Subgroups — C₂×#13093095 #15686896

Order 2944.

1328 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093095	C₂	1	16	yes	maximal, direct product factor
2	C₂×#10625050	C₂	1	2	yes	maximal, complemented
3	C₂×#10625107	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#10625054	C₂	1	2	yes	maximal, complemented
5	C₂×#10625067	C₂	1	2	yes	maximal, complemented
6	C₂×#10625108	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×C₂²⋊D₂₃	C₂	1	2	yes	maximal, complemented
8	C₂×#10625063	C₂	1	2	yes	maximal, complemented
9	C₂×C₂²⋊C₄×D₂₃	C₂	1	2	yes	maximal, complemented
10	C₂×C₂₃×#123	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁴×D₂₃	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
14	C₂²×C₂²⋊D₂₃	C₂²	1	2	yes	—
15	C₂²×C₄×D₂₃	C₂²	1	2	yes	—
16	C₂²×C₄×D₂₃	C₂²	1	2	yes	—
17	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic, complemented
18	C₂₃×#123	C₂²	1	8	yes	complemented
19	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
20	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
22	C₂×C₂²⋊Dic₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊Dic₂₃	C₂²	1	2	yes	—
24	C₂×C₂²⋊Dic₂₃	C₂²	1	2	yes	complemented
25	#10625108	C₂²	1	8	yes	—
26	#10625107	C₂²	1	8	yes	complemented
27	C₄×C₂²⋊D₂₃	C₂²	1	16	yes	complemented
28	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
29	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
30	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
31	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	complemented
32	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	complemented
33	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	complemented
34	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic, complemented
35	C₂×C₄×Dic₂₃	C₂²	1	2	yes	complemented
36	#10625067	C₂²	1	16	yes	complemented
37	#10625063	C₂²	1	16	yes	complemented
38	#10625054	C₂²	1	16	yes	complemented
39	C₂²⋊C₄×D₂₃	C₂²	1	16	yes	complemented
40	#10625050	C₂²	1	16	yes	complemented
41	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
42	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
43	C₂³×D₂₃	—	2	8	no	—
44	C₂³×D₂₃	—	4	2	no	—
45	C₂³×D₂₃	C₂³	1	4	yes	—
46	C₂³×D₂₃	C₂³	1	2	yes	—
47	C₂×D₄×C₂₃	—	4	2	no	—
48	C₂×D₄×C₂₃	C₂³	1	4	yes	—
49	C₂²×C₄×C₂₃	—	4	2	no	—
50	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
51	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
53	C₂²×C₄×C₂₃	C₂³	1	4	yes	complemented
54	C₂×C₂²⋊D₂₃	—	2	8	no	—
55	C₂×C₂²⋊D₂₃	C₂³	1	8	yes	—
56	C₂²×Dic₂₃	C₂³	1	2	yes	—
57	C₂²×Dic₂₃	C₂³	1	2	yes	—
58	C₂²×Dic₂₃	C₂³	1	2	yes	—
59	C₂×C₄×D₂₃	—	2	8	no	—
60	C₂×C₄×D₂₃	—	4	4	no	—
61	C₂×C₄×D₂₃	C₂³	1	8	yes	—
62	C₂×C₄×D₂₃	C₂³	1	8	yes	—
63	C₂×Dic₄₆	—	4	2	no	—
64	C₂×Dic₄₆	C₂³	1	4	yes	complemented
65	C₄⋊C₄×C₂₃	C₂³	1	8	yes	complemented
66	C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
67	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	complemented
68	C₂²⋊Dic₂₃	C₂³	1	4	yes	—
69	C₂²⋊Dic₂₃	C₂³	1	8	yes	—
70	C₂²⋊Dic₂₃	C₂³	1	8	yes	complemented
