Subgroups — #32016391

Order 18176.

822 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₄⋊Dic₇₁	C₂	1	2	yes	maximal
2	#22589104	C₂	1	6	yes	maximal
3	#22588468	C₂	1	3	yes	maximal
4	C₄×#14556060	C₂	1	3	yes	maximal
5	C₂×C₈×Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×Dic₇₁	—	2	2	no	—
7	C₂×#14556060	C₂²	1	6	yes	—
8	#17985898	—	2	6	no	—
9	#17985869	C₂²	1	6	yes	—
10	C₄×C₄⋊Dic₇₁	C₂²	1	6	yes	—
11	#17985762	C₂²	1	3	yes	—
12	#17985762	C₂²	1	6	yes	—
13	#17985762	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₇₁	C₄	1	8	yes	complemented
16	C₂×C₄⋊C₈×C₇₁	C₂²	1	3	yes	—
17	C₂×C₄×C₈×C₇₁	C₂²	1	3	yes	—
18	Q₈×Dic₇₁	—	4	8	no	—
19	C₂×C₄⋊Dic₇₁	—	2	6	no	—
20	C₂×C₄⋊Dic₇₁	—	2	6	no	—
21	C₂×C₄⋊Dic₇₁	C₂³	1	6	yes	—
22	C₂×C₄⋊Dic₇₁	C₂³	1	2	yes	—
23	C₂×C₄×Dic₇₁	—	2	6	no	—
24	#14556060	—	2	12	no	—
25	C₇₁⋊#95	—	2	6	no	—
26	C₂²×Q₈×C₇₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₁	C₂³	1	3	yes	—
28	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
29	C₄×Q₈×C₇₁	C₂×C₄	1	4	yes	—
30	C₄×Q₈×C₇₁	Q₈	1	4	yes	—
31	C₂×C₄⋊C₄×C₇₁	D₄	1	3	yes	—
32	C₂×C₄²×C₇₁	C₂³	1	3	yes	—
33	C₄⋊C₈×C₇₁	C₂×C₄	1	12	yes	—
34	C₄×C₈×C₇₁	C₂×C₄	1	12	yes	—
35	C₂²×Dic₇₁	—	2	2	no	—
36	C₂²×Dic₇₁	—	2	6	no	—
37	C₄⋊Dic₇₁	—	2	2	no	—
38	C₄⋊Dic₇₁	—	2	6	no	—
39	C₄⋊Dic₇₁	—	2	6	no	—
40	C₄⋊Dic₇₁	—	4	12	no	—
41	C₄⋊Dic₇₁	—	8	6	no	—
42	C₄⋊Dic₇₁	#41	1	4	yes	—
43	C₄×Dic₇₁	—	4	12	no	—
44	C₂×Q₈×C₇₁	C₄⋊C₄	1	4	yes	—
45	C₂×Q₈×C₇₁	SD₁₆	1	8	yes	—
46	C₂²×C₄×C₇₁	C₂×D₄	1	3	yes	—
47	C₂²×C₄×C₇₁	C₂×D₄	1	3	yes	—
48	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
49	C₂×C₈×C₇₁	—	2	6	no	—
50	C₂×C₈×C₇₁	—	2	2	no	—
51	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
52	C₂×C₈×C₇₁	C₂²×C₄	1	6	yes	—
53	C₄⋊C₄×C₇₁	C₄⋊C₄	1	12	yes	—
54	C₄²×C₇₁	C₂²×C₄	1	6	yes	—
55	C₄²×C₇₁	C₂×Q₈	1	6	yes	—
56	C₂³×C₇₁	C₂²×D₄	1	1	yes	characteristic
57	Q₈×C₇₁	#106	1	16	yes	complemented
58	C₂×C₄×C₇₁	—	2	3	no	—
59	C₂×C₄×C₇₁	—	2	3	no	—
60	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	6	yes	—
61	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	6	yes	—
62	C₂×C₄×C₇₁	C₂×SD₁₆	1	12	yes	—
63	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
