Subgroups — C₂×#13093042 #15686791

Order 2944.

1296 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093042	C₂	1	16	yes	maximal, direct product factor
2	C₂×#10625068	C₂	1	2	yes	maximal, complemented
3	C₂×#10625095	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#10625067	C₂	1	2	yes	maximal, complemented
5	C₂×#10625067	C₂	1	2	yes	maximal, complemented
6	C₂×#10625049	C₂	1	2	yes	maximal, complemented
7	C₂×#10625117	C₂	1	2	yes	maximal, complemented
8	C₂²×(C₂×D₂₃)⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₂×C₂²⋊C₄×D₂₃	C₂	1	2	yes	maximal
10	C₂×C₂₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁴×D₂₃	C₂²	1	1	yes	characteristic
12	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
13	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂³×Dic₂₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₂₃	C₂²	1	2	yes	—
16	C₂²×C₄×D₂₃	C₂²	1	2	yes	—
17	C₂²×Dic₄₆	C₂²	1	2	yes	complemented
18	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic, complemented
19	C₂₃×C₂²⋊Q₈	C₂²	1	8	yes	complemented
20	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
22	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
23	C₂×C₂²⋊Dic₂₃	C₂²	1	2	yes	—
24	#10625117	C₂²	1	16	yes	complemented
25	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
26	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
27	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	8	yes	—
28	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
29	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
30	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	complemented
31	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic, complemented
32	#10625095	C₂²	1	8	yes	complemented
33	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	complemented
34	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	complemented
35	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	complemented
36	#10625068	C₂²	1	16	yes	complemented
37	#10625067	C₂²	1	16	yes	complemented
38	#10625067	C₂²	1	16	yes	complemented
39	C₂²⋊C₄×D₂₃	C₂²	1	16	yes	—
40	#10625049	C₂²	1	16	yes	complemented
41	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
42	C₂³×D₂₃	—	4	2	no	—
43	C₂³×D₂₃	—	4	2	no	—
44	C₂³×D₂₃	Q₈	1	8	yes	complemented
45	C₂³×D₂₃	C₂³	1	2	yes	—
46	C₂³×D₂₃	C₂³	1	4	yes	—
47	C₂×Q₈×C₂₃	—	4	2	no	—
48	C₂×Q₈×C₂₃	C₂³	1	4	yes	complemented
49	C₂²×C₄×C₂₃	—	4	2	no	—
50	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
52	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
53	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
55	C₂²×Dic₂₃	—	4	2	no	—
56	C₂²×Dic₂₃	C₂³	1	2	yes	—
57	C₂²×Dic₂₃	C₂³	1	2	yes	—
58	C₂²×Dic₂₃	C₂³	1	2	yes	—
59	C₂²×Dic₂₃	C₂³	1	4	yes	—
60	C₂×C₄×D₂₃	—	4	4	no	—
61	C₂×C₄×D₂₃	—	4	4	no	—
62	C₂×C₄×D₂₃	C₂³	1	8	yes	—
