Subgroups — C₂×#13093025 #15686761

Order 2944.

1004 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093025	C₂	1	16	yes	maximal, direct product factor
2	C₂×#10625115	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#10625069	C₂	1	2	yes	maximal
4	C₂×#10625047	C₂	1	2	yes	maximal
5	C₂×#10625041	C₂	1	2	yes	maximal, complemented
6	C₂×#10625061	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#10625060	C₂	1	2	yes	maximal, complemented
8	C₂×#10625068	C₂	1	1	yes	maximal, characteristic
9	C₂×#10625117	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Dic₄₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#10625043	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
15	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic
16	C₄×Q₈×C₂₃	C₂²	1	8	yes	complemented
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
21	#10625117	C₂²	1	8	yes	—
22	#10625115	C₂²	1	8	yes	complemented
23	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
26	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
27	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
29	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
30	C₂×Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
31	C₂×Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
32	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
33	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
34	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
35	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
36	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
37	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
39	#10625069	C₂²	1	16	yes	—
40	#10625068	C₂²	1	8	yes	—
41	#10625061	C₂²	1	8	yes	complemented
42	#10625060	C₂²	1	16	yes	complemented
43	#10625047	C₂²	1	16	yes	—
44	#10625043	C₂²	1	8	yes	—
45	#10625041	C₂²	1	16	yes	complemented
46	C₄×Dic₄₆	C₂²	1	8	yes	complemented
47	C₂³×D₂₃	C₂³	1	1	yes	characteristic
48	C₂×Q₈×C₂₃	—	2	4	no	—
49	C₂×Q₈×C₂₃	C₂³	1	4	yes	—
50	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
53	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
54	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
55	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
56	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
57	C₂²×Dic₂₃	C₂³	1	2	yes	—
58	C₂²×Dic₂₃	C₂³	1	2	yes	—
59	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
60	C₂×C₄×D₂₃	—	2	4	no	—
61	C₂×C₄×D₂₃	C₂³	1	4	yes	—
62	C₂×Dic₄₆	—	2	4	no	—
63	C₂×Dic₄₆	C₂³	1	4	yes	—
64	C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
65	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
66	C₄²×C₂₃	C₂³	1	4	yes	—
67	C₄²×C₂₃	C₂³	1	8	yes	—
68	(C₂×D₂₃)⋊C₄	C₂³	1	4	yes	—
69	(C₂×D₂₃)⋊C₄	C₂³	1	4	yes	—
70	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
71	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
72	C₄⋊Dic₂₃	C₂³	1	4	yes	—
73	C₄⋊Dic₂₃	C₂³	1	8	yes	—
74	C₄⋊Dic₂₃	C₂³	1	8	yes	—
75	Dic₂₃⋊C₄	C₂³	1	8	yes	—
76	Dic₂₃⋊C₄	C₂³	1	4	yes	—
77	Dic₂₃⋊C₄	C₂³	1	8	yes	—
78	Dic₂₃⋊C₄	C₂³	1	8	yes	—
79	Dic₂₃⋊C₄	C₂³	1	4	yes	—
80	Dic₂₃⋊C₄	C₂³	1	8	yes	—
81	Dic₂₃⋊C₄	C₂³	1	8	yes	—
82	C₄×Dic₂₃	C₂³	1	8	yes	—
83	C₄×Dic₂₃	C₂³	1	4	yes	—
84	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
