Subgroups — #32150112

Order 18176.

716 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22601443	C₂	1	1	yes	maximal, characteristic
2	#22596766	C₂	1	1	yes	maximal, characteristic
3	#22589944	C₂	1	6	yes	maximal, complemented
4	C₂×#17985776	C₂	1	6	yes	maximal, complemented
5	C₇₁×#3407	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556135	C₄	1	1	yes	characteristic
7	#41×Dic₇₁	C₄	1	1	yes	characteristic
8	#17986416	C₄	1	8	yes	complemented
9	C₂×#14556068	C₂²	1	6	yes	—
10	#17985957	C₄	1	3	yes	—
11	#17985954	C₂²	1	3	yes	—
12	M₁₆×Dic₇₁	C₄	1	3	yes	—
13	C₂×#14556038	C₂²	1	3	yes	—
14	#17985776	C₂²	1	4	yes	—
15	#17985776	C₂²	1	12	yes	—
16	C₂×#131×C₇₁	C₂²	1	1	yes	characteristic
17	#504×C₇₁	C₄	1	8	yes	complemented
18	C₂×M₃₂×C₇₁	C₂²	1	6	yes	—
19	#14556135	—	2	4	no	—
20	C₂×#11765849	C₂×C₄	1	3	yes	—
21	C₂²×C₇₁⋊C₈	C₂×C₄	1	3	yes	—
22	Q₈×Dic₇₁	—	2	1	no	—
23	D₄×Dic₇₁	—	2	3	no	—
24	#14556087	C₂×C₄	1	3	yes	—
25	C₂×C₄×Dic₇₁	C₂×C₄	1	3	yes	—
26	#14556068	C₂×C₄	1	12	yes	complemented
27	#14556068	C₂³	1	4	yes	—
28	C₂×C₇₁⋊C₁₆	C₂×C₄	1	12	yes	complemented
29	#14556038	C₂×C₄	1	3	yes	—
30	#14556038	C₂×C₄	1	3	yes	—
31	#14556038	C₂³	1	1	yes	characteristic
32	#14556038	C₂³	1	3	yes	—
33	C₈×Dic₇₁	C₂×C₄	1	6	yes	—
34	C₂×#41×C₇₁	C₂×C₄	1	1	yes	characteristic
35	#131×C₇₁	C₂×C₄	1	4	yes	—
36	#131×C₇₁	C₂×C₄	1	4	yes	—
37	C₂×M₁₆×C₇₁	C₂×C₄	1	3	yes	—
38	C₂²×C₈×C₇₁	C₂³	1	3	yes	—
39	M₃₂×C₇₁	C₂×C₄	1	12	yes	complemented
40	M₃₂×C₇₁	C₂³	1	4	yes	—
41	C₂×C₁₆×C₇₁	C₂×C₄	1	12	yes	complemented
42	C₂²×Dic₇₁	—	2	3	no	—
43	#11765849	—	2	6	no	—
44	C₂×C₇₁⋊C₈	—	2	6	no	—
45	C₂×C₇₁⋊C₈	C₂²×C₄	1	3	yes	—
46	C₂×C₇₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
47	C₄⋊Dic₇₁	—	2	3	no	—
48	C₂²⋊Dic₇₁	—	2	3	no	—
49	C₄×Dic₇₁	—	2	3	no	—
50	C₄×Dic₇₁	C₂²×C₄	1	3	yes	—
51	C₄×Dic₇₁	C₂²×C₄	1	1	yes	characteristic
52	C₇₁⋊C₁₆	C₂²×C₄	1	8	yes	—
53	#41×C₇₁	C₄²	1	4	yes	—
54	#41×C₇₁	M₁₆	1	4	yes	—
55	C₂×Q₈×C₇₁	M₁₆	1	1	yes	characteristic
56	C₂×D₄×C₇₁	M₁₆	1	3	yes	—
57	C₂²×C₄×C₇₁	C₂²×C₄	1	3	yes	—
58	M₁₆×C₇₁	C₄²	1	12	yes	—
59	C₂×C₈×C₇₁	C₂²×C₄	1	3	yes	—
60	C₂×C₈×C₇₁	C₂²×C₄	1	6	yes	—
61	C₂×C₈×C₇₁	C₂²×C₄	1	6	yes	—
62	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
