Subgroups — C₂×#13093047 #15686797

Order 2944.

1080 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	#13093047	C₂	1	16	yes	maximal, direct product factor
2	C₂×#10625040	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#10625059	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#10625059	C₂	1	2	yes	maximal, complemented
5	C₂×#10625061	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10625060	C₂	1	2	yes	maximal, complemented
7	C₂×#10625068	C₂	1	2	yes	maximal
8	C₂×#10625067	C₂	1	2	yes	maximal
9	C₂×#10625063	C₂	1	1	yes	maximal, characteristic
10	C₂×#10625043	C₂	1	1	yes	maximal, characteristic
11	C₂×C₄⋊C₄×D₂₃	C₂	1	1	yes	maximal, characteristic
12	C₂×C₂₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×C₄×D₂₃	C₂²	1	2	yes	—
14	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
15	C₂²×Dic₄₆	C₂²	1	2	yes	—
16	C₂²×Dic₄₆	C₂²	1	2	yes	—
17	C₂₃×#125	C₂²	1	8	yes	complemented
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
26	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
29	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
30	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
31	C₂×Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
32	C₂×Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
33	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
34	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
35	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
36	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
37	#10625068	C₂²	1	16	yes	—
38	#10625067	C₂²	1	16	yes	—
39	#10625063	C₂²	1	8	yes	—
40	C₄⋊C₄×D₂₃	C₂²	1	8	yes	—
41	#10625061	C₂²	1	8	yes	complemented
42	#10625060	C₂²	1	16	yes	complemented
43	#10625059	C₂²	1	8	yes	complemented
44	#10625059	C₂²	1	16	yes	complemented
45	#10625043	C₂²	1	8	yes	—
46	#10625040	C₂²	1	8	yes	complemented
47	C₂³×D₂₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
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×Dic₂₃	C₂³	1	1	yes	characteristic
54	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
55	C₂²×Dic₂₃	C₂³	1	2	yes	—
56	C₂²×Dic₂₃	C₂³	1	2	yes	—
57	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
58	C₂×C₄×D₂₃	—	4	4	no	—
59	C₂×C₄×D₂₃	Q₈	1	8	yes	complemented
60	C₂×C₄×D₂₃	C₂³	1	8	yes	—
61	C₂×C₄×D₂₃	C₂³	1	4	yes	—
62	C₂×Dic₄₆	—	4	4	no	—
63	C₂×Dic₄₆	—	4	4	no	—
64	C₂×Dic₄₆	C₂³	1	8	yes	—
65	C₂×Dic₄₆	C₂³	1	8	yes	—
66	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
67	C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
68	C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
69	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
70	C₄²×C₂₃	C₂³	1	4	yes	—
71	(C₂×D₂₃)⋊C₄	C₂³	1	4	yes	—
72	(C₂×D₂₃)⋊C₄	C₂³	1	4	yes	—
73	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
74	C₄⋊Dic₂₃	C₂³	1	4	yes	—
75	C₄⋊Dic₂₃	C₂³	1	8	yes	—
76	C₄⋊Dic₂₃	C₂³	1	4	yes	—
77	C₄⋊Dic₂₃	C₂³	1	8	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	8	yes	—
85	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
86	C₂²×D₂₃	—	2	2	no	—
87	C₂²×D₂₃	—	2	2	no	—
88	C₂²×D₂₃	C₂×Q₈	1	4	yes	—
89	C₂²×D₂₃	C₂⁴	1	2	yes	—
90	C₂×C₄×C₂₃	—	2	2	no	—
91	C₂×C₄×C₂₃	—	2	4	no	—
92	C₂×C₄×C₂₃	—	2	4	no	—
93	C₂×C₄×C₂₃	—	2	2	no	—
94	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	—
95	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
96	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
97	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
98	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
99	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
100	C₂×Dic₂₃	—	2	2	no	—
101	C₂×Dic₂₃	—	2	4	no	—
102	C₂×Dic₂₃	—	2	4	no	—
103	C₂×Dic₂₃	—	2	2	no	—
104	C₂×Dic₂₃	C₂×Q₈	1	4	yes	—
105	C₂×Dic₂₃	C₂⁴	1	2	yes	—
106	C₂×Dic₂₃	C₂⁴	1	2	yes	—
107	C₂×Dic₂₃	C₂⁴	1	4	yes	—
108	C₂×Dic₂₃	C₂⁴	1	4	yes	—
109	C₂×Dic₂₃	C₂⁴	1	2	yes	—
110	C₄×D₂₃	—	4	8	no	—
111	C₄×D₂₃	C₂×Q₈	1	16	yes	complemented
112	Dic₄₆	—	4	8	no	—
113	Dic₄₆	—	4	8	no	—
114	C₂×#544	—	23	1	no	maximal, Sylow
115	C₂²×C₂₃	C₂²×Q₈	1	2	yes	—
116	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
117	C₂²×C₂₃	2₋¹⁺⁴	1	2	yes	—
118	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
119	C₂×D₂₃	—	2	8	no	—
120	C₂×D₂₃	—	2	2	no	—
121	C₂×D₂₃	—	2	2	no	—
122	C₂×D₂₃	C₂²×Q₈	1	4	yes	—
123	C₄×C₂₃	—	2	2	no	—
124	C₄×C₂₃	—	2	4	no	—
125	C₄×C₂₃	—	2	4	no	—
126	C₄×C₂₃	—	2	2	no	—
