Subgroups — C₂×#13093082 #15686870

Order 2944.

1568 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×D₂₃	C₂	1	2	yes	maximal, complemented
2	#13093082	C₂	1	16	yes	maximal, direct product factor
3	C₂×#10625059	C₂	1	2	yes	maximal, complemented
4	C₂×#10625067	C₂	1	4	yes	maximal, complemented
5	C₂×#10625117	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×Dic₄₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×D₉₂	C₂	1	1	yes	maximal, characteristic
8	C₂×#10625064	C₂	1	2	yes	maximal
9	C₂×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₈×C₂₃	C₂²	1	2	yes	—
11	C₂×Q₈×D₂₃	—	2	16	no	—
12	C₂×Q₈×D₂₃	C₂²	1	16	yes	complemented
13	C₂²×D₉₂	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₂₃	C₂²	1	4	yes	—
15	C₂²×C₄×D₂₃	C₂²	1	2	yes	—
16	C₂²×Dic₄₆	C₂²	1	4	yes	—
17	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic
18	C₂₃×C₄⋊Q₈	C₂²	1	8	yes	complemented
19	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
22	#10625117	C₂²	1	16	yes	complemented
23	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	4	yes	—
24	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
25	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
27	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
28	C₂×Dic₂₃⋊C₄	C₂²	1	4	yes	—
29	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
30	#10625067	C₂²	1	32	yes	complemented
31	#10625064	C₂²	1	16	yes	—
32	#10625059	C₂²	1	16	yes	complemented
33	C₄×D₉₂	C₂²	1	8	yes	—
34	C₄×Dic₄₆	C₂²	1	8	yes	complemented
35	C₂³×D₂₃	C₂³	1	2	yes	—
36	C₂×Q₈×C₂₃	—	2	8	no	—
37	C₂×Q₈×C₂₃	C₂³	1	8	yes	—
38	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
40	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
42	C₂²×Dic₂₃	C₂³	1	2	yes	—
43	C₂²×Dic₂₃	C₂³	1	2	yes	—
44	C₂²×Dic₂₃	C₂³	1	2	yes	—
45	Q₈×D₂₃	—	2	64	no	—
46	C₂×D₉₂	Q₈	1	8	yes	complemented
47	C₂×D₉₂	C₂³	1	4	yes	—
48	C₂×C₄×D₂₃	—	2	8	no	—
49	C₂×C₄×D₂₃	—	2	16	no	—
50	C₂×C₄×D₂₃	C₂³	1	8	yes	—
51	C₂×C₄×D₂₃	C₂³	1	16	yes	—
52	C₂×Dic₄₆	—	2	16	no	—
53	C₂×Dic₄₆	D₄	1	8	yes	complemented
54	C₂×Dic₄₆	C₂³	1	16	yes	—
55	C₂×Dic₄₆	C₂³	1	4	yes	—
56	C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
57	C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
58	C₄²×C₂₃	C₂³	1	4	yes	—
59	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
60	(C₂×D₂₃)⋊C₄	C₂³	1	16	yes	—
61	C₄⋊Dic₂₃	C₂³	1	4	yes	—
62	C₄⋊Dic₂₃	C₂³	1	4	yes	—
63	Dic₂₃⋊C₄	C₂³	1	8	yes	—
64	Dic₂₃⋊C₄	C₂³	1	16	yes	—
65	C₄×Dic₂₃	C₂³	1	8	yes	—
66	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
67	C₂²×D₂₃	—	2	4	no	—
68	C₂²×D₂₃	—	2	4	no	—
69	C₂²×D₂₃	C₂×Q₈	1	8	yes	complemented
70	C₂²×D₂₃	C₂⁴	1	4	yes	—
71	Q₈×C₂₃	—	2	16	no	—
72	C₂×C₄×C₂₃	—	2	8	no	—
73	C₂×C₄×C₂₃	—	2	2	no	—
74	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
75	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	complemented
