
15	#17986037	C₂²	1	2	yes	—
16	#17986005	—	2	2	no	—
17	#17985827	C₂²	1	2	yes	complemented
18	#17985825	C₂²	1	2	yes	complemented
19	#17985801	C₂²	1	8	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	C₂²×D₂₈₄	—	2	2	no	—
26	C₂×D₅₆₈	—	2	2	no	—
27	C₂×D₅₆₈	—	2	4	no	—
28	C₂×D₅₆₈	C₂×C₄	1	8	yes	complemented
29	C₂×D₅₆₈	C₂³	1	4	yes	—
30	C₄×D₂₈₄	—	2	8	no	—
31	#14556090	—	4	4	no	—
32	#14556087	—	2	2	no	—
33	C₂²⋊C₄×D₇₁	—	4	4	no	—
34	D₁₁₃₆	C₂×C₄	1	8	yes	complemented
35	D₁₁₃₆	C₂×C₄	1	8	yes	complemented
36	#14556062	C₂³	1	4	yes	—
37	#14556060	—	2	2	no	—
38	#14556050	—	2	8	no	—
39	C₂×M₁₆×C₇₁	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
41	#117×C₇₁	D₄	1	1	yes	characteristic
42	M₃₂×C₇₁	—	2	2	no	—
43	C₂×C₁₆×C₇₁	C₂×C₄	1	4	yes	complemented
44	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
45	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
47	#97×C₇₁	D₄	1	2	yes	—
48	C₄×C₈×C₇₁	C₂³	1	2	yes	—
49	C₂³×D₇₁	—	4	2	no	—
50	C₂×D₂₈₄	—	2	8	no	—
51	C₂×D₂₈₄	—	2	4	no	—
52	C₂×D₂₈₄	—	4	2	no	—
53	C₂×D₂₈₄	—	4	4	no	—
54	C₂×C₄×D₇₁	—	4	8	no	—
55	D₅₆₈	—	2	8	no	—
56	D₅₆₈	—	2	4	no	—
57	D₅₆₈	C₂²×C₄	1	8	yes	—
58	C₄⋊Dic₇₁	—	2	4	no	—
59	C₂²⋊Dic₇₁	—	4	2	no	—
60	(C₂×D₇₁)⋊C₄	—	4	8	no	—
61	C₄×Dic₇₁	—	4	2	no	—
62	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
63	M₁₆×C₇₁	—	2	2	no	—
64	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₇₁	#41	1	2	yes	—
68	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
69	C₄⋊C₄×C₇₁	D₈	1	2	yes	—
70	C₂²⋊C₄×C₇₁	D₈	1	2	yes	—
71	C₄²×C₇₁	C₂×D₄	1	2	yes	—
72	C₁₆×C₇₁	—	2	2	no	—
73	C₁₆×C₇₁	C₂²×C₄	1	2	yes	—
74	C₁₆×C₇₁	C₂²×C₄	1	2	yes	—
75	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
76	C₂²×D₇₁	—	4	4	no	—
77	C₂²×D₇₁	—	4	8	no	—
78	C₂²×D₇₁	—	16	4	no	—
79	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
80	C₂×C₄×C₇₁	C₂×D₈	1	4	yes	—
81	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₇₁	#135	1	2	yes	—
83	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₇₁	—	4	4	no	—
85	D₂₈₄	—	2	8	no	—
86	D₂₈₄	—	4	4	no	—
87	D₂₈₄	—	4	8	no	—
88	C₄×D₇₁	—	16	8	no	—
89	C₈×C₇₁	—	2	2	no	—
90	C₈×C₇₁	C₄×D₄	1	2	yes	—
91	C₈×C₇₁	C₂³×C₄	1	1	yes	characteristic
92	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
93	C₂²×C₇₁	—	2	2	no	—
94	C₂²×C₇₁	C₄×D₈	1	2	yes	—
95	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
96	C₂×D₇₁	—	4	8	no	—
97	C₂×D₇₁	—	8	8	no	—
