Subgroups — #31888667

Order 18176.

394 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	#22588086	C₂	1	1	yes	maximal, characteristic
3	C₂×#17985751	C₂	1	2	yes	maximal
4	#22586174	C₂	1	2	yes	maximal
5	C₇₁×#1767	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₇₁⋊C₈	—	2	1	no	—
8	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
10	#17985754	—	2	1	no	—
11	#17985754	C₄	1	4	yes	—
12	#17985751	C₄	1	8	yes	complemented
13	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
14	C₂×#97×C₇₁	—	2	1	no	—
15	C₄³×C₇₁	C₂²	1	1	yes	characteristic
16	#336×C₇₁	—	2	1	no	—
17	C₂²×C₇₁⋊C₈	—	2	2	no	—
18	C₂²×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
19	C₂²×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
20	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
21	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	4	yes	—
22	C₄×C₇₁⋊C₈	—	2	2	no	—
23	C₄×C₇₁⋊C₈	—	4	2	no	—
24	C₄×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₇₁	—	2	2	no	—
26	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₇₁	Q₈	1	2	yes	—
30	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₇₁	—	2	2	no	—
32	C₄⋊C₈×C₇₁	C₂×C₄	1	4	yes	—
33	#97×C₇₁	—	4	2	no	—
34	C₂×C₇₁⋊C₈	—	2	4	no	—
35	C₂×C₇₁⋊C₈	—	2	2	no	—
36	C₂×C₇₁⋊C₈	—	2	4	no	—
37	C₂×C₇₁⋊C₈	—	2	4	no	—
38	C₂×C₇₁⋊C₈	—	4	2	no	—
39	C₂×C₇₁⋊C₈	—	4	2	no	—
40	C₂×C₇₁⋊C₈	C₄²	1	4	yes	—
41	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₇₁	—	2	4	no	—
47	C₂×C₈×C₇₁	—	4	2	no	—
48	C₂×C₈×C₇₁	—	4	2	no	—
49	C₄²×C₇₁	—	2	4	no	—
50	C₄²×C₇₁	—	2	2	no	—
51	C₄²×C₇₁	—	4	2	no	—
52	C₄²×C₇₁	D₈	1	2	yes	—
53	C₄²×C₇₁	Q₁₆	1	2	yes	—
54	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
55	C₄²×C₇₁	#41	1	2	yes	—
56	C₂³×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₇₁	—	2	2	no	—
58	C₂×C₄×C₇₁	—	2	2	no	—
59	C₂×C₄×C₇₁	—	2	2	no	—
60	C₂×C₄×C₇₁	—	2	2	no	—
61	C₂×C₄×C₇₁	—	2	2	no	—
62	C₂×C₄×C₇₁	—	2	2	no	—
63	C₂×C₄×C₇₁	#107	1	4	yes	—
64	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
65	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
66	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
67	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
68	C₂×C₄×C₇₁	#131	1	4	yes	—
69	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
70	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×C₄×C₇₁	#135	1	2	yes	—
72	C₈×C₇₁	—	4	4	no	—
73	C₇₁⋊C₈	—	2	4	no	—
74	C₇₁⋊C₈	—	4	2	no	—
75	C₇₁⋊C₈	—	4	4	no	—
76	C₂²×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₇₁	#424	1	2	yes	—
78	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
79	C₂²×C₇₁	#427	1	1	yes	characteristic
80	C₂²×C₇₁	#430	1	2	yes	—
81	C₄×C₇₁	—	2	4	no	—
82	C₄×C₇₁	—	2	2	no	—
83	C₄×C₇₁	—	4	2	no	—
84	C₄×C₇₁	#405	1	2	yes	—
85	C₄×C₇₁	#433	1	2	yes	—
86	C₄×C₇₁	C₄×D₈	1	2	yes	—
87	C₄×C₇₁	C₄×Q₁₆	1	2	yes	—
88	#8278	—	71	1	no	maximal, Sylow
89	C₂×C₇₁	#1571	1	2	yes	—
90	C₂×C₇₁	#1572	1	2	yes	—
91	C₂×C₇₁	#1768	1	1	yes	characteristic
92	C₂×C₇₁	C₄×#107	1	1	yes	characteristic
