Subgroups — C₂×#13092976 #15686697

Order 2944.

1984 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₂²⋊D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092976	C₂	1	16	yes	maximal, direct product factor
3	C₂×#10625100	C₂	1	2	yes	maximal, complemented
4	C₂×#10625106	C₂	1	2	yes	maximal, complemented
5	C₂×#10625110	C₂	1	4	yes	maximal, complemented
6	C₂×#10625124	C₂	1	2	yes	maximal, complemented
7	C₂×#10625108	C₂	1	2	yes	maximal, complemented
8	C₂²×C₂²⋊Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁵×C₂₃	C₂²	1	1	yes	characteristic
11	C₂⁵×C₂₃	C₂²	1	1	yes	characteristic, complemented
12	C₂⁴×D₂₃	C₂²	1	1	yes	characteristic, complemented
13	C₂²×D₄×C₂₃	—	2	8	no	—
14	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
15	C₂²×D₄×C₂₃	C₂²	1	2	yes	complemented
16	C₂²×D₄×C₂₃	C₂²	1	8	yes	complemented
17	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
18	C₂²×C₂²⋊D₂₃	—	2	8	no	—
19	C₂²×C₂²⋊D₂₃	C₂²	1	2	yes	—
20	C₂²×C₂²⋊D₂₃	C₂²	1	2	yes	complemented
21	C₂²×C₂²⋊D₂₃	C₂²	1	4	yes	complemented
22	C₂²×C₂²⋊D₂₃	C₂²	1	8	yes	complemented
23	C₂³×Dic₂₃	C₂²	1	1	yes	characteristic, complemented
24	C₂³×Dic₂₃	C₂²	1	2	yes	complemented
25	#10625124	C₂²	1	16	yes	complemented
26	C₂×C₂²⋊Dic₂₃	C₂²	1	2	yes	complemented
27	C₂×C₂²⋊Dic₂₃	C₂²	1	2	yes	complemented
28	C₂×C₂²⋊Dic₂₃	C₂²	1	4	yes	complemented
29	C₂×C₂²⋊Dic₂₃	C₂²	1	8	yes	complemented
30	#10625110	C₂²	1	32	yes	complemented
31	#10625108	C₂²	1	16	yes	complemented
32	#10625106	C₂²	1	16	yes	complemented
33	#10625100	C₂²	1	16	yes	complemented
34	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	4	yes	complemented
35	C₂×Dic₂₃⋊C₄	C₂²	1	4	yes	complemented
36	C₂⁴×C₂₃	—	2	4	no	—
37	C₂⁴×C₂₃	—	4	4	no	—
38	C₂⁴×C₂₃	—	4	4	no	—
39	C₂⁴×C₂₃	D₄	1	8	yes	complemented
40	C₂⁴×C₂₃	C₂³	1	2	yes	—
41	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
42	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
43	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
44	C₂⁴×C₂₃	C₂³	1	4	yes	—
45	C₂⁴×C₂₃	C₂³	1	4	yes	complemented
46	C₂³×D₂₃	—	2	4	no	—
47	C₂³×D₂₃	—	4	4	no	—
48	C₂³×D₂₃	C₂³	1	2	yes	—
49	C₂³×D₂₃	C₂³	1	4	yes	complemented
50	C₂×D₄×C₂₃	—	2	8	no	—
51	C₂×D₄×C₂₃	—	2	8	no	—
52	C₂×D₄×C₂₃	—	2	32	no	—
53	C₂×D₄×C₂₃	C₂³	1	8	yes	—
