Subgroups — C₂×#22590623 #32152946

Order 18176.

518 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	C₂×#17986401	C₂	1	1	yes	maximal, characteristic
2	#22590623	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17985826	C₂	1	2	yes	maximal
4	C₂×#17985824	C₂	1	2	yes	maximal
5	C₂×#17985775	C₂	1	1	yes	maximal, characteristic
6	C₂×#500×C₇₁	C₂	1	1	yes	maximal, characteristic
7	C₂×#14556120	—	2	1	no	—
8	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
9	#17986401	—	2	2	no	—
10	#17986401	C₂²	1	4	yes	—
11	C₂×#14556062	C₂²	1	1	yes	characteristic
12	C₂×#14556062	C₂²	1	1	yes	characteristic
13	C₂×#14556060	—	2	1	no	—
14	#17985826	C₂²	1	8	yes	—
15	#17985824	C₂²	1	8	yes	—
16	#17985775	C₂²	1	4	yes	—
17	C₂×C₄⋊Q₈×C₇₁	—	2	1	no	—
18	#500×C₇₁	—	2	2	no	—
19	#500×C₇₁	C₂²	1	4	yes	—
20	C₂×#107×C₇₁	—	2	1	no	—
21	C₂×#107×C₇₁	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
23	#14556120	—	2	4	no	—
24	#14556120	—	4	2	no	—
25	C₂×C₄⋊Dic₇₁	—	2	1	no	—
26	C₂×C₄⋊Dic₇₁	—	2	1	no	—
27	C₂×Dic₇₁⋊C₄	—	4	1	no	—
28	C₂×Dic₇₁⋊C₄	—	4	1	no	—
29	C₂×C₇₁⋊C₁₆	Q₈	1	8	yes	complemented
30	C₂×C₇₁⋊C₁₆	C₂³	1	4	yes	—
31	#14556062	—	2	1	no	—
32	#14556062	—	2	1	no	—
33	#14556062	C₂³	1	2	yes	—
34	#14556062	C₂³	1	2	yes	—
35	#14556060	—	2	4	no	—
36	C₂²×Q₈×C₇₁	—	4	1	no	—
37	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
39	C₄⋊Q₈×C₇₁	—	2	4	no	—
40	C₄⋊Q₈×C₇₁	—	4	2	no	—
41	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₇₁	—	4	1	no	—
43	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
44	#107×C₇₁	—	2	4	no	—
45	#107×C₇₁	—	2	2	no	—
46	#107×C₇₁	C₂³	1	4	yes	—
47	C₄×C₈×C₇₁	D₄	1	2	yes	—
48	C₄×C₈×C₇₁	C₂³	1	2	yes	—
49	C₂²×Dic₇₁	—	4	1	no	—
50	C₂²×Dic₇₁	—	4	1	no	—
51	C₄⋊Dic₇₁	—	2	2	no	—
52	C₄⋊Dic₇₁	—	2	2	no	—
53	C₄⋊Dic₇₁	—	4	1	no	—
54	C₄⋊Dic₇₁	—	4	1	no	—
55	Dic₇₁⋊C₄	—	4	4	no	—
56	Dic₇₁⋊C₄	—	4	4	no	—
57	C₇₁⋊C₁₆	C₂×Q₈	1	8	yes	complemented
58	C₂×Q₈×C₇₁	—	4	4	no	—
59	C₂×Q₈×C₇₁	—	8	4	no	—
60	C₂²×C₄×C₇₁	—	4	2	no	—
61	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₇₁	Q₁₆	1	4	yes	—
65	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₇₁	C₂×Q₈	1	4	yes	—
68	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
69	C₄⋊C₄×C₇₁	—	2	4	no	—
70	C₄⋊C₄×C₇₁	—	4	2	no	—
71	C₄⋊C₄×C₇₁	—	4	4	no	—
72	C₄²×C₇₁	D₈	1	2	yes	—
73	C₄²×C₇₁	C₂×D₄	1	2	yes	—
74	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
75	Q₈×C₇₁	—	8	8	no	—
76	C₂×C₄×C₇₁	—	2	2	no	—
77	C₂×C₄×C₇₁	—	2	2	no	—
78	C₂×C₄×C₇₁	—	4	4	no	—
79	C₂×C₄×C₇₁	—	8	4	no	—