98	C₂×D₇₁	—	16	8	no	—
99	C₄×C₇₁	—	2	4	no	—
100	C₄×C₇₁	C₄×D₈	1	2	yes	—
101	C₄×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
102	C₄×C₇₁	C₂×#135	1	1	yes	characteristic
103	Dic₇₁	—	8	4	no	—
104	#33456	—	71	1	no	maximal, Sylow
105	C₂×C₇₁	—	2	2	no	—
106	C₂×C₇₁	C₂×C₄×D₈	1	1	yes	characteristic
107	C₂×C₇₁	#3415	1	2	yes	—
108	D₇₁	—	8	8	no	—
109	C₂²×D₁₆	—	71	1	no	—
110	#2948	—	71	2	no	—
111	#2177	—	71	4	no	—
112	C₄×D₁₆	—	71	4	no	—
113	#2158	—	71	1	no	—
114	#2140	—	71	2	no	—
115	#2108	D₇₁	1	1	yes	characteristic, complemented
116	C₇₁	#33456	1	1	yes	characteristic, complemented, Sylow
117	C₂²×D₈	—	71	2	no	—
118	#518	—	142	2	no	—
119	C₂×D₁₆	—	71	8	no	—
120	C₂×D₁₆	—	71	2	no	—
121	C₂×D₁₆	—	71	2	no	—
122	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
123	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
124	#442	—	142	4	no	—
125	C₄×D₈	—	71	8	no	—
126	#427	—	71	2	no	—
127	#418	—	142	2	no	—
128	#405	C₂×D₇₁	1	1	yes	characteristic
129	#367	—	71	2	no	—
130	#366	—	71	2	no	—
131	#357	—	71	8	no	—
132	#346	C₂×D₇₁	1	2	yes	complemented
133	C₄×C₁₆	C₂×D₇₁	1	2	yes	complemented
134	C₂²×D₄	—	142	2	no	—
135	C₂×D₈	—	71	8	no	—
136	C₂×D₈	—	71	4	no	—
137	C₂×D₈	—	142	2	no	—
138	C₂×D₈	—	142	4	no	—
139	C₂×M₁₆	D₂₈₄	1	1	yes	characteristic
140	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
141	C₄×D₄	—	284	4	no	—
142	C₄×D₄	—	142	8	no	—
143	#117	—	142	2	no	—
144	#117	D₂₈₄	1	1	yes	characteristic
145	C₂×C₂²⋊C₄	—	284	4	no	—
146	D₁₆	—	71	8	no	—
147	D₁₆	—	71	8	no	—
148	M₃₂	—	2	2	no	—
149	C₂×C₁₆	C₄×D₇₁	1	4	yes	complemented
150	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
151	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
152	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
153	#107	—	71	4	no	—
154	#106	—	142	2	no	—
155	#102	—	142	8	no	—
156	#97	D₂₈₄	1	2	yes	—
157	C₄×C₈	C₂²×D₇₁	1	2	yes	—
158	C₂⁴	—	284	2	no	—
159	C₂×D₄	—	284	2	no	—
160	C₂×D₄	—	284	4	no	—
161	C₂×D₄	—	142	4	no	—
162	C₂×D₄	—	142	8	no	—
163	C₂²×C₄	—	284	8	no	—
164	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
165	D₈	—	71	8	no	—
166	D₈	—	142	4	no	—
167	D₈	—	142	8	no	—
168	M₁₆	—	2	2	no	—
169	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
170	C₂×C₈	C₂×D₂₈₄	1	2	yes	—
171	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
172	C₂×C₈	#11765865	1	2	yes	—
