Subgroups — C₂×#13093088 #15686877

Order 2944.

1736 conjugacy classes of subgroups.

The subgroup lattice has 248 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	2	yes	maximal, complemented
2	#13093088	C₂	1	16	yes	maximal, direct product factor
3	C₂×#10625045	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#10625111	C₂	1	2	yes	maximal, complemented
5	C₂×#10625101	C₂	1	2	yes	maximal, complemented
6	C₂×#10625055	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×Dic₄₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10625064	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
12	C₂²×C₂²⋊D₂₃	C₂²	1	4	yes	—
13	C₂²×C₂²⋊D₂₃	C₂²	1	4	yes	—
14	C₂×#67288	—	2	16	no	—
15	C₂×#67288	C₂²	1	16	yes	complemented
16	C₂²×D₉₂	C₂²	1	2	yes	—
17	C₂²×D₉₂	C₂²	1	2	yes	—
18	C₂²×D₉₂	C₂²	1	2	yes	—
19	C₂²×C₄×D₂₃	C₂²	1	4	yes	—
20	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic
21	C₄×D₄×C₂₃	C₂²	1	8	yes	complemented
22	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
24	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
25	#10625111	C₂²	1	16	yes	complemented
26	#10625101	C₂²	1	16	yes	complemented
27	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	4	yes	—
28	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
29	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
30	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
31	#10625064	C₂²	1	16	yes	complemented
32	#10625055	C₂²	1	32	yes	complemented
33	#10625045	C₂²	1	8	yes	complemented
34	C₄×Dic₄₆	C₂²	1	8	yes	complemented
35	C₂⁴×C₂₃	C₂³	1	2	yes	—
36	C₂³×D₂₃	C₂³	1	4	yes	—
37	C₂×D₄×C₂₃	—	2	4	no	—
38	C₂×D₄×C₂₃	C₂³	1	4	yes	—
39	C₂²×C₄×C₂₃	—	2	8	no	—
40	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₂₃	C₂³	1	8	yes	—
43	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
45	C₂×C₂²⋊D₂₃	—	2	16	no	—
46	C₂×C₂²⋊D₂₃	—	4	8	no	—
47	C₂×C₂²⋊D₂₃	C₂³	1	16	yes	—
48	C₂×C₂²⋊D₂₃	C₂³	1	16	yes	—
49	C₂²×Dic₂₃	C₂³	1	2	yes	—
50	C₂²×Dic₂₃	C₂³	1	2	yes	—
51	#67288	—	2	64	no	—
52	C₂×D₉₂	—	2	8	no	—
53	C₂×D₉₂	—	2	8	no	—
54	C₂×D₉₂	—	4	4	no	—
55	C₂×D₉₂	C₂³	1	8	yes	—
56	C₂×D₉₂	C₂³	1	8	yes	—
57	C₂×D₉₂	C₂³	1	8	yes	—
58	C₂×C₄×D₂₃	—	2	16	no	—
59	C₂×C₄×D₂₃	C₂³	1	16	yes	—
60	C₂×Dic₄₆	D₄	1	8	yes	complemented
61	C₂×Dic₄₆	C₂³	1	4	yes	—
62	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
63	C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
64	C₄²×C₂₃	C₂³	1	4	yes	—
65	(C₂×D₂₃)⋊C₄	C₂³	1	16	yes	—
66	C₄⋊Dic₂₃	C₂³	1	4	yes	—
67	Dic₂₃⋊C₄	C₂³	1	8	yes	—
68	C₄×Dic₂₃	C₂³	1	8	yes	—
69	C₂³×C₂₃	—	2	4	no	—
70	C₂³×C₂₃	—	2	4	no	—
71	C₂³×C₂₃	—	2	4	no	—
72	C₂³×C₂₃	C₂⁴	1	4	yes	—
73	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
74	C₂²×D₂₃	—	2	8	no	—
75	C₂²×D₂₃	—	2	8	no	—
76	C₂²×D₂₃	—	2	8	no	—
77	C₂²×D₂₃	C₂⁴	1	8	yes	—
78	D₄×C₂₃	—	2	8	no	—
79	C₂×C₄×C₂₃	—	2	4	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	2	16	no	—
82	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
83	C₂×C₄×C₂₃	#41	1	4	yes	complemented
84	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