54	C₂×D₄×C₂₃	C₂³	1	8	yes	complemented
55	C₂²×C₄×C₂₃	—	2	4	no	—
56	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
57	C₂²×C₄×C₂₃	C₂³	1	4	yes	complemented
58	C₂×C₂²⋊D₂₃	—	2	8	no	—
59	C₂×C₂²⋊D₂₃	—	2	8	no	—
60	C₂×C₂²⋊D₂₃	—	2	32	no	—
61	C₂×C₂²⋊D₂₃	—	4	8	no	—
62	C₂×C₂²⋊D₂₃	C₂³	1	8	yes	—
63	C₂×C₂²⋊D₂₃	C₂³	1	8	yes	complemented
64	C₂×C₂²⋊D₂₃	C₂³	1	16	yes	complemented
65	C₂²×Dic₂₃	—	2	4	no	—
66	C₂²×Dic₂₃	—	4	4	no	—
67	C₂²×Dic₂₃	C₂³	1	2	yes	—
68	C₂²×Dic₂₃	C₂³	1	4	yes	—
69	C₂²×Dic₂₃	C₂³	1	4	yes	complemented
70	C₂²×Dic₂₃	C₂³	1	8	yes	complemented
71	C₂²⋊Dic₂₃	C₂³	1	8	yes	complemented
72	C₂²⋊Dic₂₃	C₂³	1	8	yes	complemented
73	C₂²⋊Dic₂₃	C₂³	1	16	yes	complemented
74	(C₂×D₂₃)⋊C₄	C₂³	1	16	yes	complemented
75	Dic₂₃⋊C₄	C₂³	1	16	yes	complemented
76	C₂³×C₂₃	—	2	4	no	—
77	C₂³×C₂₃	—	2	4	no	—
78	C₂³×C₂₃	—	2	4	no	—
79	C₂³×C₂₃	—	2	4	no	—
80	C₂³×C₂₃	—	2	8	no	—
81	C₂³×C₂₃	—	2	8	no	—
82	C₂³×C₂₃	—	4	16	no	—
83	C₂³×C₂₃	—	4	8	no	—
84	C₂³×C₂₃	—	4	8	no	—
85	C₂³×C₂₃	—	4	16	no	—
86	C₂³×C₂₃	C₂×D₄	1	4	yes	—
87	C₂³×C₂₃	C₂×D₄	1	4	yes	—
88	C₂³×C₂₃	C₂×D₄	1	4	yes	—
89	C₂³×C₂₃	C₂×D₄	1	16	yes	complemented
90	C₂³×C₂₃	C₂⁴	1	4	yes	—
91	C₂³×C₂₃	C₂⁴	1	2	yes	—
92	C₂³×C₂₃	C₂⁴	1	2	yes	—
93	C₂³×C₂₃	C₂⁴	1	2	yes	—
94	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
95	C₂²×D₂₃	—	2	4	no	—
96	C₂²×D₂₃	—	2	8	no	—
97	C₂²×D₂₃	—	2	8	no	—
98	C₂²×D₂₃	—	4	8	no	—
99	C₂²×D₂₃	—	4	16	no	—
100	C₂²×D₂₃	C₂⁴	1	4	yes	—
101	D₄×C₂₃	—	2	16	no	—
102	D₄×C₂₃	—	2	16	no	—
103	C₂×C₄×C₂₃	—	2	4	no	—
104	C₂×C₄×C₂₃	—	2	8	no	—
105	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
106	C₂²⋊D₂₃	—	2	16	no	—
107	C₂²⋊D₂₃	—	2	16	no	—
108	C₂²⋊D₂₃	—	4	16	no	—
109	C₂×Dic₂₃	—	2	4	no	—
110	C₂×Dic₂₃	—	2	8	no	—
111	C₂×Dic₂₃	—	2	8	no	—
112	C₂×Dic₂₃	—	4	8	no	—
113	C₂×Dic₂₃	C₂⁴	1	4	yes	—
114	C₂×Dic₂₃	C₂⁴	1	8	yes	—
115	C₂×#534	—	23	1	no	maximal, Sylow
116	C₂²×C₂₃	—	2	8	no	—
117	C₂²×C₂₃	—	2	8	no	—
118	C₂²×C₂₃	—	2	8	no	—
119	C₂²×C₂₃	—	2	4	no	—
120	C₂²×C₂₃	—	2	4	no	—
121	C₂²×C₂₃	—	2	4	no	—
122	C₂²×C₂₃	—	2	4	no	—
123	C₂²×C₂₃	—	2	8	no	—
124	C₂²×C₂₃	—	4	16	no	—
125	C₂²×C₂₃	—	4	16	no	—
126	C₂²×C₂₃	—	4	8	no	—
127	C₂²×C₂₃	C₂²×D₄	1	4	yes	—
128	C₂²×C₂₃	C₂²×D₄	1	4	yes	—
129	C₂²×C₂₃	C₂²×D₄	1	4	yes	—
130	C₂²×C₂₃	C₂²×D₄	1	2	yes	—
