Subgroups — C₂×#13093171 #15687531

Order 2944.

1024 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093171	C₂	1	16	yes	maximal, direct product factor
2	C₂×#10625059	C₂	1	2	yes	maximal, complemented
3	C₂×#10625069	C₂	1	2	yes	maximal, complemented
4	C₂×#10625069	C₂	1	2	yes	maximal, complemented
5	C₂×#10625041	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10625046	C₂	1	1	yes	maximal, characteristic
7	C₂×#10625065	C₂	1	2	yes	maximal
8	C₂×#10625068	C₂	1	2	yes	maximal, complemented
9	C₂×#10625067	C₂	1	2	yes	maximal, complemented
10	C₂×C₂₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×D₉₂	C₂²	1	1	yes	characteristic
12	C₂²×C₄×D₂₃	C₂²	1	2	yes	—
13	C₂²×Dic₄₆	C₂²	1	2	yes	—
14	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic
15	C₂₃×#125	C₂²	1	8	yes	complemented
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
22	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
23	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
24	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
25	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
26	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
28	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
29	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
30	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
31	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
32	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
33	#10625069	C₂²	1	16	yes	complemented
34	#10625069	C₂²	1	16	yes	complemented
35	#10625068	C₂²	1	16	yes	complemented
36	#10625067	C₂²	1	16	yes	complemented
37	#10625065	C₂²	1	16	yes	—
38	#10625059	C₂²	1	16	yes	complemented
39	#10625046	C₂²	1	8	yes	—
40	#10625041	C₂²	1	8	yes	complemented
41	C₂³×D₂₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₂²×Dic₂₃	C₂³	1	2	yes	—
48	C₂²×Dic₂₃	C₂³	1	2	yes	—
49	C₂²×Dic₂₃	C₂³	1	2	yes	—
50	C₂×D₉₂	—	4	2	no	—
51	C₂×D₉₂	C₂³	1	4	yes	—
52	C₂×C₄×D₂₃	—	4	4	no	—
53	C₂×C₄×D₂₃	C₂³	1	8	yes	—
54	C₂×Dic₄₆	—	4	2	no	—
55	C₂×Dic₄₆	—	4	4	no	—
56	C₂×Dic₄₆	C₂³	1	4	yes	—
57	C₂×Dic₄₆	C₂³	1	8	yes	—
58	C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
59	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
60	C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
61	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
62	C₄²×C₂₃	C₂³	1	4	yes	—
63	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
64	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
65	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
66	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
67	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
68	C₄⋊Dic₂₃	C₂³	1	4	yes	—
69	C₄⋊Dic₂₃	C₂³	1	4	yes	—
70	C₄⋊Dic₂₃	C₂³	1	8	yes	—
71	Dic₂₃⋊C₄	C₂³	1	8	yes	—
72	Dic₂₃⋊C₄	C₂³	1	8	yes	—
73	Dic₂₃⋊C₄	C₂³	1	8	yes	—
74	C₄×Dic₂₃	C₂³	1	8	yes	—
75	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
76	C₂²×D₂₃	—	2	4	no	—
77	C₂²×D₂₃	—	2	4	no	—
78	C₂²×D₂₃	—	2	4	no	—
79	C₂²×D₂₃	C₂⁴	1	4	yes	—
80	C₂×C₄×C₂₃	—	2	4	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	2	2	no	—
83	C₂×C₄×C₂₃	—	2	4	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
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	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
89	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
90	C₂×Dic₂₃	—	2	4	no	—
91	C₂×Dic₂₃	—	2	4	no	—
92	C₂×Dic₂₃	—	2	4	no	—
93	C₂×Dic₂₃	C₂⁴	1	4	yes	—
94	C₂×Dic₂₃	C₂⁴	1	4	yes	—
95	C₂×Dic₂₃	C₂⁴	1	4	yes	—
96	D₉₂	—	4	4	no	—
97	C₄×D₂₃	—	4	8	no	—
98	Dic₄₆	—	4	4	no	—
99	Dic₄₆	—	4	8	no	—
100	C₂×#563	—	23	1	no	maximal, Sylow
101	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
102	C₂²×C₂₃	2₋¹⁺⁴	1	2	yes	—
103	C₂²×C₂₃	2₋¹⁺⁴	1	2	yes	—
104	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
105	C₂×D₂₃	—	2	4	no	—
106	C₂×D₂₃	—	2	4	no	—
107	C₂×D₂₃	—	2	4	no	—
108	C₂×D₂₃	—	4	8	no	—
109	C₄×C₂₃	—	2	4	no	—
110	C₄×C₂₃	—	2	2	no	—
111	C₄×C₂₃	—	2	4	no	—
112	C₄×C₂₃	—	2	2	no	—
