Subgroups — C₂×#13093038 #15686775

Order 2944.

1132 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	#13093038	C₂	1	16	yes	maximal, direct product factor
2	C₂×#10625040	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#10625069	C₂	1	2	yes	maximal, complemented
4	C₂×#10625046	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10625056	C₂	1	2	yes	maximal, complemented
6	C₂×#10625054	C₂	1	2	yes	maximal, complemented
7	C₂×#10625095	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10625067	C₂	1	2	yes	maximal, complemented
9	C₂×#10625101	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Dic₄₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×D₉₂	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂²×C₂²⋊D₂₃	C₂²	1	2	yes	—
15	C₂²×D₉₂	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₂₃	C₂²	1	2	yes	—
17	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic
18	C₂²×Dic₄₆	C₂²	1	2	yes	—
19	C₂₃×#117	C₂²	1	8	yes	complemented
20	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊Dic₂₃	C₂²	1	2	yes	—
27	#10625101	C₂²	1	8	yes	—
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×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
35	#10625095	C₂²	1	8	yes	complemented
36	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
37	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
38	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
39	#10625069	C₂²	1	16	yes	complemented
40	#10625067	C₂²	1	16	yes	complemented
41	#10625056	C₂²	1	16	yes	complemented
42	#10625054	C₂²	1	16	yes	complemented
43	#10625046	C₂²	1	8	yes	complemented
44	C₄×D₉₂	C₂²	1	8	yes	complemented
45	#10625040	C₂²	1	8	yes	complemented
46	C₄×Dic₄₆	C₂²	1	8	yes	complemented
47	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
48	C₂³×D₂₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₂₃	—	2	4	no	—
50	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	C₂³	1	4	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	1	yes	characteristic
57	C₂×C₂²⋊D₂₃	—	4	4	no	—
58	C₂×C₂²⋊D₂₃	C₂³	1	8	yes	—
59	C₂²×Dic₂₃	C₂³	1	2	yes	—
60	C₂²×Dic₂₃	C₂³	1	2	yes	—
61	C₂²×Dic₂₃	C₂³	1	2	yes	—
62	C₂×D₉₂	—	2	4	no	—
63	C₂×D₉₂	C₂³	1	4	yes	—
64	C₂×C₄×D₂₃	—	4	4	no	—
65	C₂×C₄×D₂₃	C₂³	1	8	yes	—
66	C₂×Dic₄₆	—	2	4	no	—
67	C₂×Dic₄₆	—	4	4	no	—
68	C₂×Dic₄₆	C₂³	1	4	yes	—
69	C₂×Dic₄₆	C₂³	1	8	yes	—
70	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
71	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
72	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
73	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
74	C₄²×C₂₃	C₂³	1	4	yes	—
75	C₄²×C₂₃	C₂³	1	4	yes	—
76	C₂²⋊Dic₂₃	C₂³	1	8	yes	—
77	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
78	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
79	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
80	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
81	C₄⋊Dic₂₃	C₂³	1	4	yes	—
82	C₄⋊Dic₂₃	C₂³	1	4	yes	—
83	C₄⋊Dic₂₃	C₂³	1	8	yes	—
84	Dic₂₃⋊C₄	C₂³	1	8	yes	—
85	Dic₂₃⋊C₄	C₂³	1	8	yes	—
86	C₄×Dic₂₃	C₂³	1	8	yes	—
87	C₂³×C₂₃	—	2	2	no	—
88	C₂³×C₂₃	—	2	2	no	—
89	C₂³×C₂₃	—	2	2	no	—
90	C₂³×C₂₃	C₂⁴	1	2	yes	—
91	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
92	C₂²×D₂₃	—	2	4	no	—
93	C₂²×D₂₃	—	2	4	no	—
94	C₂²×D₂₃	—	2	4	no	—
95	C₂²×D₂₃	C₂⁴	1	4	yes	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	2	no	—
98	C₂×C₄×C₂₃	—	2	2	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	2	no	—
