Subgroups — C₂×#13093004 #15686737

Order 2944.

1336 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093004	C₂	1	16	yes	maximal, direct product factor
2	C₂×#10625069	C₂	1	2	yes	maximal, complemented
3	C₂×#10625056	C₂	1	2	yes	maximal, complemented
4	C₂×#10625057	C₂	1	2	yes	maximal, complemented
5	C₂×#10625095	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10625110	C₂	1	2	yes	maximal, complemented
7	C₂×#10625108	C₂	1	2	yes	maximal, complemented
8	C₂×D₄×Dic₂₃	C₂	1	2	yes	maximal, complemented
9	C₂²×(C₂×D₂₃)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×#121	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁴×D₂₃	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
13	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
14	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂²×C₂²⋊D₂₃	C₂²	1	2	yes	—
16	C₂²×C₂²⋊D₂₃	C₂²	1	2	yes	—
17	C₂³×Dic₂₃	C₂²	1	1	yes	characteristic
18	C₂³×Dic₂₃	C₂²	1	2	yes	complemented
19	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic, complemented
20	C₂₃×#121	C₂²	1	8	yes	complemented
21	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
22	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	complemented
23	C₂×C₂²⋊Dic₂₃	C₂²	1	2	yes	—
24	C₂×C₂²⋊Dic₂₃	C₂²	1	2	yes	complemented
25	#10625110	C₂²	1	16	yes	complemented
26	#10625108	C₂²	1	16	yes	complemented
27	D₄×Dic₂₃	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×(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
32	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	8	yes	complemented
33	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	complemented
34	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic, complemented
35	#10625095	C₂²	1	8	yes	complemented
36	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	complemented
37	C₂×C₄×Dic₂₃	C₂²	1	2	yes	complemented
38	#10625069	C₂²	1	16	yes	complemented
39	#10625057	C₂²	1	16	yes	complemented
40	#10625056	C₂²	1	16	yes	complemented
41	C₂⁴×C₂₃	C₂³	1	2	yes	—
42	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
43	C₂³×D₂₃	—	2	4	no	—
44	C₂³×D₂₃	—	4	2	no	—
45	C₂³×D₂₃	—	4	2	no	—
46	C₂³×D₂₃	C₂³	1	2	yes	—
47	C₂³×D₂₃	C₂³	1	4	yes	—
48	C₂×D₄×C₂₃	—	2	8	no	—
49	C₂×D₄×C₂₃	—	4	2	no	—
50	C₂×D₄×C₂₃	C₂³	1	8	yes	—
51	C₂×D₄×C₂₃	C₂³	1	4	yes	complemented
52	C₂²×C₄×C₂₃	—	4	2	no	—
53	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
54	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
57	C₂×C₂²⋊D₂₃	—	4	4	no	—
58	C₂×C₂²⋊D₂₃	—	4	4	no	—
59	C₂×C₂²⋊D₂₃	C₂³	1	8	yes	—
60	C₂×C₂²⋊D₂₃	C₂³	1	8	yes	—
61	C₂²×Dic₂₃	—	4	2	no	—
62	C₂²×Dic₂₃	—	4	4	no	—
63	C₂²×Dic₂₃	C₂³	1	2	yes	—
64	C₂²×Dic₂₃	C₂³	1	2	yes	—
65	C₂²×Dic₂₃	C₂³	1	2	yes	—
66	C₂²×Dic₂₃	C₂³	1	4	yes	—
67	C₂²×Dic₂₃	C₂³	1	8	yes	complemented
68	C₂×Dic₄₆	—	4	2	no	—
69	C₂×Dic₄₆	C₂³	1	4	yes	complemented
70	C₄⋊C₄×C₂₃	C₂³	1	4	yes	complemented
71	C₂²⋊C₄×C₂₃	C₂³	1	8	yes	complemented
72	C₂²⋊Dic₂₃	C₂³	1	8	yes	—
73	C₂²⋊Dic₂₃	C₂³	1	8	yes	complemented
74	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
75	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
76	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
