Subgroups — #31891197

Order 18176.

422 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	#22588457	C₂	1	2	yes	maximal
2	C₂×#17985752	C₂	1	2	yes	maximal
3	C₂×#17985750	C₂	1	2	yes	maximal
4	C₄²×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	4	yes	—
6	#17985754	—	2	4	no	—
7	#17985752	—	2	4	no	—
8	#17985750	—	2	4	no	—
9	C₂×C₄×C₈×C₇₁	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₇₁	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₇₁	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₇₁	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₄³×C₇₁	C₂²	1	1	yes	characteristic
16	C₂²×C₇₁⋊C₈	—	2	4	no	—
17	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
18	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
19	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₇₁	D₄	1	2	yes	—
26	C₂×C₄²×C₇₁	Q₈	1	2	yes	—
27	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
28	C₄×C₈×C₇₁	—	2	2	no	—
29	C₄×C₈×C₇₁	—	2	2	no	—
30	C₄×C₈×C₇₁	—	2	2	no	—
31	C₄×C₈×C₇₁	C₈	1	4	yes	complemented
32	C₄×C₈×C₇₁	D₄	1	2	yes	—
33	C₄×C₈×C₇₁	D₄	1	2	yes	—
34	C₄×C₈×C₇₁	Q₈	1	2	yes	—
35	C₄×C₈×C₇₁	Q₈	1	2	yes	—
36	C₂×C₇₁⋊C₈	—	4	4	no	—
37	C₂×C₇₁⋊C₈	—	4	8	no	—
38	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₇₁	—	2	1	no	—
44	C₂×C₈×C₇₁	—	2	1	no	—
45	C₂×C₈×C₇₁	—	2	1	no	—
46	C₂×C₈×C₇₁	—	2	1	no	—
47	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
50	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
51	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
52	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
53	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
54	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
55	C₄²×C₇₁	—	2	4	no	—
56	C₄²×C₇₁	—	2	1	no	—
57	C₄²×C₇₁	—	2	1	no	—
58	C₄²×C₇₁	—	2	4	no	—
59	C₄²×C₇₁	C₂×C₈	1	2	yes	—
60	C₄²×C₇₁	M₁₆	1	2	yes	—
61	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
64	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
65	C₂³×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₇₁	—	2	2	no	—
67	C₂×C₄×C₇₁	—	2	2	no	—
68	C₂×C₄×C₇₁	—	2	2	no	—
69	C₂×C₄×C₇₁	—	2	2	no	—
70	C₂×C₄×C₇₁	—	2	2	no	—
71	C₂×C₄×C₇₁	C₄⋊C₈	1	4	yes	—
72	C₂×C₄×C₇₁	#108	1	4	yes	—
73	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
76	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
83	C₂×C₄×C₇₁	#135	1	4	yes	—
84	C₈×C₇₁	—	2	2	no	—
85	C₈×C₇₁	—	2	2	no	—
86	C₈×C₇₁	C₄⋊C₈	1	4	yes	—
87	C₈×C₇₁	C₄⋊C₈	1	4	yes	—
88	C₇₁⋊C₈	—	8	4	no	—
89	C₂²×C₇₁	#384	1	1	yes	characteristic
90	C₂²×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
91	C₂²×C₇₁	#423	1	1	yes	characteristic
92	C₂²×C₇₁	#427	1	2	yes	—
93	C₂²×C₇₁	C₂×#108	1	2	yes	—
94	C₄×C₇₁	—	2	4	no	—
95	C₄×C₇₁	—	2	4	no	—
96	C₄×C₇₁	—	2	1	no	—
97	C₄×C₇₁	—	2	1	no	—
98	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
99	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
100	C₄×C₇₁	#423	1	1	yes	characteristic
101	C₄×C₇₁	#423	1	1	yes	characteristic
102	C₄×C₇₁	#495	1	2	yes	—
103	C₄×C₇₁	#499	1	2	yes	—
104	#8807	—	71	1	no	maximal, Sylow
105	C₂×C₇₁	#1564	1	2	yes	—
106	C₂×C₇₁	#1568	1	2	yes	—
107	C₂×C₇₁	#1842	1	1	yes	characteristic
108	C₂×C₇₁	#1846	1	1	yes	characteristic
109	C₂×C₇₁	#1848	1	1	yes	characteristic
110	#1831	—	71	2	no	—
111	C₄²×C₈	D₇₁	1	1	yes	characteristic
112	C₂×#335	—	71	2	no	—
113	C₂×#334	—	71	2	no	—
114	C₇₁	#8807	1	1	yes	characteristic, complemented, Sylow
115	C₂×C₄⋊C₈	—	71	4	no	—
116	C₂×C₄×C₈	Dic₇₁	1	1	yes	characteristic
117	C₂×C₄×C₈	Dic₇₁	1	1	yes	characteristic
118	C₂×C₄×C₈	Dic₇₁	1	1	yes	characteristic
119	C₂×C₄×C₈	Dic₇₁	1	1	yes	characteristic
120	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
121	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
122	C₄³	C₂×D₇₁	1	1	yes	characteristic
123	#336	—	142	4	no	—
124	#335	—	142	4	no	—
