Subgroups — C₂×#13092972 #15686686

Order 2944.

1168 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092972	C₂	1	16	yes	maximal, direct product factor
2	C₂×#10625040	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#10625061	C₂	1	2	yes	maximal, complemented
4	C₂×#10625068	C₂	1	2	yes	maximal
5	C₂×#10625117	C₂	1	2	yes	maximal
6	C₂×Q₈×Dic₂₃	C₂	1	2	yes	maximal, complemented
7	C₂×#10625058	C₂	1	2	yes	maximal, complemented
8	C₂×#10625063	C₂	1	2	yes	maximal
9	C₂×C₄²×D₂₃	C₂	1	1	yes	maximal, characteristic
10	C₂×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×Q₈×C₂₃	C₂²	1	2	yes	—
12	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
15	C₂²×Dic₄₆	C₂²	1	2	yes	—
16	C₂₃×C₄⋊Q₈	C₂²	1	8	yes	complemented
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
20	#10625117	C₂²	1	16	yes	—
21	Q₈×Dic₂₃	C₂²	1	16	yes	complemented
22	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	4	yes	—
23	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
24	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
25	C₂×C₄⋊Dic₂₃	C₂²	1	4	yes	—
26	C₂×Dic₂₃⋊C₄	C₂²	1	4	yes	—
27	C₂×C₄×Dic₂₃	C₂²	1	4	yes	—
28	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
31	#10625068	C₂²	1	16	yes	—
32	#10625063	C₂²	1	16	yes	—
33	#10625061	C₂²	1	16	yes	complemented
34	#10625058	C₂²	1	16	yes	complemented
35	C₄²×D₂₃	C₂²	1	8	yes	—
36	#10625040	C₂²	1	8	yes	complemented
37	C₂³×D₂₃	C₂³	1	1	yes	characteristic
38	C₂×Q₈×C₂₃	—	2	8	no	—
39	C₂×Q₈×C₂₃	C₂³	1	8	yes	—
40	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
41	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
44	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
45	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
46	C₂²×Dic₂₃	C₂³	1	4	yes	—
47	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
48	C₂×C₄×D₂₃	—	2	4	no	—
49	C₂×C₄×D₂₃	—	2	4	no	—
50	C₂×C₄×D₂₃	Q₈	1	8	yes	complemented
51	C₂×C₄×D₂₃	C₂³	1	4	yes	—
52	C₂×C₄×D₂₃	C₂³	1	4	yes	—
53	C₂×C₄×D₂₃	C₂³	1	4	yes	—
54	C₂×Dic₄₆	—	2	8	no	—
55	C₂×Dic₄₆	C₂³	1	8	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	16	yes	—
60	C₄⋊Dic₂₃	C₂³	1	8	yes	—
61	C₄⋊Dic₂₃	C₂³	1	8	yes	—
62	C₄⋊Dic₂₃	C₂³	1	16	yes	—
63	Dic₂₃⋊C₄	C₂³	1	16	yes	—
64	C₄×Dic₂₃	C₂³	1	4	yes	—
65	C₄×Dic₂₃	C₂³	1	4	yes	—
66	C₄×Dic₂₃	C₂³	1	4	yes	—
67	C₄×Dic₂₃	C₂³	1	16	yes	—
68	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
69	C₂²×D₂₃	—	2	2	no	—
70	C₂²×D₂₃	—	2	2	no	—
71	C₂²×D₂₃	C₂×Q₈	1	4	yes	—
72	C₂²×D₂₃	C₂⁴	1	2	yes	—
73	Q₈×C₂₃	—	2	16	no	—
74	C₂×C₄×C₂₃	—	2	8	no	—
75	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	—
76	C₂×C₄×C₂₃	#41	1	4	yes	—
77	C₂×C₄×C₂₃	#41	1	4	yes	—
78	C₂×C₄×C₂₃	C₂⁴	1	8	yes	—
