Subgroups — #32150456

Order 18176.

684 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22601443	C₂	1	2	yes	maximal
2	#22590106	C₂	1	6	yes	maximal, complemented
3	C₂×#17985820	C₂	1	6	yes	maximal, complemented
4	C₄×#131×C₇₁	C₂	1	1	yes	maximal, characteristic
5	#17986416	—	2	8	no	—
6	C₂×#14556068	C₂²	1	12	yes	—
7	#17985820	C₂²	1	4	yes	—
8	#17985820	C₂²	1	12	yes	—
9	C₂×#131×C₇₁	C₄	1	1	yes	characteristic
10	C₂×#131×C₇₁	C₂²	1	1	yes	characteristic
11	C₄×#41×C₇₁	C₄	1	1	yes	characteristic
12	#405×C₇₁	C₄	1	3	yes	—
13	#405×C₇₁	C₂²	1	3	yes	—
14	C₄×M₁₆×C₇₁	C₄	1	3	yes	—
15	C₂×C₄×C₈×C₇₁	C₂²	1	3	yes	—
16	#14556068	—	2	12	no	—
17	#14556068	C₂³	1	8	yes	—
18	C₂×C₇₁⋊C₁₆	—	2	12	no	—
19	C₂×#41×C₇₁	C₂×C₄	1	1	yes	characteristic
20	#131×C₇₁	C₈	1	8	yes	—
21	#131×C₇₁	D₄	1	4	yes	—
22	#131×C₇₁	Q₈	1	4	yes	—
23	C₂×M₁₆×C₇₁	C₂×C₄	1	3	yes	—
24	C₂×M₁₆×C₇₁	C₂×C₄	1	3	yes	—
25	C₂²×C₈×C₇₁	C₂×C₄	1	3	yes	—
26	C₂²×C₈×C₇₁	C₂³	1	3	yes	—
27	C₄×Q₈×C₇₁	—	2	1	no	—
28	C₄×D₄×C₇₁	—	2	3	no	—
29	#117×C₇₁	C₂×C₄	1	3	yes	—
30	C₂×C₄²×C₇₁	C₂×C₄	1	3	yes	—
31	#97×C₇₁	C₂×C₄	1	6	yes	—
32	C₄×C₈×C₇₁	C₂×C₄	1	3	yes	—
33	C₄×C₈×C₇₁	C₂×C₄	1	3	yes	—
34	C₄×C₈×C₇₁	C₂³	1	3	yes	—
35	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
36	C₇₁⋊C₁₆	—	2	8	no	—
37	#41×C₇₁	C₄⋊C₄	1	4	yes	—
38	#41×C₇₁	C₂×C₈	1	4	yes	—
39	C₂×Q₈×C₇₁	M₁₆	1	1	yes	characteristic
40	C₂×D₄×C₇₁	M₁₆	1	3	yes	—
41	C₂²×C₄×C₇₁	—	2	3	no	—
42	C₂²×C₄×C₇₁	C₂²×C₄	1	3	yes	—
43	M₁₆×C₇₁	C₄⋊C₄	1	12	yes	—
44	M₁₆×C₇₁	C₂×C₈	1	12	yes	—
45	C₂×C₈×C₇₁	C₂×C₈	1	12	yes	—
46	C₂×C₈×C₇₁	C₂²×C₄	1	3	yes	—
47	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₇₁	C₂²×C₄	1	3	yes	—
49	C₂×C₈×C₇₁	C₂×D₄	1	6	yes	—
50	C₂×C₈×C₇₁	C₂×Q₈	1	6	yes	—
51	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
52	C₄⋊C₄×C₇₁	—	2	3	no	—
53	C₂²⋊C₄×C₇₁	—	2	3	no	—
54	C₄²×C₇₁	—	2	3	no	—
55	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₇₁	C₂²×C₄	1	3	yes	—
57	C₂³×C₇₁	C₂×M₁₆	1	3	yes	—
58	Q₈×C₇₁	C₄⋊C₈	1	4	yes	—
59	D₄×C₇₁	C₄⋊C₈	1	12	yes	—
60	C₂×C₄×C₇₁	—	2	1	no	—
61	C₂×C₄×C₇₁	—	2	3	no	—
62	C₂×C₄×C₇₁	—	4	6	no	—
63	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	6	yes	—
64	C₂×C₄×C₇₁	C₂²×C₈	1	6	yes	—
65	C₂×C₄×C₇₁	C₂×M₁₆	1	3	yes	—
66	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
67	C₈×C₇₁	C₂×C₄⋊C₄	1	6	yes	—
68	C₈×C₇₁	C₂²×C₈	1	2	yes	—
69	C₈×C₇₁	C₂²×C₈	1	6	yes	—
70	C₈×C₇₁	C₂²×D₄	1	1	yes	characteristic
71	C₈×C₇₁	C₂²×Q₈	1	1	yes	characteristic
72	C₂²×C₇₁	—	4	3	no	—
73	C₂²×C₇₁	C₂×C₄⋊C₈	1	6	yes	—
74	C₂²×C₇₁	C₂²×M₁₆	1	1	yes	characteristic
75	C₄×C₇₁	—	2	2	no	—
76	C₄×C₇₁	—	4	3	no	—
77	C₄×C₇₁	C₂×C₄⋊C₈	1	6	yes	—
78	C₄×C₇₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
79	C₄×C₇₁	C₂³×C₈	1	1	yes	characteristic
80	#33401	—	71	1	no	maximal, Sylow
81	C₂×C₇₁	—	2	1	no	—
82	C₂×C₇₁	—	2	6	no	—
83	C₂×C₇₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
84	#3407	—	71	2	no	—
85	C₄×#131	D₇₁	1	1	yes	characteristic
86	#2153	—	71	6	no	—
87	C₂×#364	—	71	6	no	—
88	C₇₁	#33401	1	1	yes	characteristic, complemented, Sylow
89	C₂×#131	Dic₇₁	1	1	yes	characteristic