77	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
78	C₄⋊Dic₂₃	C₂³	1	4	yes	complemented
79	C₄⋊Dic₂₃	C₂³	1	8	yes	complemented
80	Dic₂₃⋊C₄	C₂³	1	8	yes	complemented
81	C₄×Dic₂₃	C₂³	1	8	yes	complemented
82	C₂³×C₂₃	—	2	2	no	—
83	C₂³×C₂₃	—	2	2	no	—
84	C₂³×C₂₃	—	2	4	no	—
85	C₂³×C₂₃	—	2	4	no	—
86	C₂³×C₂₃	—	2	4	no	—
87	C₂³×C₂₃	#41	1	4	yes	—
88	C₂³×C₂₃	C₂⁴	1	4	yes	—
89	C₂³×C₂₃	C₂⁴	1	2	yes	—
90	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
91	C₂²×D₂₃	—	2	8	no	—
92	C₂²×D₂₃	—	2	4	no	—
93	C₂²×D₂₃	—	2	4	no	—
94	C₂²×D₂₃	—	2	4	no	—
95	C₂²×D₂₃	—	4	4	no	—
96	C₂²×D₂₃	—	4	4	no	—
97	C₂²×D₂₃	—	4	16	no	—
98	C₂²×D₂₃	C₂⁴	1	4	yes	—
99	D₄×C₂₃	—	2	16	no	—
100	D₄×C₂₃	—	4	4	no	—
101	C₂×C₄×C₂₃	—	2	4	no	—
102	C₂×C₄×C₂₃	—	2	2	no	—
103	C₂×C₄×C₂₃	—	2	2	no	—
104	C₂×C₄×C₂₃	—	4	4	no	—
105	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
106	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
107	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
108	C₂²⋊D₂₃	—	4	8	no	—
109	C₂²⋊D₂₃	—	4	8	no	—
110	C₂×Dic₂₃	—	2	4	no	—
111	C₂×Dic₂₃	—	2	4	no	—
112	C₂×Dic₂₃	—	2	4	no	—
113	C₂×Dic₂₃	—	4	4	no	—
114	C₂×Dic₂₃	—	4	8	no	—
115	C₂×Dic₂₃	C₂⁴	1	4	yes	—
116	C₂×Dic₂₃	C₂⁴	1	4	yes	—
117	C₂×Dic₂₃	C₂⁴	1	4	yes	—
118	Dic₄₆	—	4	4	no	—
119	C₂×#538	—	23	1	no	maximal, Sylow
120	C₂²×C₂₃	—	2	8	no	—
121	C₂²×C₂₃	—	2	2	no	—
122	C₂²×C₂₃	—	2	2	no	—
123	C₂²×C₂₃	—	2	4	no	—
124	C₂²×C₂₃	—	2	4	no	—
125	C₂²×C₂₃	—	2	4	no	—
126	C₂²×C₂₃	—	4	8	no	—
127	C₂²×C₂₃	C₂×#41	1	4	yes	—
128	C₂²×C₂₃	C₂×#41	1	2	yes	—
129	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
130	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
131	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
132	C₂×D₂₃	—	2	4	no	—
133	C₂×D₂₃	—	2	4	no	—
134	C₂×D₂₃	—	2	4	no	—
135	C₂×D₂₃	—	4	16	no	—
136	C₂×D₂₃	—	4	8	no	—
137	C₄×C₂₃	—	2	4	no	—
138	C₄×C₂₃	—	2	2	no	—
139	C₄×C₂₃	—	2	2	no	—
140	Dic₂₃	—	2	4	no	—
141	Dic₂₃	—	2	4	no	—
142	Dic₂₃	—	2	4	no	—
143	#538	—	23	16	no	—
144	C₂×#126	—	23	2	no	—
145	C₂×#124	—	23	2	no	—
146	C₂×#123	—	23	2	no	—
147	C₂×C₂²⋊Q₈	—	23	1	no	—
148	C₂×#121	—	23	2	no	—
149	C₂×#121	D₂₃	1	1	yes	characteristic, complemented
150	C₂×#120	—	23	2	no	—
151	C₂×C₄×D₄	—	23	2	no	—
152	C₂²×C₂²⋊C₄	—	23	1	no	—
153	C₂×C₂₃	—	2	4	no	—
154	C₂×C₂₃	—	4	4	no	—
155	C₂×C₂₃	#538	1	4	yes	complemented
156	C₂×C₂₃	C₂²×#41	1	1	yes	characteristic
157	C₂×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
158	C₂×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
159	D₂₃	—	4	4	no	—
160	C₂⁵	—	23	1	no	—
161	C₂²×Q₈	—	23	1	no	—
162	C₂²×D₄	—	23	2	no	—
163	C₂²×D₄	—	23	2	no	—
164	C₂²×D₄	C₂×D₂₃	1	2	yes	—
165	C₂²×D₄	C₂×D₂₃	1	1	yes	characteristic, complemented
166	C₂³×C₄	—	23	1	no	—
167	C₂³×C₄	—	23	2	no	—
168	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
169	#126	—	23	16	no	—
170	#124	—	23	16	no	—
171	#123	—	23	16	no	—
172	C₂²⋊Q₈	—	23	8	no	—
173	#121	—	23	16	no	—
174	#121	C₂×D₂₃	1	8	yes	complemented
175	#120	—	23	16	no	—
176	C₄×D₄	—	23	16	no	—
177	C₂×C₄⋊C₄	—	23	2	no	—
178	C₂×C₄⋊C₄	—	23	2	no	—
179	C₂×C₄⋊C₄	—	23	1	no	—
180	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic, complemented
181	C₂×C₂²⋊C₄	—	23	2	no	—
182	C₂×C₂²⋊C₄	—	23	2	no	—
