Subgroups — #31907159

Order 18176.

450 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590096	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#14556068	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊(C₇₁⋊C₁₆)	C₂	1	4	yes	maximal, complemented
4	C₂×C₈×D₄×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556068	—	2	1	no	—
6	C₂×#14556068	—	2	1	no	—
7	C₄⋊(C₇₁⋊C₁₆)	—	2	1	no	—
8	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	complemented
9	#17985775	—	2	1	no	—
10	C₄×C₇₁⋊C₁₆	C₂²	1	2	yes	complemented
11	C₂³×C₈×C₇₁	—	2	1	no	—
12	C₂×C₄×D₄×C₇₁	C₄	1	1	yes	characteristic
13	C₈×D₄×C₇₁	—	2	1	no	—
14	C₈×D₄×C₇₁	—	2	1	no	—
15	C₈×D₄×C₇₁	C₄	1	2	yes	—
16	C₈×D₄×C₇₁	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₇₁	C₄	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₇₁	—	2	1	no	—
19	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
20	#14556068	—	2	4	no	—
21	#14556068	—	4	2	no	—
22	C₂×C₇₁⋊C₁₆	—	2	2	no	—
23	C₂×C₇₁⋊C₁₆	—	2	2	no	—
24	C₂²×D₄×C₇₁	C₈	1	1	yes	characteristic
25	C₂³×C₄×C₇₁	—	2	1	no	—
26	C₂²×C₈×C₇₁	—	2	1	no	—
27	C₂²×C₈×C₇₁	—	2	2	no	—
28	C₂²×C₈×C₇₁	—	2	2	no	—
29	C₂²×C₈×C₇₁	—	4	2	no	—
30	C₂²×C₈×C₇₁	—	4	2	no	—
31	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
33	C₄×D₄×C₇₁	—	2	1	no	—
34	C₄×D₄×C₇₁	—	2	1	no	—
35	C₄×D₄×C₇₁	C₈	1	2	yes	—
36	C₄×D₄×C₇₁	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₇₁	C₈	1	1	yes	characteristic
38	C₂×C₂²⋊C₄×C₇₁	—	2	1	no	—
39	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
40	C₄⋊C₈×C₇₁	—	2	1	no	—
41	C₄⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
42	C₂²⋊C₈×C₇₁	—	2	2	no	—
43	C₂²⋊C₈×C₇₁	—	2	2	no	—
44	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
46	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
47	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
48	C₇₁⋊C₁₆	—	2	4	no	—
49	C₇₁⋊C₁₆	—	4	2	no	—
50	C₂⁴×C₇₁	—	2	1	no	—
51	C₂×D₄×C₇₁	—	2	4	no	—
52	C₂×D₄×C₇₁	—	2	1	no	—
53	C₂×D₄×C₇₁	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₇₁	—	2	1	no	—
55	C₂²×C₄×C₇₁	—	2	2	no	—
56	C₂²×C₄×C₇₁	—	2	2	no	—
57	C₂²×C₄×C₇₁	—	4	2	no	—
58	C₂²×C₄×C₇₁	—	4	2	no	—
59	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₇₁	M₁₆	1	1	yes	characteristic
62	C₂×C₈×C₇₁	—	2	2	no	—
63	C₂×C₈×C₇₁	—	2	2	no	—
64	C₂×C₈×C₇₁	—	4	2	no	—
65	C₂×C₈×C₇₁	—	4	4	no	—
66	C₂×C₈×C₇₁	—	4	2	no	—
67	C₂×C₈×C₇₁	—	4	2	no	—
68	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₇₁	D₈	1	2	yes	—
71	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
72	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
74	C₄⋊C₄×C₇₁	—	2	1	no	—
75	C₄⋊C₄×C₇₁	C₂×C₈	1	2	yes	—
76	C₂²⋊C₄×C₇₁	—	2	2	no	—
77	C₂²⋊C₄×C₇₁	—	2	2	no	—
78	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
79	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
80	C₄²×C₇₁	M₁₆	1	1	yes	characteristic
81	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
82	C₂³×C₇₁	—	2	4	no	—
83	C₂³×C₇₁	—	2	2	no	—
84	C₂³×C₇₁	—	4	2	no	—
85	C₂³×C₇₁	—	4	2	no	—