113	C₄×C₂₃	—	2	2	no	—
114	Dic₂₃	—	2	4	no	—
115	Dic₂₃	—	2	4	no	—
116	Dic₂₃	—	2	4	no	—
117	#563	—	23	16	no	—
118	C₂×C₄⋊Q₈	—	23	2	no	—
119	C₂×#126	—	23	2	no	—
120	C₂×#126	—	23	2	no	—
121	C₂×#125	—	23	1	no	—
122	C₂×#125	D₂₃	1	1	yes	characteristic, complemented
123	C₂×#124	—	23	1	no	—
124	C₂×#123	—	23	2	no	—
125	C₂×C₂²⋊Q₈	—	23	2	no	—
126	C₂×C₂²⋊Q₈	—	23	2	no	—
127	C₂×C₂₃	#563	1	4	yes	complemented
128	C₂×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
129	C₂×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
130	C₂×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
131	D₂₃	—	4	4	no	—
132	C₂²×Q₈	—	23	2	no	—
133	C₂²×Q₈	—	23	1	no	—
134	C₂²×D₄	—	23	1	no	—
135	C₂³×C₄	—	23	2	no	—
136	C₄⋊Q₈	—	23	16	no	—
137	#126	—	23	16	no	—
138	#126	—	23	16	no	—
139	#125	—	23	8	no	—
140	#125	C₂×D₂₃	1	8	yes	complemented
141	#124	—	23	8	no	—
142	#123	—	23	16	no	—
143	C₂²⋊Q₈	—	23	16	no	—
144	C₂²⋊Q₈	—	23	16	no	—
145	C₂×C₄⋊C₄	—	23	1	no	—
146	C₂×C₄⋊C₄	—	23	2	no	—
147	C₂×C₄⋊C₄	—	23	2	no	—
148	C₂×C₄⋊C₄	—	23	2	no	—
149	C₂×C₄⋊C₄	—	23	2	no	—
150	C₂×C₄⋊C₄	—	23	1	no	—
151	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
152	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
153	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
154	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
155	C₂×C₂²⋊C₄	—	23	2	no	—
156	C₂×C₂²⋊C₄	—	23	2	no	—
157	C₂×C₂²⋊C₄	—	23	2	no	—
158	C₂×C₂²⋊C₄	—	23	2	no	—
159	C₂×C₂²⋊C₄	—	23	2	no	—
160	C₂×C₄²	—	23	2	no	—
161	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
162	C₂₃	C₂×#563	1	1	yes	characteristic, complemented, Sylow
163	C₂⁴	—	23	2	no	—
164	C₂×Q₈	—	23	8	no	—
165	C₂×Q₈	—	23	4	no	—
166	C₂×Q₈	—	92	4	no	—
167	C₂×Q₈	—	92	2	no	—
168	C₂×D₄	—	23	4	no	—
169	C₂×D₄	—	92	2	no	—
170	C₂²×C₄	—	23	2	no	—
171	C₂²×C₄	—	23	2	no	—
172	C₂²×C₄	—	23	2	no	—
173	C₂²×C₄	—	23	8	no	—
174	C₂²×C₄	—	92	4	no	—
175	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
176	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
177	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
178	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
179	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
180	C₄⋊C₄	—	23	4	no	—
181	C₄⋊C₄	—	23	8	no	—
182	C₄⋊C₄	—	23	8	no	—
183	C₄⋊C₄	—	23	8	no	—
184	C₄⋊C₄	—	23	8	no	—
185	C₄⋊C₄	—	23	4	no	—
186	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
187	C₄⋊C₄	C₂²×D₂₃	1	8	yes	—
188	C₄⋊C₄	C₂²×D₂₃	1	8	yes	—
189	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
190	C₂²⋊C₄	—	23	8	no	—
191	C₂²⋊C₄	—	23	8	no	—
192	C₂²⋊C₄	—	23	8	no	—
193	C₂²⋊C₄	—	23	8	no	—
194	C₂²⋊C₄	—	23	8	no	—
195	C₄²	—	23	8	no	—
196	C₄²	C₂²×D₂₃	1	4	yes	—
197	C₂³	—	23	4	no	—
198	C₂³	—	46	4	no	—
199	C₂³	—	46	4	no	—
200	C₂³	—	46	4	no	—
201	C₂³	C₂³×D₂₃	1	1	yes	characteristic
202	Q₈	—	92	8	no	—
203	Q₈	—	92	4	no	—
204	D₄	—	92	4	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	23	4	no	—
211	C₂×C₄	—	23	4	no	—
212	C₂×C₄	—	23	4	no	—
213	C₂×C₄	—	46	4	no	—
214	C₂×C₄	—	46	4	no	—
215	C₂×C₄	—	46	4	no	—
216	C₂×C₄	—	92	8	no	—
217	C₂×C₄	C₂³×D₂₃	1	4	yes	—
218	C₂×C₄	C₂³×D₂₃	1	2	yes	—
219	C₂×C₄	C₂³×D₂₃	1	2	yes	—
220	C₂×C₄	C₂³×D₂₃	1	2	yes	—
221	C₂×C₄	C₂³×D₂₃	1	4	yes	—
222	C₂²	—	46	4	no	—
223	C₂²	—	46	4	no	—
224	C₂²	—	46	4	no	—
225	C₂²	—	92	8	no	—
226	C₂²	#10625158	1	2	yes	—
227	C₂²	#10625160	1	2	yes	—
228	C₂²	#10625161	1	2	yes	—
229	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
230	C₄	—	2	4	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	4	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	46	4	no	—
236	C₄	—	46	4	no	—
237	C₄	—	46	4	no	—
238	C₂	—	92	4	no	—
239	C₂	#13093171	1	4	yes	direct product factor
240	C₂	C₂×#10625160	1	1	yes	characteristic
241	C₂	C₂×#10625161	1	1	yes	characteristic
242	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×#125	C₂
socle	C₂³×C₂₃	C₂⁴
prime radical	C₂²×D₉₂	C₂²
hypercenter	C₂×#125	D₂₃
hypocenter	C₂₃	C₂×#563
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×#125

