Subgroups — C₂×#13093078 #15686865

Order 2944.

1620 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	C₂²×#67292	C₂	1	1	yes	maximal, characteristic, complemented
2	#13093078	C₂	1	16	yes	maximal, direct product factor
3	C₂×#10625118	C₂	1	3	yes	maximal, complemented
4	C₂×#10625117	C₂	1	3	yes	maximal, complemented
5	C₂×#10625101	C₂	1	3	yes	maximal, complemented
6	C₂×Q₈×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₂²⋊D₂₃	C₂	1	3	yes	maximal, complemented
8	C₂³×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₈×C₂₃	—	2	8	no	—
10	C₂²×Q₈×C₂₃	C₂²	1	3	yes	—
11	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂²×Q₈×C₂₃	C₂²	1	8	yes	complemented
13	C₂³×C₄×C₂₃	C₂²	1	3	yes	—
14	C₂²×C₂²⋊D₂₃	C₂²	1	3	yes	—
15	C₂×#67292	—	2	8	no	—
16	C₂×#67292	C₂²	1	8	yes	complemented
17	C₂²×D₉₂	C₂²	1	3	yes	—
18	C₂²×C₄×D₂₃	C₂²	1	3	yes	—
19	C₂×C₂²⋊Dic₂₃	C₂²	1	1	yes	characteristic
20	#10625118	C₂²	1	24	yes	complemented
21	#10625117	C₂²	1	24	yes	complemented
22	Q₈×Dic₂₃	C₂²	1	8	yes	complemented
23	#10625101	C₂²	1	24	yes	complemented
24	C₄×C₂²⋊D₂₃	C₂²	1	24	yes	complemented
25	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	3	yes	—
26	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	6	yes	—
27	C₂×C₄⋊Dic₂₃	C₂²	1	3	yes	—
28	C₂×Dic₂₃⋊C₄	C₂²	1	3	yes	—
29	C₂×C₄×Dic₂₃	C₂²	1	3	yes	—
30	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
31	C₂³×D₂₃	C₂³	1	3	yes	—
32	C₂×Q₈×C₂₃	—	2	32	no	—
33	C₂×Q₈×C₂₃	—	2	12	no	—
34	C₂×Q₈×C₂₃	D₄	1	8	yes	complemented
35	C₂×Q₈×C₂₃	C₂³	1	12	yes	—
36	C₂×Q₈×C₂₃	C₂³	1	4	yes	—
37	C₂²×C₄×C₂₃	—	2	12	no	—
38	C₂²×C₄×C₂₃	C₂³	1	3	yes	—
39	C₂²×C₄×C₂₃	C₂³	1	12	yes	—
40	C₂²×C₄×C₂₃	C₂³	1	3	yes	—
41	C₂×C₂²⋊D₂₃	—	2	12	no	—
42	C₂×C₂²⋊D₂₃	C₂³	1	12	yes	—
43	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
44	C₂²×Dic₂₃	C₂³	1	3	yes	—
45	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
46	#67292	—	2	32	no	—
47	C₂×D₉₂	—	2	12	no	—
48	C₂×D₉₂	C₂³	1	12	yes	—
49	C₂×C₄×D₂₃	—	2	12	no	—
50	C₂×C₄×D₂₃	C₂³	1	12	yes	—
51	C₂²⋊Dic₂₃	C₂³	1	4	yes	—
52	(C₂×D₂₃)⋊C₄	C₂³	1	12	yes	—
53	(C₂×D₂₃)⋊C₄	C₂³	1	24	yes	—
54	C₄⋊Dic₂₃	C₂³	1	12	yes	—
55	Dic₂₃⋊C₄	C₂³	1	12	yes	—
56	C₄×Dic₂₃	C₂³	1	12	yes	—
57	C₂³×C₂₃	—	2	2	no	—
58	C₂³×C₂₃	—	2	2	no	—
59	C₂³×C₂₃	#41	1	4	yes	complemented
60	C₂³×C₂₃	C₂⁴	1	2	yes	—
61	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
62	C₂²×D₂₃	—	2	6	no	—
63	C₂²×D₂₃	—	2	6	no	—
64	C₂²×D₂₃	—	2	6	no	—
65	C₂²×D₂₃	C₂⁴	1	6	yes	—
66	Q₈×C₂₃	—	2	24	no	—
67	Q₈×C₂₃	C₂×D₄	1	16	yes	complemented
68	C₂×C₄×C₂₃	—	2	6	no	—
69	C₂×C₄×C₂₃	—	2	24	no	—
70	C₂×C₄×C₂₃	C₂×D₄	1	12	yes	—
71	C₂×C₄×C₂₃	C₂⁴	1	6	yes	—
