Subgroups — C₂×#22586395 #31908730

Order 18176.

350 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985752	C₂	1	1	yes	maximal, characteristic
3	#22586395	C₂	1	4	yes	maximal, direct product factor
4	C₂×#500×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₁⋊C₁₆	—	2	1	no	—
6	C₂×C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
7	C₄⋊(C₇₁⋊C₁₆)	—	2	1	no	—
8	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	—
9	#17985752	—	2	1	no	—
10	#17985752	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₇₁	—	2	1	no	—
12	#500×C₇₁	—	2	1	no	—
13	#500×C₇₁	—	2	1	no	—
14	#500×C₇₁	C₄	1	2	yes	—
15	#500×C₇₁	C₂²	1	2	yes	—
16	C₂×#107×C₇₁	—	2	1	no	—
17	C₂×#107×C₇₁	C₄	1	2	yes	—
18	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
19	C₂²×C₇₁⋊C₈	—	4	1	no	—
20	C₂×C₇₁⋊C₁₆	—	2	2	no	—
21	C₂×C₇₁⋊C₁₆	—	4	2	no	—
22	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
23	C₄⋊(C₇₁⋊C₈)	—	4	1	no	—
24	C₂²×Q₈×C₇₁	—	4	1	no	—
25	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
27	C₄⋊Q₈×C₇₁	—	2	2	no	—
28	C₄⋊Q₈×C₇₁	—	2	2	no	—
29	C₄⋊Q₈×C₇₁	—	4	1	no	—
30	C₄⋊Q₈×C₇₁	—	4	1	no	—
31	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
32	C₂×C₄⋊C₄×C₇₁	—	4	1	no	—
33	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
34	#107×C₇₁	—	2	2	no	—
35	#107×C₇₁	—	2	2	no	—
36	#107×C₇₁	—	2	2	no	—
37	#107×C₇₁	C₂×C₄	1	4	yes	—
38	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
40	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
41	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
42	C₂×C₇₁⋊C₈	—	4	2	no	—
43	C₂×C₇₁⋊C₈	—	8	2	no	—
44	C₇₁⋊C₁₆	—	4	2	no	—
45	C₂×Q₈×C₇₁	—	4	2	no	—
46	C₂×Q₈×C₇₁	—	4	2	no	—
47	C₂×Q₈×C₇₁	—	8	2	no	—
48	C₂×Q₈×C₇₁	—	8	2	no	—
49	C₂²×C₄×C₇₁	—	4	2	no	—
50	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
53	C₂×C₈×C₇₁	—	2	2	no	—
54	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
57	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
58	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
60	C₄⋊C₄×C₇₁	—	2	4	no	—
61	C₄⋊C₄×C₇₁	—	4	2	no	—
62	C₄⋊C₄×C₇₁	—	4	2	no	—
63	C₄⋊C₄×C₇₁	—	4	2	no	—
64	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
65	C₄²×C₇₁	D₈	1	1	yes	characteristic
66	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
67	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
68	C₂³×C₇₁	#102	1	1	yes	characteristic
69	Q₈×C₇₁	—	8	4	no	—
70	Q₈×C₇₁	—	8	2	no	—
71	Q₈×C₇₁	—	8	2	no	—
72	C₂×C₄×C₇₁	—	2	2	no	—
73	C₂×C₄×C₇₁	—	4	4	no	—
74	C₂×C₄×C₇₁	—	8	4	no	—
75	C₂×C₄×C₇₁	#101	1	2	yes	—
76	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
77	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
78	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
79	C₂×C₄×C₇₁	#103	1	2	yes	—
80	C₂×C₄×C₇₁	D₁₆	1	2	yes	—
81	C₂×C₄×C₇₁	SD₃₂	1	2	yes	—
82	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
85	C₈×C₇₁	—	2	1	no	—
86	C₈×C₇₁	—	2	1	no	—
87	C₈×C₇₁	#103	1	2	yes	—
88	C₈×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
89	C₈×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
90	C₇₁⋊C₈	—	8	2	no	—
91	C₂²×C₇₁	#332	1	2	yes	—
92	C₂²×C₇₁	#357	1	2	yes	—
93	C₂²×C₇₁	#362	1	2	yes	—
94	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
95	C₄×C₇₁	—	2	1	no	—
96	C₄×C₇₁	—	2	1	no	—
97	C₄×C₇₁	—	8	4	no	—
98	C₄×C₇₁	#332	1	2	yes	—
99	C₄×C₇₁	#357	1	2	yes	—
100	C₄×C₇₁	C₂×#101	1	1	yes	characteristic
101	C₄×C₇₁	C₂×#103	1	1	yes	characteristic
102	C₄×C₇₁	C₂×D₁₆	1	1	yes	characteristic
103	C₄×C₇₁	C₂×SD₃₂	1	1	yes	characteristic
104	C₂×#1364	—	71	1	no	maximal, Sylow
105	C₂×C₇₁	#1364	1	4	yes	complemented
106	C₂×C₇₁	C₂×#332	1	1	yes	characteristic
107	C₂×C₇₁	C₂×#357	1	1	yes	characteristic
108	C₂×C₇₁	C₂×#362	1	1	yes	characteristic
109	C₂×#500	D₇₁	1	1	yes	characteristic
110	C₂×C₄⋊C₁₆	—	71	1	no	—
111	C₂×#335	—	71	1	no	—
112	#1364	—	71	4	no	—
113	C₇₁	C₂×#1364	1	1	yes	characteristic, complemented, Sylow
114	C₂×C₄⋊Q₈	—	2	1	no	—