90	C₂×#131	C₂×D₇₁	1	1	yes	characteristic
91	C₄×#41	Dic₇₁	1	1	yes	characteristic
92	#504	—	142	8	no	—
93	C₂×M₃₂	—	71	12	no	—
94	#405	Dic₇₁	1	3	yes	—
95	#405	C₂×D₇₁	1	3	yes	—
96	C₄×M₁₆	Dic₇₁	1	3	yes	—
97	C₂×C₄×C₈	C₂×D₇₁	1	3	yes	—
98	#364	—	71	4	no	—
99	#364	—	71	12	no	—
100	C₂×#41	C₂×Dic₇₁	1	1	yes	characteristic
101	#131	C₇₁⋊C₈	1	8	yes	—
102	#131	Dic₁₄₂	1	4	yes	—
103	#131	D₂₈₄	1	4	yes	—
104	C₂×M₁₆	C₂×Dic₇₁	1	3	yes	—
105	C₂×M₁₆	C₂×Dic₇₁	1	3	yes	—
106	C₂²×C₈	C₂×Dic₇₁	1	3	yes	—
107	C₂²×C₈	C₂²×D₇₁	1	3	yes	—
108	C₄×Q₈	—	2	1	no	—
109	C₄×D₄	—	2	3	no	—
110	#117	C₂×Dic₇₁	1	3	yes	—
111	C₂×C₄²	C₂×Dic₇₁	1	3	yes	—
112	M₃₂	—	71	8	no	—
113	M₃₂	—	142	12	no	—
114	C₂×C₁₆	—	142	12	no	—
115	#97	C₂×Dic₇₁	1	6	yes	—
116	C₄×C₈	C₂×Dic₇₁	1	3	yes	—
117	C₄×C₈	C₂×Dic₇₁	1	3	yes	—
118	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
119	C₄×C₈	C₂²×D₇₁	1	3	yes	—
120	#41	C₄⋊Dic₇₁	1	4	yes	—
121	#41	C₂×C₇₁⋊C₈	1	4	yes	—
122	C₂×Q₈	#11765849	1	1	yes	characteristic
123	C₂×D₄	#11765849	1	3	yes	—
124	C₂²×C₄	—	2	3	no	—
125	C₂²×C₄	C₂²×Dic₇₁	1	3	yes	—
126	M₁₆	C₄⋊Dic₇₁	1	12	yes	—
127	M₁₆	C₂×C₇₁⋊C₈	1	12	yes	—
128	C₂×C₈	C₂×C₇₁⋊C₈	1	12	yes	—
129	C₂×C₈	C₂²×Dic₇₁	1	3	yes	—
130	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
131	C₂×C₈	C₂²×Dic₇₁	1	3	yes	—
132	C₂×C₈	C₂×D₂₈₄	1	6	yes	—
133	C₂×C₈	C₂×Dic₁₄₂	1	6	yes	—
134	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
135	C₄⋊C₄	—	2	3	no	—
136	C₂²⋊C₄	—	2	3	no	—
137	C₄²	—	2	3	no	—
138	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
139	C₄²	C₂²×Dic₇₁	1	3	yes	—
140	C₁₆	—	142	8	no	—
141	C₂³	C₂×#11765849	1	3	yes	—
142	Q₈	C₄⋊(C₇₁⋊C₈)	1	4	yes	—
143	D₄	C₄⋊(C₇₁⋊C₈)	1	12	yes	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	3	no	—
146	C₂×C₄	—	4	6	no	—
147	C₂×C₄	C₂×C₄⋊Dic₇₁	1	6	yes	—
148	C₂×C₄	C₂²×C₇₁⋊C₈	1	6	yes	—
149	C₂×C₄	C₂×#11765849	1	3	yes	—
150	C₂×C₄	C₂³×Dic₇₁	1	1	yes	characteristic
151	C₈	C₂×C₄⋊Dic₇₁	1	6	yes	—
152	C₈	C₂²×C₇₁⋊C₈	1	2	yes	—
153	C₈	C₂²×C₇₁⋊C₈	1	6	yes	—
154	C₈	C₂²×D₂₈₄	1	1	yes	characteristic
155	C₈	C₂²×Dic₁₄₂	1	1	yes	characteristic
156	C₂²	—	4	3	no	—
157	C₂²	C₂×C₄⋊(C₇₁⋊C₈)	1	6	yes	—
158	C₂²	C₂²×#11765849	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	4	3	no	—
161	C₄	C₂×C₄⋊(C₇₁⋊C₈)	1	6	yes	—
162	C₄	C₂²×C₄⋊Dic₇₁	1	1	yes	characteristic
163	C₄	C₂³×C₇₁⋊C₈	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	2	6	no	—
166	C₂	C₂²×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²×D₂₈₄
commutator	C₄×C₇₁	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₄×#131×C₇₁	C₂
socle	C₂×C₇₁	C₂²×C₄⋊C₈
prime radical	C₂×#41×C₇₁	C₂×C₄
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	C₄×#131	D₇₁
hypocenter	C₇₁	#33401
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₈ ◁ C₂×#131 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. #22590106	C₂	semidirect
3. C₂×#17985820	C₂	semidirect
88. C₇₁	#33401	semidirect

Maximal subgroups

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

subgroup	index
#22601443	2
#22590106	2
C₂×#17985820	2
C₄×#131×C₇₁	2
#33401	71