64	C₂×C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
65	C₂×C₄×C₇₁	C₂×#41	1	2	yes	—
66	C₂×C₄×C₇₁	2₋¹⁺⁴	1	2	yes	—
67	C₂×Dic₇₁	—	2	4	no	—
68	C₂×Dic₇₁	—	4	2	no	—
69	C₂×Dic₇₁	—	4	2	no	—
70	C₂×Dic₇₁	—	4	6	no	—
71	C₂×Dic₇₁	—	4	6	no	—
72	C₂×Dic₇₁	—	4	6	no	—
73	C₈×C₇₁	—	2	6	no	—
74	C₈×C₇₁	—	2	2	no	—
75	C₂²×C₇₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₇₁	#548	1	2	yes	—
77	C₂²×C₇₁	C₂²×SD₁₆	1	2	yes	—
78	C₂²×C₇₁	#576	1	1	yes	characteristic
79	C₂²×C₇₁	#578	1	1	yes	characteristic
80	C₄×C₇₁	—	2	6	no	—
81	C₄×C₇₁	C₂×#106	1	12	yes	—
82	C₄×C₇₁	C₄×#41	1	1	yes	characteristic
83	C₄×C₇₁	#519	1	1	yes	characteristic
84	C₄×C₇₁	#557	1	2	yes	—
85	Dic₇₁	—	4	4	no	—
86	Dic₇₁	—	8	6	no	—
87	#22738	—	71	1	no	maximal, Sylow
88	C₂×C₇₁	#2351	1	1	yes	characteristic
89	C₂×C₇₁	C₂²×#106	1	1	yes	characteristic
90	C₂×C₇₁	#2913	1	1	yes	characteristic
91	C₂×C₇₁	#3359	1	2	yes	—
92	C₂×C₇₁	#3362	1	2	yes	—
93	C₂×C₈×Q₈	D₇₁	1	1	yes	characteristic
94	Q₈×C₄⋊C₄	—	71	2	no	—
95	#1948	—	71	6	no	—
96	#1834	—	71	3	no	—
97	C₄×#106	—	71	3	no	—
98	C₇₁	#22738	1	1	yes	characteristic, complemented, Sylow
99	C₂×C₄×Q₈	—	142	2	no	—
100	C₂×C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
101	C₈×Q₈	Dic₇₁	1	8	yes	complemented
102	C₂×#106	—	71	6	no	—
103	C₂×C₄⋊C₈	C₂×D₇₁	1	3	yes	—
104	C₂×C₄×C₈	C₂×D₇₁	1	3	yes	—
105	#389	—	142	6	no	—
106	#384	—	71	6	no	—
107	C₄×C₄⋊C₄	—	71	6	no	—
108	#340	—	71	3	no	—
109	#340	—	71	6	no	—
110	#340	—	71	1	no	—
111	C₂²×Q₈	D₂₈₄	1	1	yes	characteristic
112	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
113	C₂²×C₈	C₂²×D₇₁	1	3	yes	—
114	C₄×Q₈	—	284	8	no	—
115	C₄×Q₈	Dic₁₄₂	1	4	yes	—
116	C₄×Q₈	C₂×Dic₇₁	1	4	yes	—
117	C₂×C₄⋊C₄	—	71	2	no	—
118	C₂×C₄⋊C₄	—	71	6	no	—
119	C₂×C₄⋊C₄	—	142	6	no	—
120	C₂×C₄⋊C₄	—	142	6	no	—
121	C₂×C₄⋊C₄	D₂₈₄	1	3	yes	—
122	C₂×C₄²	—	142	6	no	—
123	C₂×C₄²	C₂²×D₇₁	1	3	yes	—
124	#106	—	142	12	no	—
125	C₄⋊C₈	C₂×Dic₇₁	1	12	yes	—
126	C₄×C₈	C₂×Dic₇₁	1	12	yes	—
127	#95	—	142	6	no	—
128	C₂×Q₈	C₄⋊Dic₇₁	1	4	yes	—
129	C₂×Q₈	#11765854	1	8	yes	—
130	C₂²×C₄	—	142	6	no	—
131	C₂²×C₄	—	142	2	no	—