85	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
86	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
87	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
88	C₂²⋊D₂₃	—	2	32	no	—
89	C₂²⋊D₂₃	—	4	16	no	—
90	C₂×Dic₂₃	—	2	4	no	—
91	C₂×Dic₂₃	C₂×D₄	1	8	yes	—
92	C₂×Dic₂₃	C₂⁴	1	4	yes	—
93	C₂×Dic₂₃	C₂⁴	1	4	yes	—
94	D₉₂	—	2	16	no	—
95	D₉₂	—	2	16	no	—
96	D₉₂	—	4	8	no	—
97	C₄×D₂₃	—	2	32	no	—
98	Dic₄₆	C₂×D₄	1	16	yes	complemented
99	C₂×#550	—	23	1	no	maximal, Sylow
100	C₂²×C₂₃	—	2	4	no	—
101	C₂²×C₂₃	—	2	4	no	—
102	C₂²×C₂₃	—	2	4	no	—
103	C₂²×C₂₃	—	4	8	no	—
104	C₂²×C₂₃	C₂²×D₄	1	2	yes	—
105	C₂²×C₂₃	C₂×#41	1	2	yes	—
106	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
107	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
108	C₂×D₂₃	—	2	8	no	—
109	C₂×D₂₃	—	2	8	no	—
110	C₂×D₂₃	—	2	8	no	—
111	C₂×D₂₃	—	4	16	no	—
112	C₄×C₂₃	—	2	4	no	—
113	C₄×C₂₃	—	2	2	no	—
114	C₄×C₂₃	C₂²×D₄	1	4	yes	—
115	C₄×C₂₃	C₂×#41	1	4	yes	complemented
116	Dic₂₃	—	2	4	no	—
117	Dic₂₃	C₂²×D₄	1	8	yes	—
118	C₂²×#41	—	23	2	no	—
119	#550	—	23	16	no	—
120	C₂×#127	—	23	2	no	—
121	C₂×#127	—	23	1	no	—
122	C₂×#121	—	23	2	no	—
123	C₂×#121	—	23	4	no	—
124	C₂×C₄×Q₈	—	23	1	no	—
125	C₂×C₄×D₄	—	23	2	no	—
126	C₂×C₄×D₄	D₂₃	1	1	yes	characteristic, complemented
127	C₂×C₂₃	—	4	4	no	—
128	C₂×C₂₃	#550	1	4	yes	complemented
129	C₂×C₂₃	C₂³×D₄	1	1	yes	characteristic
130	C₂×C₂₃	C₂²×#41	1	1	yes	characteristic
131	C₂×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
132	D₂₃	—	4	8	no	—
133	C₂×#41	—	23	16	no	—
134	C₂×#41	—	46	16	no	—
135	C₂²×Q₈	—	23	1	no	—
136	C₂²×D₄	—	23	4	no	—
137	C₂²×D₄	—	23	4	no	—
138	C₂²×D₄	—	23	2	no	—
139	C₂²×D₄	—	23	2	no	—
140	C₂²×D₄	—	23	2	no	—
141	C₂²×D₄	C₂×D₂₃	1	1	yes	characteristic
142	C₂³×C₄	—	23	4	no	—
143	C₂³×C₄	C₂×D₂₃	1	2	yes	—
144	#127	—	23	16	no	—
145	#127	—	23	8	no	—
146	#121	—	23	16	no	—
147	#121	—	23	32	no	—
148	C₄×Q₈	—	23	8	no	—
149	C₄×D₄	—	23	16	no	—
150	C₄×D₄	C₂×D₂₃	1	8	yes	complemented
151	C₂×C₄⋊C₄	—	23	2	no	—
152	C₂×C₄⋊C₄	—	23	1	no	—
153	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
154	C₂×C₂²⋊C₄	—	23	4	no	—
155	C₂×C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
156	C₂×C₄²	—	23	2	no	—
157	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
158	C₂₃	C₂×#550	1	1	yes	characteristic, complemented, Sylow
159	C₂⁴	—	23	4	no	—
160	C₂⁴	C₂²×D₂₃	1	2	yes	—
161	#41	—	46	64	no	—
162	C₂×Q₈	—	23	8	no	—
163	C₂×Q₈	—	23	4	no	—
164	C₂×D₄	—	2	4	no	—
165	C₂×D₄	—	23	16	no	—
166	C₂×D₄	—	23	16	no	—
167	C₂×D₄	—	23	8	no	—
168	C₂×D₄	—	23	8	no	—
169	C₂×D₄	—	23	8	no	—
170	C₂×D₄	—	46	16	no	—
171	C₂×D₄	—	46	8	no	—
172	C₂×D₄	—	46	8	no	—
173	C₂×D₄	—	92	4	no	—
174	C₂×D₄	—	92	8	no	—
175	C₂×D₄	C₂²×D₂₃	1	4	yes	—
176	C₂²×C₄	—	2	8	no	—
177	C₂²×C₄	—	23	2	no	—
178	C₂²×C₄	—	23	2	no	—
179	C₂²×C₄	—	23	16	no	—
180	C₂²×C₄	—	46	16	no	—
181	C₂²×C₄	C₂²×D₂₃	1	8	yes	—
182	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
183	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
184	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