76	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
77	C₂×C₄×C₂₃	C₂⁴	1	8	yes	—
78	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
79	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
80	C₂×Dic₂₃	—	2	4	no	—
81	C₂×Dic₂₃	—	2	4	no	—
82	C₂×Dic₂₃	C₂×D₄	1	8	yes	—
83	C₂×Dic₂₃	C₂⁴	1	4	yes	—
84	C₂×Dic₂₃	C₂⁴	1	4	yes	—
85	C₂×Dic₂₃	C₂⁴	1	4	yes	—
86	D₉₂	C₂×Q₈	1	16	yes	complemented
87	C₄×D₂₃	—	2	16	no	—
88	C₄×D₂₃	—	2	32	no	—
89	Dic₄₆	—	2	32	no	—
90	Dic₄₆	C₂×D₄	1	16	yes	complemented
91	C₂×D₄×Q₈	—	23	1	no	maximal, Sylow
92	C₂²×C₂₃	C₂²×D₄	1	2	yes	—
93	C₂²×C₂₃	C₂²×Q₈	1	2	yes	—
94	C₂²×C₂₃	2₋¹⁺⁴	1	2	yes	—
95	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
96	C₂×D₂₃	—	2	16	no	—
97	C₂×D₂₃	—	2	4	no	—
98	C₂×D₂₃	—	2	4	no	—
99	C₂×D₂₃	C₂²×Q₈	1	8	yes	complemented
100	C₄×C₂₃	—	2	8	no	—
101	C₄×C₂₃	—	2	2	no	—
102	C₄×C₂₃	C₂²×D₄	1	4	yes	—
103	C₄×C₂₃	C₂²×Q₈	1	4	yes	complemented
104	Dic₂₃	—	2	4	no	—
105	Dic₂₃	—	2	4	no	—
106	Dic₂₃	C₂²×D₄	1	8	yes	—
107	C₂³×Q₈	—	23	2	no	—
108	D₄×Q₈	—	23	16	no	—
109	C₂×C₄⋊Q₈	—	23	2	no	—
110	C₂×C₄⋊Q₈	D₂₃	1	1	yes	characteristic, complemented
111	C₂×C₂²⋊Q₈	—	23	2	no	—
112	C₂×C₂²⋊Q₈	—	23	4	no	—
113	C₂×C₄×Q₈	—	23	1	no	—
114	C₂×C₄×D₄	—	23	2	no	—
115	C₂×C₄×D₄	—	23	1	no	—
116	C₂×C₂₃	D₄×Q₈	1	4	yes	complemented
117	C₂×C₂₃	C₂³×D₄	1	1	yes	characteristic
118	C₂×C₂₃	C₂³×Q₈	1	1	yes	characteristic
119	C₂×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
120	D₂₃	—	2	8	no	—
121	C₂²×Q₈	—	23	16	no	—
122	C₂²×Q₈	—	23	1	no	—
123	C₂²×Q₈	—	23	4	no	—
124	C₂²×Q₈	—	46	16	no	—
125	C₂²×Q₈	C₂×D₂₃	1	2	yes	—
126	C₂²×D₄	—	23	1	no	—
127	C₂³×C₄	—	23	4	no	—
128	C₂³×C₄	—	23	2	no	—
129	C₄⋊Q₈	—	23	16	no	—
130	C₄⋊Q₈	C₂×D₂₃	1	8	yes	complemented
131	C₂²⋊Q₈	—	23	16	no	—
132	C₂²⋊Q₈	—	23	32	no	—
133	C₄×Q₈	—	23	8	no	—
134	C₄×D₄	—	23	16	no	—
135	C₄×D₄	—	23	8	no	—
136	C₂×C₄⋊C₄	—	23	1	no	—
137	C₂×C₄⋊C₄	—	23	2	no	—
138	C₂×C₄⋊C₄	—	23	1	no	—
139	C₂×C₄⋊C₄	—	23	4	no	—
140	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
141	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
142	C₂×C₂²⋊C₄	—	23	4	no	—
143	C₂×C₂²⋊C₄	—	23	2	no	—
144	C₂×C₄²	—	23	2	no	—
145	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
146	C₂₃	C₂×D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
147	C₂⁴	—	23	2	no	—
148	C₂×Q₈	—	2	8	no	—
149	C₂×Q₈	—	23	8	no	—
150	C₂×Q₈	—	23	4	no	—
151	C₂×Q₈	—	23	16	no	—
152	C₂×Q₈	—	46	16	no	—
153	C₂×Q₈	—	46	64	no	—
154	C₂×Q₈	C₂²×D₂₃	1	8	yes	—
155	C₂×D₄	—	23	8	no	—
156	C₂×D₄	—	23	4	no	—
157	C₂²×C₄	—	23	2	no	—
158	C₂²×C₄	—	23	2	no	—
159	C₂²×C₄	—	23	2	no	—
160	C₂²×C₄	—	23	16	no	—
161	C₂²×C₄	—	23	8	no	—
162	C₂²×C₄	—	46	16	no	—
163	C₂²×C₄	—	46	8	no	—
164	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
165	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
166	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
167	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