101	C₂×C₄×C₂₃	—	2	8	no	—
102	C₂×C₄×C₂₃	#41	1	4	yes	—
103	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
104	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
105	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
106	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
107	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
108	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
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	C₂⁴	1	4	yes	—
114	C₂×Dic₂₃	C₂⁴	1	4	yes	—
115	C₂×Dic₂₃	C₂⁴	1	4	yes	—
116	D₉₂	—	2	8	no	—
117	C₄×D₂₃	—	4	8	no	—
118	Dic₄₆	—	2	8	no	—
119	Dic₄₆	—	4	8	no	—
120	C₂×#542	—	23	1	no	maximal, Sylow
121	C₂²×C₂₃	—	2	2	no	—
122	C₂²×C₂₃	—	2	2	no	—
123	C₂²×C₂₃	—	2	2	no	—
124	C₂²×C₂₃	—	4	4	no	—
125	C₂²×C₂₃	C₂×#41	1	2	yes	—
126	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
127	C₂²×C₂₃	2₋¹⁺⁴	1	2	yes	—
128	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
129	C₂×D₂₃	—	2	4	no	—
130	C₂×D₂₃	—	2	4	no	—
131	C₂×D₂₃	—	2	4	no	—
132	C₂×D₂₃	—	4	8	no	—
133	C₄×C₂₃	—	2	2	no	—
134	C₄×C₂₃	—	2	2	no	—
135	C₄×C₂₃	—	2	2	no	—
136	C₄×C₂₃	—	2	2	no	—
137	C₄×C₂₃	—	2	2	no	—
138	C₄×C₂₃	C₂×#41	1	4	yes	—
139	Dic₂₃	—	2	4	no	—
140	Dic₂₃	—	2	4	no	—
141	Dic₂₃	—	2	4	no	—
142	#542	—	23	16	no	—
143	C₂×C₄⋊Q₈	—	23	1	no	—
144	C₂×#126	—	23	2	no	—
145	C₂×#124	—	23	2	no	—
146	C₂×#124	—	23	1	no	—
147	C₂×#123	—	23	2	no	—
148	C₂×C₂²⋊Q₈	—	23	1	no	—
149	C₂×C₂²⋊Q₈	—	23	2	no	—
150	C₂×#121	—	23	1	no	—
151	C₂×C₄×Q₈	—	23	1	no	—
152	C₂×C₄×D₄	—	23	1	no	—
153	C₂×#117	D₂₃	1	1	yes	characteristic, complemented
154	C₂×C₂₃	—	4	2	no	—
155	C₂×C₂₃	#542	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	C₂×2₋¹⁺⁴	1	1	yes	characteristic
159	D₂₃	—	4	4	no	—
160	C₂²×Q₈	—	23	1	no	—
161	C₂²×Q₈	—	23	2	no	—
162	C₂²×D₄	—	23	2	no	—
163	C₂²×D₄	—	23	1	no	—
164	C₂³×C₄	—	23	2	no	—
165	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
166	C₄⋊Q₈	—	23	8	no	—
167	#126	—	23	16	no	—
168	#124	—	23	16	no	—
169	#124	—	23	8	no	—
170	#123	—	23	16	no	—
171	C₂²⋊Q₈	—	23	8	no	—
172	C₂²⋊Q₈	—	23	16	no	—
173	#121	—	23	8	no	—
174	C₄×Q₈	—	23	8	no	—
175	C₄×D₄	—	23	8	no	—
176	#117	C₂×D₂₃	1	8	yes	complemented
177	C₂×C₄⋊C₄	—	23	1	no	—
178	C₂×C₄⋊C₄	—	23	1	no	—
179	C₂×C₄⋊C₄	—	23	2	no	—
180	C₂×C₄⋊C₄	—	23	2	no	—
181	C₂×C₄⋊C₄	—	23	2	no	—
182	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
183	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
184	C₂×C₂²⋊C₄	—	23	2	no	—
185	C₂×C₂²⋊C₄	—	23	2	no	—
186	C₂×C₂²⋊C₄	—	23	2	no	—
187	C₂×C₂²⋊C₄	—	23	2	no	—
188	C₂×C₂²⋊C₄	—	23	2	no	—
189	C₂×C₂²⋊C₄	C₂×D₂₃	1	1	yes	characteristic
190	C₂×C₂²⋊C₄	C₂×D₂₃	1	1	yes	characteristic
191	C₂×C₄²	—	23	2	no	—
192	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
193	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
194	C₂₃	C₂×#542	1	1	yes	characteristic, complemented, Sylow
195	C₂⁴	—	23	2	no	—
196	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
197	C₂×Q₈	—	23	4	no	—
198	C₂×Q₈	—	23	8	no	—
199	C₂×Q₈	—	46	4	no	—
200	C₂×Q₈	—	92	4	no	—
201	C₂×D₄	—	23	8	no	—
202	C₂×D₄	—	23	4	no	—
203	C₂×D₄	—	46	4	no	—
204	C₂×D₄	—	92	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	23	2	no	—
207	C₂²×C₄	—	23	2	no	—
208	C₂²×C₄	—	23	2	no	—
209	C₂²×C₄	—	23	8	no	—
210	C₂²×C₄	—	92	4	no	—
211	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
212	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