63	C₁₆×C₇₁	C₂²×C₄	1	8	yes	—
64	C₂³×C₇₁	C₂×M₁₆	1	3	yes	—
65	Q₈×C₇₁	#97	1	4	yes	—
66	D₄×C₇₁	#97	1	12	yes	—
67	C₂×C₄×C₇₁	C₂×C₄²	1	6	yes	—
68	C₂×C₄×C₇₁	C₂×M₁₆	1	6	yes	—
69	C₂×C₄×C₇₁	C₂×M₁₆	1	3	yes	—
70	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
71	C₂×Dic₇₁	—	2	1	no	—
72	C₂×Dic₇₁	—	2	3	no	—
73	C₂×Dic₇₁	—	8	3	no	—
74	C₈×C₇₁	C₂×C₄²	1	6	yes	—
75	C₈×C₇₁	C₂³×C₄	1	1	yes	characteristic
76	C₈×C₇₁	C₂³×C₄	1	1	yes	characteristic
77	C₇₁⋊C₈	—	2	1	no	—
78	C₇₁⋊C₈	—	2	3	no	—
79	C₂²×C₇₁	—	4	3	no	—
80	C₂²×C₇₁	C₂×#97	1	6	yes	—
81	C₂²×C₇₁	C₂²×M₁₆	1	1	yes	characteristic
82	C₄×C₇₁	C₂×#97	1	6	yes	—
83	C₄×C₇₁	C₂²×C₄²	1	1	yes	characteristic
84	C₄×C₇₁	C₂²×M₁₆	1	1	yes	characteristic
85	Dic₇₁	—	4	3	no	—
86	Dic₇₁	—	4	1	no	—
87	#33329	—	71	1	no	maximal, Sylow
88	C₂×C₇₁	—	2	1	no	—
89	C₂×C₇₁	—	2	6	no	—
90	C₂×C₇₁	C₂²×#97	1	1	yes	characteristic
91	#3407	—	71	1	no	—
92	#3407	D₇₁	1	1	yes	characteristic
93	#2875	—	71	1	no	—
94	#2110	—	71	6	no	—
95	C₂×#347	—	71	6	no	—
96	C₇₁	#33329	1	1	yes	characteristic, complemented, Sylow
97	C₂×#131	—	71	1	no	—
98	C₂×#131	C₂×D₇₁	1	1	yes	characteristic
99	C₄×#41	—	71	1	no	—
100	#504	—	71	8	no	—
101	#504	Dic₇₁	1	8	yes	complemented
102	C₂×M₃₂	—	71	6	no	—
103	C₂×M₃₂	C₂×D₇₁	1	6	yes	—
104	#405	—	71	3	no	—
105	#405	—	71	3	no	—
106	C₄×M₁₆	—	71	3	no	—
107	C₂×#97	—	71	3	no	—
108	#347	—	71	4	no	—
109	#347	—	71	12	no	—
110	C₂×#41	C₄×D₇₁	1	1	yes	characteristic
111	#131	—	142	4	no	—
112	#131	C₄×D₇₁	1	4	yes	—
113	#131	C₂×Dic₇₁	1	4	yes	—
114	C₂×M₁₆	—	71	3	no	—
115	C₂×M₁₆	C₄×D₇₁	1	3	yes	—
116	C₂²×C₈	—	71	3	no	—
117	C₂²×C₈	C₂²×D₇₁	1	3	yes	—
118	C₄×Q₈	—	142	1	no	—
119	C₄×D₄	—	142	3	no	—
120	#117	—	71	3	no	—
121	C₂×C₄²	—	71	3	no	—
122	M₃₂	—	71	12	no	—
123	M₃₂	—	71	4	no	—
124	M₃₂	C₂×Dic₇₁	1	12	yes	complemented
125	M₃₂	C₂²×D₇₁	1	4	yes	—
126	C₂×C₁₆	—	71	12	no	—
127	C₂×C₁₆	C₂×Dic₇₁	1	12	yes	complemented
128	#97	—	71	3	no	—
129	#97	—	71	3	no	—
130	#97	—	71	1	no	—
131	#97	—	71	3	no	—
132	C₄×C₈	—	71	6	no	—
133	#41	C₄×Dic₇₁	1	4	yes	—
134	#41	#11765848	1	4	yes	—
135	C₂×Q₈	#11765848	1	1	yes	characteristic
136	C₂×D₄	#11765848	1	3	yes	—
137	C₂²×C₄	—	142	3	no	—
138	C₂²×C₄	C₂×C₄×D₇₁	1	3	yes	—
