Subgroups — C₂×#13092984 #15686706

Order 2944.

1896 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	C₂²×#67288	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092984	C₂	1	16	yes	maximal, direct product factor
3	C₂×#10625111	C₂	1	2	yes	maximal, complemented
4	C₂×#10625107	C₂	1	2	yes	maximal, complemented
5	C₂×#10625109	C₂	1	4	yes	maximal, complemented
6	C₂×#10625124	C₂	1	4	yes	maximal, complemented
7	C₂×#10625097	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×C₂₃	C₂²	1	2	yes	—
10	C₂²×D₄×C₂₃	—	2	8	no	—
11	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
12	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
13	C₂²×D₄×C₂₃	C₂²	1	8	yes	complemented
14	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂²×C₂²⋊D₂₃	C₂²	1	2	yes	—
16	C₂²×C₂²⋊D₂₃	C₂²	1	8	yes	complemented
17	C₂×#67288	—	2	8	no	—
18	C₂×#67288	C₂²	1	8	yes	complemented
19	C₂²×D₉₂	C₂²	1	1	yes	characteristic, complemented
20	C₂²×C₄×D₂₃	C₂²	1	2	yes	complemented
21	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic, complemented
22	#10625124	C₂²	1	32	yes	complemented
23	C₂×C₂²⋊Dic₂₃	C₂²	1	2	yes	—
24	C₂×C₂²⋊Dic₂₃	C₂²	1	8	yes	complemented
25	#10625111	C₂²	1	16	yes	complemented
26	#10625109	C₂²	1	32	yes	complemented
27	#10625107	C₂²	1	16	yes	complemented
28	#10625097	C₂²	1	8	yes	complemented
29	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	complemented
30	C₂×C₄×Dic₂₃	C₂²	1	2	yes	complemented
31	C₂⁴×C₂₃	—	2	8	no	—
32	C₂⁴×C₂₃	—	4	8	no	—
33	C₂⁴×C₂₃	C₂³	1	4	yes	—
34	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
35	C₂⁴×C₂₃	C₂³	1	8	yes	—
36	C₂³×D₂₃	C₂³	1	2	yes	—
37	C₂×D₄×C₂₃	—	2	8	no	—
38	C₂×D₄×C₂₃	—	2	8	no	—
39	C₂×D₄×C₂₃	—	2	32	no	—
40	C₂×D₄×C₂₃	C₂³	1	8	yes	—
41	C₂×D₄×C₂₃	C₂³	1	8	yes	—
42	C₂²×C₄×C₂₃	D₄	1	8	yes	complemented
43	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
46	C₂×C₂²⋊D₂₃	—	2	8	no	—
47	C₂×C₂²⋊D₂₃	—	4	16	no	—
48	C₂×C₂²⋊D₂₃	C₂³	1	8	yes	—
49	C₂×C₂²⋊D₂₃	C₂³	1	32	yes	complemented
50	C₂²×Dic₂₃	C₂³	1	4	yes	—
51	C₂²×Dic₂₃	C₂³	1	2	yes	—
52	#67288	—	2	32	no	—
53	C₂×D₉₂	—	2	4	no	—
54	C₂×D₉₂	C₂³	1	4	yes	complemented
55	C₂×C₄×D₂₃	—	2	8	no	—
56	C₂×C₄×D₂₃	C₂³	1	8	yes	complemented
57	C₂×Dic₄₆	—	2	4	no	—
