Subgroups — C₂×#13092988 #15686711

Order 2944.

1512 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₄₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092988	C₂	1	16	yes	maximal, direct product factor
3	C₂×#10625115	C₂	1	2	yes	maximal, complemented
4	C₂×#10625107	C₂	1	2	yes	maximal, complemented
5	C₂×#10625106	C₂	1	4	yes	maximal, complemented
6	C₂×#10625095	C₂	1	4	yes	maximal, complemented
7	C₂×#10625097	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#41×C₂₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#41×C₂₃	—	2	8	no	—
10	C₂×#41×C₂₃	C₂²	1	8	yes	—
11	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
13	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
15	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂³×Dic₂₃	C₂²	1	2	yes	—
17	C₂²×Dic₄₆	—	2	8	no	—
18	C₂²×Dic₄₆	C₂²	1	2	yes	—
19	C₂²×Dic₄₆	C₂²	1	2	yes	—
20	C₂²×Dic₄₆	C₂²	1	8	yes	complemented
21	C₂×C₂²⋊Dic₂₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊Dic₂₃	C₂²	1	8	yes	—
23	#10625115	C₂²	1	16	yes	complemented
24	#10625107	C₂²	1	16	yes	complemented
25	#10625106	C₂²	1	32	yes	complemented
26	#10625097	C₂²	1	8	yes	complemented
27	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
28	#10625095	C₂²	1	32	yes	complemented
29	C₂×Dic₂₃⋊C₄	C₂²	1	8	yes	—
30	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
31	C₂⁴×C₂₃	C₂³	1	2	yes	—
32	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
33	#41×C₂₃	—	2	32	no	—
34	C₂×Q₈×C₂₃	—	2	4	no	—
35	C₂×Q₈×C₂₃	C₂³	1	4	yes	—
36	C₂×D₄×C₂₃	—	2	8	no	—
37	C₂×D₄×C₂₃	—	2	4	no	—
38	C₂×D₄×C₂₃	C₂³	1	8	yes	—
39	C₂×D₄×C₂₃	C₂³	1	4	yes	—
40	C₂²×C₄×C₂₃	—	2	8	no	—
41	C₂²×C₄×C₂₃	D₄	1	8	yes	—
42	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂³	1	8	yes	—
44	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
47	C₂²×Dic₂₃	—	2	8	no	—
48	C₂²×Dic₂₃	C₂³	1	4	yes	—
49	C₂²×Dic₂₃	C₂³	1	4	yes	—
50	C₂²×Dic₂₃	C₂³	1	8	yes	—
51	C₂×Dic₄₆	—	2	8	no	—
52	C₂×Dic₄₆	—	2	8	no	—
53	C₂×Dic₄₆	—	2	32	no	—
54	C₂×Dic₄₆	C₂³	1	8	yes	—
55	C₂×Dic₄₆	C₂³	1	8	yes	—
56	C₂²⋊Dic₂₃	C₂³	1	32	yes	—
57	C₂²⋊Dic₂₃	C₂³	1	8	yes	—
58	C₄⋊Dic₂₃	C₂³	1	8	yes	—
59	Dic₂₃⋊C₄	C₂³	1	32	yes	—
60	C₄×Dic₂₃	C₂³	1	8	yes	—
61	C₂³×C₂₃	—	2	4	no	—
62	C₂³×C₂₃	—	2	4	no	—
63	C₂³×C₂₃	—	2	8	no	—
64	C₂³×C₂₃	C₂×D₄	1	4	yes	—
65	C₂³×C₂₃	C₂⁴	1	4	yes	—
66	C₂³×C₂₃	C₂⁴	1	2	yes	—
67	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
68	Q₈×C₂₃	—	2	8	no	—
69	D₄×C₂₃	—	2	16	no	—
70	D₄×C₂₃	—	2	8	no	—
71	C₂×C₄×C₂₃	—	2	4	no	—
72	C₂×C₄×C₂₃	—	2	16	no	—
