Subgroups — C₂×#22590626 #32152949

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×#17986404	C₂	1	1	yes	maximal, characteristic
2	C₂×#17986401	C₂	1	1	yes	maximal, characteristic
3	#22590626	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17985827	C₂	1	2	yes	maximal
5	C₂×#17985825	C₂	1	2	yes	maximal
6	C₂×#346×C₇₁	C₂	1	1	yes	maximal, characteristic
7	C₂×#14556128	—	2	1	no	—
8	C₂×#14556120	—	2	1	no	—
9	#17986404	—	2	2	no	—
10	#17986404	C₂²	1	4	yes	—
11	#17986401	—	2	2	no	—
12	#17986401	C₂²	1	4	yes	—
13	C₂×#14556062	—	2	1	no	—
14	C₂×#14556062	C₂²	1	1	yes	characteristic
15	C₂×#14556062	C₂²	1	2	yes	—
16	C₂×#14556062	C₂²	1	1	yes	characteristic
17	C₂×#14556060	—	2	1	no	—
18	#17985827	C₂²	1	8	yes	—
19	#17985825	C₂²	1	8	yes	—
20	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
22	#346×C₇₁	C₂²	1	4	yes	—
23	C₂²×Dic₁₄₂	—	4	1	no	—
24	#14556128	—	2	4	no	—
25	#14556128	—	4	2	no	—
26	#14556120	—	2	4	no	—
27	#14556120	—	4	2	no	—
28	C₂×C₄⋊Dic₇₁	—	2	1	no	—
29	C₂×C₄⋊Dic₇₁	—	2	1	no	—
30	C₂×C₄⋊Dic₇₁	—	2	2	no	—
31	C₂×C₄⋊Dic₇₁	—	4	1	no	—
32	C₂×Dic₇₁⋊C₄	—	4	1	no	—
33	C₂×Dic₇₁⋊C₄	—	4	1	no	—
34	#14556062	—	2	1	no	—
35	#14556062	—	2	1	no	—
36	#14556062	—	2	2	no	—
37	#14556062	—	2	4	no	—
38	#14556062	C₂³	1	2	yes	—
39	#14556062	C₂³	1	4	yes	—
40	#14556062	C₂³	1	2	yes	—
41	#14556060	—	2	4	no	—
42	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
45	C₂×C₁₆×C₇₁	Q₈	1	8	yes	complemented
46	C₂×C₁₆×C₇₁	C₂³	1	4	yes	—
47	C₄×C₈×C₇₁	D₄	1	2	yes	—
48	C₄×C₈×C₇₁	C₂³	1	2	yes	—
49	C₂×Dic₁₄₂	—	4	4	no	—
50	C₂×Dic₁₄₂	—	8	4	no	—
51	C₂²×Dic₇₁	—	4	2	no	—
52	C₂²×Dic₇₁	—	4	1	no	—
53	C₂²×Dic₇₁	—	4	1	no	—
54	C₄⋊Dic₇₁	—	2	2	no	—
55	C₄⋊Dic₇₁	—	2	2	no	—
56	C₄⋊Dic₇₁	—	2	4	no	—
57	C₄⋊Dic₇₁	—	4	1	no	—
58	C₄⋊Dic₇₁	—	4	1	no	—
59	C₄⋊Dic₇₁	—	4	2	no	—
60	C₄⋊Dic₇₁	—	4	4	no	—
61	Dic₇₁⋊C₄	—	4	4	no	—
62	Dic₇₁⋊C₄	—	4	4	no	—
63	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₇₁	Q₁₆	1	4	yes	—
67	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₇₁	C₂×Q₈	1	4	yes	—
70	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
71	C₄²×C₇₁	D₈	1	2	yes	—
72	C₄²×C₇₁	C₂×D₄	1	2	yes	—
73	C₁₆×C₇₁	C₂×Q₈	1	8	yes	complemented
74	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₇₁	—	2	2	no	—
76	C₂×C₄×C₇₁	—	2	2	no	—
77	C₂×C₄×C₇₁	C₄⋊Q₈	1	2	yes	—
78	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
79	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
80	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
82	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
83	C₂×Dic₇₁	—	4	2	no	—
84	C₂×Dic₇₁	—	4	2	no	—
85	C₂×Dic₇₁	—	4	4	no	—
86	C₂×Dic₇₁	—	8	2	no	—
87	C₂×Dic₇₁	—	8	4	no	—
88	C₂×Dic₇₁	—	8	2	no	—
89	Dic₁₄₂	—	8	8	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	#500	1	2	yes	—
100	C₄×C₇₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₄×C₇₁	C₂²×Q₁₆	1	1	yes	characteristic
102	Dic₇₁	—	8	4	no	—
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	—	71	1	no	—
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	D₇₁	1	1	yes	characteristic
116	C₇₁	C₂×#2255	1	1	yes	characteristic, complemented, Sylow
117	C₂×C₄⋊Q₈	—	142	1	no	—
118	C₂×#125	—	142	1	no	—
119	C₂²×C₁₆	C₂×D₇₁	1	2	yes	—
120	#500	—	71	4	no	—
121	#500	—	142	2	no	—
122	#499	—	71	4	no	—
123	#499	—	142	2	no	—
124	C₂×#107	—	71	1	no	—
125	C₂×#107	—	71	1	no	—
126	C₂×#107	—	71	2	no	—
127	C₂×#107	—	142	1	no	—