63	C₂×C₄×D₂₃	C₂³	1	8	yes	—
64	C₂×Dic₄₆	—	4	2	no	—
65	C₂×Dic₄₆	—	4	4	no	—
66	C₂×Dic₄₆	C₂³	1	8	yes	complemented
67	C₂×Dic₄₆	C₂³	1	4	yes	complemented
68	C₄⋊C₄×C₂₃	C₂³	1	4	yes	complemented
69	C₄⋊C₄×C₂₃	C₂³	1	8	yes	complemented
70	C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
71	C₂²⋊Dic₂₃	C₂³	1	8	yes	—
72	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
73	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
74	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
75	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
76	C₄⋊Dic₂₃	C₂³	1	8	yes	complemented
77	C₄⋊Dic₂₃	C₂³	1	4	yes	complemented
78	Dic₂₃⋊C₄	C₂³	1	8	yes	complemented
79	Dic₂₃⋊C₄	C₂³	1	8	yes	complemented
80	Dic₂₃⋊C₄	C₂³	1	8	yes	complemented
81	C₂³×C₂₃	—	2	2	no	—
82	C₂³×C₂₃	—	2	2	no	—
83	C₂³×C₂₃	C₂×Q₈	1	4	yes	—
84	C₂³×C₂₃	C₂⁴	1	2	yes	—
85	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
86	C₂²×D₂₃	—	2	4	no	—
87	C₂²×D₂₃	—	2	4	no	—
88	C₂²×D₂₃	—	4	16	no	—
89	C₂²×D₂₃	—	4	4	no	—
90	C₂²×D₂₃	—	4	4	no	—
91	C₂²×D₂₃	C₂×Q₈	1	8	yes	—
92	C₂²×D₂₃	C₂×Q₈	1	16	yes	complemented
93	C₂²×D₂₃	C₂⁴	1	4	yes	—
94	Q₈×C₂₃	—	4	4	no	—
95	C₂×C₄×C₂₃	—	2	4	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	4	no	—
98	C₂×C₄×C₂₃	—	2	2	no	—
99	C₂×C₄×C₂₃	—	4	4	no	—
100	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
101	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
102	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
103	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
104	C₂×Dic₂₃	—	2	4	no	—
105	C₂×Dic₂₃	—	2	4	no	—
106	C₂×Dic₂₃	—	2	4	no	—
107	C₂×Dic₂₃	—	4	4	no	—
108	C₂×Dic₂₃	C₂⁴	1	4	yes	—
109	C₂×Dic₂₃	C₂⁴	1	4	yes	—
110	C₂×Dic₂₃	C₂⁴	1	4	yes	—
111	C₄×D₂₃	—	4	8	no	—
112	C₄×D₂₃	—	4	8	no	—
113	Dic₄₆	—	4	4	no	—
114	Dic₄₆	—	4	8	no	—
115	C₂×#543	—	23	1	no	maximal, Sylow
116	C₂²×C₂₃	—	2	8	no	—
117	C₂²×C₂₃	—	2	2	no	—
118	C₂²×C₂₃	—	2	2	no	—
119	C₂²×C₂₃	C₂²×Q₈	1	4	yes	—
120	C₂²×C₂₃	C₂²×Q₈	1	2	yes	—
121	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
122	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
123	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
124	C₂×D₂₃	—	2	16	no	—
125	C₂×D₂₃	—	2	4	no	—
126	C₂×D₂₃	—	2	4	no	—
127	C₂×D₂₃	—	4	16	no	—
128	C₂×D₂₃	C₂²×Q₈	1	8	yes	—
129	C₄×C₂₃	—	2	4	no	—
130	C₄×C₂₃	—	2	2	no	—
131	C₄×C₂₃	—	2	4	no	—
132	C₄×C₂₃	—	2	2	no	—
133	Dic₂₃	—	2	4	no	—
134	Dic₂₃	—	2	4	no	—
135	Dic₂₃	—	2	4	no	—
136	#543	—	23	16	no	—
137	C₂×C₂²⋊Q₈	—	23	2	no	—
138	C₂×C₂²⋊Q₈	—	23	1	no	—
139	C₂×C₂²⋊Q₈	—	23	2	no	—
140	C₂×C₂²⋊Q₈	—	23	2	no	—
141	C₂×C₂²⋊Q₈	—	23	2	no	—
142	C₂×C₂²⋊Q₈	—	23	2	no	—
143	C₂×C₂²⋊Q₈	D₂₃	1	1	yes	characteristic, complemented
144	C₂²×C₂²⋊C₄	—	23	1	no	—
145	C₂²×C₂²⋊C₄	—	23	2	no	—
146	C₂×C₂₃	—	2	4	no	—
147	C₂×C₂₃	#543	1	4	yes	complemented
148	C₂×C₂₃	C₂³×Q₈	1	1	yes	characteristic