183	C₂×C₂²⋊C₄	—	23	2	no	—
184	C₂×C₂²⋊C₄	—	23	2	no	—
185	C₂×C₂²⋊C₄	—	23	8	no	—
186	C₂×C₂²⋊C₄	—	23	2	no	—
187	C₂×C₂²⋊C₄	—	23	2	no	—
188	C₂×C₂²⋊C₄	C₂×D₂₃	1	2	yes	complemented
189	C₂×C₄²	—	23	2	no	—
190	C₂₃	C₂×#538	1	1	yes	characteristic, complemented, Sylow
191	C₂⁴	—	23	2	no	—
192	C₂⁴	—	23	4	no	—
193	C₂⁴	—	46	4	no	—
194	C₂⁴	—	92	2	no	—
195	C₂⁴	—	92	2	no	—
196	C₂⁴	C₂²×D₂₃	1	2	yes	—
197	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
198	C₂×Q₈	—	23	4	no	—
199	C₂×Q₈	—	92	2	no	—
200	C₂×D₄	—	2	8	no	—
201	C₂×D₄	—	4	2	no	—
202	C₂×D₄	—	23	8	no	—
203	C₂×D₄	—	23	8	no	—
204	C₂×D₄	—	92	4	no	—
205	C₂×D₄	—	92	4	no	—
206	C₂×D₄	C₂²×D₂₃	1	8	yes	—
207	C₂×D₄	C₂²×D₂₃	1	4	yes	complemented
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	—	23	2	no	—
210	C₂²×C₄	—	23	2	no	—
211	C₂²×C₄	—	23	2	no	—
212	C₂²×C₄	—	23	4	no	—
213	C₂²×C₄	—	23	8	no	—
214	C₂²×C₄	—	92	2	no	—
215	C₂²×C₄	—	92	4	no	—
216	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
217	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
218	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
219	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
220	C₄⋊C₄	—	23	8	no	—
221	C₄⋊C₄	—	23	8	no	—
222	C₄⋊C₄	—	23	4	no	—
223	C₄⋊C₄	C₂²×D₂₃	1	4	yes	complemented
224	C₂²⋊C₄	—	23	8	no	—
225	C₂²⋊C₄	—	23	8	no	—
226	C₂²⋊C₄	—	23	8	no	—
227	C₂²⋊C₄	—	23	8	no	—
228	C₂²⋊C₄	—	23	8	no	—
229	C₂²⋊C₄	—	23	8	no	—
230	C₂²⋊C₄	C₂²×D₂₃	1	8	yes	complemented
231	C₄²	—	23	8	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	2	4	no	—
235	C₂³	—	2	4	no	—
236	C₂³	—	2	4	no	—
237	C₂³	—	23	4	no	—
238	C₂³	—	46	8	no	—
239	C₂³	—	46	4	no	—
240	C₂³	—	46	4	no	—
241	C₂³	—	46	4	no	—
242	C₂³	—	92	4	no	—
243	C₂³	—	92	4	no	—
244	C₂³	—	92	16	no	—
245	C₂³	#67290	1	4	yes	—
246	C₂³	C₂³×D₂₃	1	4	yes	—
247	C₂³	C₂³×D₂₃	1	2	yes	—
248	C₂³	C₂³×D₂₃	1	1	yes	characteristic
249	Q₈	—	92	4	no	—
250	D₄	—	2	16	no	—
251	D₄	—	4	4	no	—
252	D₄	—	92	8	no	—
253	D₄	—	92	8	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	23	4	no	—
259	C₂×C₄	—	23	4	no	—
260	C₂×C₄	—	23	4	no	—
261	C₂×C₄	—	46	4	no	—
262	C₂×C₄	—	46	4	no	—
263	C₂×C₄	—	46	4	no	—
264	C₂×C₄	—	92	4	no	—
265	C₂×C₄	—	92	8	no	—
266	C₂×C₄	C₂³×D₂₃	1	4	yes	—
267	C₂×C₄	C₂³×D₂₃	1	2	yes	—
268	C₂×C₄	C₂³×D₂₃	1	2	yes	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	8	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	4	8	no	—
276	C₂²	—	46	4	no	—
277	C₂²	—	46	4	no	—
278	C₂²	—	46	4	no	—
279	C₂²	—	92	16	no	—
280	C₂²	—	92	8	no	—
281	C₂²	C₂×#67290	1	4	yes	—
282	C₂²	C₂×#67290	1	2	yes	—
283	C₂²	#10625155	1	2	yes	—
284	C₂²	#10625160	1	2	yes	—
285	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
286	C₄	—	2	4	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	46	4	no	—
290	C₄	—	46	4	no	—
291	C₄	—	46	4	no	—
292	C₂	—	2	4	no	—
293	C₂	—	4	4	no	—
294	C₂	—	92	4	no	—
295	C₂	#13093004	1	4	yes	direct product factor
296	C₂	C₂²×#67290	1	1	yes	characteristic
297	C₂	C₂×#10625155	1	1	yes	characteristic
298	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×#121	C₂
socle	C₂³×C₂₃	C₂⁴
hypercenter	C₂×#121	D₂₃
hypocenter	C₂₃	C₂×#538
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×#121