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	C₂×D₇₁	1	4	yes	—
133	C₂²×Q₈	—	284	1	no	—
134	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
136	C₄⋊Q₈	—	284	2	no	—
137	C₄⋊Q₈	—	142	4	no	—
138	#125	—	284	2	no	—
139	#125	—	142	4	no	—
140	C₂×C₄⋊C₄	—	284	1	no	—
141	C₂×C₄⋊C₄	—	284	1	no	—
142	C₂×C₄⋊C₄	—	284	1	no	—
143	C₂×C₄⋊C₄	—	142	1	no	—
144	C₂×C₄⋊C₄	—	142	1	no	—
145	C₂×C₄⋊C₄	—	142	2	no	—
146	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
147	C₂×C₁₆	Dic₁₄₂	1	8	yes	complemented
148	C₂×C₁₆	C₂²×D₇₁	1	4	yes	—
149	#107	—	71	2	no	—
150	#107	—	71	4	no	—
151	#107	—	71	2	no	—
152	#107	—	142	1	no	—
153	#107	—	142	1	no	—
154	#107	—	142	2	no	—
155	#107	—	142	4	no	—
156	#106	—	142	4	no	—
157	C₄×C₈	D₂₈₄	1	2	yes	—
158	C₄×C₈	C₂²×D₇₁	1	2	yes	—
159	C₂×Q₈	—	284	4	no	—
160	C₂×Q₈	—	568	4	no	—
161	C₂²×C₄	—	284	1	no	—
162	C₂²×C₄	—	284	1	no	—
163	C₂²×C₄	—	284	2	no	—
164	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
165	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
167	C₂×C₈	Dic₂₈₄	1	4	yes	—
168	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
169	C₂×C₈	C₂×D₂₈₄	1	2	yes	—
170	C₂×C₈	C₂×Dic₁₄₂	1	4	yes	—
171	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
172	C₄⋊C₄	—	284	4	no	—
173	C₄⋊C₄	—	284	4	no	—
174	C₄⋊C₄	—	284	4	no	—
175	C₄⋊C₄	—	284	1	no	—
176	C₄⋊C₄	—	284	1	no	—
177	C₄⋊C₄	—	284	2	no	—
178	C₄⋊C₄	—	142	2	no	—
179	C₄⋊C₄	—	142	2	no	—
180	C₄⋊C₄	—	142	4	no	—
181	C₄²	D₅₆₈	1	2	yes	—
182	C₄²	C₂×D₂₈₄	1	2	yes	—
183	C₁₆	C₂×Dic₁₄₂	1	8	yes	complemented
184	C₂³	C₂×D₅₆₈	1	1	yes	characteristic
185	Q₈	—	568	8	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	284	2	no	—
189	C₂×C₄	—	284	2	no	—
190	C₂×C₄	—	284	4	no	—
191	C₂×C₄	—	568	4	no	—
192	C₂×C₄	—	568	2	no	—
193	C₂×C₄	—	568	2	no	—
194	C₂×C₄	#14556128	1	2	yes	—
195	C₂×C₄	C₂×D₅₆₈	1	2	yes	—
196	C₂×C₄	C₂×D₅₆₈	1	2	yes	—
197	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
198	C₂×C₄	C₂×Dic₂₈₄	1	2	yes	—
199	C₂×C₄	C₂²×D₂₈₄	1	1	yes	characteristic
200	C₈	#14556128	1	2	yes	—
201	C₈	C₂×Dic₂₈₄	1	4	yes	—
202	C₈	C₂²×Dic₁₄₂	1	2	yes	—
203	C₂²	#17986404	1	2	yes	—
204	C₂²	#17986437	1	2	yes	—
205	C₂²	#17986442	1	2	yes	—
206	C₂²	C₂²×D₅₆₈	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	568	4	no	—
210	C₄	—	568	2	no	—
211	C₄	—	568	2	no	—
212	C₄	#17986404	1	2	yes	—
213	C₄	C₂×#14556128	1	1	yes	characteristic
214	C₄	C₂²×Dic₂₈₄	1	1	yes	characteristic
215	C₂	#22590626	1	4	yes	direct product factor
216	C₂	C₂×#17986404	1	1	yes	characteristic
217	C₂	C₂×#17986437	1	1	yes	characteristic
218	C₂	C₂×#17986442	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₆₈
commutator, cosocle	C₂×C₈×C₇₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₂×#346×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×D₈
hypercenter	C₂×#346	D₇₁
hypocenter	C₇₁	C₂×#2255
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
3. #22590626	C₂	direct (N × H)
45. C₂×C₁₆×C₇₁	Q₈	semidirect
73. C₁₆×C₇₁	C₂×Q₈	semidirect
106. C₂×C₇₁	#2255	semidirect
116. C₇₁	C₂×#2255	semidirect
147. C₂×C₁₆	Dic₁₄₂	semidirect
183. C₁₆	C₂×Dic₁₄₂	semidirect
215. C₂	#22590626	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×#17986404	2
C₂×#17986401	2
#22590626	2
C₂×#17985827	2
C₂×#17985825	2
C₂×#346×C₇₁	2
C₂×#2255	71

Subgroups — C2×#22590626 #32152949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22590626 #32152949