71	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
72	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
73	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
74	C₄⋊Dic₂₃	C₂³	1	8	yes	complemented
75	Dic₂₃⋊C₄	C₂³	1	8	yes	complemented
76	Dic₂₃⋊C₄	C₂³	1	8	yes	complemented
77	C₄×Dic₂₃	C₂³	1	8	yes	complemented
78	C₄×Dic₂₃	C₂³	1	4	yes	complemented
79	C₂³×C₂₃	—	2	4	no	—
80	C₂³×C₂₃	—	2	4	no	—
81	C₂³×C₂₃	—	2	2	no	—
82	C₂³×C₂₃	—	2	2	no	—
83	C₂³×C₂₃	C₂⁴	1	2	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	—	2	16	no	—
89	C₂²×D₂₃	—	4	4	no	—
90	C₂²×D₂₃	—	4	16	no	—
91	C₂²×D₂₃	#41	1	8	yes	—
92	C₂²×D₂₃	C₂⁴	1	4	yes	—
93	D₄×C₂₃	—	4	4	no	—
94	C₂×C₄×C₂₃	—	2	4	no	—
95	C₂×C₄×C₂₃	—	2	2	no	—
96	C₂×C₄×C₂₃	—	2	4	no	—
97	C₂×C₄×C₂₃	—	4	4	no	—
98	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
99	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
100	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
101	C₂²⋊D₂₃	—	2	16	no	—
102	C₂×Dic₂₃	—	2	4	no	—
103	C₂×Dic₂₃	—	2	4	no	—
104	C₂×Dic₂₃	—	2	4	no	—
105	C₂×Dic₂₃	C₂⁴	1	4	yes	—
106	C₂×Dic₂₃	C₂⁴	1	4	yes	—
107	C₂×Dic₂₃	C₂⁴	1	4	yes	—
108	C₄×D₂₃	—	2	16	no	—
109	C₄×D₂₃	—	4	8	no	—
110	Dic₄₆	—	4	4	no	—
111	C₂×#551	—	23	1	no	maximal, Sylow
112	C₂²×C₂₃	—	2	4	no	—
113	C₂²×C₂₃	—	2	4	no	—
114	C₂²×C₂₃	—	2	2	no	—
115	C₂²×C₂₃	—	2	2	no	—
116	C₂²×C₂₃	—	4	4	no	—
117	C₂²×C₂₃	—	4	4	no	—
118	C₂²×C₂₃	C₂×#41	1	4	yes	—
119	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
120	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
121	C₂×D₂₃	—	2	16	no	—
122	C₂×D₂₃	—	2	4	no	—
123	C₂×D₂₃	—	2	4	no	—
124	C₂×D₂₃	—	4	16	no	—
125	C₂×D₂₃	C₂×#41	1	8	yes	—
126	C₄×C₂₃	—	2	4	no	—
127	C₄×C₂₃	—	2	2	no	—
128	C₄×C₂₃	—	2	4	no	—
129	Dic₂₃	—	2	4	no	—
130	Dic₂₃	—	2	4	no	—
131	Dic₂₃	—	2	4	no	—
132	#551	—	23	16	no	—
133	C₂×#126	—	23	2	no	—
134	C₂×#124	—	23	1	no	—
135	C₂×#123	—	23	2	no	—
136	C₂×#123	D₂₃	1	1	yes	characteristic, complemented
137	C₂×C₂²⋊Q₈	—	23	2	no	—
138	C₂×#120	—	23	1	no	—
139	C₂×C₄×D₄	—	23	2	no	—
140	C₂×#117	—	23	2	no	—
141	C₂²×C₂²⋊C₄	—	23	2	no	—
142	C₂×C₂₃	—	4	2	no	—
143	C₂×C₂₃	—	4	2	no	—
144	C₂×C₂₃	#551	1	4	yes	complemented
145	C₂×C₂₃	C₂²×#41	1	2	yes	—
146	C₂×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
147	D₂₃	—	2	8	no	—
148	C₂⁵	—	23	1	no	—
149	C₂²×Q₈	—	23	1	no	—
150	C₂²×D₄	—	23	2	no	—
151	C₂²×D₄	C₂×D₂₃	1	1	yes	characteristic
152	C₂³×C₄	—	23	2	no	—
153	C₂³×C₄	—	23	2	no	—
154	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic, complemented
155	#126	—	23	16	no	—
156	#124	—	23	8	no	—
157	#123	—	23	16	no	—
158	#123	C₂×D₂₃	1	8	yes	complemented
159	C₂²⋊Q₈	—	23	16	no	—
160	#120	—	23	8	no	—
161	C₄×D₄	—	23	16	no	—
162	#117	—	23	16	no	—
163	C₂×C₄⋊C₄	—	23	2	no	—
164	C₂×C₄⋊C₄	—	23	2	no	—
165	C₂×C₄⋊C₄	—	23	2	no	—
166	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	complemented
167	C₂×C₂²⋊C₄	—	23	2	no	—
168	C₂×C₂²⋊C₄	—	23	2	no	—
169	C₂×C₂²⋊C₄	—	23	1	no	—
170	C₂×C₂²⋊C₄	—	23	2	no	—
171	C₂×C₂²⋊C₄	—	23	2	no	—
172	C₂×C₂²⋊C₄	—	23	2	no	—