80	C₂×C₄×C₇₁	C₄⋊Q₈	1	2	yes	—
81	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
82	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
83	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
85	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
86	C₂×Dic₇₁	—	4	2	no	—
87	C₂×Dic₇₁	—	4	2	no	—
88	C₂×Dic₇₁	—	8	2	no	—
89	C₂×Dic₇₁	—	8	2	no	—
90	C₈×C₇₁	C₄⋊Q₈	1	2	yes	—
91	C₈×C₇₁	C₂×Q₁₆	1	4	yes	—
92	C₈×C₇₁	C₂²×Q₈	1	2	yes	—
93	C₂²×C₇₁	#500	1	2	yes	—
94	C₂²×C₇₁	#509	1	2	yes	—
95	C₂²×C₇₁	#510	1	2	yes	—
96	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
97	C₄×C₇₁	—	2	2	no	—
98	C₄×C₇₁	—	2	2	no	—
99	C₄×C₇₁	—	8	4	no	—
100	C₄×C₇₁	#500	1	2	yes	—
101	C₄×C₇₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
102	C₄×C₇₁	C₂²×Q₁₆	1	1	yes	characteristic
103	Dic₇₁	—	8	2	no	—
104	Dic₇₁	—	8	2	no	—
105	C₂×#2255	—	71	1	no	maximal, Sylow
106	C₂×C₇₁	#2255	1	4	yes	complemented
107	C₂×C₇₁	C₂×#500	1	1	yes	characteristic
108	C₂×C₇₁	C₂×#509	1	1	yes	characteristic
109	C₂×C₇₁	C₂×#510	1	1	yes	characteristic
110	C₂×#500	D₇₁	1	1	yes	characteristic
111	C₂×#499	—	71	1	no	—
112	#2255	—	71	8	no	—
113	C₂×#367	—	71	2	no	—
114	C₂×#366	—	71	2	no	—
115	C₂×#346	—	71	1	no	—
116	C₇₁	C₂×#2255	1	1	yes	characteristic, complemented, Sylow
117	C₂×C₄⋊Q₈	—	2	1	no	—
118	C₂×#125	—	142	1	no	—
119	C₂²×C₁₆	—	71	2	no	—
120	#500	—	2	2	no	—
121	#500	C₂×D₇₁	1	4	yes	—
122	#499	—	71	4	no	—
123	#499	—	142	2	no	—
124	C₂×#107	—	2	1	no	—
125	C₂×#107	—	71	1	no	—
126	C₂×#107	—	71	1	no	—
127	C₂×#107	C₂×D₇₁	1	2	yes	—
128	C₂×#106	—	142	1	no	—
129	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
130	#367	—	71	8	no	—
131	#366	—	71	8	no	—
132	#346	—	71	4	no	—
133	C₂²×Q₈	—	4	1	no	—
134	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
136	C₄⋊Q₈	—	2	4	no	—
137	C₄⋊Q₈	—	4	2	no	—
138	#125	—	284	2	no	—
139	#125	—	142	4	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	—	4	1	no	—
142	C₂×C₄⋊C₄	—	284	1	no	—
143	C₂×C₄⋊C₄	—	284	1	no	—
144	C₂×C₄⋊C₄	—	142	1	no	—
145	C₂×C₄⋊C₄	—	142	1	no	—
146	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
147	C₂×C₁₆	—	71	8	no	—
148	C₂×C₁₆	—	71	4	no	—
149	#107	—	2	4	no	—
150	#107	—	2	2	no	—
151	#107	—	71	2	no	—
152	#107	—	71	2	no	—
153	#107	—	142	1	no	—
154	#107	—	142	1	no	—
155	#107	C₂²×D₇₁	1	4	yes	—
156	#106	—	142	4	no	—
157	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
158	C₄×C₈	C₂²×D₇₁	1	2	yes	—
159	C₂×Q₈	—	4	4	no	—
160	C₂×Q₈	—	8	4	no	—
161	C₂²×C₄	—	4	2	no	—