58	C₂×Dic₄₆	C₂³	1	4	yes	complemented
59	C₂²⋊Dic₂₃	C₂³	1	8	yes	—
60	C₂²⋊Dic₂₃	C₂³	1	32	yes	complemented
61	C₄⋊Dic₂₃	C₂³	1	8	yes	complemented
62	C₄×Dic₂₃	C₂³	1	8	yes	complemented
63	C₂³×C₂₃	—	2	4	no	—
64	C₂³×C₂₃	—	2	16	no	—
65	C₂³×C₂₃	—	2	8	no	—
66	C₂³×C₂₃	—	2	16	no	—
67	C₂³×C₂₃	—	4	16	no	—
68	C₂³×C₂₃	—	4	32	no	—
69	C₂³×C₂₃	C₂×D₄	1	4	yes	—
70	C₂³×C₂₃	C₂⁴	1	8	yes	—
71	C₂³×C₂₃	C₂⁴	1	2	yes	—
72	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
73	C₂²×D₂₃	—	2	4	no	—
74	C₂²×D₂₃	—	2	8	no	—
75	C₂²×D₂₃	C₂⁴	1	4	yes	—
76	D₄×C₂₃	—	2	16	no	—
77	D₄×C₂₃	—	2	16	no	—
78	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
79	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
80	C₂×C₄×C₂₃	C₂×D₄	1	16	yes	complemented
81	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
82	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
83	C₂²⋊D₂₃	—	2	16	no	—
84	C₂²⋊D₂₃	—	4	32	no	—
85	C₂×Dic₂₃	—	2	4	no	—
86	C₂×Dic₂₃	—	2	8	no	—
87	C₂×Dic₂₃	C₂⁴	1	8	yes	—
88	C₂×Dic₂₃	C₂⁴	1	4	yes	—
89	D₉₂	—	2	8	no	—
90	C₄×D₂₃	—	2	16	no	—
91	Dic₄₆	—	2	8	no	—
92	C₂×#535	—	23	1	no	maximal, Sylow
93	C₂²×C₂₃	—	2	8	no	—
94	C₂²×C₂₃	—	2	4	no	—
95	C₂²×C₂₃	—	2	8	no	—
96	C₂²×C₂₃	—	2	16	no	—
97	C₂²×C₂₃	—	4	32	no	—
98	C₂²×C₂₃	—	4	16	no	—
99	C₂²×C₂₃	C₂²×D₄	1	4	yes	—
100	C₂²×C₂₃	C₂²×D₄	1	2	yes	—
101	C₂²×C₂₃	H(5,𝔽₂)	1	4	yes	—
102	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
103	C₂×D₂₃	—	2	4	no	—
104	C₂×D₂₃	—	2	8	no	—
105	C₂×D₂₃	—	4	8	no	—
106	C₄×C₂₃	C₂²×D₄	1	4	yes	—
107	C₄×C₂₃	C₂²×D₄	1	4	yes	—
108	Dic₂₃	—	2	4	no	—
109	Dic₂₃	—	2	8	no	—
110	C₂²×#41	—	23	1	no	—
111	C₂³×D₄	D₂₃	1	1	yes	characteristic, complemented
112	#535	—	23	16	no	—
113	C₂×#127	—	23	2	no	—
114	C₂×#124	—	23	2	no	—
115	C₂×#121	—	23	4	no	—
116	C₂×#120	—	23	4	no	—
117	C₂×#117	—	23	1	no	—
118	C₂×C₂₃	—	2	4	no	—
119	C₂×C₂₃	—	4	8	no	—
120	C₂×C₂₃	#535	1	4	yes	complemented
121	C₂×C₂₃	C₂³×D₄	1	1	yes	characteristic
122	C₂×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
123	D₂₃	—	4	4	no	—
124	C₂⁵	C₂×D₂₃	1	2	yes	—
125	C₂×#41	—	23	8	no	—
126	C₂×#41	—	46	8	no	—
127	C₂²×Q₈	—	23	1	no	—
128	C₂²×D₄	—	2	8	no	—
129	C₂²×D₄	—	23	2	no	—
130	C₂²×D₄	—	23	8	no	—
131	C₂²×D₄	—	23	1	no	—
132	C₂²×D₄	C₂×D₂₃	1	2	yes	—