132	C₂²×C₄	C₂×D₂₈₄	1	3	yes	—
133	C₂²×C₄	C₂×D₂₈₄	1	3	yes	—
134	C₂²×C₄	C₂³×D₇₁	1	1	yes	characteristic
135	C₂×C₈	—	2	6	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	C₂²×Dic₇₁	1	2	yes	—
138	C₂×C₈	C₂²×Dic₇₁	1	6	yes	—
139	C₄⋊C₄	—	284	12	no	—
140	C₄⋊C₄	—	568	6	no	—
141	C₄⋊C₄	—	71	4	no	—
142	C₄⋊C₄	—	142	6	no	—
143	C₄⋊C₄	—	142	6	no	—
144	C₄⋊C₄	—	142	2	no	—
145	C₄⋊C₄	C₄⋊Dic₇₁	1	12	yes	—
146	C₄²	—	284	12	no	—
147	C₄²	C₂²×Dic₇₁	1	6	yes	—
148	C₄²	C₂×Dic₁₄₂	1	6	yes	—
149	C₂³	C₂²×D₂₈₄	1	1	yes	characteristic
150	Q₈	#14556060	1	16	yes	complemented
151	C₂×C₄	—	2	3	no	—
152	C₂×C₄	—	2	3	no	—
153	C₂×C₄	—	284	2	no	—
154	C₂×C₄	—	284	2	no	—
155	C₂×C₄	—	284	6	no	—
156	C₂×C₄	—	284	6	no	—
157	C₂×C₄	—	284	6	no	—
158	C₂×C₄	—	142	4	no	—
159	C₂×C₄	C₂×C₄⋊Dic₇₁	1	6	yes	—
160	C₂×C₄	C₂×C₄⋊Dic₇₁	1	6	yes	—
161	C₂×C₄	C₂×#11765854	1	12	yes	—
162	C₂×C₄	C₂³×Dic₇₁	1	1	yes	characteristic
163	C₂×C₄	C₂²×Dic₁₄₂	1	1	yes	characteristic
164	C₂×C₄	#41×D₇₁	1	2	yes	—
165	C₂×C₄	#14556176	1	2	yes	—
166	C₈	—	2	6	no	—
167	C₈	—	2	2	no	—
168	C₂²	C₂²×C₄⋊Dic₇₁	1	1	yes	characteristic
169	C₂²	#17986661	1	2	yes	—
170	C₂²	C₂²×#11765854	1	2	yes	—
171	C₂²	#17986804	1	1	yes	characteristic
172	C₂²	#17986816	1	1	yes	characteristic
173	C₄	—	2	6	no	—
174	C₄	—	284	4	no	—
175	C₄	—	568	6	no	—
176	C₄	C₂×#14556060	1	12	yes	—
177	C₄	#41×Dic₇₁	1	1	yes	characteristic
178	C₄	#17986477	1	1	yes	characteristic
179	C₄	#17986725	1	2	yes	—
180	C₂	#22591261	1	1	yes	characteristic
181	C₂	C₂²×#14556060	1	1	yes	characteristic
182	C₂	#22596940	1	1	yes	characteristic
183	C₂	#22600983	1	2	yes	—
184	C₂	#22601012	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₁
Fitting	C₂×C₈×Q₈×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂²×D₄
cosocle	C₂²×C₄×C₇₁	C₂⁴
hypercenter	C₂×C₈×Q₈	D₇₁
hypocenter	C₇₁	#22738
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
15. C₈×Q₈×C₇₁	C₄	semidirect
57. Q₈×C₇₁	#106	semidirect
98. C₇₁	#22738	semidirect
101. C₈×Q₈	Dic₇₁	semidirect
150. Q₈	#14556060	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₄⋊Dic₇₁	2
#22589104	2
#22588468	2
C₄×#14556060	2
C₂×C₈×Q₈×C₇₁	2
#22738	71