93	C₂×C₇₁	#1778	1	1	yes	characteristic
94	#1767	D₇₁	1	1	yes	characteristic
95	C₄×C₄⋊C₈	—	71	1	no	—
96	#1744	—	71	1	no	—
97	C₂×#335	—	71	2	no	—
98	#1277	—	71	2	no	—
99	C₇₁	#8278	1	1	yes	characteristic, complemented, Sylow
100	C₂×C₄⋊C₈	—	71	2	no	—
101	C₂×C₄⋊C₈	C₂×D₇₁	1	2	yes	—
102	C₂×#97	—	2	1	no	—
103	C₂×C₄×C₈	—	71	1	no	—
104	C₂×C₄×C₈	—	71	1	no	—
105	C₂×C₄×C₈	—	142	1	no	—
106	C₄³	C₂×D₇₁	1	1	yes	characteristic
107	#336	—	2	1	no	—
108	#336	—	71	4	no	—
109	#336	—	142	1	no	—
110	#335	—	71	8	no	—
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	—	71	2	no	—
113	C₂²×C₈	—	71	2	no	—
114	C₂²×C₈	—	142	2	no	—
115	C₂×C₄²	Dic₁₄₂	1	2	yes	—
116	C₂×C₄²	C₄×D₇₁	1	2	yes	—
117	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
118	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
119	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	71	4	no	—
122	C₄⋊C₈	—	142	2	no	—
123	C₄⋊C₈	C₄×D₇₁	1	4	yes	—
124	#97	—	4	2	no	—
125	C₄×C₈	—	284	2	no	—
126	C₄×C₈	—	71	4	no	—
127	C₄×C₈	—	142	2	no	—
128	C₂²×C₄	Dic₇₁⋊C₄	1	2	yes	—
129	C₂²×C₄	Dic₇₁⋊C₄	1	2	yes	—
130	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
131	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
132	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	284	2	no	—
137	C₂×C₈	—	284	2	no	—
138	C₂×C₈	—	71	4	no	—
139	C₂×C₈	—	142	2	no	—
140	C₂×C₈	—	142	4	no	—
141	C₂×C₈	—	142	4	no	—
142	C₂×C₈	—	142	4	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	#11765850	1	2	yes	—
147	C₄²	#11765855	1	2	yes	—
148	C₄²	C₂×C₄×D₇₁	1	2	yes	—
149	C₄²	#11765865	1	2	yes	—
150	C₂³	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	#14556061	1	4	yes	—
158	C₂×C₄	C₄²×D₇₁	1	1	yes	characteristic
159	C₂×C₄	#14556085	1	1	yes	characteristic
160	C₂×C₄	C₄×C₂²⋊D₇₁	1	2	yes	—
161	C₂×C₄	C₄×Dic₁₄₂	1	2	yes	—
162	C₂×C₄	#14556136	1	4	yes	—
163	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
164	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
165	C₂×C₄	#14556148	1	2	yes	—
166	C₈	—	4	4	no	—
167	C₈	—	284	2	no	—
168	C₈	—	284	4	no	—
169	C₈	—	142	4	no	—
170	C₂²	C₄×Dic₇₁⋊C₄	1	1	yes	characteristic
171	C₂²	#17986027	1	2	yes	—
172	C₂²	C₂×#14556061	1	1	yes	characteristic
173	C₂²	#17986035	1	1	yes	characteristic
174	C₂²	#17986047	1	2	yes	—
175	C₄	—	2	2	no	—
176	C₄	—	2	4	no	—
177	C₄	—	4	2	no	—
178	C₄	#17985956	1	2	yes	—
179	C₄	#17986061	1	2	yes	—
180	C₄	C₄×#11765850	1	2	yes	—
181	C₄	C₄×#11765855	1	2	yes	—
182	C₂	#22587116	1	2	yes	—
183	C₂	#22587121	1	2	yes	—
184	C₂	#22588200	1	1	yes	characteristic
185	C₂	C₄×#14556061	1	1	yes	characteristic
186	C₂	#22588247	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1767	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1767	D₇₁
hypocenter	C₇₁	#8278
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
12. #17985751	C₄	semidirect
99. C₇₁	#8278	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₇₁⋊C₈)	2
#22588086	2
C₂×#17985751	2
#22586174	2
C₇₁×#1767	2
#8278	71