85	C₂²×D₂₃	—	2	2	no	—
86	C₂²×D₂₃	—	2	2	no	—
87	C₂²×D₂₃	—	2	2	no	—
88	C₂²×D₂₃	C₂⁴	1	2	yes	—
89	Q₈×C₂₃	—	2	8	no	—
90	C₂×C₄×C₂₃	—	2	4	no	—
91	C₂×C₄×C₂₃	—	2	2	no	—
92	C₂×C₄×C₂₃	—	2	2	no	—
93	C₂×C₄×C₂₃	—	2	4	no	—
94	C₂×C₄×C₂₃	#41	1	4	yes	—
95	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
96	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
97	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
98	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
99	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
100	C₂×Dic₂₃	—	2	4	no	—
101	C₂×Dic₂₃	—	2	2	no	—
102	C₂×Dic₂₃	—	2	2	no	—
103	C₂×Dic₂₃	—	2	4	no	—
104	C₂×Dic₂₃	—	2	2	no	—
105	C₂×Dic₂₃	C₂⁴	1	2	yes	—
106	C₂×Dic₂₃	C₂⁴	1	4	yes	—
107	C₂×Dic₂₃	C₂⁴	1	2	yes	—
108	C₂×Dic₂₃	C₂⁴	1	4	yes	—
109	C₂×Dic₂₃	C₂⁴	1	2	yes	—
110	C₄×D₂₃	—	2	8	no	—
111	Dic₄₆	—	2	8	no	—
112	C₂×#541	—	23	1	no	maximal, Sylow
113	C₂²×C₂₃	C₂×#41	1	2	yes	—
114	C₂²×C₂₃	2₋¹⁺⁴	1	2	yes	—
115	C₂²×C₂₃	2₋¹⁺⁴	1	2	yes	—
116	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
117	C₂×D₂₃	—	2	2	no	—
118	C₂×D₂₃	—	2	2	no	—
119	C₂×D₂₃	—	2	2	no	—
120	C₂×D₂₃	—	4	4	no	—
121	C₄×C₂₃	—	2	4	no	—
122	C₄×C₂₃	—	2	2	no	—
123	C₄×C₂₃	—	2	4	no	—
124	C₄×C₂₃	—	2	2	no	—
125	C₄×C₂₃	C₂×#41	1	4	yes	—
126	Dic₂₃	—	2	4	no	—
127	Dic₂₃	—	2	2	no	—
128	Dic₂₃	—	2	2	no	—
129	Dic₂₃	—	2	4	no	—
130	Dic₂₃	—	2	2	no	—
131	#541	—	23	16	no	—
132	C₂×C₄⋊Q₈	—	23	1	no	—
133	C₂×#126	—	23	2	no	—
134	C₂×#126	—	23	2	no	—
135	C₂×#125	—	23	2	no	—
136	C₂×#125	—	23	1	no	—
137	C₂×#125	—	23	2	no	—
138	C₂×C₂²⋊Q₈	—	23	1	no	—
139	C₂×C₂²⋊Q₈	—	23	1	no	—
140	C₂×C₄×Q₈	—	23	1	no	—
141	C₂×C₄×Q₈	D₂₃	1	1	yes	characteristic, complemented
142	C₂×#117	—	23	1	no	—
143	C₂×C₂₃	#541	1	4	yes	complemented
144	C₂×C₂₃	C₂²×#41	1	1	yes	characteristic
145	C₂×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
146	C₂×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
147	D₂₃	—	4	2	no	—
148	C₂²×Q₈	—	23	1	no	—
149	C₂²×Q₈	C₂×D₂₃	1	1	yes	characteristic
150	C₂³×C₄	—	23	1	no	—
151	C₄⋊Q₈	—	23	8	no	—
152	#126	—	23	16	no	—
153	#126	—	23	16	no	—
154	#125	—	23	16	no	—
155	#125	—	23	8	no	—
156	#125	—	23	16	no	—
157	C₂²⋊Q₈	—	23	8	no	—
158	C₂²⋊Q₈	—	23	8	no	—
159	C₄×Q₈	—	23	8	no	—
160	C₄×Q₈	C₂×D₂₃	1	8	yes	complemented
161	#117	—	23	8	no	—
162	C₂×C₄⋊C₄	—	23	1	no	—
163	C₂×C₄⋊C₄	—	23	1	no	—
164	C₂×C₄⋊C₄	—	23	1	no	—
165	C₂×C₄⋊C₄	—	23	2	no	—
166	C₂×C₄⋊C₄	—	23	2	no	—
167	C₂×C₄⋊C₄	—	23	2	no	—
168	C₂×C₄⋊C₄	—	23	2	no	—
169	C₂×C₄⋊C₄	—	23	2	no	—
170	C₂×C₄⋊C₄	—	23	2	no	—
171	C₂×C₄⋊C₄	—	23	2	no	—
172	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
173	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
174	C₂×C₂²⋊C₄	—	23	1	no	—
175	C₂×C₂²⋊C₄	—	23	1	no	—
176	C₂×C₂²⋊C₄	—	23	2	no	—
177	C₂×C₂²⋊C₄	—	23	2	no	—
178	C₂×C₄²	—	23	1	no	—
179	C₂×C₄²	—	23	2	no	—
180	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
181	C₂×C₄²	C₂×D₂₃	1	2	yes	—
182	C₂₃	C₂×#541	1	1	yes	characteristic, complemented, Sylow
183	C₂⁴	—	23	1	no	—
184	C₂×Q₈	—	2	4	no	—
185	C₂×Q₈	—	23	4	no	—
186	C₂×Q₈	—	46	4	no	—
187	C₂×Q₈	C₂²×D₂₃	1	4	yes	—
188	C₂²×C₄	—	23	1	no	—
189	C₂²×C₄	—	23	2	no	—
190	C₂²×C₄	—	23	1	no	—
191	C₂²×C₄	—	23	2	no	—
192	C₂²×C₄	—	23	1	no	—
193	C₂²×C₄	—	23	4	no	—