72	C₂×C₄×C₂₃	C₂⁴	1	6	yes	—
73	C₂²⋊D₂₃	—	2	24	no	—
74	C₂×Dic₂₃	—	2	2	no	—
75	C₂×Dic₂₃	—	2	6	no	—
76	C₂×Dic₂₃	—	2	2	no	—
77	C₂×Dic₂₃	C₂⁴	1	2	yes	—
78	C₂×Dic₂₃	C₂⁴	1	6	yes	—
79	C₂×Dic₂₃	C₂⁴	1	2	yes	—
80	D₉₂	—	2	24	no	—
81	C₄×D₂₃	—	2	24	no	—
82	C₂×#548	—	23	1	no	maximal, Sylow
83	C₂²×C₂₃	—	2	8	no	—
84	C₂²×C₂₃	—	2	2	no	—
85	C₂²×C₂₃	—	2	2	no	—
86	C₂²×C₂₃	C₂²×D₄	1	2	yes	—
87	C₂²×C₂₃	C₂×#41	1	2	yes	—
88	C₂²×C₂₃	C₂×#41	1	4	yes	complemented
89	C₂²×C₂₃	2₋¹⁺⁴	1	2	yes	—
90	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
91	C₂×D₂₃	—	2	6	no	—
92	C₂×D₂₃	—	2	6	no	—
93	C₂×D₂₃	—	2	6	no	—
94	C₂×D₂₃	—	4	12	no	—
95	C₄×C₂₃	—	2	6	no	—
96	C₄×C₂₃	C₂²×D₄	1	12	yes	—
97	Dic₂₃	—	2	2	no	—
98	Dic₂₃	—	2	6	no	—
99	Dic₂₃	—	2	2	no	—
100	C₂²×#41	—	23	1	no	—
101	C₂³×Q₈	D₂₃	1	1	yes	characteristic, complemented
102	#548	—	23	16	no	—
103	C₂×#124	—	23	3	no	—
104	C₂×C₂²⋊Q₈	—	23	3	no	—
105	C₂×#121	—	23	3	no	—
106	C₂×C₄×Q₈	—	23	1	no	—
107	C₂×C₄×D₄	—	23	3	no	—
108	C₂×C₂₃	—	2	4	no	—
109	C₂×C₂₃	#548	1	4	yes	complemented
110	C₂×C₂₃	C₂³×D₄	1	1	yes	characteristic
111	C₂×C₂₃	C₂²×#41	1	1	yes	characteristic
112	C₂×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
113	D₂₃	—	4	6	no	—
114	C₂×#41	—	23	8	no	—
115	C₂×#41	—	46	8	no	—
116	C₂²×Q₈	—	2	8	no	—
117	C₂²×Q₈	C₂×D₂₃	1	3	yes	—
118	C₂²×Q₈	C₂×D₂₃	1	1	yes	characteristic
119	C₂²×Q₈	C₂×D₂₃	1	8	yes	complemented
120	C₂²×D₄	—	23	3	no	—
121	C₂²×D₄	—	23	3	no	—
122	C₂³×C₄	—	23	3	no	—
123	C₂³×C₄	C₂×D₂₃	1	3	yes	—
124	#124	—	23	24	no	—
125	C₂²⋊Q₈	—	23	24	no	—
126	#121	—	23	24	no	—
127	C₄×Q₈	—	23	8	no	—
128	C₄×D₄	—	23	24	no	—
129	C₂×C₄⋊C₄	—	23	3	no	—
130	C₂×C₄⋊C₄	—	23	3	no	—
131	C₂×C₂²⋊C₄	—	23	1	no	—
132	C₂×C₂²⋊C₄	—	23	3	no	—
133	C₂×C₂²⋊C₄	—	23	6	no	—
134	C₂×C₄²	—	23	3	no	—
135	C₂₃	C₂×#548	1	1	yes	characteristic, complemented, Sylow
136	C₂⁴	—	23	3	no	—
137	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
138	#41	—	46	32	no	—
139	C₂×Q₈	—	2	32	no	—
140	C₂×Q₈	—	2	12	no	—
141	C₂×Q₈	C₂²⋊D₂₃	1	8	yes	complemented
142	C₂×Q₈	C₂²×D₂₃	1	12	yes	—
143	C₂×Q₈	C₂²×D₂₃	1	4	yes	—
144	C₂×D₄	—	23	12	no	—
145	C₂×D₄	—	23	12	no	—
146	C₂×D₄	—	46	12	no	—
147	C₂×D₄	—	46	12	no	—
148	C₂²×C₄	—	2	12	no	—
149	C₂²×C₄	—	23	1	no	—
150	C₂²×C₄	—	23	3	no	—
151	C₂²×C₄	—	23	1	no	—
152	C₂²×C₄	—	23	12	no	—
153	C₂²×C₄	—	46	12	no	—
154	C₂²×C₄	C₂²×D₂₃	1	12	yes	—
155	C₂²×C₄	C₂²×D₂₃	1	3	yes	—
156	C₂²×C₄	C₂²×D₂₃	1	3	yes	—
157	C₄⋊C₄	—	23	12	no	—
158	C₄⋊C₄	—	23	12	no	—
159	C₂²⋊C₄	—	23	4	no	—
160	C₂²⋊C₄	—	23	12	no	—
161	C₂²⋊C₄	—	23	24	no	—
162	C₄²	—	23	12	no	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	2	no	—
