Subgroups — C₂×#13093139 #15686945

Order 2944.

1112 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093139	C₂	1	16	yes	maximal, direct product factor
2	C₂×#10625115	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#10625069	C₂	1	2	yes	maximal, complemented
4	C₂×#10625047	C₂	1	2	yes	maximal, complemented
5	C₂×#10625118	C₂	1	1	yes	maximal, characteristic
6	C₂×#10625046	C₂	1	1	yes	maximal, characteristic
7	C₂×#10625067	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×Dic₄₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#10625058	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10625064	C₂	1	1	yes	maximal, characteristic
11	C₂×#10625043	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂²×D₉₂	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₂₃	C₂²	1	2	yes	—
16	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic
17	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₂₃	C₂²	1	8	yes	complemented
19	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
23	#10625118	C₂²	1	8	yes	—
24	#10625115	C₂²	1	8	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	2	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	1	yes	characteristic
31	C₂×Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
32	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
33	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
34	C₂×Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
35	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
36	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
37	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
39	#10625069	C₂²	1	16	yes	complemented
40	#10625067	C₂²	1	16	yes	complemented
41	#10625064	C₂²	1	8	yes	—
42	#10625058	C₂²	1	8	yes	complemented
43	#10625047	C₂²	1	16	yes	complemented
44	#10625046	C₂²	1	8	yes	—
45	#10625043	C₂²	1	16	yes	complemented
46	C₄×Dic₄₆	C₂²	1	8	yes	complemented
47	C₂³×D₂₃	C₂³	1	2	yes	—
48	C₂×Q₈×C₂₃	—	2	4	no	—
49	C₂×Q₈×C₂₃	C₂³	1	4	yes	—
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	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
55	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
56	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
57	C₂²×Dic₂₃	C₂³	1	2	yes	—
58	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
59	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
60	C₂×D₉₂	—	2	4	no	—
61	C₂×D₉₂	C₂³	1	4	yes	—
62	C₂×C₄×D₂₃	—	2	8	no	—
63	C₂×C₄×D₂₃	C₂³	1	8	yes	—
64	C₂×Dic₄₆	—	2	4	no	—
65	C₂×Dic₄₆	—	2	4	no	—
66	C₂×Dic₄₆	C₂³	1	4	yes	—
67	C₂×Dic₄₆	C₂³	1	4	yes	—
68	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
69	C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
70	C₄²×C₂₃	C₂³	1	4	yes	—
71	C₄²×C₂₃	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	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
77	C₄⋊Dic₂₃	C₂³	1	4	yes	—
78	Dic₂₃⋊C₄	C₂³	1	4	yes	—
79	Dic₂₃⋊C₄	C₂³	1	4	yes	—
80	Dic₂₃⋊C₄	C₂³	1	8	yes	—
81	Dic₂₃⋊C₄	C₂³	1	8	yes	—
82	Dic₂₃⋊C₄	C₂³	1	8	yes	—
83	C₄×Dic₂₃	C₂³	1	4	yes	—
84	C₄×Dic₂₃	C₂³	1	4	yes	—
85	C₄×Dic₂₃	C₂³	1	8	yes	—
86	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
87	C₂²×D₂₃	—	2	4	no	—
88	C₂²×D₂₃	—	2	4	no	—
89	C₂²×D₂₃	—	2	4	no	—
90	C₂²×D₂₃	C₂⁴	1	4	yes	—
