Subgroups — #31913919

Order 18176.

438 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556068	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₇₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	#22586418	C₂	1	2	yes	maximal, complemented
5	C₈×C₂²⋊C₄×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556068	—	2	4	no	—
7	C₂×#14556068	—	2	2	no	—
8	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
11	C₄⋊(C₇₁⋊C₁₆)	—	2	4	no	—
12	C₂²⋊(C₇₁⋊C₁₆)	—	2	2	no	—
13	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊C₈×C₇₁	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
16	C₄×C₂²⋊C₄×C₇₁	C₄	1	1	yes	characteristic
17	#336×C₇₁	C₄	1	2	yes	—
18	#14556068	—	2	8	no	—
19	C₂×C₇₁⋊C₁₆	—	2	2	no	—
20	C₂×C₇₁⋊C₁₆	—	2	2	no	—
21	C₂×C₇₁⋊C₁₆	—	2	2	no	—
22	C₂×C₇₁⋊C₁₆	—	2	1	no	—
23	C₂×C₇₁⋊C₁₆	—	2	1	no	—
24	C₂×C₇₁⋊C₁₆	—	2	2	no	—
25	C₂×C₇₁⋊C₁₆	D₄	1	4	yes	—
26	C₂³×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₁	—	2	2	no	—
28	C₂²×C₈×C₇₁	—	2	2	no	—
29	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₁	D₄	1	2	yes	—
32	C₂²×C₈×C₇₁	Q₈	1	2	yes	—
33	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₇₁	C₈	1	2	yes	—
35	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
36	C₂²⋊C₈×C₇₁	—	2	2	no	—
37	C₄×C₈×C₇₁	—	2	2	no	—
38	C₄×C₈×C₇₁	—	2	2	no	—
39	#95×C₇₁	C₈	1	1	yes	characteristic
40	#95×C₇₁	C₈	1	1	yes	characteristic
41	C₇₁⋊C₁₆	—	2	4	no	—
42	C₇₁⋊C₁₆	—	4	1	no	—
43	C₇₁⋊C₁₆	—	4	1	no	—
44	C₂⁴×C₇₁	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₇₁	—	2	2	no	—
46	C₂²×C₄×C₇₁	—	2	2	no	—
47	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₇₁	M₁₆	1	2	yes	—
52	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₇₁	—	2	2	no	—
54	C₂×C₈×C₇₁	—	2	2	no	—
55	C₂×C₈×C₇₁	—	2	4	no	—
56	C₂×C₈×C₇₁	—	2	2	no	—
57	C₂×C₈×C₇₁	—	2	2	no	—
58	C₂×C₈×C₇₁	—	2	2	no	—
59	C₂×C₈×C₇₁	—	2	2	no	—
60	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
61	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₈×C₇₁	#41	1	2	yes	—
65	C₂²⋊C₄×C₇₁	—	2	4	no	—
66	C₄²×C₇₁	—	2	2	no	—
67	C₄²×C₇₁	—	2	2	no	—
68	C₂³×C₇₁	—	2	2	no	—
69	C₂³×C₇₁	—	2	2	no	—
70	C₂³×C₇₁	—	2	1	no	—
71	C₂³×C₇₁	—	2	1	no	—
72	C₂³×C₇₁	C₄⋊C₈	1	2	yes	—
73	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₇₁	—	2	2	no	—
75	C₂×C₄×C₇₁	—	2	2	no	—
76	C₂×C₄×C₇₁	—	2	2	no	—
77	C₂×C₄×C₇₁	—	2	2	no	—
78	C₂×C₄×C₇₁	—	2	2	no	—
79	C₂×C₄×C₇₁	—	2	4	no	—
80	C₂×C₄×C₇₁	—	2	2	no	—
81	C₂×C₄×C₇₁	—	2	2	no	—
82	C₂×C₄×C₇₁	—	2	2	no	—
83	C₂×C₄×C₇₁	C₄⋊C₈	1	2	yes	—
84	C₂×C₄×C₇₁	M₃₂	1	4	yes	complemented
85	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
87	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
88	C₂×C₄×C₇₁	#131	1	2	yes	—
89	C₈×C₇₁	—	2	2	no	—
90	C₈×C₇₁	—	4	2	no	—
91	C₈×C₇₁	#120	1	1	yes	characteristic
92	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
93	C₈×C₇₁	#123	1	1	yes	characteristic
94	C₂²×C₇₁	—	2	2	no	—
95	C₂²×C₇₁	—	2	2	no	—
96	C₂²×C₇₁	—	2	2	no	—
97	C₂²×C₇₁	—	4	2	no	—
98	C₂²×C₇₁	—	4	2	no	—
99	C₂²×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
100	C₂²×C₇₁	C₈×D₄	1	2	yes	—
101	C₂²×C₇₁	C₂×M₃₂	1	2	yes	—
102	C₂²×C₇₁	#504	1	1	yes	characteristic
103	C₂²×C₇₁	#504	1	1	yes	characteristic
104	C₄×C₇₁	—	2	2	no	—
105	C₄×C₇₁	—	2	4	no	—
106	C₄×C₇₁	—	4	2	no	—
107	C₄×C₇₁	#385	1	1	yes	characteristic
108	C₄×C₇₁	#424	1	1	yes	characteristic
109	C₄×C₇₁	#435	1	2	yes	—
110	#12913	—	71	1	no	maximal, Sylow
111	C₂×C₇₁	—	4	2	no	—
112	C₂×C₇₁	#1850	1	1	yes	characteristic
113	C₂×C₇₁	#2150	1	1	yes	characteristic
114	C₂×C₇₁	#2151	1	1	yes	characteristic
115	C₂×C₇₁	#2168	1	2	yes	—
116	C₂×C₇₁	#2169	1	2	yes	—
117	C₂²×M₃₂	—	71	1	no	—
118	C₂×C₄⋊C₁₆	—	71	2	no	—
119	C₂×C₂²⋊C₁₆	—	71	1	no	—
120	C₈×C₂²⋊C₄	D₇₁	1	1	yes	characteristic
121	#1374	—	71	2	no	—
