Subgroups — #31909024

Order 18176.

242 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590094	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₁₆×D₇₁	C₂	1	1	yes	maximal, characteristic
3	#22586622	C₂	1	1	yes	maximal, characteristic
4	#22586410	C₂	1	1	yes	maximal, characteristic, complemented
5	#22586409	C₂	1	1	yes	maximal, characteristic, complemented
6	#22586409	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#1370	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14556067	—	2	1	no	—
9	C₂×C₁₆×D₇₁	—	2	1	no	—
10	#17986050	C₄	1	1	yes	characteristic
11	#17986050	C₄	1	1	yes	characteristic
12	C₄×C₈×D₇₁	C₂²	1	1	yes	characteristic
13	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	1	yes	characteristic
14	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	1	yes	characteristic
15	Dic₇₁⋊C₁₆	—	2	1	no	—
16	Dic₇₁⋊C₁₆	—	2	1	no	—
17	#17985721	C₄	1	1	yes	characteristic
18	#17985721	C₄	1	1	yes	characteristic
19	C₈×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊C₁₆×C₇₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₁₆×C₇₁	C₂²	1	1	yes	characteristic
22	C₈²×C₇₁	C₂²	1	1	yes	characteristic
23	C₂×C₈×D₇₁	D₄	1	1	yes	characteristic
24	C₂×C₈×D₇₁	Q₈	1	1	yes	characteristic
25	C₄²×D₇₁	C₂×C₄	1	1	yes	characteristic
26	#14556067	—	4	2	no	—
27	C₂×C₇₁⋊C₁₆	—	2	1	no	—
28	C₂×C₇₁⋊C₁₆	—	2	1	no	—
29	C₁₆×D₇₁	—	4	2	no	—
30	Dic₇₁⋊C₈	—	2	1	no	—
31	Dic₇₁⋊C₈	C₈	1	1	yes	characteristic
32	Dic₇₁⋊C₈	C₈	1	1	yes	characteristic
33	(C₂×D₇₁)⋊C₈	—	2	1	no	—
34	(C₂×D₇₁)⋊C₈	C₈	1	1	yes	characteristic
35	(C₂×D₇₁)⋊C₈	C₈	1	1	yes	characteristic
36	C₄×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₄×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₄×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₇₁	D₄	1	1	yes	characteristic
40	C₈×Dic₇₁	Q₈	1	1	yes	characteristic
41	C₂×C₁₆×C₇₁	—	2	1	no	—
42	C₂×C₁₆×C₇₁	—	2	1	no	—
43	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
46	C₂×C₄×D₇₁	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×D₇₁	C₂×C₈	1	1	yes	characteristic
48	C₂×C₄×D₇₁	M₁₆	1	1	yes	characteristic
49	C₂×C₇₁⋊C₈	—	2	1	no	—
50	C₂×C₇₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₇₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₇₁⋊C₈	C₂×C₈	1	1	yes	characteristic
53	C₂×C₇₁⋊C₈	C₂×C₈	1	1	yes	characteristic
54	C₈×D₇₁	—	2	2	no	—
55	C₈×D₇₁	D₈	1	2	yes	—
56	C₈×D₇₁	Q₁₆	1	2	yes	—
57	C₄×Dic₇₁	C₄⋊C₄	1	1	yes	characteristic
58	C₄×Dic₇₁	C₂×C₈	1	1	yes	characteristic
59	C₄×Dic₇₁	M₁₆	1	1	yes	characteristic
60	C₇₁⋊C₁₆	—	4	1	no	—
61	C₇₁⋊C₁₆	—	4	1	no	—
62	C₂×C₈×C₇₁	—	2	1	no	—
63	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