Complements

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

N	H ≅ G/N	type
1. #13093171	C₂	direct (N × H)
2. C₂×#10625059	C₂	semidirect
3. C₂×#10625069	C₂	semidirect
4. C₂×#10625069	C₂	semidirect
5. C₂×#10625041	C₂	semidirect
8. C₂×#10625068	C₂	semidirect
9. C₂×#10625067	C₂	semidirect
10. C₂×C₂₃×#125	C₂	semidirect
15. C₂₃×#125	C₂²	semidirect
33. #10625069	C₂²	semidirect
34. #10625069	C₂²	semidirect
35. #10625068	C₂²	semidirect
36. #10625067	C₂²	semidirect
38. #10625059	C₂²	semidirect
40. #10625041	C₂²	semidirect
122. C₂×#125	D₂₃	semidirect
127. C₂×C₂₃	#563	semidirect
140. #125	C₂×D₂₃	semidirect
162. C₂₃	C₂×#563	semidirect
239. C₂	#13093171	direct (N × H)

Maximal subgroups

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

subgroup	index
#13093171	2
C₂×#10625059	2
C₂×#10625069	2
C₂×#10625041	2
C₂×#10625046	2
C₂×#10625065	2
C₂×#10625068	2
C₂×#10625067	2
C₂×C₂₃×#125	2
C₂×#563	23

Subgroups — C2×#13093171 #15687531

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13093171 #15687531