149	C₂×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
150	C₂×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
151	D₂₃	—	2	8	no	—
152	C₂⁵	—	23	1	no	—
153	C₂²×Q₈	—	23	1	no	—
154	C₂²×Q₈	—	23	2	no	—
155	C₂²×Q₈	C₂×D₂₃	1	1	yes	characteristic, complemented
156	C₂³×C₄	—	23	1	no	—
157	C₂³×C₄	—	23	2	no	—
158	C₂³×C₄	—	23	2	no	—
159	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
160	C₂²⋊Q₈	—	23	16	no	—
161	C₂²⋊Q₈	—	23	8	no	—
162	C₂²⋊Q₈	—	23	16	no	—
163	C₂²⋊Q₈	—	23	16	no	—
164	C₂²⋊Q₈	—	23	16	no	—
165	C₂²⋊Q₈	—	23	16	no	—
166	C₂²⋊Q₈	C₂×D₂₃	1	8	yes	complemented
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	1	no	—
171	C₂×C₄⋊C₄	—	23	2	no	—
172	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic, complemented
173	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	complemented
174	C₂×C₂²⋊C₄	—	23	2	no	—
175	C₂×C₂²⋊C₄	—	23	2	no	—
176	C₂×C₂²⋊C₄	—	23	2	no	—
177	C₂×C₂²⋊C₄	—	23	2	no	—
178	C₂×C₂²⋊C₄	—	23	2	no	—
179	C₂×C₂²⋊C₄	—	23	8	no	—
180	C₂×C₂²⋊C₄	—	23	16	no	—
181	C₂×C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
182	C₂₃	C₂×#543	1	1	yes	characteristic, complemented, Sylow
183	C₂⁴	—	23	2	no	—
184	C₂⁴	—	23	8	no	—
185	C₂⁴	—	23	4	no	—
186	C₂⁴	—	92	2	no	—
187	C₂⁴	—	92	2	no	—
188	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
189	C₂×Q₈	—	4	2	no	—
190	C₂×Q₈	—	23	4	no	—
191	C₂×Q₈	—	23	8	no	—
192	C₂×Q₈	—	92	2	no	—
193	C₂×Q₈	—	92	4	no	—
194	C₂×Q₈	C₂²×D₂₃	1	4	yes	complemented
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	23	2	no	—
197	C₂²×C₄	—	23	2	no	—
198	C₂²×C₄	—	23	2	no	—
199	C₂²×C₄	—	23	4	no	—
200	C₂²×C₄	—	23	8	no	—
201	C₂²×C₄	—	23	8	no	—
202	C₂²×C₄	—	92	4	no	—
203	C₂²×C₄	—	92	4	no	—
204	C₂²×C₄	—	92	2	no	—
205	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
206	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
208	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
209	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
210	C₄⋊C₄	—	23	8	no	—
211	C₄⋊C₄	—	23	8	no	—
212	C₄⋊C₄	—	23	8	no	—
213	C₄⋊C₄	—	23	4	no	—
214	C₄⋊C₄	—	23	8	no	—
215	C₄⋊C₄	C₂²×D₂₃	1	4	yes	complemented
216	C₄⋊C₄	C₂²×D₂₃	1	8	yes	complemented
217	C₂²⋊C₄	—	23	8	no	—
218	C₂²⋊C₄	—	23	8	no	—
219	C₂²⋊C₄	—	23	8	no	—
220	C₂²⋊C₄	—	23	8	no	—
221	C₂²⋊C₄	—	23	8	no	—
222	C₂²⋊C₄	C₂²×D₂₃	1	8	yes	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	23	8	no	—
226	C₂³	—	23	4	no	—
227	C₂³	—	23	16	no	—
228	C₂³	—	46	4	no	—
229	C₂³	—	46	4	no	—
230	C₂³	—	92	4	no	—
231	C₂³	—	92	4	no	—
232	C₂³	—	92	16	no	—
233	C₂³	Q₈×D₂₃	1	4	yes	—
234	C₂³	C₂³×D₂₃	1	2	yes	—
235	C₂³	C₂³×D₂₃	1	1	yes	characteristic
236	Q₈	—	4	4	no	—
237	Q₈	—	92	4	no	—
238	Q₈	—	92	8	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	23	4	no	—
245	C₂×C₄	—	23	4	no	—
246	C₂×C₄	—	23	4	no	—
247	C₂×C₄	—	46	4	no	—
248	C₂×C₄	—	46	4	no	—
249	C₂×C₄	—	46	4	no	—
250	C₂×C₄	—	92	8	no	—