133	C₂²×D₄	C₂×D₂₃	1	2	yes	—
134	C₂²×D₄	C₂×D₂₃	1	8	yes	complemented
135	C₂³×C₄	—	23	2	no	—
136	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
137	#127	—	23	16	no	—
138	#124	—	23	16	no	—
139	#121	—	23	32	no	—
140	#120	—	23	32	no	—
141	#117	—	23	8	no	—
142	C₂×C₄⋊C₄	—	23	2	no	—
143	C₂×C₂²⋊C₄	—	23	2	no	—
144	C₂×C₂²⋊C₄	—	23	8	no	—
145	C₂×C₄²	—	23	2	no	—
146	C₂₃	C₂×#535	1	1	yes	characteristic, complemented, Sylow
147	C₂⁴	—	2	8	no	—
148	C₂⁴	—	4	8	no	—
149	C₂⁴	—	23	2	no	—
150	C₂⁴	C₂²×D₂₃	1	4	yes	—
151	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
152	C₂⁴	C₂²×D₂₃	1	8	yes	—
153	#41	—	46	32	no	—
154	C₂×Q₈	—	23	4	no	—
155	C₂×Q₈	—	46	4	no	—
156	C₂×D₄	—	2	8	no	—
157	C₂×D₄	—	2	8	no	—
158	C₂×D₄	—	2	32	no	—
159	C₂×D₄	—	23	8	no	—
160	C₂×D₄	—	23	32	no	—
161	C₂×D₄	—	23	4	no	—
162	C₂×D₄	—	46	8	no	—
163	C₂×D₄	—	46	4	no	—
164	C₂×D₄	—	92	16	no	—
165	C₂×D₄	C₂²×D₂₃	1	8	yes	—
166	C₂×D₄	C₂²×D₂₃	1	8	yes	—
167	C₂²×C₄	—	23	4	no	—
168	C₂²×C₄	—	23	2	no	—
169	C₂²×C₄	—	23	8	no	—
170	C₂²×C₄	—	46	8	no	—
171	C₂²×C₄	C₂²⋊D₂₃	1	8	yes	complemented
172	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
173	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
174	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
175	C₄⋊C₄	—	23	8	no	—
176	C₂²⋊C₄	—	23	8	no	—
177	C₂²⋊C₄	—	23	32	no	—
178	C₄²	—	23	8	no	—
179	C₂³	—	2	4	no	—
180	C₂³	—	2	16	no	—
181	C₂³	—	2	16	no	—
182	C₂³	—	2	8	no	—
183	C₂³	—	4	16	no	—
184	C₂³	—	4	32	no	—
185	C₂³	—	23	4	no	—
186	C₂³	—	46	8	no	—
187	C₂³	—	46	4	no	—
188	C₂³	C₂×C₂²⋊D₂₃	1	4	yes	—
189	C₂³	C₂³×D₂₃	1	8	yes	—
190	C₂³	C₂³×D₂₃	1	2	yes	—
191	C₂³	C₂³×D₂₃	1	1	yes	characteristic
192	Q₈	—	46	8	no	—
193	D₄	—	2	16	no	—
194	D₄	—	2	16	no	—
195	D₄	—	46	16	no	—
196	D₄	—	46	8	no	—
197	D₄	—	92	32	no	—
198	C₂×C₄	—	23	8	no	—
199	C₂×C₄	—	23	4	no	—
200	C₂×C₄	—	46	8	no	—
201	C₂×C₄	—	46	4	no	—
202	C₂×C₄	—	46	16	no	—
203	C₂×C₄	C₂×C₂²⋊D₂₃	1	4	yes	—
204	C₂×C₄	C₂×C₂²⋊D₂₃	1	4	yes	—
205	C₂×C₄	C₂×C₂²⋊D₂₃	1	16	yes	complemented
206	C₂×C₄	C₂³×D₂₃	1	2	yes	—
207	C₂×C₄	C₂³×D₂₃	1	2	yes	—
208	C₂²	—	2	8	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	2	8	no	—
211	C₂²	—	2	16	no	—
212	C₂²	—	4	32	no	—
213	C₂²	—	4	16	no	—
214	C₂²	—	46	4	no	—