64	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
65	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
68	C₁₆×C₇₁	—	4	1	no	—
69	C₁₆×C₇₁	—	4	1	no	—
70	C₂²×D₇₁	C₄⋊C₈	1	1	yes	characteristic
71	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
74	C₂×Dic₇₁	C₄⋊C₈	1	1	yes	characteristic
75	C₂×Dic₇₁	C₄⋊C₈	1	1	yes	characteristic
76	C₂×Dic₇₁	C₄⋊C₈	1	1	yes	characteristic
77	C₄×D₇₁	—	2	1	no	—
78	C₄×D₇₁	—	2	1	no	—
79	C₄×D₇₁	—	4	1	no	—
80	C₄×D₇₁	#107	1	2	yes	—
81	C₄×D₇₁	#108	1	2	yes	—
82	C₈×C₇₁	—	2	1	no	—
83	C₈×C₇₁	—	2	1	no	—
84	C₈×C₇₁	—	2	1	no	—
85	C₈×C₇₁	—	4	1	no	—
86	C₈×C₇₁	C₂×D₈	1	1	yes	characteristic
87	C₈×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
88	C₇₁⋊C₈	—	2	1	no	—
89	C₇₁⋊C₈	—	2	1	no	—
90	C₇₁⋊C₈	—	2	1	no	—
91	C₇₁⋊C₈	—	4	1	no	—
92	C₇₁⋊C₈	#107	1	2	yes	—
93	C₂²×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
94	C₂×D₇₁	—	2	1	no	—
95	C₂×D₇₁	—	2	1	no	—
96	C₂×D₇₁	#335	1	2	yes	—
97	C₄×C₇₁	—	2	1	no	—
98	C₄×C₇₁	C₂×#107	1	1	yes	characteristic
99	C₄×C₇₁	C₂×#108	1	1	yes	characteristic
100	C₄×C₇₁	#504	1	1	yes	characteristic
101	C₄×C₇₁	#504	1	1	yes	characteristic
102	Dic₇₁	—	2	1	no	—
103	Dic₇₁	—	2	1	no	—
104	Dic₇₁	#335	1	2	yes	—
105	#12005	—	71	1	no	maximal, Sylow
106	C₂×C₇₁	C₂×#335	1	1	yes	characteristic
107	C₂×C₇₁	#2151	1	1	yes	characteristic
108	C₂×C₇₁	#2153	1	1	yes	characteristic
109	D₇₁	—	2	2	no	—
110	#2150	—	71	1	no	—
111	C₂×C₄⋊C₁₆	—	71	1	no	—
112	#1450	—	71	1	no	—
113	#1370	—	71	1	no	—
114	#1370	—	71	1	no	—
115	#1370	—	71	1	no	—
116	#1370	D₇₁	1	1	yes	characteristic, complemented
117	C₇₁	#12005	1	1	yes	characteristic, complemented, Sylow
118	C₂×M₃₂	—	142	1	no	—
119	C₂²×C₁₆	—	142	1	no	—
120	#431	—	71	1	no	—
121	#431	—	71	1	no	—
122	C₂×C₄×C₈	—	71	1	no	—
123	C₄⋊C₁₆	—	71	1	no	—
124	C₄⋊C₁₆	—	71	1	no	—
125	C₄⋊C₁₆	—	142	1	no	—
126	C₄⋊C₁₆	—	142	1	no	—
127	C₄⋊C₁₆	C₂×D₇₁	1	1	yes	characteristic
128	C₄⋊C₁₆	C₂×D₇₁	1	1	yes	characteristic
129	#322	—	71	1	no	—
130	#322	—	71	1	no	—
131	C₈²	—	71	1	no	—
132	C₈²	C₂×D₇₁	1	1	yes	characteristic
133	C₂²×C₈	—	71	1	no	—
134	C₂²×C₈	—	71	1	no	—
135	C₂×C₄²	—	71	1	no	—
136	M₃₂	—	284	2	no	—
137	C₂×C₁₆	—	2	1	no	—
138	C₂×C₁₆	—	2	1	no	—
139	C₂×C₁₆	—	284	2	no	—
140	C₂×C₁₆	—	142	1	no	—
141	C₂×C₁₆	—	142	1	no	—
142	C₄⋊C₈	—	71	1	no	—
143	C₄⋊C₈	—	71	1	no	—
144	C₄⋊C₈	—	142	1	no	—
145	C₂²⋊C₈	—	71	1	no	—
146	C₂²⋊C₈	—	71	1	no	—
147	C₂²⋊C₈	—	142	1	no	—
148	C₄×C₈	—	71	1	no	—
149	C₄×C₈	—	71	1	no	—
150	C₄×C₈	—	71	1	no	—
151	C₄×C₈	—	71	1	no	—
152	C₄×C₈	—	71	1	no	—
153	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
154	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