91	Q₈×C₂₃	—	2	8	no	—
92	C₂×C₄×C₂₃	—	2	2	no	—
93	C₂×C₄×C₂₃	—	2	4	no	—
94	C₂×C₄×C₂₃	—	2	4	no	—
95	C₂×C₄×C₂₃	—	2	2	no	—
96	C₂×C₄×C₂₃	#41	1	4	yes	—
97	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
98	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
99	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
100	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
101	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
102	C₂×Dic₂₃	—	2	4	no	—
103	C₂×Dic₂₃	—	2	2	no	—
104	C₂×Dic₂₃	—	2	2	no	—
105	C₂×Dic₂₃	—	2	2	no	—
106	C₂×Dic₂₃	#41	1	4	yes	—
107	C₂×Dic₂₃	C₂⁴	1	2	yes	—
108	C₂×Dic₂₃	C₂⁴	1	2	yes	—
109	C₂×Dic₂₃	C₂⁴	1	2	yes	—
110	C₂×Dic₂₃	C₂⁴	1	4	yes	—
111	C₂×Dic₂₃	C₂⁴	1	2	yes	—
112	D₉₂	—	2	8	no	—
113	C₄×D₂₃	—	2	16	no	—
114	Dic₄₆	—	2	8	no	—
115	Dic₄₆	—	2	8	no	—
116	C₂×#556	—	23	1	no	maximal, Sylow
117	C₂²×C₂₃	C₂×#41	1	2	yes	—
118	C₂²×C₂₃	C₂×#41	1	2	yes	—
119	C₂²×C₂₃	2₋¹⁺⁴	1	2	yes	—
120	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
121	C₂×D₂₃	—	2	4	no	—
122	C₂×D₂₃	—	2	4	no	—
123	C₂×D₂₃	—	2	4	no	—
124	C₂×D₂₃	—	4	8	no	—
125	C₄×C₂₃	—	2	2	no	—
126	C₄×C₂₃	—	2	4	no	—
127	C₄×C₂₃	—	2	4	no	—
128	C₄×C₂₃	—	2	2	no	—
129	C₄×C₂₃	C₂×#41	1	4	yes	—
130	Dic₂₃	—	2	2	no	—
131	Dic₂₃	—	2	4	no	—
132	Dic₂₃	—	2	2	no	—
133	Dic₂₃	—	2	2	no	—
134	Dic₂₃	C₂×#41	1	4	yes	—
135	#556	—	23	16	no	—
136	C₂×C₄⋊Q₈	—	23	1	no	—
137	C₂×#126	—	23	2	no	—
138	C₂×#126	—	23	2	no	—
139	C₂×#124	—	23	1	no	—
140	C₂×#124	—	23	1	no	—
141	C₂×C₂²⋊Q₈	—	23	2	no	—
142	C₂×C₄×Q₈	—	23	1	no	—
143	C₂×C₄×Q₈	—	23	1	no	—
144	C₂×C₄×Q₈	D₂₃	1	1	yes	characteristic, complemented
145	C₂×C₄×D₄	—	23	1	no	—
146	C₂×#117	—	23	2	no	—
147	C₂×C₂₃	#556	1	4	yes	complemented
148	C₂×C₂₃	C₂²×#41	1	1	yes	characteristic
149	C₂×C₂₃	C₂²×#41	1	1	yes	characteristic
150	C₂×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
151	D₂₃	—	4	4	no	—
152	C₂²×Q₈	—	23	1	no	—
153	C₂²×Q₈	—	23	1	no	—
154	C₂²×Q₈	C₂×D₂₃	1	1	yes	characteristic
155	C₂²×D₄	—	23	1	no	—
156	C₂³×C₄	—	23	2	no	—
157	C₄⋊Q₈	—	23	8	no	—
158	#126	—	23	16	no	—
159	#126	—	23	16	no	—
160	#124	—	23	8	no	—
161	#124	—	23	8	no	—
162	C₂²⋊Q₈	—	23	16	no	—
163	C₄×Q₈	—	23	8	no	—
164	C₄×Q₈	—	23	8	no	—
165	C₄×Q₈	C₂×D₂₃	1	8	yes	complemented
166	C₄×D₄	—	23	8	no	—
167	#117	—	23	16	no	—
168	C₂×C₄⋊C₄	—	23	1	no	—
169	C₂×C₄⋊C₄	—	23	2	no	—
170	C₂×C₄⋊C₄	—	23	2	no	—
171	C₂×C₄⋊C₄	—	23	1	no	—
172	C₂×C₄⋊C₄	—	23	1	no	—
173	C₂×C₄⋊C₄	—	23	2	no	—
174	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
175	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
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	2	no	—
180	C₂×C₂²⋊C₄	—	23	2	no	—
181	C₂×C₄²	—	23	2	no	—
182	C₂×C₄²	—	23	1	no	—
183	C₂×C₄²	—	23	1	no	—
184	C₂×C₄²	C₂×D₂₃	1	2	yes	—
185	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
186	C₂₃	C₂×#556	1	1	yes	characteristic, complemented, Sylow
187	C₂⁴	—	23	2	no	—
188	C₂×Q₈	—	2	4	no	—
189	C₂×Q₈	—	23	4	no	—
190	C₂×Q₈	—	23	4	no	—
191	C₂×Q₈	—	46	4	no	—
192	C₂×Q₈	—	46	4	no	—
193	C₂×Q₈	C₂²×D₂₃	1	4	yes	—
194	C₂×D₄	—	23	4	no	—
195	C₂×D₄	—	46	4	no	—
196	C₂²×C₄	—	23	1	no	—
197	C₂²×C₄	—	23	1	no	—
198	C₂²×C₄	—	23	1	no	—
199	C₂²×C₄	—	23	2	no	—
200	C₂²×C₄	—	23	1	no	—
201	C₂²×C₄	—	23	8	no	—
202	C₂²×C₄	—	46	8	no	—
203	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