131	C₂²×C₂₃	H(5,𝔽₂)	1	4	yes	—
132	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
133	C₂×D₂₃	—	2	4	no	—
134	C₂×D₂₃	—	2	8	no	—
135	C₂×D₂₃	—	4	16	no	—
136	C₂×D₂₃	—	4	8	no	—
137	C₄×C₂₃	—	2	4	no	—
138	Dic₂₃	—	2	4	no	—
139	Dic₂₃	—	2	8	no	—
140	C₂³×D₄	—	23	1	no	—
141	C₂³×D₄	D₂₃	1	1	yes	characteristic, complemented
142	#534	—	23	16	no	—
143	C₂×#123	—	23	2	no	—
144	C₂×#123	—	23	2	no	—
145	C₂×#121	—	23	4	no	—
146	C₂×#120	—	23	2	no	—
147	C₂×#120	—	23	2	no	—
148	C₂²×C₂²⋊C₄	—	23	1	no	—
149	C₂×C₂₃	—	2	4	no	—
150	C₂×C₂₃	—	2	4	no	—
151	C₂×C₂₃	—	2	4	no	—
152	C₂×C₂₃	—	4	4	no	—
153	C₂×C₂₃	#534	1	4	yes	complemented
154	C₂×C₂₃	C₂³×D₄	1	1	yes	characteristic
155	C₂×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
156	D₂₃	—	4	4	no	—
157	C₂⁵	—	23	1	no	—
158	C₂⁵	C₂×D₂₃	1	1	yes	characteristic
159	C₂⁵	C₂×D₂₃	1	1	yes	characteristic, complemented
160	C₂²×D₄	—	2	8	no	—
161	C₂²×D₄	—	23	2	no	—
162	C₂²×D₄	—	23	2	no	—
163	C₂²×D₄	—	23	8	no	—
164	C₂²×D₄	—	23	4	no	—
165	C₂²×D₄	—	46	8	no	—
166	C₂²×D₄	C₂×D₂₃	1	2	yes	—
167	C₂²×D₄	C₂×D₂₃	1	2	yes	complemented
168	C₂²×D₄	C₂×D₂₃	1	8	yes	complemented
169	C₂³×C₄	—	23	1	no	—
170	C₂³×C₄	—	23	2	no	—
171	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic, complemented
172	#123	—	23	16	no	—
173	#123	—	23	16	no	—
174	#121	—	23	32	no	—
175	#120	—	23	16	no	—
176	#120	—	23	16	no	—
177	C₂×C₄⋊C₄	—	23	4	no	—
178	C₂×C₂²⋊C₄	—	23	2	no	—
179	C₂×C₂²⋊C₄	—	23	2	no	—
180	C₂×C₂²⋊C₄	—	23	8	no	—
181	C₂×C₂²⋊C₄	—	23	4	no	—
182	C₂×C₂²⋊C₄	—	23	4	no	—
183	C₂₃	C₂×#534	1	1	yes	characteristic, complemented, Sylow
184	C₂⁴	—	2	4	no	—
185	C₂⁴	—	4	4	no	—
186	C₂⁴	—	4	4	no	—
187	C₂⁴	—	23	2	no	—
188	C₂⁴	—	23	4	no	—
189	C₂⁴	—	46	4	no	—
190	C₂⁴	—	92	4	no	—
191	C₂⁴	C₂²⋊D₂₃	1	8	yes	complemented
192	C₂⁴	C₂²×D₂₃	1	2	yes	—
193	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
194	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
195	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
196	C₂⁴	C₂²×D₂₃	1	4	yes	—
197	C₂⁴	C₂²×D₂₃	1	4	yes	complemented
198	C₂×D₄	—	2	8	no	—
199	C₂×D₄	—	2	8	no	—
200	C₂×D₄	—	2	32	no	—
201	C₂×D₄	—	23	8	no	—
202	C₂×D₄	—	23	8	no	—
203	C₂×D₄	—	23	16	no	—
204	C₂×D₄	—	46	8	no	—
205	C₂×D₄	—	46	8	no	—
206	C₂×D₄	—	46	32	no	—
207	C₂×D₄	—	92	8	no	—