86	C₂³×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
87	D₄×C₇₁	—	2	4	no	—
88	D₄×C₇₁	—	2	2	no	—
89	D₄×C₇₁	M₃₂	1	4	yes	complemented
90	C₂×C₄×C₇₁	—	2	2	no	—
91	C₂×C₄×C₇₁	—	2	2	no	—
92	C₂×C₄×C₇₁	—	4	1	no	—
93	C₂×C₄×C₇₁	—	4	2	no	—
94	C₂×C₄×C₇₁	—	4	4	no	—
95	C₂×C₄×C₇₁	—	4	2	no	—
96	C₂×C₄×C₇₁	—	4	2	no	—
97	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
98	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
99	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
100	C₂×C₄×C₇₁	#102	1	2	yes	—
101	C₂×C₄×C₇₁	#104	1	2	yes	—
102	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
105	C₈×C₇₁	—	2	1	no	—
106	C₈×C₇₁	—	2	1	no	—
107	C₈×C₇₁	—	4	2	no	—
108	C₈×C₇₁	#102	1	2	yes	—
109	C₈×C₇₁	C₂×D₈	1	1	yes	characteristic
110	C₈×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
111	C₂²×C₇₁	—	2	1	no	—
112	C₂²×C₇₁	—	2	1	no	—
113	C₂²×C₇₁	—	2	4	no	—
114	C₂²×C₇₁	—	4	8	no	—
115	C₂²×C₇₁	—	4	2	no	—
116	C₂²×C₇₁	—	4	2	no	—
117	C₂²×C₇₁	D₄⋊C₈	1	2	yes	—
118	C₂²×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
119	C₄×C₇₁	—	2	1	no	—
120	C₄×C₇₁	—	2	1	no	—
121	C₄×C₇₁	—	2	1	no	—
122	C₄×C₇₁	—	4	2	no	—
123	C₄×C₇₁	D₄⋊C₈	1	2	yes	—
124	C₄×C₇₁	C₂×#102	1	1	yes	characteristic
125	C₄×C₇₁	C₂×#104	1	1	yes	characteristic
126	C₄×C₇₁	C₂×M₃₂	1	2	yes	—
127	#11583	—	71	1	no	maximal, Sylow
128	C₂×C₇₁	—	2	2	no	—
129	C₂×C₇₁	—	4	4	no	—
130	C₂×C₇₁	C₂×D₄⋊C₈	1	1	yes	characteristic
131	C₂×C₇₁	#2112	1	1	yes	characteristic
132	C₂×C₇₁	#2117	1	1	yes	characteristic
133	C₂×C₈×D₄	D₇₁	1	1	yes	characteristic
134	#2150	—	71	1	no	—
135	C₄×M₃₂	—	71	1	no	—
136	D₄⋊C₁₆	—	71	4	no	—
137	C₇₁	#11583	1	1	yes	characteristic, complemented, Sylow
138	C₂³×C₈	—	2	1	no	—
139	C₂×C₄×D₄	Dic₇₁	1	1	yes	characteristic
140	C₂×M₃₂	—	142	1	no	—
141	C₂×M₃₂	—	142	1	no	—
142	C₈×D₄	—	2	1	no	—
143	C₈×D₄	—	2	1	no	—
144	C₈×D₄	Dic₇₁	1	2	yes	—
145	C₈×D₄	C₂×D₇₁	1	2	yes	—
146	C₂×C₄⋊C₈	Dic₇₁	1	1	yes	characteristic
147	C₂×C₂²⋊C₈	—	2	1	no	—
148	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
149	C₄⋊C₁₆	—	71	2	no	—
150	C₄⋊C₁₆	—	142	1	no	—
151	#346	—	142	1	no	—
152	C₄×C₁₆	—	71	2	no	—
153	C₂²×D₄	C₇₁⋊C₈	1	1	yes	characteristic
154	C₂³×C₄	—	2	1	no	—
155	C₂²×C₈	—	2	1	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	4	2	no	—
159	C₂²×C₈	—	4	2	no	—
160	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
161	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
162	C₄×D₄	—	2	1	no	—
163	C₄×D₄	—	2	1	no	—
164	C₄×D₄	C₇₁⋊C₈	1	2	yes	—
165	C₄×D₄	C₂×Dic₇₁	1	2	yes	—
166	C₂×C₄⋊C₄	C₇₁⋊C₈	1	1	yes	characteristic
167	C₂×C₂²⋊C₄	—	2	1	no	—
168	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
169	M₃₂	—	284	2	no	—
170	M₃₂	—	142	4	no	—
171	C₂×C₁₆	—	142	2	no	—
172	C₂×C₁₆	—	142	2	no	—
173	C₄⋊C₈	—	2	1	no	—
174	C₄⋊C₈	C₂×Dic₇₁	1	2	yes	—
175	C₂²⋊C₈	—	2	2	no	—
176	C₂²⋊C₈	—	2	2	no	—
177	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
178	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
179	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