127	C₄×C₂₃	C₂²×Q₈	1	4	yes	—
128	Dic₂₃	—	2	2	no	—
129	Dic₂₃	—	2	4	no	—
130	Dic₂₃	—	2	4	no	—
131	Dic₂₃	—	2	2	no	—
132	Dic₂₃	C₂²×Q₈	1	4	yes	—
133	#544	—	23	16	no	—
134	C₂×C₄⋊Q₈	—	23	2	no	—
135	C₂×C₄⋊Q₈	—	23	1	no	—
136	C₂×C₄⋊Q₈	—	23	1	no	—
137	C₂×#125	—	23	2	no	—
138	C₂×#125	—	23	1	no	—
139	C₂×#125	D₂₃	1	1	yes	characteristic, complemented
140	C₂×C₂²⋊Q₈	—	23	2	no	—
141	C₂×C₂²⋊Q₈	—	23	2	no	—
142	C₂×#117	—	23	1	no	—
143	C₂×#117	—	23	1	no	—
144	C₂²×C₄⋊C₄	—	23	1	no	—
145	C₂×C₂₃	#544	1	4	yes	complemented
146	C₂×C₂₃	C₂³×Q₈	1	1	yes	characteristic
147	C₂×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
148	C₂×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
149	D₂₃	—	2	4	no	—
150	C₂²×Q₈	—	23	2	no	—
151	C₂²×Q₈	—	23	2	no	—
152	C₂³×C₄	—	23	1	no	—
153	C₂³×C₄	—	23	2	no	—
154	C₄⋊Q₈	—	23	16	no	—
155	C₄⋊Q₈	—	23	8	no	—
156	C₄⋊Q₈	—	23	8	no	—
157	#125	—	23	16	no	—
158	#125	—	23	8	no	—
159	#125	C₂×D₂₃	1	8	yes	complemented
160	C₂²⋊Q₈	—	23	16	no	—
161	C₂²⋊Q₈	—	23	16	no	—
162	#117	—	23	8	no	—
163	#117	—	23	8	no	—
164	C₂×C₄⋊C₄	—	23	2	no	—
165	C₂×C₄⋊C₄	—	23	1	no	—
166	C₂×C₄⋊C₄	—	23	1	no	—
167	C₂×C₄⋊C₄	—	23	1	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	2	no	—
172	C₂×C₄⋊C₄	—	23	2	no	—
173	C₂×C₄⋊C₄	—	23	8	no	—
174	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
175	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
176	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
177	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
178	C₂×C₂²⋊C₄	—	23	1	no	—
179	C₂×C₂²⋊C₄	—	23	2	no	—
180	C₂×C₂²⋊C₄	—	23	1	no	—
181	C₂×C₄²	—	23	2	no	—
182	C₂×C₄²	—	23	1	no	—
183	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
184	C₂₃	C₂×#544	1	1	yes	characteristic, complemented, Sylow
185	C₂⁴	—	23	1	no	—
186	C₂×Q₈	—	23	8	no	—
187	C₂×Q₈	—	23	8	no	—
188	C₂×Q₈	—	92	4	no	—
189	C₂×Q₈	—	92	4	no	—
190	C₂²×C₄	—	23	1	no	—
191	C₂²×C₄	—	23	8	no	—
192	C₂²×C₄	—	23	4	no	—
193	C₂²×C₄	—	23	1	no	—
194	C₂²×C₄	—	23	1	no	—
195	C₂²×C₄	—	23	2	no	—
196	C₂²×C₄	—	23	2	no	—
197	C₂²×C₄	—	23	8	no	—
198	C₂²×C₄	—	92	4	no	—
199	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
200	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
201	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
202	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
203	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
204	C₄⋊C₄	—	23	8	no	—
205	C₄⋊C₄	—	23	4	no	—
206	C₄⋊C₄	—	23	4	no	—
207	C₄⋊C₄	—	23	4	no	—
208	C₄⋊C₄	—	23	4	no	—
209	C₄⋊C₄	—	23	8	no	—
210	C₄⋊C₄	—	23	8	no	—
211	C₄⋊C₄	—	23	8	no	—
212	C₄⋊C₄	—	23	8	no	—
213	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
214	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
215	C₄⋊C₄	C₂²×D₂₃	1	8	yes	—
216	C₄⋊C₄	C₂²×D₂₃	1	8	yes	—
217	C₂²⋊C₄	—	23	4	no	—
218	C₂²⋊C₄	—	23	8	no	—
219	C₂²⋊C₄	—	23	4	no	—
220	C₄²	—	23	8	no	—
221	C₄²	—	23	4	no	—
222	C₄²	C₂²×D₂₃	1	4	yes	—
223	C₂³	—	23	4	no	—
224	C₂³	—	23	2	no	—
225	C₂³	—	46	2	no	—
226	C₂³	—	46	2	no	—
227	C₂³	C₂³×D₂₃	1	1	yes	characteristic
228	Q₈	—	92	8	no	—
229	Q₈	—	92	8	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	23	4	no	—
235	C₂×C₄	—	23	2	no	—
236	C₂×C₄	—	23	16	no	—
237	C₂×C₄	—	23	2	no	—
238	C₂×C₄	—	23	2	no	—
239	C₂×C₄	—	23	4	no	—
240	C₂×C₄	—	23	4	no	—
241	C₂×C₄	—	46	4	no	—
242	C₂×C₄	—	46	2	no	—
243	C₂×C₄	—	46	4	no	—
244	C₂×C₄	—	46	2	no	—
245	C₂×C₄	—	92	8	no	—
246	C₂×C₄	Q₈×D₂₃	1	4	yes	—
247	C₂×C₄	C₂³×D₂₃	1	4	yes	—
248	C₂×C₄	C₂³×D₂₃	1	2	yes	—
249	C₂×C₄	C₂³×D₂₃	1	2	yes	—
250	C₂×C₄	C₂³×D₂₃	1	4	yes	—
251	C₂×C₄	C₂³×D₂₃	1	2	yes	—
252	C₂²	—	23	4	no	—
253	C₂²	—	46	2	no	—
254	C₂²	—	46	2	no	—
255	C₂²	—	46	8	no	—
256	C₂²	C₂×Q₈×D₂₃	1	2	yes	—
257	C₂²	#10625160	1	2	yes	—
258	C₂²	#10625161	1	2	yes	—
259	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
260	C₄	—	2	4	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	4	no	—
263	C₄	—	2	2	no	—
264	C₄	—	23	4	no	—
265	C₄	—	46	4	no	—
266	C₄	—	46	2	no	—
267	C₄	—	46	4	no	—
268	C₄	—	46	2	no	—
269	C₄	C₂×Q₈×D₂₃	1	4	yes	—
270	C₂	—	46	4	no	—
271	C₂	#13093047	1	4	yes	direct product factor
272	C₂	C₂²×Q₈×D₂₃	1	1	yes	characteristic
273	C₂	C₂×#10625160	1	1	yes	characteristic
274	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×#125	C₂
socle	C₂³×C₂₃	C₂⁴
prime radical	C₂³×D₂₃	C₂³
hypercenter	C₂×#125	D₂₃
hypocenter	C₂₃	C₂×#544
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×#125