173	C₂×C₂²⋊C₄	—	23	16	no	—
174	C₂×C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
175	C₂×C₂²⋊C₄	C₂×D₂₃	1	1	yes	characteristic, complemented
176	C₂×C₄²	—	23	1	no	—
177	C₂×C₄²	—	23	2	no	—
178	C₂₃	C₂×#551	1	1	yes	characteristic, complemented, Sylow
179	C₂⁴	—	23	2	no	—
180	C₂⁴	—	23	4	no	—
181	C₂⁴	—	46	8	no	—
182	C₂⁴	—	92	2	no	—
183	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
184	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
185	C₂×Q₈	—	23	4	no	—
186	C₂×Q₈	—	92	2	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂×D₄	—	23	8	no	—
189	C₂×D₄	—	46	8	no	—
190	C₂×D₄	C₂²×D₂₃	1	4	yes	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	—	23	2	no	—
193	C₂²×C₄	—	23	2	no	—
194	C₂²×C₄	—	23	2	no	—
195	C₂²×C₄	—	23	8	no	—
196	C₂²×C₄	—	23	8	no	—
197	C₂²×C₄	—	46	8	no	—
198	C₂²×C₄	—	92	4	no	—
199	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
200	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
201	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
202	C₂²×C₄	C₂²×D₂₃	1	4	yes	complemented
203	C₄⋊C₄	—	23	8	no	—
204	C₄⋊C₄	—	23	8	no	—
205	C₄⋊C₄	—	23	8	no	—
206	C₄⋊C₄	C₂²×D₂₃	1	8	yes	complemented
207	C₂²⋊C₄	—	23	8	no	—
208	C₂²⋊C₄	—	23	8	no	—
209	C₂²⋊C₄	—	23	4	no	—
210	C₂²⋊C₄	—	23	8	no	—
211	C₂²⋊C₄	—	23	8	no	—
212	C₂²⋊C₄	—	23	8	no	—
213	C₂²⋊C₄	C₂²×D₂₃	1	8	yes	—
214	C₂²⋊C₄	C₂²×D₂₃	1	4	yes	complemented
215	C₄²	—	23	4	no	—
216	C₄²	—	23	8	no	—
217	C₂³	—	2	2	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	4	no	—
220	C₂³	—	2	4	no	—
221	C₂³	—	23	8	no	—
222	C₂³	—	23	4	no	—
223	C₂³	—	46	16	no	—
224	C₂³	—	46	4	no	—
225	C₂³	—	46	4	no	—
226	C₂³	—	92	4	no	—
227	C₂³	—	92	16	no	—
228	C₂³	C₂³×D₂₃	1	2	yes	—
229	C₂³	C₂³×D₂₃	1	2	yes	—
230	C₂³	C₂³×D₂₃	1	1	yes	characteristic
231	Q₈	—	92	4	no	—
232	D₄	—	4	4	no	—
233	D₄	—	46	16	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	23	4	no	—
239	C₂×C₄	—	23	4	no	—
240	C₂×C₄	—	23	4	no	—
241	C₂×C₄	—	46	16	no	—
242	C₂×C₄	—	46	4	no	—
243	C₂×C₄	—	46	4	no	—
244	C₂×C₄	—	46	4	no	—
245	C₂×C₄	—	92	8	no	—
246	C₂×C₄	C₂³×D₂₃	1	2	yes	—
247	C₂×C₄	C₂³×D₂₃	1	4	yes	—
248	C₂×C₄	C₂³×D₂₃	1	4	yes	—
249	C₂²	—	2	4	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	4	4	no	—
254	C₂²	—	4	4	no	—
255	C₂²	—	23	8	no	—
256	C₂²	—	46	4	no	—
257	C₂²	—	46	4	no	—
258	C₂²	—	46	16	no	—
259	C₂²	—	92	16	no	—
260	C₂²	#10625155	1	2	yes	—
261	C₂²	#41×D₂₃	1	4	yes	—
262	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
263	C₄	—	2	4	no	—
264	C₄	—	2	4	no	—
265	C₄	—	2	2	no	—
266	C₄	—	46	4	no	—
267	C₄	—	46	4	no	—
268	C₄	—	46	4	no	—
269	C₂	—	4	2	no	—
270	C₂	—	4	2	no	—
271	C₂	—	46	8	no	—
272	C₂	#13093095	1	4	yes	direct product factor
273	C₂	C₂×#41×D₂₃	1	2	yes	—
274	C₂	C₂×#10625155	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₂₃
commutator, cosocle	C₂²×C₂₃	C₂⁵
Frattini	C₂²	C₂⁴×D₂₃
Fitting	C₂×C₂₃×#123	C₂
socle	C₂³×C₂₃	C₂⁴
prime radical	C₂×#10625108	C₂
hypercenter	C₂×#123	D₂₃
hypocenter	C₂₃	C₂×#551
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×#123