155	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
156	C₂²×C₄	—	71	1	no	—
157	C₂²×C₄	—	71	1	no	—
158	C₂²×C₄	—	71	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	71	1	no	—
161	C₂×C₈	—	71	2	no	—
162	C₂×C₈	—	71	2	no	—
163	C₂×C₈	—	71	1	no	—
164	C₂×C₈	—	71	1	no	—
165	C₂×C₈	—	71	1	no	—
166	C₂×C₈	—	142	2	no	—
167	C₂×C₈	—	142	1	no	—
168	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
169	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
170	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
171	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
172	C₄²	—	71	1	no	—
173	C₄²	—	71	1	no	—
174	C₄²	—	71	1	no	—
175	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
176	C₁₆	—	4	1	no	—
177	C₁₆	—	4	1	no	—
178	C₁₆	—	284	1	no	—
179	C₁₆	—	284	1	no	—
180	C₂³	—	71	1	no	—
181	C₂×C₄	—	284	1	no	—
182	C₂×C₄	—	71	1	no	—
183	C₂×C₄	—	71	2	no	—
184	C₂×C₄	—	71	2	no	—
185	C₂×C₄	—	71	1	no	—
186	C₂×C₄	—	71	1	no	—
187	C₂×C₄	—	142	1	no	—
188	C₂×C₄	—	142	1	no	—
189	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
190	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
191	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
192	C₈	—	2	1	no	—
193	C₈	—	2	1	no	—
194	C₈	—	2	1	no	—
195	C₈	—	4	1	no	—
196	C₈	—	284	1	no	—
197	C₈	—	71	2	no	—
198	C₈	—	142	1	no	—
199	C₈	—	142	1	no	—
200	C₈	—	142	1	no	—
201	C₈	D₈×D₇₁	1	1	yes	characteristic
202	C₈	Q₁₆×D₇₁	1	1	yes	characteristic
203	C₂²	—	71	2	no	—
204	C₂²	—	142	1	no	—
205	C₂²	—	142	1	no	—
206	C₂²	C₄⋊C₈×D₇₁	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	71	2	no	—
209	C₄	—	142	1	no	—
210	C₄	—	142	1	no	—
211	C₄	#107×D₇₁	1	1	yes	characteristic
212	C₄	#108×D₇₁	1	1	yes	characteristic
213	C₄	#17986415	1	1	yes	characteristic
214	C₄	#17986417	1	1	yes	characteristic
215	C₂	—	142	2	no	—
216	C₂	#335×D₇₁	1	1	yes	characteristic
217	C₂	#22590097	1	1	yes	characteristic
218	C₂	#22590104	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1370	C₂
socle	C₂²×C₇₁	C₂×C₄⋊C₈
prime radical	C₂²×D₇₁	C₄⋊C₈
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	#1370	D₇₁
hypocenter	C₇₁	#12005
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1370

Complements

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

N	H ≅ G/N	type
4. #22586410	C₂	semidirect
5. #22586409	C₂	semidirect
6. #22586409	C₂	semidirect
7. C₇₁×#1370	C₂	semidirect
116. #1370	D₇₁	semidirect
117. C₇₁	#12005	semidirect

Maximal subgroups

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

subgroup	index
#22590094	2
C₄⋊C₁₆×D₇₁	2
#22586622	2
#22586410	2
#22586409	2
C₇₁×#1370	2
#12005	71