208	C₂×D₄	C₂²×D₂₃	1	8	yes	—
209	C₂×D₄	C₂²×D₂₃	1	8	yes	complemented
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	23	4	no	—
212	C₂²×C₄	—	23	4	no	—
213	C₂²×C₄	—	23	2	no	—
214	C₂²×C₄	—	23	8	no	—
215	C₂²×C₄	—	46	4	no	—
216	C₂²×C₄	—	92	4	no	—
217	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
218	C₂²×C₄	C₂²×D₂₃	1	4	yes	complemented
219	C₄⋊C₄	—	23	16	no	—
220	C₂²⋊C₄	—	23	8	no	—
221	C₂²⋊C₄	—	23	8	no	—
222	C₂²⋊C₄	—	23	16	no	—
223	C₂²⋊C₄	—	23	16	no	—
224	C₂³	—	2	4	no	—
225	C₂³	—	2	4	no	—
226	C₂³	—	2	4	no	—
227	C₂³	—	2	8	no	—
228	C₂³	—	2	4	no	—
229	C₂³	—	2	8	no	—
230	C₂³	—	4	16	no	—
231	C₂³	—	4	8	no	—
232	C₂³	—	4	8	no	—
233	C₂³	—	4	16	no	—
234	C₂³	—	23	4	no	—
235	C₂³	—	46	8	no	—
236	C₂³	—	46	8	no	—
237	C₂³	—	46	4	no	—
238	C₂³	—	92	8	no	—
239	C₂³	—	92	16	no	—
240	C₂³	C₂×C₂²⋊D₂₃	1	4	yes	—
241	C₂³	C₂×C₂²⋊D₂₃	1	4	yes	—
242	C₂³	C₂×C₂²⋊D₂₃	1	4	yes	—
243	C₂³	C₂×C₂²⋊D₂₃	1	16	yes	complemented
244	C₂³	C₂³×D₂₃	1	4	yes	—
245	C₂³	C₂³×D₂₃	1	2	yes	—
246	C₂³	C₂³×D₂₃	1	2	yes	—
247	C₂³	C₂³×D₂₃	1	2	yes	—
248	C₂³	C₂³×D₂₃	1	1	yes	characteristic
249	D₄	—	2	16	no	—
250	D₄	—	2	16	no	—
251	D₄	—	46	16	no	—
252	D₄	—	46	16	no	—
253	D₄	—	92	16	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	23	8	no	—
257	C₂×C₄	—	23	4	no	—
258	C₂×C₄	—	46	8	no	—
259	C₂×C₄	—	46	8	no	—
260	C₂×C₄	—	46	4	no	—
261	C₂×C₄	—	92	8	no	—
262	C₂×C₄	C₂³×D₂₃	1	4	yes	—
263	C₂²	—	2	8	no	—
264	C₂²	—	2	8	no	—
265	C₂²	—	2	8	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	8	no	—
271	C₂²	—	4	16	no	—
272	C₂²	—	4	16	no	—
273	C₂²	—	4	8	no	—
274	C₂²	—	46	4	no	—
275	C₂²	—	46	8	no	—
276	C₂²	—	92	16	no	—
277	C₂²	—	92	8	no	—
278	C₂²	#10625155	1	4	yes	—
279	C₂²	C₂²×C₂²⋊D₂₃	1	4	yes	—
280	C₂²	C₂²×C₂²⋊D₂₃	1	4	yes	—
281	C₂²	C₂²×C₂²⋊D₂₃	1	4	yes	—
282	C₂²	C₂²×C₂²⋊D₂₃	1	2	yes	—
283	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
284	C₄	—	2	4	no	—
285	C₄	—	46	4	no	—
286	C₄	—	46	8	no	—
287	C₂	—	2	4	no	—
288	C₂	—	2	4	no	—
289	C₂	—	2	4	no	—
290	C₂	—	4	4	no	—
291	C₂	—	92	4	no	—
292	C₂	#13092976	1	4	yes	direct product factor
293	C₂	C₂³×C₂²⋊D₂₃	1	1	yes	characteristic
294	C₂	C₂×#10625155	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₂₃
commutator, cosocle	C₂²×C₂₃	C₂⁵
Frattini	C₂²	C₂⁴×D₂₃