Complements

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

N	H ≅ G/N	type
1. #13093095	C₂	direct (N × H)
2. C₂×#10625050	C₂	semidirect
3. C₂×#10625107	C₂	semidirect
4. C₂×#10625054	C₂	semidirect
5. C₂×#10625067	C₂	semidirect
7. C₂×C₄×C₂²⋊D₂₃	C₂	semidirect
8. C₂×#10625063	C₂	semidirect
9. C₂×C₂²⋊C₄×D₂₃	C₂	semidirect
10. C₂×C₂₃×#123	C₂	semidirect
13. C₂³×C₄×C₂₃	C₂²	semidirect
17. C₂²×Dic₄₆	C₂²	semidirect
18. C₂₃×#123	C₂²	semidirect
19. C₂×C₄⋊C₄×C₂₃	C₂²	semidirect
21. C₂×C₂²⋊C₄×C₂₃	C₂²	semidirect
24. C₂×C₂²⋊Dic₂₃	C₂²	semidirect
26. #10625107	C₂²	semidirect
27. C₄×C₂²⋊D₂₃	C₂²	semidirect
31. C₂×C₄⋊Dic₂₃	C₂²	semidirect
32. C₂×Dic₂₃⋊C₄	C₂²	semidirect
33. C₂×Dic₂₃⋊C₄	C₂²	semidirect
34. C₂×C₄×Dic₂₃	C₂²	semidirect
35. C₂×C₄×Dic₂₃	C₂²	semidirect
36. #10625067	C₂²	semidirect
37. #10625063	C₂²	semidirect
38. #10625054	C₂²	semidirect
39. C₂²⋊C₄×D₂₃	C₂²	semidirect
40. #10625050	C₂²	semidirect
53. C₂²×C₄×C₂₃	C₂³	semidirect
64. C₂×Dic₄₆	C₂³	semidirect
65. C₄⋊C₄×C₂₃	C₂³	semidirect
67. C₂²⋊C₄×C₂₃	C₂³	semidirect
70. C₂²⋊Dic₂₃	C₂³	semidirect
74. C₄⋊Dic₂₃	C₂³	semidirect
75. Dic₂₃⋊C₄	C₂³	semidirect
76. Dic₂₃⋊C₄	C₂³	semidirect
77. C₄×Dic₂₃	C₂³	semidirect
78. C₄×Dic₂₃	C₂³	semidirect
136. C₂×#123	D₂₃	semidirect
144. C₂×C₂₃	#551	semidirect
154. C₂³×C₄	C₂×D₂₃	semidirect
158. #123	C₂×D₂₃	semidirect
166. C₂×C₄⋊C₄	C₂×D₂₃	semidirect
175. C₂×C₂²⋊C₄	C₂×D₂₃	semidirect
178. C₂₃	C₂×#551	semidirect
202. C₂²×C₄	C₂²×D₂₃	semidirect
206. C₄⋊C₄	C₂²×D₂₃	semidirect
214. C₂²⋊C₄	C₂²×D₂₃	semidirect
272. C₂	#13093095	direct (N × H)

Maximal subgroups

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

subgroup	index
#13093095	2
C₂×#10625050	2
C₂×#10625107	2
C₂×#10625054	2
C₂×#10625067	2
C₂×#10625108	2
C₂×C₄×C₂²⋊D₂₃	2
C₂×#10625063	2
C₂×C₂²⋊C₄×D₂₃	2
C₂×C₂₃×#123	2
C₂×#551	23

Subgroups — C2×#13093095 #15686896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13093095 #15686896