Subgroups — #31912775

Order 18176.

358 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17985819	C₂	1	2	yes	maximal
2	C₄×#14556068	C₂	1	1	yes	maximal, characteristic
3	#22588026	C₂	1	1	yes	maximal, characteristic
4	#22586421	C₂	1	2	yes	maximal
5	C₄×M₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556068	C₂²	1	2	yes	—
7	C₂×#14556039	C₄	1	1	yes	characteristic
8	C₂×#14556038	C₄	1	1	yes	characteristic
9	C₂×#14556038	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
12	#17985845	C₄	1	1	yes	characteristic
13	#17985819	C₄	1	8	yes	complemented
14	#17985775	—	2	1	no	—
15	C₄×C₇₁⋊C₁₆	—	2	1	no	—
16	C₂×M₃₂×C₇₁	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
18	#346×C₇₁	—	2	1	no	—
19	C₄×C₁₆×C₇₁	—	2	1	no	—
20	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
24	#14556068	—	2	2	no	—
25	#14556068	C₂×C₄	1	4	yes	—
26	C₂×C₇₁⋊C₁₆	—	2	2	no	—
27	#14556039	C₈	1	4	yes	—
28	#14556038	—	2	2	no	—
29	#14556038	C₈	1	4	yes	—
30	C₈×Dic₇₁	—	2	2	no	—
31	C₈×Dic₇₁	C₂×C₄	1	4	yes	—
32	C₇₁⋊#95	—	2	2	no	—
33	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
36	M₃₂×C₇₁	—	2	2	no	—
37	M₃₂×C₇₁	C₂×C₄	1	4	yes	—
38	C₂×C₁₆×C₇₁	—	2	2	no	—
39	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
40	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
41	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
42	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
43	C₂²×Dic₇₁	—	2	1	no	—
44	C₂²×Dic₇₁	—	2	1	no	—
45	C₂×C₇₁⋊C₈	—	2	1	no	—
46	C₂×C₇₁⋊C₈	—	2	1	no	—
47	C₂×C₇₁⋊C₈	C₄²	1	2	yes	—
48	C₂×C₇₁⋊C₈	C₂×C₈	1	2	yes	—
49	C₂×C₇₁⋊C₈	C₂×C₈	1	2	yes	—
50	C₂×C₇₁⋊C₈	C₂×C₈	1	2	yes	—
51	C₄×Dic₇₁	—	2	1	no	—
52	C₄×Dic₇₁	—	2	1	no	—
53	C₄×Dic₇₁	C₄²	1	2	yes	—
54	C₄×Dic₇₁	C₂×C₈	1	2	yes	—
55	C₇₁⋊C₁₆	—	2	4	no	—
56	C₂²×C₄×C₇₁	M₁₆	1	2	yes	—
57	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
61	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
62	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
63	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₈×C₇₁	#41	1	2	yes	—
66	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
67	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
68	C₄²×C₇₁	C₂×C₈	1	2	yes	—
69	C₁₆×C₇₁	—	2	4	no	—
70	C₂³×C₇₁	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₄×C₇₁	—	4	2	no	—
72	C₂×C₄×C₇₁	C₄⋊C₈	1	2	yes	—
73	C₂×C₄×C₇₁	C₄⋊C₈	1	2	yes	—
74	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
77	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
80	C₂×Dic₇₁	—	2	2	no	—
81	C₂×Dic₇₁	—	2	2	no	—
82	C₂×Dic₇₁	—	4	2	no	—
83	C₂×Dic₇₁	—	4	2	no	—
84	C₈×C₇₁	C₄⋊C₈	1	2	yes	—
85	C₈×C₇₁	C₄⋊C₈	1	2	yes	—
86	C₈×C₇₁	C₄×D₄	1	2	yes	—
87	C₈×C₇₁	C₄×Q₈	1	2	yes	—
88	C₇₁⋊C₈	—	2	2	no	—
89	C₇₁⋊C₈	C₄×C₈	1	4	yes	—
90	C₂²×C₇₁	—	2	2	no	—
91	C₂²×C₇₁	C₄×M₁₆	1	1	yes	characteristic
92	C₂²×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
93	C₂²×C₇₁	#431	1	1	yes	characteristic
94	C₄×C₇₁	—	2	2	no	—
95	C₄×C₇₁	—	2	2	no	—
96	C₄×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
97	C₄×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
98	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
99	C₄×C₇₁	#431	1	1	yes	characteristic
100	Dic₇₁	—	4	2	no	—
101	Dic₇₁	—	4	2	no	—
102	#12656	—	71	1	no	maximal, Sylow
103	C₂×C₇₁	—	2	1	no	—
104	C₂×C₇₁	—	2	1	no	—
105	C₂×C₇₁	C₄×C₄⋊C₈	1	1	yes	characteristic
106	C₂×C₇₁	#2153	1	2	yes	—
107	C₂×#364	—	71	2	no	—
108	C₄×M₃₂	—	71	1	no	—
109	C₄×M₃₂	D₇₁	1	1	yes	characteristic
110	#1726	—	71	1	no	—
111	#1376	—	71	2	no	—
112	C₇₁	#12656	1	1	yes	characteristic, complemented, Sylow
113	C₂×M₃₂	—	71	2	no	—
114	C₂×M₃₂	C₂×D₇₁	1	2	yes	—
115	C₂×#97	—	71	1	no	—
116	C₂×#97	—	71	1	no	—