73	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
74	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
75	C₂×C₄×C₂₃	C₂×D₄	1	16	yes	—
76	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
77	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
78	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
79	C₂×Dic₂₃	—	2	16	no	—
80	C₂×Dic₂₃	—	2	8	no	—
81	C₂×Dic₂₃	—	2	8	no	—
82	C₂×Dic₂₃	C₂⁴	1	8	yes	—
83	C₂×Dic₂₃	C₂⁴	1	8	yes	—
84	Dic₄₆	—	2	16	no	—
85	Dic₄₆	—	2	16	no	—
86	C₂×#536	—	23	1	no	maximal, Sylow
87	C₂²×C₂₃	—	2	8	no	—
88	C₂²×C₂₃	—	2	4	no	—
89	C₂²×C₂₃	—	2	4	no	—
90	C₂²×C₂₃	—	2	8	no	—
91	C₂²×C₂₃	—	4	8	no	—
92	C₂²×C₂₃	C₂²×D₄	1	4	yes	—
93	C₂²×C₂₃	C₂²×D₄	1	2	yes	—
94	C₂²×C₂₃	2₋¹⁺⁴	1	4	yes	—
95	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
96	C₄×C₂₃	—	2	4	no	—
97	C₄×C₂₃	C₂²×D₄	1	4	yes	—
98	C₄×C₂₃	C₂²×D₄	1	4	yes	—
99	Dic₂₃	—	2	8	no	—
100	Dic₂₃	—	2	8	no	—
101	C₂²×#41	D₂₃	1	1	yes	characteristic
102	C₂³×Q₈	—	23	1	no	—
103	#536	—	23	16	no	—
104	C₂×C₄⋊Q₈	—	23	2	no	—
105	C₂×#124	—	23	2	no	—
106	C₂×#123	—	23	4	no	—
107	C₂×C₂²⋊Q₈	—	23	4	no	—
108	C₂×#117	—	23	1	no	—
109	C₂×C₂₃	—	2	4	no	—
110	C₂×C₂₃	—	4	4	no	—
111	C₂×C₂₃	#536	1	4	yes	complemented
112	C₂×C₂₃	C₂³×D₄	1	1	yes	characteristic
113	C₂×C₂₃	C₂×2₋¹⁺⁴	1	2	yes	—
114	C₂×#41	—	2	8	no	—
115	C₂×#41	C₂×D₂₃	1	8	yes	—
116	C₂²×Q₈	—	23	2	no	—
117	C₂²×Q₈	—	23	2	no	—
118	C₂²×Q₈	—	23	8	no	—
119	C₂²×Q₈	—	46	8	no	—
120	C₂²×Q₈	C₂×D₂₃	1	1	yes	characteristic
121	C₂²×D₄	C₂×D₂₃	1	2	yes	—
122	C₂²×D₄	C₂×D₂₃	1	1	yes	characteristic
123	C₂³×C₄	—	23	2	no	—
124	C₂³×C₄	C₂×D₂₃	1	2	yes	—
125	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
126	C₄⋊Q₈	—	23	16	no	—
127	#124	—	23	16	no	—
128	#123	—	23	32	no	—
129	C₂²⋊Q₈	—	23	32	no	—
130	#117	—	23	8	no	—
131	C₂×C₄⋊C₄	—	23	2	no	—
132	C₂×C₄⋊C₄	—	23	8	no	—
133	C₂×C₂²⋊C₄	—	23	2	no	—
134	C₂×C₂²⋊C₄	—	23	8	no	—
135	C₂×C₄²	—	23	2	no	—
136	C₂₃	C₂×#536	1	1	yes	characteristic, complemented, Sylow
137	C₂⁴	C₂²×D₂₃	1	2	yes	—
138	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
139	#41	—	2	32	no	—
140	C₂×Q₈	—	2	4	no	—
141	C₂×Q₈	—	23	8	no	—
142	C₂×Q₈	—	23	8	no	—
143	C₂×Q₈	—	46	8	no	—
144	C₂×Q₈	—	46	8	no	—
145	C₂×Q₈	—	46	32	no	—
146	C₂×Q₈	C₂²×D₂₃	1	4	yes	—
147	C₂×D₄	—	2	8	no	—
148	C₂×D₄	—	2	4	no	—
149	C₂×D₄	C₂²×D₂₃	1	8	yes	—
150	C₂×D₄	C₂²×D₂₃	1	4	yes	—
151	C₂²×C₄	—	2	8	no	—
152	C₂²×C₄	—	23	8	no	—
153	C₂²×C₄	—	23	4	no	—
154	C₂²×C₄	—	23	4	no	—
155	C₂²×C₄	—	46	8	no	—
156	C₂²×C₄	C₂²⋊D₂₃	1	8	yes	—
157	C₂²×C₄	C₂²×D₂₃	1	8	yes	—