115	C₂²×C₁₆	—	142	1	no	—
116	#500	—	2	1	no	—
117	#500	—	2	1	no	—
118	#500	Dic₇₁	1	2	yes	—
119	#500	C₂×D₇₁	1	2	yes	—
120	C₂×#107	—	2	1	no	—
121	C₂×#107	Dic₇₁	1	2	yes	—
122	C₂×C₄⋊C₈	—	142	1	no	—
123	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
124	C₄⋊C₁₆	—	71	2	no	—
125	C₄⋊C₁₆	—	142	1	no	—
126	#335	—	71	2	no	—
127	#335	—	142	1	no	—
128	C₂²×Q₈	—	4	1	no	—
129	C₂²×C₈	—	284	1	no	—
130	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
131	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	—	2	2	no	—
134	C₄⋊Q₈	—	4	1	no	—
135	C₄⋊Q₈	—	4	1	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	4	1	no	—
138	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
139	C₂×C₁₆	—	284	2	no	—
140	C₂×C₁₆	—	142	2	no	—
141	#107	—	2	2	no	—
142	#107	—	2	2	no	—
143	#107	—	2	2	no	—
144	#107	C₂×Dic₇₁	1	4	yes	—
145	C₄⋊C₈	—	284	1	no	—
146	C₄⋊C₈	—	142	2	no	—
147	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
148	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
149	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
150	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	4	2	no	—
153	C₂×Q₈	—	8	2	no	—
154	C₂×Q₈	—	8	2	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
157	C₂²×C₄	#11765850	1	1	yes	characteristic
158	C₂²×C₄	#11765853	1	1	yes	characteristic
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	284	2	no	—
161	C₂×C₈	—	568	2	no	—
162	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
163	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
164	C₂×C₈	#11765852	1	2	yes	—
165	C₂×C₈	#11765855	1	2	yes	—
166	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
167	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
173	C₄²	#11765850	1	1	yes	characteristic
174	C₄²	#11765853	1	1	yes	characteristic
175	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
176	C₁₆	—	284	2	no	—
177	C₂³	#14556049	1	1	yes	characteristic
178	Q₈	—	8	4	no	—
179	Q₈	—	8	2	no	—
180	Q₈	—	8	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	8	4	no	—
184	C₂×C₄	#14556047	1	2	yes	—
185	C₂×C₄	#14556049	1	1	yes	characteristic
186	C₂×C₄	#14556049	1	1	yes	characteristic
187	C₂×C₄	#14556049	1	1	yes	characteristic
188	C₂×C₄	#14556052	1	2	yes	—
189	C₂×C₄	#14556069	1	2	yes	—
190	C₂×C₄	#14556072	1	2	yes	—
191	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
192	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
193	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
194	C₈	—	2	1	no	—
195	C₈	—	2	1	no	—
196	C₈	—	568	2	no	—
197	C₈	#14556052	1	2	yes	—
198	C₈	C₂×#11765852	1	1	yes	characteristic
199	C₈	C₂×#11765855	1	1	yes	characteristic
200	C₂²	#17985745	1	2	yes	—
201	C₂²	#17985800	1	2	yes	—
202	C₂²	#17985815	1	2	yes	—
203	C₂²	C₂×#14556049	1	1	yes	characteristic
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	8	4	no	—
207	C₄	#17985745	1	2	yes	—
208	C₄	#17985800	1	2	yes	—
209	C₄	C₂×#14556047	1	1	yes	characteristic
210	C₄	C₂×#14556052	1	1	yes	characteristic
211	C₄	C₂×#14556069	1	1	yes	characteristic
212	C₄	C₂×#14556072	1	1	yes	characteristic
213	C₂	#22586395	1	4	yes	direct product factor
214	C₂	C₂×#17985745	1	1	yes	characteristic
215	C₂	C₂×#17985800	1	1	yes	characteristic
216	C₂	C₂×#17985815	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556049
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₂×#500×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	#102
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	C₂×#500	D₇₁
hypocenter	C₇₁	C₂×#1364
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
3. #22586395	C₂	direct (N × H)
105. C₂×C₇₁	#1364	semidirect
113. C₇₁	C₂×#1364	semidirect
213. C₂	#22586395	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₇₁⋊C₁₆)	2
C₂×#17985752	2
#22586395	2
C₂×#500×C₇₁	2
C₂×#1364	71

Subgroups — C2×#22586395 #31908730

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22586395 #31908730