180	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
181	C₂⁴	—	2	1	no	—
182	C₂×D₄	—	2	4	no	—
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	C₂×C₇₁⋊C₈	1	2	yes	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
192	C₂²×C₄	#11765849	1	1	yes	characteristic
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	4	4	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
200	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
201	C₂×C₈	#11765850	1	2	yes	—
202	C₂×C₈	#11765853	1	2	yes	—
203	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
204	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	C₂×C₇₁⋊C₈	1	2	yes	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	2	2	no	—
209	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
210	C₄²	C₂×C₇₁⋊C₈	1	1	yes	characteristic
211	C₄²	#11765849	1	1	yes	characteristic
212	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
213	C₁₆	—	284	2	no	—
214	C₁₆	—	142	4	no	—
215	C₂³	—	2	4	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	4	2	no	—
218	C₂³	—	4	2	no	—
219	C₂³	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
220	D₄	—	2	4	no	—
221	D₄	—	2	2	no	—
222	D₄	#14556068	1	4	yes	complemented
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
231	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
232	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
233	C₂×C₄	#14556049	1	2	yes	—
234	C₂×C₄	#14556055	1	2	yes	—
235	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
236	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
237	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	4	2	no	—
241	C₈	#14556049	1	2	yes	—
242	C₈	C₂×#11765850	1	1	yes	characteristic
243	C₈	C₂×#11765853	1	1	yes	characteristic
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	4	no	—
247	C₂²	—	4	8	no	—
248	C₂²	—	4	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	D₄⋊(C₇₁⋊C₈)	1	2	yes	—
251	C₂²	C₂×C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	4	2	no	—
256	C₄	D₄⋊(C₇₁⋊C₈)	1	2	yes	—
257	C₄	C₂×#14556049	1	1	yes	characteristic
258	C₄	C₂×#14556055	1	1	yes	characteristic
259	C₄	C₂×#14556068	1	2	yes	—
260	C₂	—	2	2	no	—
261	C₂	—	4	4	no	—
262	C₂	C₂×D₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
263	C₂	#22589951	1	1	yes	characteristic
264	C₂	#22589968	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₂×C₈×D₄×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₂²⋊C₈
prime radical	C₂²×D₄×C₇₁	C₈
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	C₂×C₈×D₄	D₇₁
hypocenter	C₇₁	#11583
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #22590096	C₂	semidirect
2. C₄×#14556068	C₂	semidirect
3. D₄⋊(C₇₁⋊C₁₆)	C₂	semidirect
8. C₄⋊(C₇₁⋊C₁₆)	C₂²	semidirect
10. C₄×C₇₁⋊C₁₆	C₂²	semidirect
89. D₄×C₇₁	M₃₂	semidirect
137. C₇₁	#11583	semidirect
222. D₄	#14556068	semidirect

Maximal subgroups

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

subgroup	index
#22590096	2
C₄×#14556068	2
D₄⋊(C₇₁⋊C₁₆)	2
C₂×C₈×D₄×C₇₁	2
#11583	71