185	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
186	C₄⋊C₄	—	23	8	no	—
187	C₄⋊C₄	—	23	4	no	—
188	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
189	C₂²⋊C₄	—	23	16	no	—
190	C₂²⋊C₄	C₂²×D₂₃	1	8	yes	—
191	C₄²	—	23	8	no	—
192	C₄²	C₂²×D₂₃	1	4	yes	—
193	C₂³	—	2	4	no	—
194	C₂³	—	2	4	no	—
195	C₂³	—	2	4	no	—
196	C₂³	—	23	8	no	—
197	C₂³	—	46	8	no	—
198	C₂³	—	46	8	no	—
199	C₂³	—	46	8	no	—
200	C₂³	C₂³×D₂₃	1	4	yes	—
201	C₂³	C₂³×D₂₃	1	1	yes	characteristic
202	Q₈	—	23	16	no	—
203	D₄	—	2	8	no	—
204	D₄	—	46	32	no	—
205	D₄	—	46	16	no	—
206	D₄	—	46	16	no	—
207	D₄	—	92	8	no	—
208	D₄	—	92	16	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	16	no	—
212	C₂×C₄	—	23	8	no	—
213	C₂×C₄	—	23	4	no	—
214	C₂×C₄	—	23	4	no	—
215	C₂×C₄	—	46	32	no	—
216	C₂×C₄	—	46	4	no	—
217	C₂×C₄	#67288	1	4	yes	complemented
218	C₂×C₄	D₄×D₂₃	1	4	yes	—
219	C₂×C₄	C₂³×D₂₃	1	2	yes	—
220	C₂×C₄	C₂³×D₂₃	1	4	yes	—
221	C₂×C₄	C₂³×D₂₃	1	2	yes	—
222	C₂×C₄	C₂³×D₂₃	1	2	yes	—
223	C₂²	—	2	4	no	—
224	C₂²	—	2	4	no	—
225	C₂²	—	2	4	no	—
226	C₂²	—	4	8	no	—
227	C₂²	—	46	8	no	—
228	C₂²	—	46	8	no	—
229	C₂²	—	46	8	no	—
230	C₂²	—	92	16	no	—
231	C₂²	C₂×#67288	1	2	yes	—
232	C₂²	C₂×D₄×D₂₃	1	2	yes	—
233	C₂²	#10625160	1	2	yes	—
234	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
235	C₄	—	2	4	no	—
236	C₄	—	2	2	no	—
237	C₄	—	23	8	no	—
238	C₄	—	46	4	no	—
239	C₄	C₂×#67288	1	4	yes	complemented
240	C₄	C₂×D₄×D₂₃	1	4	yes	—
241	C₂	—	4	4	no	—
242	C₂	—	92	8	no	—
243	C₂	#13093088	1	4	yes	direct product factor
244	C₂	C₂²×D₄×D₂₃	1	1	yes	characteristic
245	C₂	C₂²×#67288	1	1	yes	characteristic
246	C₂	C₂×#10625160	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₂₃
commutator, cosocle	C₂²×C₂₃	C₂⁵
Frattini	C₂²	C₂⁴×D₂₃
Fitting	C₂×C₄×D₄×C₂₃	C₂
socle	C₂³×C₂₃	C₂⁴
hypercenter	C₂×C₄×D₄	D₂₃
hypocenter	C₂₃	C₂×#550
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#67288	C₂	semidirect
2. #13093088	C₂	direct (N × H)
3. C₂×#10625045	C₂	semidirect
4. C₂×#10625111	C₂	semidirect
5. C₂×#10625101	C₂	semidirect
6. C₂×#10625055	C₂	semidirect
7. C₂×C₄×Dic₄₆	C₂	semidirect
8. C₂×#10625064	C₂	semidirect
9. C₂×C₄×D₄×C₂₃	C₂	semidirect
15. C₂×#67288	C₂²	semidirect
21. C₄×D₄×C₂₃	C₂²	semidirect
25. #10625111	C₂²	semidirect
26. #10625101	C₂²	semidirect
31. #10625064	C₂²	semidirect
32. #10625055	C₂²	semidirect
33. #10625045	C₂²	semidirect
34. C₄×Dic₄₆	C₂²	semidirect
60. C₂×Dic₄₆	D₄	semidirect
83. C₂×C₄×C₂₃	#41	semidirect
98. Dic₄₆	C₂×D₄	semidirect
115. C₄×C₂₃	C₂×#41	semidirect
126. C₂×C₄×D₄	D₂₃	semidirect
128. C₂×C₂₃	#550	semidirect
150. C₄×D₄	C₂×D₂₃	semidirect
158. C₂₃	C₂×#550	semidirect
217. C₂×C₄	#67288	semidirect
239. C₄	C₂×#67288	semidirect
243. C₂	#13093088	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
#13093088	2
C₂×#10625045	2
C₂×#10625111	2
C₂×#10625101	2
C₂×#10625055	2
C₂×C₄×Dic₄₆	2
C₂×#10625064	2
C₂×C₄×D₄×C₂₃	2
C₂×#550	23

Subgroups — C2×#13093088 #15686877

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13093088 #15686877