162	C₂²×C₄	—	284	1	no	—
163	C₂²×C₄	—	284	1	no	—
164	C₂²×C₄	#11765850	1	1	yes	characteristic
165	C₂²×C₄	#11765850	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
167	C₂×C₈	#11765855	1	4	yes	—
168	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
169	C₂×C₈	C₂×C₂²⋊D₇₁	1	2	yes	—
170	C₂×C₈	Q₈×D₇₁	1	4	yes	—
171	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	4	4	no	—
175	C₄⋊C₄	—	284	4	no	—
176	C₄⋊C₄	—	284	4	no	—
177	C₄⋊C₄	—	284	1	no	—
178	C₄⋊C₄	—	284	1	no	—
179	C₄⋊C₄	—	142	2	no	—
180	C₄⋊C₄	—	142	2	no	—
181	C₄²	#11765850	1	2	yes	—
182	C₄²	C₂×C₂²⋊D₇₁	1	2	yes	—
183	C₁₆	—	71	8	no	—
184	C₂³	C₂×#11765850	1	1	yes	characteristic
185	Q₈	—	8	8	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	4	no	—
189	C₂×C₄	—	8	4	no	—
190	C₂×C₄	—	284	2	no	—
191	C₂×C₄	—	284	2	no	—
192	C₂×C₄	—	568	2	no	—
193	C₂×C₄	—	568	2	no	—
194	C₂×C₄	#14556126	1	2	yes	—
195	C₂×C₄	C₂×#11765850	1	2	yes	—
196	C₂×C₄	C₂×#11765850	1	2	yes	—
197	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
198	C₂×C₄	C₂×#11765855	1	2	yes	—
199	C₂×C₄	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
200	C₈	#14556126	1	2	yes	—
201	C₈	C₂×#11765855	1	4	yes	—
202	C₈	C₂×Q₈×D₇₁	1	2	yes	—
203	C₂²	#17986402	1	2	yes	—
204	C₂²	#17986435	1	2	yes	—
205	C₂²	#17986440	1	2	yes	—
206	C₂²	C₂²×#11765850	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	8	4	no	—
210	C₄	—	568	2	no	—
211	C₄	—	568	2	no	—
212	C₄	#17986402	1	2	yes	—
213	C₄	C₂×#14556126	1	1	yes	characteristic
214	C₄	C₂²×#11765855	1	1	yes	characteristic
215	C₂	#22590623	1	4	yes	direct product factor
216	C₂	C₂×#17986402	1	1	yes	characteristic
217	C₂	C₂×#17986435	1	1	yes	characteristic
218	C₂	C₂×#17986440	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11765850
commutator, cosocle	C₂×C₈×C₇₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₂×#500×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×D₈
hypercenter	C₂×#500	D₇₁
hypocenter	C₇₁	C₂×#2255
LCS	G ▷ C₂×C₈×C₇₁ ▷ 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
2. #22590623	C₂	direct (N × H)
29. C₂×C₇₁⋊C₁₆	Q₈	semidirect
57. C₇₁⋊C₁₆	C₂×Q₈	semidirect
106. C₂×C₇₁	#2255	semidirect
116. C₇₁	C₂×#2255	semidirect
215. C₂	#22590623	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×#17986401	2
#22590623	2
C₂×#17985826	2
C₂×#17985824	2
C₂×#17985775	2
C₂×#500×C₇₁	2
C₂×#2255	71

Subgroups — C2×#22590623 #32152946

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22590623 #32152946