Complements

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

N	H ≅ G/N	type
1. #13093004	C₂	direct (N × H)
2. C₂×#10625069	C₂	semidirect
3. C₂×#10625056	C₂	semidirect
4. C₂×#10625057	C₂	semidirect
5. C₂×#10625095	C₂	semidirect
6. C₂×#10625110	C₂	semidirect
7. C₂×#10625108	C₂	semidirect
8. C₂×D₄×Dic₂₃	C₂	semidirect
9. C₂²×(C₂×D₂₃)⋊C₄	C₂	semidirect
10. C₂×C₂₃×#121	C₂	semidirect
13. C₂²×D₄×C₂₃	C₂²	semidirect
18. C₂³×Dic₂₃	C₂²	semidirect
19. C₂²×Dic₄₆	C₂²	semidirect
20. C₂₃×#121	C₂²	semidirect
21. C₂×C₄⋊C₄×C₂₃	C₂²	semidirect
22. C₂×C₂²⋊C₄×C₂₃	C₂²	semidirect
24. C₂×C₂²⋊Dic₂₃	C₂²	semidirect
25. #10625110	C₂²	semidirect
26. #10625108	C₂²	semidirect
27. D₄×Dic₂₃	C₂²	semidirect
32. C₂×(C₂×D₂₃)⋊C₄	C₂²	semidirect
33. C₂×C₄⋊Dic₂₃	C₂²	semidirect
34. C₂×C₄⋊Dic₂₃	C₂²	semidirect
35. #10625095	C₂²	semidirect
36. C₂×Dic₂₃⋊C₄	C₂²	semidirect
37. C₂×C₄×Dic₂₃	C₂²	semidirect
38. #10625069	C₂²	semidirect
39. #10625057	C₂²	semidirect
40. #10625056	C₂²	semidirect
51. C₂×D₄×C₂₃	C₂³	semidirect
67. C₂²×Dic₂₃	C₂³	semidirect
69. C₂×Dic₄₆	C₂³	semidirect
70. C₄⋊C₄×C₂₃	C₂³	semidirect
71. C₂²⋊C₄×C₂₃	C₂³	semidirect
73. C₂²⋊Dic₂₃	C₂³	semidirect
78. C₄⋊Dic₂₃	C₂³	semidirect
79. C₄⋊Dic₂₃	C₂³	semidirect
80. Dic₂₃⋊C₄	C₂³	semidirect
81. C₄×Dic₂₃	C₂³	semidirect
149. C₂×#121	D₂₃	semidirect
155. C₂×C₂₃	#538	semidirect
165. C₂²×D₄	C₂×D₂₃	semidirect
174. #121	C₂×D₂₃	semidirect
180. C₂×C₄⋊C₄	C₂×D₂₃	semidirect
188. C₂×C₂²⋊C₄	C₂×D₂₃	semidirect
190. C₂₃	C₂×#538	semidirect
207. C₂×D₄	C₂²×D₂₃	semidirect
223. C₄⋊C₄	C₂²×D₂₃	semidirect
230. C₂²⋊C₄	C₂²×D₂₃	semidirect
295. C₂	#13093004	direct (N × H)

Maximal subgroups

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

subgroup	index
#13093004	2
C₂×#10625069	2
C₂×#10625056	2
C₂×#10625057	2
C₂×#10625095	2
C₂×#10625110	2
C₂×#10625108	2
C₂×D₄×Dic₂₃	2
C₂²×(C₂×D₂₃)⋊C₄	2
C₂×C₂₃×#121	2
C₂×#538	23

Subgroups — C2×#13093004 #15686737

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13093004 #15686737