122	C₇₁	#12913	1	1	yes	characteristic, complemented, Sylow
123	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
124	C₂×M₃₂	—	142	4	no	—
125	C₂×M₃₂	—	142	2	no	—
126	C₂²×C₁₆	—	71	2	no	—
127	C₂²×C₁₆	—	71	1	no	—
128	C₂²×C₁₆	—	71	1	no	—
129	C₂×C₂²⋊C₈	Dic₇₁	1	1	yes	characteristic
130	C₂×C₄×C₈	C₂×D₇₁	1	2	yes	—
131	C₄×C₂²⋊C₄	Dic₇₁	1	1	yes	characteristic
132	C₄⋊C₁₆	—	142	4	no	—
133	C₂²⋊C₁₆	—	142	2	no	—
134	#336	Dic₇₁	1	2	yes	—
135	C₂³×C₄	C₂×Dic₇₁	1	1	yes	characteristic
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	—	2	2	no	—
138	C₂²×C₈	Dic₁₄₂	1	2	yes	—
139	C₂²×C₈	D₂₈₄	1	2	yes	—
140	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
141	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
142	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
143	C₂×C₂²⋊C₄	C₇₁⋊C₈	1	2	yes	—
144	C₂×C₄²	C₂×Dic₇₁	1	2	yes	—
145	M₃₂	—	142	8	no	—
146	C₂×C₁₆	—	71	4	no	—
147	C₂×C₁₆	—	142	2	no	—
148	C₂×C₁₆	—	142	1	no	—
149	C₂×C₁₆	—	142	1	no	—
150	C₂×C₁₆	—	142	2	no	—
151	C₂×C₁₆	—	142	2	no	—
152	C₂×C₁₆	—	142	2	no	—
153	C₂²⋊C₈	—	2	2	no	—
154	C₄×C₈	—	2	2	no	—
155	C₄×C₈	—	2	2	no	—
156	#95	C₇₁⋊C₈	1	1	yes	characteristic
157	#95	C₇₁⋊C₈	1	1	yes	characteristic
158	C₂⁴	C₂×C₇₁⋊C₈	1	1	yes	characteristic
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	C₄⋊Dic₇₁	1	2	yes	—
162	C₂²×C₄	C₂×C₇₁⋊C₈	1	2	yes	—
163	C₂²×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
164	C₂²×C₄	C₂×C₇₁⋊C₈	1	2	yes	—
165	C₂²×C₄	#11765849	1	2	yes	—
166	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	4	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
175	C₂×C₈	D₄×D₇₁	1	2	yes	—
176	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
177	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
178	C₂×C₈	#11765864	1	2	yes	—
179	C₂²⋊C₄	—	2	4	no	—
180	C₄²	—	2	2	no	—
181	C₄²	—	2	2	no	—
182	C₁₆	—	284	1	no	—
183	C₁₆	—	284	1	no	—
184	C₁₆	—	142	4	no	—
185	C₂³	—	2	2	no	—
186	C₂³	—	2	2	no	—
187	C₂³	—	2	1	no	—
188	C₂³	—	2	1	no	—
189	C₂³	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
190	C₂³	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
201	C₂×C₄	#14556068	1	4	yes	complemented
202	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
203	C₂×C₄	D₄×Dic₇₁	1	2	yes	—
204	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
205	C₂×C₄	#14556135	1	2	yes	—
206	C₈	—	2	2	no	—
207	C₈	—	4	2	no	—
208	C₈	#14556098	1	1	yes	characteristic
209	C₈	#14556105	1	2	yes	—
210	C₈	#14556112	1	1	yes	characteristic
211	C₂²	—	2	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	4	2	no	—
215	C₂²	—	4	2	no	—
216	C₂²	C₂×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
217	C₂²	D₄×C₇₁⋊C₈	1	2	yes	—
218	C₂²	C₂×#14556068	1	2	yes	—
219	C₂²	#17986416	1	1	yes	characteristic
220	C₂²	#17986416	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	2	4	no	—
223	C₄	—	4	2	no	—
224	C₄	#17985874	1	1	yes	characteristic
225	C₄	#17986026	1	1	yes	characteristic
226	C₄	#17986071	1	2	yes	—
227	C₂	—	4	2	no	—
228	C₂	#22588520	1	1	yes	characteristic
229	C₂	#22590096	1	1	yes	characteristic
230	C₂	#22590099	1	1	yes	characteristic
231	C₂	#22590166	1	2	yes	—
232	C₂	#22590172	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₁
commutator, socle	C₂³×C₇₁	C₂²×C₈
Fitting	C₈×C₂²⋊C₄×C₇₁	C₂
prime radical	C₂⁴×C₇₁	C₂×C₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₈×C₂²⋊C₄	D₇₁
hypocenter	C₇₁	#12913
LCS	G ▷ C₂³×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₈×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	semidirect
4. #22586418	C₂	semidirect
84. C₂×C₄×C₇₁	M₃₂	semidirect
122. C₇₁	#12913	semidirect
201. C₂×C₄	#14556068	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14556068	2
C₂×C₄⋊(C₇₁⋊C₁₆)	2
C₂×C₂²⋊(C₇₁⋊C₁₆)	2
#22586418	2
C₈×C₂²⋊C₄×C₇₁	2
#12913	71