Complements

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

N	H ≅ G/N	type
1. #13093047	C₂	direct (N × H)
2. C₂×#10625040	C₂	semidirect
3. C₂×#10625059	C₂	semidirect
4. C₂×#10625059	C₂	semidirect
5. C₂×#10625061	C₂	semidirect
6. C₂×#10625060	C₂	semidirect
12. C₂×C₂₃×#125	C₂	semidirect
17. C₂₃×#125	C₂²	semidirect
41. #10625061	C₂²	semidirect
42. #10625060	C₂²	semidirect
43. #10625059	C₂²	semidirect
44. #10625059	C₂²	semidirect
46. #10625040	C₂²	semidirect
59. C₂×C₄×D₂₃	Q₈	semidirect
111. C₄×D₂₃	C₂×Q₈	semidirect
139. C₂×#125	D₂₃	semidirect
145. C₂×C₂₃	#544	semidirect
159. #125	C₂×D₂₃	semidirect
184. C₂₃	C₂×#544	semidirect
271. C₂	#13093047	direct (N × H)

Maximal subgroups

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

subgroup	index
#13093047	2
C₂×#10625040	2
C₂×#10625059	2
C₂×#10625061	2
C₂×#10625060	2
C₂×#10625068	2
C₂×#10625067	2
C₂×#10625063	2
C₂×#10625043	2
C₂×C₄⋊C₄×D₂₃	2
C₂×C₂₃×#125	2
C₂×#544	23

Subgroups — C2×#13093047 #15686797

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13093047 #15686797