125	#334	—	142	4	no	—
126	C₂²×C₈	—	142	4	no	—
127	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
128	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
129	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
130	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
131	C₂×C₄²	Dic₁₄₂	1	2	yes	—
132	C₂×C₄²	D₂₈₄	1	2	yes	—
133	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
134	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
135	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
136	C₄⋊C₈	—	142	4	no	—
137	C₄⋊C₈	—	142	4	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	—	2	2	no	—
140	C₄×C₈	—	2	2	no	—
141	C₄×C₈	C₇₁⋊C₈	1	4	yes	complemented
142	C₄×C₈	Dic₁₄₂	1	2	yes	—
143	C₄×C₈	Dic₁₄₂	1	2	yes	—
144	C₄×C₈	D₂₈₄	1	2	yes	—
145	C₄×C₈	D₂₈₄	1	2	yes	—
146	C₂²×C₄	C₄⋊Dic₇₁	1	2	yes	—
147	C₂²×C₄	C₄⋊Dic₇₁	1	2	yes	—
148	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
149	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
150	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	284	4	no	—
156	C₂×C₈	—	284	8	no	—
157	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
158	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
159	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
160	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
161	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
162	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
163	C₂×C₈	#11765849	1	2	yes	—
164	C₂×C₈	#11765849	1	2	yes	—
165	C₄²	—	2	4	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	2	1	no	—
168	C₄²	—	2	4	no	—
169	C₄²	C₂×C₇₁⋊C₈	1	2	yes	—
170	C₄²	#11765849	1	2	yes	—
171	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
172	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
173	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
174	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
175	C₂³	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	4	yes	—
182	C₂×C₄	#14556064	1	4	yes	—
183	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
184	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
185	C₂×C₄	#14556125	1	1	yes	characteristic
186	C₂×C₄	#14556128	1	1	yes	characteristic
187	C₂×C₄	#14556128	1	1	yes	characteristic
188	C₂×C₄	#14556128	1	1	yes	characteristic
189	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
190	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
191	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
192	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
193	C₂×C₄	#14556150	1	4	yes	—
194	C₈	—	2	2	no	—
195	C₈	—	2	2	no	—
196	C₈	—	568	4	no	—
197	C₈	C₄⋊(C₇₁⋊C₈)	1	4	yes	—
198	C₈	C₄⋊(C₇₁⋊C₈)	1	4	yes	—
199	C₂²	#17985869	1	1	yes	characteristic
200	C₂²	C₂×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
201	C₂²	#17986023	1	1	yes	characteristic
202	C₂²	#17986037	1	2	yes	—
203	C₂²	C₂×#14556064	1	2	yes	—
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	4	no	—
207	C₄	—	2	4	no	—
208	C₄	C₂×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
209	C₄	C₂×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
210	C₄	#17986023	1	1	yes	characteristic
211	C₄	#17986023	1	1	yes	characteristic
212	C₄	#17986383	1	2	yes	—
213	C₄	#17986401	1	2	yes	—
214	C₂	#22587088	1	2	yes	—
215	C₂	#22587103	1	2	yes	—
216	C₂	#22588492	1	1	yes	characteristic
217	C₂	#22588504	1	1	yes	characteristic
218	C₂	#22588512	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₄²×C₈×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₄²×C₈	D₇₁
hypocenter	C₇₁	#8807
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
31. C₄×C₈×C₇₁	C₈	semidirect
114. C₇₁	#8807	semidirect
141. C₄×C₈	C₇₁⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22588457	2
C₂×#17985752	2
C₂×#17985750	2
C₄²×C₈×C₇₁	2
#8807	71