215	C₂²	—	46	8	no	—
216	C₂²	—	92	8	no	—
217	C₂²	#10625155	1	4	yes	—
218	C₂²	C₂²×C₂²⋊D₂₃	1	4	yes	—
219	C₂²	C₂²×C₂²⋊D₂₃	1	2	yes	—
220	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
221	C₄	—	46	4	no	—
222	C₄	—	46	8	no	—
223	C₄	C₂²×C₂²⋊D₂₃	1	4	yes	—
224	C₄	C₂²×C₂²⋊D₂₃	1	4	yes	—
225	C₂	—	2	4	no	—
226	C₂	—	4	8	no	—
227	C₂	—	92	4	no	—
228	C₂	#13092984	1	4	yes	direct product factor
229	C₂	C₂³×C₂²⋊D₂₃	1	1	yes	characteristic
230	C₂	C₂×#10625155	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₂₃
commutator, cosocle	C₂²×C₂₃	C₂⁵
Frattini	C₂²	C₂⁴×D₂₃
Fitting	C₂³×D₄×C₂₃	C₂
socle	C₂³×C₂₃	C₂⁴
hypercenter	C₂³×D₄	D₂₃
hypocenter	C₂₃	C₂×#535
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂³×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#67288	C₂	semidirect
2. #13092984	C₂	direct (N × H)
3. C₂×#10625111	C₂	semidirect
4. C₂×#10625107	C₂	semidirect
5. C₂×#10625109	C₂	semidirect
6. C₂×#10625124	C₂	semidirect
7. C₂×#10625097	C₂	semidirect
8. C₂³×D₄×C₂₃	C₂	semidirect
13. C₂²×D₄×C₂₃	C₂²	semidirect
16. C₂²×C₂²⋊D₂₃	C₂²	semidirect
18. C₂×#67288	C₂²	semidirect
19. C₂²×D₉₂	C₂²	semidirect
20. C₂²×C₄×D₂₃	C₂²	semidirect
21. C₂²×Dic₄₆	C₂²	semidirect
22. #10625124	C₂²	semidirect
24. C₂×C₂²⋊Dic₂₃	C₂²	semidirect
25. #10625111	C₂²	semidirect
26. #10625109	C₂²	semidirect
27. #10625107	C₂²	semidirect
28. #10625097	C₂²	semidirect
29. C₂×C₄⋊Dic₂₃	C₂²	semidirect
30. C₂×C₄×Dic₂₃	C₂²	semidirect
42. C₂²×C₄×C₂₃	D₄	semidirect
49. C₂×C₂²⋊D₂₃	C₂³	semidirect
54. C₂×D₉₂	C₂³	semidirect
56. C₂×C₄×D₂₃	C₂³	semidirect
58. C₂×Dic₄₆	C₂³	semidirect
60. C₂²⋊Dic₂₃	C₂³	semidirect
61. C₄⋊Dic₂₃	C₂³	semidirect
62. C₄×Dic₂₃	C₂³	semidirect
80. C₂×C₄×C₂₃	C₂×D₄	semidirect
111. C₂³×D₄	D₂₃	semidirect
120. C₂×C₂₃	#535	semidirect
134. C₂²×D₄	C₂×D₂₃	semidirect
146. C₂₃	C₂×#535	semidirect
171. C₂²×C₄	C₂²⋊D₂₃	semidirect
205. C₂×C₄	C₂×C₂²⋊D₂₃	semidirect
228. C₂	#13092984	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×#67288	2
#13092984	2
C₂×#10625111	2
C₂×#10625107	2
C₂×#10625109	2
C₂×#10625124	2
C₂×#10625097	2
C₂³×D₄×C₂₃	2
C₂×#535	23

Subgroups — C2×#13092984 #15686706

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13092984 #15686706