213	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
214	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
215	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
216	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
217	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
218	C₄⋊C₄	—	23	4	no	—
219	C₄⋊C₄	—	23	4	no	—
220	C₄⋊C₄	—	23	8	no	—
221	C₄⋊C₄	—	23	8	no	—
222	C₄⋊C₄	—	23	8	no	—
223	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
224	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
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	4	yes	—
231	C₂²⋊C₄	C₂²×D₂₃	1	4	yes	—
232	C₄²	—	23	8	no	—
233	C₄²	C₂²×D₂₃	1	4	yes	—
234	C₄²	C₂²×D₂₃	1	4	yes	—
235	C₂³	—	2	2	no	—
236	C₂³	—	2	2	no	—
237	C₂³	—	2	2	no	—
238	C₂³	—	23	4	no	—
239	C₂³	—	46	4	no	—
240	C₂³	—	46	4	no	—
241	C₂³	—	46	4	no	—
242	C₂³	C₂³×D₂₃	1	2	yes	—
243	C₂³	C₂³×D₂₃	1	1	yes	characteristic
244	Q₈	—	46	8	no	—
245	Q₈	—	92	8	no	—
246	D₄	—	46	8	no	—
247	D₄	—	92	8	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	8	no	—
254	C₂×C₄	—	23	4	no	—
255	C₂×C₄	—	23	4	no	—
256	C₂×C₄	—	23	4	no	—
257	C₂×C₄	—	46	4	no	—
258	C₂×C₄	—	46	4	no	—
259	C₂×C₄	—	46	4	no	—
260	C₂×C₄	—	92	8	no	—
261	C₂×C₄	#67288	1	4	yes	—
262	C₂×C₄	C₂³×D₂₃	1	2	yes	—
263	C₂×C₄	C₂³×D₂₃	1	2	yes	—
264	C₂×C₄	C₂³×D₂₃	1	2	yes	—
265	C₂×C₄	C₂³×D₂₃	1	2	yes	—
266	C₂×C₄	C₂³×D₂₃	1	2	yes	—
267	C₂×C₄	C₂³×D₂₃	1	2	yes	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	4	4	no	—
272	C₂²	—	46	4	no	—
273	C₂²	—	46	4	no	—
274	C₂²	—	46	4	no	—
275	C₂²	—	92	8	no	—
276	C₂²	C₂×#67288	1	2	yes	—
277	C₂²	#10625160	1	2	yes	—
278	C₂²	#10625161	1	2	yes	—
279	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	46	4	no	—
286	C₄	—	46	4	no	—
287	C₄	—	46	4	no	—
288	C₄	C₂×#67288	1	4	yes	—
289	C₂	—	4	2	no	—
290	C₂	—	92	4	no	—
291	C₂	#13093038	1	4	yes	direct product factor
292	C₂	C₂²×#67288	1	1	yes	characteristic
293	C₂	C₂×#10625160	1	1	yes	characteristic
294	C₂	C₂×#10625161	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₂₃
commutator, cosocle	C₂²×C₂₃	C₂⁵
Frattini	C₂²	C₂⁴×D₂₃
Fitting	C₂×C₂₃×#117	C₂
socle	C₂³×C₂₃	C₂⁴
prime radical	C₂×#10625101	C₂
hypercenter	C₂×#117	D₂₃
hypocenter	C₂₃	C₂×#542
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #13093038	C₂	direct (N × H)
2. C₂×#10625040	C₂	semidirect
3. C₂×#10625069	C₂	semidirect
4. C₂×#10625046	C₂	semidirect
5. C₂×#10625056	C₂	semidirect
6. C₂×#10625054	C₂	semidirect
7. C₂×#10625095	C₂	semidirect
8. C₂×#10625067	C₂	semidirect
10. C₂×C₄×Dic₄₆	C₂	semidirect
11. C₂×C₄×D₉₂	C₂	semidirect
12. C₂×C₂₃×#117	C₂	semidirect
19. C₂₃×#117	C₂²	semidirect
35. #10625095	C₂²	semidirect
39. #10625069	C₂²	semidirect
40. #10625067	C₂²	semidirect
41. #10625056	C₂²	semidirect
42. #10625054	C₂²	semidirect
43. #10625046	C₂²	semidirect
44. C₄×D₉₂	C₂²	semidirect
45. #10625040	C₂²	semidirect
46. C₄×Dic₄₆	C₂²	semidirect
153. C₂×#117	D₂₃	semidirect
155. C₂×C₂₃	#542	semidirect
176. #117	C₂×D₂₃	semidirect
194. C₂₃	C₂×#542	semidirect
291. C₂	#13093038	direct (N × H)

Maximal subgroups

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

subgroup	index
#13093038	2
C₂×#10625040	2
C₂×#10625069	2
C₂×#10625046	2
C₂×#10625056	2
C₂×#10625054	2
C₂×#10625095	2
C₂×#10625067	2
C₂×#10625101	2
C₂×C₄×Dic₄₆	2
C₂×C₄×D₉₂	2
C₂×C₂₃×#117	2
C₂×#542	23

Subgroups — C2×#13093038 #15686775

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13093038 #15686775