79	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
80	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
81	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
82	C₂×Dic₂₃	—	2	8	no	—
83	C₂×Dic₂₃	—	2	2	no	—
84	C₂×Dic₂₃	—	2	2	no	—
85	C₂×Dic₂₃	C₂×Q₈	1	4	yes	—
86	C₂×Dic₂₃	C₂⁴	1	2	yes	—
87	C₂×Dic₂₃	C₂⁴	1	2	yes	—
88	C₂×Dic₂₃	C₂⁴	1	8	yes	—
89	C₂×Dic₂₃	C₂⁴	1	2	yes	—
90	C₄×D₂₃	—	2	8	no	—
91	C₄×D₂₃	—	2	8	no	—
92	C₄×D₂₃	C₂×Q₈	1	16	yes	complemented
93	Dic₄₆	—	2	16	no	—
94	C₂×#533	—	23	1	no	maximal, Sylow
95	C₂²×C₂₃	C₂²×Q₈	1	2	yes	—
96	C₂²×C₂₃	C₂×#41	1	2	yes	—
97	C₂²×C₂₃	C₂×#41	1	2	yes	—
98	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
99	C₂×D₂₃	—	2	8	no	—
100	C₂×D₂₃	—	2	2	no	—
101	C₂×D₂₃	—	2	2	no	—
102	C₂×D₂₃	C₂²×Q₈	1	4	yes	—
103	C₄×C₂₃	—	2	8	no	—
104	C₄×C₂₃	C₂²×Q₈	1	4	yes	—
105	C₄×C₂₃	C₂×#41	1	4	yes	—
106	C₄×C₂₃	C₂×#41	1	4	yes	—
107	Dic₂₃	—	2	8	no	—
108	Dic₂₃	—	2	2	no	—
109	Dic₂₃	—	2	2	no	—
110	Dic₂₃	C₂²×Q₈	1	4	yes	—
111	#533	—	23	16	no	—
112	C₂×C₄⋊Q₈	—	23	1	no	—
113	C₂×C₄⋊Q₈	D₂₃	1	1	yes	characteristic, complemented
114	C₂×#125	—	23	2	no	—
115	C₂×C₂²⋊Q₈	—	23	2	no	—
116	C₂×C₂²⋊Q₈	—	23	2	no	—
117	C₂×C₄×Q₈	—	23	2	no	—
118	C₂×C₄×Q₈	—	23	2	no	—
119	C₂×#117	—	23	2	no	—
120	C₂²×C₄²	—	23	1	no	—
121	C₂×C₂₃	#533	1	4	yes	complemented
122	C₂×C₂₃	C₂³×Q₈	1	1	yes	characteristic
123	C₂×C₂₃	C₂²×#41	1	1	yes	characteristic
124	C₂×C₂₃	C₂²×#41	1	1	yes	characteristic
125	D₂₃	—	2	4	no	—
126	C₂²×Q₈	—	23	2	no	—
127	C₂²×Q₈	C₂×D₂₃	1	2	yes	—
128	C₂³×C₄	—	23	1	no	—
129	C₂³×C₄	—	23	1	no	—
130	C₂³×C₄	—	23	1	no	—
131	C₄⋊Q₈	—	23	8	no	—
132	C₄⋊Q₈	C₂×D₂₃	1	8	yes	complemented
133	#125	—	23	16	no	—
134	C₂²⋊Q₈	—	23	16	no	—
135	C₂²⋊Q₈	—	23	16	no	—
136	C₄×Q₈	—	23	16	no	—
137	C₄×Q₈	—	23	16	no	—
138	#117	—	23	16	no	—
139	C₂×C₄⋊C₄	—	23	2	no	—
140	C₂×C₄⋊C₄	—	23	4	no	—
141	C₂×C₄⋊C₄	—	23	4	no	—
142	C₂×C₄⋊C₄	—	23	2	no	—
143	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
144	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
145	C₂×C₂²⋊C₄	—	23	4	no	—
146	C₂×C₄²	—	23	4	no	—
147	C₂×C₄²	—	23	1	no	—
148	C₂×C₄²	—	23	1	no	—
149	C₂×C₄²	—	23	1	no	—
150	C₂×C₄²	—	23	8	no	—
151	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
152	C₂₃	C₂×#533	1	1	yes	characteristic, complemented, Sylow
153	C₂⁴	—	23	1	no	—
154	C₂×Q₈	—	2	8	no	—
155	C₂×Q₈	—	23	8	no	—
156	C₂×Q₈	—	46	8	no	—
157	C₂×Q₈	C₂²×D₂₃	1	8	yes	—
158	C₂²×C₄	—	23	1	no	—
159	C₂²×C₄	—	23	8	no	—
160	C₂²×C₄	—	23	4	no	—
161	C₂²×C₄	—	23	1	no	—
162	C₂²×C₄	—	23	1	no	—
163	C₂²×C₄	—	23	4	no	—
164	C₂²×C₄	—	23	4	no	—
165	C₂²×C₄	—	23	4	no	—
166	C₂²×C₄	—	46	4	no	—
167	C₂²×C₄	—	46	4	no	—
168	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
169	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