117	C₂×#97	—	71	1	no	—
118	C₂×C₄×C₈	—	71	1	no	—
119	C₂×C₄×C₈	—	71	1	no	—
120	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
121	#376	—	71	1	no	—
122	#364	—	71	8	no	—
123	#346	—	2	1	no	—
124	#346	—	142	1	no	—
125	C₄×C₁₆	—	2	1	no	—
126	C₄×C₁₆	—	142	1	no	—
127	C₂²×C₈	—	71	1	no	—
128	C₂²×C₈	—	71	1	no	—
129	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
131	C₂×C₄²	—	71	1	no	—
132	C₂×C₄²	—	71	1	no	—
133	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
134	M₃₂	—	2	2	no	—
135	M₃₂	—	71	4	no	—
136	M₃₂	—	142	2	no	—
137	M₃₂	C₄×D₇₁	1	4	yes	—
138	C₂×C₁₆	—	2	2	no	—
139	C₂×C₁₆	—	142	2	no	—
140	#97	—	71	4	no	—
141	#97	—	71	4	no	—
142	#97	—	142	2	no	—
143	C₄×C₈	—	71	4	no	—
144	C₄×C₈	—	142	2	no	—
145	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
146	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
147	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
148	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
149	#95	—	142	2	no	—
150	C₂²×C₄	—	142	1	no	—
151	C₂²×C₄	—	142	1	no	—
152	C₂²×C₄	#11765848	1	2	yes	—
153	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
154	C₂×C₈	—	71	2	no	—
155	C₂×C₈	—	71	2	no	—
156	C₂×C₈	—	71	2	no	—
157	C₂×C₈	—	71	2	no	—
158	C₂×C₈	—	142	1	no	—
159	C₂×C₈	—	142	1	no	—
160	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
161	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
162	C₂×C₈	C₈×D₇₁	1	2	yes	—
163	C₂×C₈	#11765848	1	2	yes	—
164	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
165	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
166	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
167	C₂×C₈	#11765865	1	2	yes	—
168	C₄²	—	71	2	no	—
169	C₄²	—	71	2	no	—
170	C₄²	—	142	1	no	—
171	C₄²	—	142	1	no	—
172	C₄²	Dic₇₁⋊C₄	1	1	yes	characteristic
173	C₄²	Dic₇₁⋊C₄	1	1	yes	characteristic
174	C₄²	C₈×D₇₁	1	2	yes	—
175	C₁₆	—	2	4	no	—
176	C₁₆	—	142	4	no	—
177	C₂³	C₂×#11765848	1	1	yes	characteristic
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	284	2	no	—
180	C₂×C₄	—	284	2	no	—
181	C₂×C₄	—	142	2	no	—
182	C₂×C₄	—	142	2	no	—
183	C₂×C₄	Dic₇₁⋊C₈	1	2	yes	—
184	C₂×C₄	Dic₇₁⋊C₈	1	2	yes	—
185	C₂×C₄	C₄²×D₇₁	1	1	yes	characteristic
186	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
187	C₂×C₄	#14556085	1	1	yes	characteristic
188	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
189	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
190	C₂×C₄	C₂×#11765848	1	1	yes	characteristic
191	C₈	—	71	4	no	—
192	C₈	—	142	2	no	—
193	C₈	Dic₇₁⋊C₈	1	2	yes	—
194	C₈	Dic₇₁⋊C₈	1	2	yes	—
195	C₈	C₄×C₂²⋊D₇₁	1	2	yes	—
196	C₈	C₄×Dic₁₄₂	1	2	yes	—
197	C₂²	—	2	2	no	—
198	C₂²	C₄×#11765848	1	1	yes	characteristic
199	C₂²	C₂×Dic₇₁⋊C₈	1	1	yes	characteristic
200	C₂²	#17986050	1	1	yes	characteristic
201	C₄	—	2	2	no	—
202	C₄	—	2	2	no	—
203	C₄	—	284	2	no	—
204	C₄	—	284	2	no	—
205	C₄	C₄×Dic₇₁⋊C₄	1	1	yes	characteristic
206	C₄	C₄×C₈×D₇₁	1	1	yes	characteristic
207	C₄	C₂×Dic₇₁⋊C₈	1	1	yes	characteristic
208	C₄	#17986050	1	1	yes	characteristic
209	C₂	—	2	1	no	—
210	C₂	—	2	1	no	—
211	C₂	C₄×Dic₇₁⋊C₈	1	1	yes	characteristic
212	C₂	#22590105	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₇₁
commutator	C₄×C₇₁	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₄×M₃₂×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₄⋊C₈
prime radical	C₂³×C₇₁	C₂×M₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₄×M₃₂	D₇₁
hypocenter	C₇₁	#12656
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
13. #17985819	C₄	semidirect
112. C₇₁	#12656	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17985819	2
C₄×#14556068	2
#22588026	2
#22586421	2
C₄×M₃₂×C₇₁	2
#12656	71