173	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
174	C₄⋊C₄	—	142	4	no	—
175	C₄⋊C₄	D₅₆₈	1	2	yes	—
176	C₂²⋊C₄	—	284	2	no	—
177	C₂²⋊C₄	—	284	8	no	—
178	C₂²⋊C₄	D₅₆₈	1	2	yes	—
179	C₄²	—	284	2	no	—
180	C₄²	C₂×D₂₈₄	1	2	yes	—
181	C₁₆	—	2	2	no	—
182	C₁₆	C₂×C₄×D₇₁	1	2	yes	—
183	C₁₆	C₂×C₄×D₇₁	1	2	yes	—
184	C₂³	—	284	4	no	—
185	C₂³	—	284	8	no	—
186	C₂³	—	1136	4	no	—
187	C₂³	C₂×D₅₆₈	1	1	yes	characteristic
188	D₄	—	284	8	no	—
189	D₄	—	284	4	no	—
190	D₄	—	142	8	no	—
191	C₂×C₄	—	284	4	no	—
192	C₂×C₄	—	1136	8	no	—
193	C₂×C₄	C₄×D₂₈₄	1	2	yes	—
194	C₂×C₄	C₂×D₅₆₈	1	4	yes	—
195	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
196	C₂×C₄	#14556150	1	2	yes	—
197	C₂×C₄	C₂²×D₂₈₄	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	C₄×D₂₈₄	1	2	yes	—
200	C₈	C₂²×C₄×D₇₁	1	1	yes	characteristic
201	C₈	C₂×#11765865	1	1	yes	characteristic
202	C₂²	—	2	2	no	—
203	C₂²	—	284	8	no	—
204	C₂²	—	568	8	no	—
205	C₂²	—	1136	8	no	—
206	C₂²	C₄×D₅₆₈	1	2	yes	—
207	C₂²	C₂²×D₅₆₈	1	1	yes	characteristic
208	C₄	—	2	4	no	—
209	C₄	—	568	4	no	—
210	C₄	C₄×D₅₆₈	1	2	yes	—
211	C₄	C₂×C₄×D₂₈₄	1	1	yes	characteristic
212	C₄	C₂×#14556150	1	1	yes	characteristic
213	C₂	—	2	2	no	—
214	C₂	—	568	8	no	—
215	C₂	C₂×C₄×D₅₆₈	1	1	yes	characteristic
216	C₂	#22601481	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₅₆₈
commutator	C₈×C₇₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₇₁×#2108	C₂
socle	C₂²×C₇₁	C₂²×D₈
prime radical	C₂²×D₁₁₃₆	C₂
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	#2108	D₇₁
hypocenter	C₇₁	#33456
LCS	G ▷ 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. #22597193	C₂	semidirect
3. #22590217	C₂	semidirect
4. C₄×D₁₁₃₆	C₂	semidirect
5. #22590124	C₂	semidirect
6. #22590057	C₂	semidirect
7. C₇₁×#2108	C₂	semidirect
10. C₂×D₁₁₃₆	C₄	semidirect
14. C₄×D₅₆₈	C₂²	semidirect
17. #17985827	C₂²	semidirect
18. #17985825	C₂²	semidirect
19. #17985801	C₂²	semidirect
23. #346×C₇₁	C₂²	semidirect
24. C₄×C₁₆×C₇₁	C₂²	semidirect
28. C₂×D₅₆₈	C₂×C₄	semidirect
34. D₁₁₃₆	C₂×C₄	semidirect
35. D₁₁₃₆	C₂×C₄	semidirect
43. C₂×C₁₆×C₇₁	C₂×C₄	semidirect
115. #2108	D₇₁	semidirect
116. C₇₁	#33456	semidirect
132. #346	C₂×D₇₁	semidirect
133. C₄×C₁₆	C₂×D₇₁	semidirect
149. C₂×C₁₆	C₄×D₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₁₁₃₆	2
#22597193	2
#22590217	2
C₄×D₁₁₃₆	2
#22590124	2
#22590057	2
C₇₁×#2108	2
#33456	71