139	M₁₆	—	142	6	no	—
140	M₁₆	C₄×Dic₇₁	1	12	yes	—
141	C₂×C₈	—	71	3	no	—
142	C₂×C₈	—	71	1	no	—
143	C₂×C₈	—	142	6	no	—
144	C₂×C₈	C₂×C₄×D₇₁	1	6	yes	—
145	C₂×C₈	C₂×C₄×D₇₁	1	3	yes	—
146	C₂×C₈	C₂²×Dic₇₁	1	6	yes	—
147	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
148	C₄⋊C₄	—	142	3	no	—
149	C₂²⋊C₄	—	142	3	no	—
150	C₄²	—	71	3	no	—
151	C₄²	—	71	1	no	—
152	C₄²	—	142	3	no	—
153	C₁₆	—	71	8	no	—
154	C₁₆	C₂²×Dic₇₁	1	8	yes	—
155	C₂³	C₂×#11765848	1	3	yes	—
156	Q₈	#14556038	1	4	yes	—
157	D₄	#14556038	1	12	yes	—
158	C₂×C₄	—	568	3	no	—
159	C₂×C₄	—	142	1	no	—
160	C₂×C₄	—	142	3	no	—
161	C₂×C₄	C₂×C₄×Dic₇₁	1	6	yes	—
162	C₂×C₄	C₂×#11765848	1	3	yes	—
163	C₂×C₄	C₂×#11765848	1	6	yes	—
164	C₂×C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
165	C₈	—	142	1	no	—
166	C₈	—	142	3	no	—
167	C₈	C₂×C₄×Dic₇₁	1	6	yes	—
168	C₈	C₂²×C₄×D₇₁	1	1	yes	characteristic
169	C₈	C₂³×Dic₇₁	1	1	yes	characteristic
170	C₂²	—	4	3	no	—
171	C₂²	C₂×#14556038	1	6	yes	—
172	C₂²	C₂²×#11765848	1	1	yes	characteristic
173	C₄	—	284	1	no	—
174	C₄	—	284	3	no	—
175	C₄	C₂×#14556038	1	6	yes	—
176	C₄	C₂²×C₄×Dic₇₁	1	1	yes	characteristic
177	C₄	C₂²×#11765848	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	2	6	no	—
180	C₂	C₂²×#14556038	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₄×Dic₇₁
commutator	C₄×C₇₁	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₇₁×#3407	C₂
socle	C₂×C₇₁	C₂²×#97
prime radical	C₂×#41×C₇₁	C₂×C₄
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	#3407	D₇₁
hypocenter	C₇₁	#33329
LCS	G ▷ C₄×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ #3407

Complements

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

N	H ≅ G/N	type
3. #22589944	C₂	semidirect
4. C₂×#17985776	C₂	semidirect
8. #17986416	C₄	semidirect
17. #504×C₇₁	C₄	semidirect
26. #14556068	C₂×C₄	semidirect
28. C₂×C₇₁⋊C₁₆	C₂×C₄	semidirect
39. M₃₂×C₇₁	C₂×C₄	semidirect
41. C₂×C₁₆×C₇₁	C₂×C₄	semidirect
96. C₇₁	#33329	semidirect
101. #504	Dic₇₁	semidirect
124. M₃₂	C₂×Dic₇₁	semidirect
127. C₂×C₁₆	C₂×Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
#22601443	2
#22596766	2
#22589944	2
C₂×#17985776	2
C₇₁×#3407	2
#33329	71