158	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
159	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
160	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
161	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
162	C₄⋊C₄	—	23	8	no	—
163	C₄⋊C₄	—	23	32	no	—
164	C₂²⋊C₄	—	23	8	no	—
165	C₂²⋊C₄	—	23	32	no	—
166	C₄²	—	23	8	no	—
167	C₂³	—	2	4	no	—
168	C₂³	—	2	8	no	—
169	C₂³	—	2	4	no	—
170	C₂³	C₂×C₂²⋊D₂₃	1	4	yes	—
171	C₂³	C₂³×D₂₃	1	4	yes	—
172	C₂³	C₂³×D₂₃	1	2	yes	—
173	C₂³	C₂³×D₂₃	1	1	yes	characteristic
174	Q₈	—	2	8	no	—
175	Q₈	—	46	16	no	—
176	Q₈	—	46	16	no	—
177	D₄	—	2	16	no	—
178	D₄	—	2	8	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	16	no	—
181	C₂×C₄	—	23	8	no	—
182	C₂×C₄	—	23	8	no	—
183	C₂×C₄	—	46	8	no	—
184	C₂×C₄	—	46	16	no	—
185	C₂×C₄	—	46	8	no	—
186	C₂×C₄	C₂×C₂²⋊D₂₃	1	4	yes	—
187	C₂×C₄	C₂×C₂²⋊D₂₃	1	4	yes	—
188	C₂×C₄	C₂×C₂²⋊D₂₃	1	16	yes	—
189	C₂×C₄	C₂³×D₂₃	1	4	yes	—
190	C₂×C₄	C₂³×D₂₃	1	2	yes	—
191	C₂×C₄	C₂³×D₂₃	1	2	yes	—
192	C₂²	—	2	8	no	—
193	C₂²	—	2	4	no	—
194	C₂²	—	2	4	no	—
195	C₂²	—	2	8	no	—
196	C₂²	—	4	8	no	—
197	C₂²	#10625161	1	4	yes	—
198	C₂²	C₂²×C₂²⋊D₂₃	1	4	yes	—
199	C₂²	C₂²×C₂²⋊D₂₃	1	2	yes	—
200	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
201	C₄	—	2	4	no	—
202	C₄	—	46	8	no	—
203	C₄	—	46	8	no	—
204	C₄	C₂²×C₂²⋊D₂₃	1	4	yes	—
205	C₄	C₂²×C₂²⋊D₂₃	1	4	yes	—
206	C₂	—	2	4	no	—
207	C₂	—	4	4	no	—
208	C₂	#13092988	1	4	yes	direct product factor
209	C₂	C₂³×C₂²⋊D₂₃	1	1	yes	characteristic
210	C₂	C₂×#10625161	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₂₃
commutator, cosocle	C₂²×C₂₃	C₂⁵
Frattini	C₂²	C₂⁴×D₂₃
Fitting, prime radical	C₂²×#41×C₂₃	C₂
socle	C₂³×C₂₃	C₂⁴
hypercenter	C₂²×#41	D₂₃
hypocenter	C₂₃	C₂×#536
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂³×Dic₄₆	C₂	semidirect
2. #13092988	C₂	direct (N × H)
3. C₂×#10625115	C₂	semidirect
4. C₂×#10625107	C₂	semidirect
5. C₂×#10625106	C₂	semidirect
6. C₂×#10625095	C₂	semidirect
7. C₂×#10625097	C₂	semidirect
20. C₂²×Dic₄₆	C₂²	semidirect
23. #10625115	C₂²	semidirect
24. #10625107	C₂²	semidirect
25. #10625106	C₂²	semidirect
26. #10625097	C₂²	semidirect
28. #10625095	C₂²	semidirect
111. C₂×C₂₃	#536	semidirect
136. C₂₃	C₂×#536	semidirect
208. C₂	#13092988	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Dic₄₆	2
#13092988	2
C₂×#10625115	2
C₂×#10625107	2
C₂×#10625106	2
C₂×#10625095	2
C₂×#10625097	2
C₂²×#41×C₂₃	2
C₂×#536	23

Subgroups — C2×#13092988 #15686711

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13092988 #15686711