171	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
172	C₄⋊C₄	—	23	8	no	—
173	C₄⋊C₄	—	23	16	no	—
174	C₄⋊C₄	—	23	16	no	—
175	C₄⋊C₄	—	23	8	no	—
176	C₄⋊C₄	C₂²×D₂₃	1	8	yes	—
177	C₄⋊C₄	C₂²×D₂₃	1	8	yes	—
178	C₂²⋊C₄	—	23	16	no	—
179	C₄²	—	23	4	no	—
180	C₄²	—	23	4	no	—
181	C₄²	—	23	4	no	—
182	C₄²	—	23	16	no	—
183	C₄²	C₂²×D₂₃	1	4	yes	—
184	C₂³	—	23	4	no	—
185	C₂³	—	23	2	no	—
186	C₂³	—	46	2	no	—
187	C₂³	—	46	2	no	—
188	C₂³	C₂³×D₂₃	1	1	yes	characteristic
189	Q₈	—	2	16	no	—
190	Q₈	—	46	16	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	23	4	no	—
193	C₂×C₄	—	23	2	no	—
194	C₂×C₄	—	23	16	no	—
195	C₂×C₄	—	23	2	no	—
196	C₂×C₄	—	23	2	no	—
197	C₂×C₄	—	23	8	no	—
198	C₂×C₄	—	46	2	no	—
199	C₂×C₄	—	46	2	no	—
200	C₂×C₄	—	46	8	no	—
201	C₂×C₄	—	46	8	no	—
202	C₂×C₄	—	46	8	no	—
203	C₂×C₄	#67290	1	4	yes	—
204	C₂×C₄	Q₈×D₂₃	1	4	yes	—
205	C₂×C₄	#67292	1	4	yes	—
206	C₂×C₄	C₂³×D₂₃	1	8	yes	—
207	C₂×C₄	C₂³×D₂₃	1	2	yes	—
208	C₂×C₄	C₂³×D₂₃	1	2	yes	—
209	C₂×C₄	C₂³×D₂₃	1	2	yes	—
210	C₂²	—	23	4	no	—
211	C₂²	—	46	8	no	—
212	C₂²	—	46	2	no	—
213	C₂²	—	46	2	no	—
214	C₂²	C₂×#67290	1	2	yes	—
215	C₂²	C₂×Q₈×D₂₃	1	2	yes	—
216	C₂²	C₂×#67292	1	2	yes	—
217	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
218	C₄	—	2	8	no	—
219	C₄	—	23	4	no	—
220	C₄	—	46	2	no	—
221	C₄	—	46	2	no	—
222	C₄	—	46	8	no	—
223	C₄	C₂×#67290	1	4	yes	—
224	C₄	C₂×Q₈×D₂₃	1	4	yes	—
225	C₄	C₂×#67292	1	4	yes	—
226	C₂	—	46	4	no	—
227	C₂	#13092972	1	4	yes	direct product factor
228	C₂	C₂²×Q₈×D₂₃	1	1	yes	characteristic
229	C₂	C₂²×#67290	1	1	yes	characteristic
230	C₂	C₂²×#67292	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	C₂³×D₂₃	C₂³
hypercenter	C₂×C₄⋊Q₈	D₂₃
hypocenter	C₂₃	C₂×#533
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. #13092972	C₂	direct (N × H)
2. C₂×#10625040	C₂	semidirect
3. C₂×#10625061	C₂	semidirect
6. C₂×Q₈×Dic₂₃	C₂	semidirect
7. C₂×#10625058	C₂	semidirect
10. C₂×C₂₃×C₄⋊Q₈	C₂	semidirect
16. C₂₃×C₄⋊Q₈	C₂²	semidirect
21. Q₈×Dic₂₃	C₂²	semidirect
33. #10625061	C₂²	semidirect
34. #10625058	C₂²	semidirect
36. #10625040	C₂²	semidirect
50. C₂×C₄×D₂₃	Q₈	semidirect
92. C₄×D₂₃	C₂×Q₈	semidirect
113. C₂×C₄⋊Q₈	D₂₃	semidirect
121. C₂×C₂₃	#533	semidirect
132. C₄⋊Q₈	C₂×D₂₃	semidirect
152. C₂₃	C₂×#533	semidirect
227. C₂	#13092972	direct (N × H)

Maximal subgroups

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

subgroup	index
#13092972	2
C₂×#10625040	2
C₂×#10625061	2
C₂×#10625068	2
C₂×#10625117	2
C₂×Q₈×Dic₂₃	2
C₂×#10625058	2
C₂×#10625063	2
C₂×C₄²×D₂₃	2
C₂×C₂₃×C₄⋊Q₈	2
C₂×#533	23

Subgroups — C2×#13092972 #15686686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13092972 #15686686