Fitting	C₂³×D₄×C₂₃	C₂
socle	C₂³×C₂₃	C₂⁴
hypercenter	C₂³×D₄	D₂₃
hypocenter	C₂₃	C₂×#534
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂³×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×C₂²⋊D₂₃	C₂	semidirect
2. #13092976	C₂	direct (N × H)
3. C₂×#10625100	C₂	semidirect
4. C₂×#10625106	C₂	semidirect
5. C₂×#10625110	C₂	semidirect
6. C₂×#10625124	C₂	semidirect
7. C₂×#10625108	C₂	semidirect
8. C₂²×C₂²⋊Dic₂₃	C₂	semidirect
9. C₂³×D₄×C₂₃	C₂	semidirect
11. C₂⁵×C₂₃	C₂²	semidirect
12. C₂⁴×D₂₃	C₂²	semidirect
15. C₂²×D₄×C₂₃	C₂²	semidirect
16. C₂²×D₄×C₂₃	C₂²	semidirect
17. C₂³×C₄×C₂₃	C₂²	semidirect
20. C₂²×C₂²⋊D₂₃	C₂²	semidirect
21. C₂²×C₂²⋊D₂₃	C₂²	semidirect
22. C₂²×C₂²⋊D₂₃	C₂²	semidirect
23. C₂³×Dic₂₃	C₂²	semidirect
24. C₂³×Dic₂₃	C₂²	semidirect
25. #10625124	C₂²	semidirect
26. C₂×C₂²⋊Dic₂₃	C₂²	semidirect
27. C₂×C₂²⋊Dic₂₃	C₂²	semidirect
28. C₂×C₂²⋊Dic₂₃	C₂²	semidirect
29. C₂×C₂²⋊Dic₂₃	C₂²	semidirect
30. #10625110	C₂²	semidirect
31. #10625108	C₂²	semidirect
32. #10625106	C₂²	semidirect
33. #10625100	C₂²	semidirect
34. C₂×(C₂×D₂₃)⋊C₄	C₂²	semidirect
35. C₂×Dic₂₃⋊C₄	C₂²	semidirect
39. C₂⁴×C₂₃	D₄	semidirect
45. C₂⁴×C₂₃	C₂³	semidirect
49. C₂³×D₂₃	C₂³	semidirect
54. C₂×D₄×C₂₃	C₂³	semidirect
57. C₂²×C₄×C₂₃	C₂³	semidirect
63. C₂×C₂²⋊D₂₃	C₂³	semidirect
64. C₂×C₂²⋊D₂₃	C₂³	semidirect
69. C₂²×Dic₂₃	C₂³	semidirect
70. C₂²×Dic₂₃	C₂³	semidirect
71. C₂²⋊Dic₂₃	C₂³	semidirect
72. C₂²⋊Dic₂₃	C₂³	semidirect
73. C₂²⋊Dic₂₃	C₂³	semidirect
74. (C₂×D₂₃)⋊C₄	C₂³	semidirect
75. Dic₂₃⋊C₄	C₂³	semidirect
89. C₂³×C₂₃	C₂×D₄	semidirect
141. C₂³×D₄	D₂₃	semidirect
153. C₂×C₂₃	#534	semidirect
159. C₂⁵	C₂×D₂₃	semidirect
167. C₂²×D₄	C₂×D₂₃	semidirect
168. C₂²×D₄	C₂×D₂₃	semidirect
171. C₂³×C₄	C₂×D₂₃	semidirect
183. C₂₃	C₂×#534	semidirect
191. C₂⁴	C₂²⋊D₂₃	semidirect
197. C₂⁴	C₂²×D₂₃	semidirect
209. C₂×D₄	C₂²×D₂₃	semidirect
218. C₂²×C₄	C₂²×D₂₃	semidirect
243. C₂³	C₂×C₂²⋊D₂₃	semidirect
292. C₂	#13092976	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₂²⋊D₂₃	2
#13092976	2
C₂×#10625100	2
C₂×#10625106	2
C₂×#10625110	2
C₂×#10625124	2
C₂×#10625108	2
C₂²×C₂²⋊Dic₂₃	2
C₂³×D₄×C₂₃	2
C₂×#534	23

Subgroups — C2×#13092976 #15686697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13092976 #15686697