165	C₂³	—	23	6	no	—
166	C₂³	—	46	6	no	—
167	C₂³	—	46	6	no	—
168	C₂³	—	46	6	no	—
169	C₂³	#67292	1	4	yes	complemented
170	C₂³	C₂³×D₂₃	1	2	yes	—
171	C₂³	C₂³×D₂₃	1	1	yes	characteristic
172	Q₈	—	2	24	no	—
173	Q₈	C₂×C₂²⋊D₂₃	1	16	yes	complemented
174	D₄	—	46	24	no	—
175	D₄	—	46	24	no	—
176	C₂×C₄	—	2	6	no	—
177	C₂×C₄	—	2	24	no	—
178	C₂×C₄	—	23	2	no	—
179	C₂×C₄	—	23	6	no	—
180	C₂×C₄	—	23	2	no	—
181	C₂×C₄	—	46	6	no	—
182	C₂×C₄	—	46	2	no	—
183	C₂×C₄	—	46	2	no	—
184	C₂×C₄	—	46	24	no	—
185	C₂×C₄	C₂×C₂²⋊D₂₃	1	12	yes	—
186	C₂×C₄	C₂³×D₂₃	1	6	yes	—
187	C₂×C₄	C₂³×D₂₃	1	6	yes	—
188	C₂²	—	2	2	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	8	no	—
191	C₂²	—	46	6	no	—
192	C₂²	—	46	6	no	—
193	C₂²	—	46	6	no	—
194	C₂²	—	92	12	no	—
195	C₂²	C₂×#67292	1	2	yes	—
196	C₂²	C₂×#67292	1	4	yes	complemented
197	C₂²	#10625158	1	2	yes	—
198	C₂²	C₂²×C₂²⋊D₂₃	1	2	yes	—
199	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
200	C₄	—	2	6	no	—
201	C₄	—	46	2	no	—
202	C₄	—	46	6	no	—
203	C₄	—	46	2	no	—
204	C₄	C₂²×C₂²⋊D₂₃	1	12	yes	—
205	C₂	—	2	4	no	—
206	C₂	—	92	6	no	—
207	C₂	#13093078	1	4	yes	direct product factor
208	C₂	C₂³×C₂²⋊D₂₃	1	1	yes	characteristic
209	C₂	C₂²×#67292	1	1	yes	characteristic
210	C₂	C₂×#10625158	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₂₃
commutator, cosocle	C₂²×C₂₃	C₂⁵
Frattini	C₂²	C₂⁴×D₂₃
Fitting	C₂³×Q₈×C₂₃	C₂
socle	C₂³×C₂₃	C₂⁴
hypercenter	C₂³×Q₈	D₂₃
hypocenter	C₂₃	C₂×#548
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#67292	C₂	semidirect
2. #13093078	C₂	direct (N × H)
3. C₂×#10625118	C₂	semidirect
4. C₂×#10625117	C₂	semidirect
5. C₂×#10625101	C₂	semidirect
6. C₂×Q₈×Dic₂₃	C₂	semidirect
7. C₂×C₄×C₂²⋊D₂₃	C₂	semidirect
8. C₂³×Q₈×C₂₃	C₂	semidirect
12. C₂²×Q₈×C₂₃	C₂²	semidirect
16. C₂×#67292	C₂²	semidirect
20. #10625118	C₂²	semidirect
21. #10625117	C₂²	semidirect
22. Q₈×Dic₂₃	C₂²	semidirect
23. #10625101	C₂²	semidirect
24. C₄×C₂²⋊D₂₃	C₂²	semidirect
34. C₂×Q₈×C₂₃	D₄	semidirect
59. C₂³×C₂₃	#41	semidirect
67. Q₈×C₂₃	C₂×D₄	semidirect
88. C₂²×C₂₃	C₂×#41	semidirect
101. C₂³×Q₈	D₂₃	semidirect
109. C₂×C₂₃	#548	semidirect
119. C₂²×Q₈	C₂×D₂₃	semidirect
135. C₂₃	C₂×#548	semidirect
141. C₂×Q₈	C₂²⋊D₂₃	semidirect
169. C₂³	#67292	semidirect
173. Q₈	C₂×C₂²⋊D₂₃	semidirect
196. C₂²	C₂×#67292	semidirect
207. C₂	#13093078	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×#67292	2
#13093078	2
C₂×#10625118	2
C₂×#10625117	2
C₂×#10625101	2
C₂×Q₈×Dic₂₃	2
C₂×C₄×C₂²⋊D₂₃	2
C₂³×Q₈×C₂₃	2
C₂×#548	23

Subgroups — C2×#13093078 #15686865

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13093078 #15686865