251	C₂×C₄	—	92	8	no	—
252	C₂×C₄	—	92	4	no	—
253	C₂×C₄	C₂³×D₂₃	1	2	yes	—
254	C₂×C₄	C₂³×D₂₃	1	2	yes	—
255	C₂×C₄	C₂³×D₂₃	1	4	yes	—
256	C₂×C₄	C₂³×D₂₃	1	4	yes	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	8	no	—
260	C₂²	—	23	8	no	—
261	C₂²	—	46	4	no	—
262	C₂²	—	46	4	no	—
263	C₂²	—	46	16	no	—
264	C₂²	—	92	16	no	—
265	C₂²	#10625155	1	2	yes	—
266	C₂²	C₂×Q₈×D₂₃	1	4	yes	—
267	C₂²	C₂×Q₈×D₂₃	1	2	yes	—
268	C₂²	#10625160	1	2	yes	—
269	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	2	4	no	—
273	C₄	—	2	4	no	—
274	C₄	—	46	4	no	—
275	C₄	—	46	4	no	—
276	C₄	—	46	4	no	—
277	C₂	—	2	4	no	—
278	C₂	—	46	8	no	—
279	C₂	#13093042	1	4	yes	direct product factor
280	C₂	C₂²×Q₈×D₂₃	1	1	yes	characteristic
281	C₂	C₂×#10625155	1	1	yes	characteristic
282	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×C₂₃×C₂²⋊Q₈	C₂
socle	C₂³×C₂₃	C₂⁴
prime radical	C₂⁴×D₂₃	C₂²
hypercenter	C₂×C₂²⋊Q₈	D₂₃
hypocenter	C₂₃	C₂×#543
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #13093042	C₂	direct (N × H)
2. C₂×#10625068	C₂	semidirect
3. C₂×#10625095	C₂	semidirect
4. C₂×#10625067	C₂	semidirect
5. C₂×#10625067	C₂	semidirect
6. C₂×#10625049	C₂	semidirect
7. C₂×#10625117	C₂	semidirect
10. C₂×C₂₃×C₂²⋊Q₈	C₂	semidirect
12. C₂²×Q₈×C₂₃	C₂²	semidirect
17. C₂²×Dic₄₆	C₂²	semidirect
18. C₂²×Dic₄₆	C₂²	semidirect
19. C₂₃×C₂²⋊Q₈	C₂²	semidirect
20. C₂×C₄⋊C₄×C₂₃	C₂²	semidirect
21. C₂×C₄⋊C₄×C₂₃	C₂²	semidirect
24. #10625117	C₂²	semidirect
30. C₂×C₄⋊Dic₂₃	C₂²	semidirect
31. C₂×C₄⋊Dic₂₃	C₂²	semidirect
32. #10625095	C₂²	semidirect
33. C₂×Dic₂₃⋊C₄	C₂²	semidirect
34. C₂×Dic₂₃⋊C₄	C₂²	semidirect
35. C₂×Dic₂₃⋊C₄	C₂²	semidirect
36. #10625068	C₂²	semidirect
37. #10625067	C₂²	semidirect
38. #10625067	C₂²	semidirect
40. #10625049	C₂²	semidirect
44. C₂³×D₂₃	Q₈	semidirect
48. C₂×Q₈×C₂₃	C₂³	semidirect
66. C₂×Dic₄₆	C₂³	semidirect
67. C₂×Dic₄₆	C₂³	semidirect
68. C₄⋊C₄×C₂₃	C₂³	semidirect
69. C₄⋊C₄×C₂₃	C₂³	semidirect
76. C₄⋊Dic₂₃	C₂³	semidirect
77. C₄⋊Dic₂₃	C₂³	semidirect
78. Dic₂₃⋊C₄	C₂³	semidirect
79. Dic₂₃⋊C₄	C₂³	semidirect
80. Dic₂₃⋊C₄	C₂³	semidirect
92. C₂²×D₂₃	C₂×Q₈	semidirect
143. C₂×C₂²⋊Q₈	D₂₃	semidirect
147. C₂×C₂₃	#543	semidirect
155. C₂²×Q₈	C₂×D₂₃	semidirect
166. C₂²⋊Q₈	C₂×D₂₃	semidirect
172. C₂×C₄⋊C₄	C₂×D₂₃	semidirect
173. C₂×C₄⋊C₄	C₂×D₂₃	semidirect
182. C₂₃	C₂×#543	semidirect
194. C₂×Q₈	C₂²×D₂₃	semidirect
215. C₄⋊C₄	C₂²×D₂₃	semidirect
216. C₄⋊C₄	C₂²×D₂₃	semidirect
279. C₂	#13093042	direct (N × H)

Maximal subgroups

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

subgroup	index
#13093042	2
C₂×#10625068	2
C₂×#10625095	2
C₂×#10625067	2
C₂×#10625049	2
C₂×#10625117	2
C₂²×(C₂×D₂₃)⋊C₄	2
C₂×C₂²⋊C₄×D₂₃	2
C₂×C₂₃×C₂²⋊Q₈	2
C₂×#543	23

Subgroups — C2×#13093042 #15686791

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13093042 #15686791