168	C₄⋊C₄	—	23	4	no	—
169	C₄⋊C₄	—	23	8	no	—
170	C₄⋊C₄	—	23	4	no	—
171	C₄⋊C₄	—	23	16	no	—
172	C₄⋊C₄	C₂²×D₂₃	1	8	yes	—
173	C₄⋊C₄	C₂²×D₂₃	1	8	yes	—
174	C₂²⋊C₄	—	23	16	no	—
175	C₂²⋊C₄	—	23	8	no	—
176	C₄²	—	23	8	no	—
177	C₄²	C₂²×D₂₃	1	4	yes	—
178	C₂³	—	23	8	no	—
179	C₂³	—	23	4	no	—
180	C₂³	—	46	4	no	—
181	C₂³	—	46	4	no	—
182	C₂³	C₂³×D₂₃	1	1	yes	characteristic
183	Q₈	—	2	16	no	—
184	Q₈	—	23	16	no	—
185	Q₈	—	46	32	no	—
186	D₄	—	23	16	no	—
187	C₂×C₄	—	2	8	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	23	8	no	—
190	C₂×C₄	—	23	4	no	—
191	C₂×C₄	—	23	4	no	—
192	C₂×C₄	—	23	4	no	—
193	C₂×C₄	—	46	32	no	—
194	C₂×C₄	—	46	4	no	—
195	C₂×C₄	—	46	4	no	—
196	C₂×C₄	—	46	16	no	—
197	C₂×C₄	D₄×D₂₃	1	4	yes	—
198	C₂×C₄	Q₈×D₂₃	1	4	yes	complemented
199	C₂×C₄	C₂³×D₂₃	1	2	yes	—
200	C₂×C₄	C₂³×D₂₃	1	8	yes	—
201	C₂×C₄	C₂³×D₂₃	1	2	yes	—
202	C₂×C₄	C₂³×D₂₃	1	2	yes	—
203	C₂²	—	23	8	no	—
204	C₂²	—	46	4	no	—
205	C₂²	—	46	4	no	—
206	C₂²	—	46	16	no	—
207	C₂²	C₂×D₄×D₂₃	1	2	yes	—
208	C₂²	C₂×Q₈×D₂₃	1	2	yes	—
209	C₂²	#10625158	1	2	yes	—
210	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
211	C₄	—	2	8	no	—
212	C₄	—	2	2	no	—
213	C₄	—	23	8	no	—
214	C₄	—	46	4	no	—
215	C₄	—	46	4	no	—
216	C₄	C₂×D₄×D₂₃	1	4	yes	—
217	C₄	C₂×Q₈×D₂₃	1	4	yes	complemented
218	C₂	—	46	8	no	—
219	C₂	#13093082	1	4	yes	direct product factor
220	C₂	C₂²×D₄×D₂₃	1	1	yes	characteristic
221	C₂	C₂²×Q₈×D₂₃	1	1	yes	characteristic
222	C₂	C₂×#10625158	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₂₃
commutator, cosocle	C₂²×C₂₃	C₂⁵
Frattini	C₂²	C₂⁴×D₂₃
Fitting	C₂×C₂₃×C₄⋊Q₈	C₂
socle	C₂³×C₂₃	C₂⁴
prime radical	C₂²×D₉₂	C₂²
hypercenter	C₂×C₄⋊Q₈	D₂₃
hypocenter	C₂₃	C₂×D₄×Q₈
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₈×D₂₃	C₂	semidirect
2. #13093082	C₂	direct (N × H)
3. C₂×#10625059	C₂	semidirect
4. C₂×#10625067	C₂	semidirect
5. C₂×#10625117	C₂	semidirect
6. C₂×C₄×Dic₄₆	C₂	semidirect
9. C₂×C₂₃×C₄⋊Q₈	C₂	semidirect
12. C₂×Q₈×D₂₃	C₂²	semidirect
18. C₂₃×C₄⋊Q₈	C₂²	semidirect
22. #10625117	C₂²	semidirect
30. #10625067	C₂²	semidirect
32. #10625059	C₂²	semidirect
34. C₄×Dic₄₆	C₂²	semidirect
46. C₂×D₉₂	Q₈	semidirect
53. C₂×Dic₄₆	D₄	semidirect
69. C₂²×D₂₃	C₂×Q₈	semidirect
75. C₂×C₄×C₂₃	C₂×Q₈	semidirect
86. D₉₂	C₂×Q₈	semidirect
90. Dic₄₆	C₂×D₄	semidirect
99. C₂×D₂₃	C₂²×Q₈	semidirect
103. C₄×C₂₃	C₂²×Q₈	semidirect
110. C₂×C₄⋊Q₈	D₂₃	semidirect
116. C₂×C₂₃	D₄×Q₈	semidirect
130. C₄⋊Q₈	C₂×D₂₃	semidirect
146. C₂₃	C₂×D₄×Q₈	semidirect
198. C₂×C₄	Q₈×D₂₃	semidirect
217. C₄	C₂×Q₈×D₂₃	semidirect
219. C₂	#13093082	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Q₈×D₂₃	2
#13093082	2
C₂×#10625059	2
C₂×#10625067	2
C₂×#10625117	2
C₂×C₄×Dic₄₆	2
C₂×C₄×D₉₂	2
C₂×#10625064	2
C₂×C₂₃×C₄⋊Q₈	2
C₂×D₄×Q₈	23

Subgroups — C2×#13093082 #15686870

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13093082 #15686870