194	C₂²×C₄	—	46	4	no	—
195	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
196	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
197	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
198	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
199	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
200	C₄⋊C₄	—	23	4	no	—
201	C₄⋊C₄	—	23	4	no	—
202	C₄⋊C₄	—	23	4	no	—
203	C₄⋊C₄	—	23	8	no	—
204	C₄⋊C₄	—	23	8	no	—
205	C₄⋊C₄	—	23	8	no	—
206	C₄⋊C₄	—	23	8	no	—
207	C₄⋊C₄	—	23	8	no	—
208	C₄⋊C₄	—	23	8	no	—
209	C₄⋊C₄	—	23	8	no	—
210	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
211	C₄⋊C₄	C₂²×D₂₃	1	8	yes	—
212	C₂²⋊C₄	—	23	4	no	—
213	C₂²⋊C₄	—	23	4	no	—
214	C₂²⋊C₄	—	23	8	no	—
215	C₂²⋊C₄	—	23	8	no	—
216	C₄²	—	23	4	no	—
217	C₄²	—	23	8	no	—
218	C₄²	C₂²×D₂₃	1	4	yes	—
219	C₄²	C₂²×D₂₃	1	8	yes	—
220	C₂³	—	23	2	no	—
221	C₂³	—	46	2	no	—
222	C₂³	—	46	2	no	—
223	C₂³	—	46	2	no	—
224	C₂³	C₂³×D₂₃	1	1	yes	characteristic
225	Q₈	—	2	8	no	—
226	Q₈	—	46	8	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	23	2	no	—
232	C₂×C₄	—	23	2	no	—
233	C₂×C₄	—	23	4	no	—
234	C₂×C₄	—	23	4	no	—
235	C₂×C₄	—	23	2	no	—
236	C₂×C₄	—	46	2	no	—
237	C₂×C₄	—	46	4	no	—
238	C₂×C₄	—	46	2	no	—
239	C₂×C₄	—	46	4	no	—
240	C₂×C₄	—	46	2	no	—
241	C₂×C₄	—	46	8	no	—
242	C₂×C₄	#67288	1	4	yes	—
243	C₂×C₄	C₂³×D₂₃	1	2	yes	—
244	C₂×C₄	C₂³×D₂₃	1	2	yes	—
245	C₂×C₄	C₂³×D₂₃	1	4	yes	—
246	C₂×C₄	C₂³×D₂₃	1	4	yes	—
247	C₂×C₄	C₂³×D₂₃	1	2	yes	—
248	C₂²	—	46	2	no	—
249	C₂²	—	46	2	no	—
250	C₂²	—	46	2	no	—
251	C₂²	—	92	4	no	—
252	C₂²	C₂×#67288	1	2	yes	—
253	C₂²	#10625158	1	2	yes	—
254	C₂²	#10625161	1	2	yes	—
255	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
256	C₄	—	2	4	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	4	no	—
259	C₄	—	2	2	no	—
260	C₄	—	46	2	no	—
261	C₄	—	46	4	no	—
262	C₄	—	46	2	no	—
263	C₄	—	46	2	no	—
264	C₄	—	46	4	no	—
265	C₄	C₂×#67288	1	4	yes	—
266	C₂	—	92	2	no	—
267	C₂	#13093025	1	4	yes	direct product factor
268	C₂	C₂²×#67288	1	1	yes	characteristic
269	C₂	C₂×#10625161	1	1	yes	characteristic
270	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×C₄×Q₈×C₂₃	C₂
socle	C₂³×C₂₃	C₂⁴
prime radical	C₂³×D₂₃	C₂³
hypercenter	C₂×C₄×Q₈	D₂₃
hypocenter	C₂₃	C₂×#541
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. #13093025	C₂	direct (N × H)
2. C₂×#10625115	C₂	semidirect
5. C₂×#10625041	C₂	semidirect
6. C₂×#10625061	C₂	semidirect
7. C₂×#10625060	C₂	semidirect
10. C₂×C₄×Dic₄₆	C₂	semidirect
12. C₂×C₄×Q₈×C₂₃	C₂	semidirect
16. C₄×Q₈×C₂₃	C₂²	semidirect
22. #10625115	C₂²	semidirect
41. #10625061	C₂²	semidirect
42. #10625060	C₂²	semidirect
45. #10625041	C₂²	semidirect
46. C₄×Dic₄₆	C₂²	semidirect
141. C₂×C₄×Q₈	D₂₃	semidirect
143. C₂×C₂₃	#541	semidirect
160. C₄×Q₈	C₂×D₂₃	semidirect
182. C₂₃	C₂×#541	semidirect
267. C₂	#13093025	direct (N × H)

Maximal subgroups

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

subgroup	index
#13093025	2
C₂×#10625115	2
C₂×#10625069	2
C₂×#10625047	2
C₂×#10625041	2
C₂×#10625061	2
C₂×#10625060	2
C₂×#10625068	2
C₂×#10625117	2
C₂×C₄×Dic₄₆	2
C₂×#10625043	2
C₂×C₄×Q₈×C₂₃	2
C₂×#541	23

Subgroups — C2×#13093025 #15686761

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13093025 #15686761