204	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
205	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
206	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
208	C₄⋊C₄	—	23	4	no	—
209	C₄⋊C₄	—	23	8	no	—
210	C₄⋊C₄	—	23	8	no	—
211	C₄⋊C₄	—	23	4	no	—
212	C₄⋊C₄	—	23	4	no	—
213	C₄⋊C₄	—	23	8	no	—
214	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
215	C₄⋊C₄	C₂²×D₂₃	1	8	yes	—
216	C₂²⋊C₄	—	23	8	no	—
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	C₄²	—	23	8	no	—
222	C₄²	—	23	4	no	—
223	C₄²	—	23	4	no	—
224	C₄²	C₂²×D₂₃	1	8	yes	—
225	C₄²	C₂²×D₂₃	1	4	yes	—
226	C₂³	—	23	4	no	—
227	C₂³	—	46	4	no	—
228	C₂³	—	46	4	no	—
229	C₂³	—	46	4	no	—
230	C₂³	C₂³×D₂₃	1	1	yes	characteristic
231	Q₈	—	2	8	no	—
232	Q₈	—	46	8	no	—
233	Q₈	—	46	8	no	—
234	D₄	—	46	8	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	23	4	no	—
240	C₂×C₄	—	23	2	no	—
241	C₂×C₄	—	23	2	no	—
242	C₂×C₄	—	23	2	no	—
243	C₂×C₄	—	23	2	no	—
244	C₂×C₄	—	23	4	no	—
245	C₂×C₄	—	46	16	no	—
246	C₂×C₄	—	46	4	no	—
247	C₂×C₄	—	46	2	no	—
248	C₂×C₄	—	46	2	no	—
249	C₂×C₄	—	46	2	no	—
250	C₂×C₄	#67288	1	4	yes	—
251	C₂×C₄	C₂³×D₂₃	1	4	yes	—
252	C₂×C₄	C₂³×D₂₃	1	2	yes	—
253	C₂×C₄	C₂³×D₂₃	1	4	yes	—
254	C₂×C₄	C₂³×D₂₃	1	2	yes	—
255	C₂×C₄	C₂³×D₂₃	1	2	yes	—
256	C₂²	—	46	4	no	—
257	C₂²	—	46	4	no	—
258	C₂²	—	46	4	no	—
259	C₂²	—	92	8	no	—
260	C₂²	C₂×#67288	1	2	yes	—
261	C₂²	#10625158	1	2	yes	—
262	C₂²	#41×D₂₃	1	2	yes	—
263	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
264	C₄	—	2	4	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	4	no	—
267	C₄	—	2	2	no	—
268	C₄	—	23	4	no	—
269	C₄	—	46	2	no	—
270	C₄	—	46	4	no	—
271	C₄	—	46	2	no	—
272	C₄	—	46	2	no	—
273	C₄	C₂×#67288	1	4	yes	—
274	C₂	—	92	4	no	—
275	C₂	#13093139	1	4	yes	direct product factor
276	C₂	C₂×#41×D₂₃	1	1	yes	characteristic
277	C₂	C₂²×#67288	1	1	yes	characteristic
278	C₂	C₂×#10625158	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₂₃
commutator, cosocle	C₂²×C₂₃	C₂⁵
Frattini	C₂²	C₂⁴×D₂₃
Fitting	C₂×C₄×Q₈×C₂₃	C₂
socle	C₂³×C₂₃	C₂⁴
prime radical	C₂²×D₉₂	C₂²
hypercenter	C₂×C₄×Q₈	D₂₃
hypocenter	C₂₃	C₂×#556
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #13093139	C₂	direct (N × H)
2. C₂×#10625115	C₂	semidirect
3. C₂×#10625069	C₂	semidirect
4. C₂×#10625047	C₂	semidirect
7. C₂×#10625067	C₂	semidirect
8. C₂×C₄×Dic₄₆	C₂	semidirect
9. C₂×#10625058	C₂	semidirect
11. C₂×#10625043	C₂	semidirect
12. C₂×C₄×Q₈×C₂₃	C₂	semidirect
18. C₄×Q₈×C₂₃	C₂²	semidirect
24. #10625115	C₂²	semidirect
39. #10625069	C₂²	semidirect
40. #10625067	C₂²	semidirect
42. #10625058	C₂²	semidirect
43. #10625047	C₂²	semidirect
45. #10625043	C₂²	semidirect
46. C₄×Dic₄₆	C₂²	semidirect
144. C₂×C₄×Q₈	D₂₃	semidirect
147. C₂×C₂₃	#556	semidirect
165. C₄×Q₈	C₂×D₂₃	semidirect
186. C₂₃	C₂×#556	semidirect
275. C₂	#13093139	direct (N × H)

Maximal subgroups

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

subgroup	index
#13093139	2
C₂×#10625115	2
C₂×#10625069	2
C₂×#10625047	2
C₂×#10625118	2
C₂×#10625046	2
C₂×#10625067	2
C₂×C₄×Dic₄₆	2
C₂×#10625058	2
C₂×#10625064	2
C₂×#10625043	2
C₂×C₄×Q₈×C₂₃	2
C₂×#556	23

Subgroups — C2×#13093139 #15686945

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13093139 #15686945