Subgroups — C₂×#13093123 #15686927

Order 2944.

1200 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093123	C₂	1	16	yes	maximal, direct product factor
2	C₂×#10625059	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#10625061	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#10625068	C₂	1	2	yes	maximal, complemented
5	C₂×#10625067	C₂	1	2	yes	maximal, complemented
6	C₂×#10625067	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×Dic₄₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₉₂	C₂	1	1	yes	maximal, characteristic
9	C₂×#10625064	C₂	1	1	yes	maximal, characteristic
10	C₂×#10625064	C₂	1	1	yes	maximal, characteristic
11	C₂×C₄⋊C₄×D₂₃	C₂	1	2	yes	maximal, complemented
12	C₂×C₂₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×D₉₂	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₂₃	C₂²	1	2	yes	—
15	C₂²×C₄×D₂₃	C₂²	1	2	yes	—
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×#125	C₂²	1	8	yes	complemented
20	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
24	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
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×C₄⋊Dic₂₃	C₂²	1	2	yes	—
29	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
34	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
35	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
36	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
37	#10625068	C₂²	1	16	yes	complemented
38	#10625067	C₂²	1	16	yes	complemented
39	#10625067	C₂²	1	16	yes	complemented
40	#10625064	C₂²	1	8	yes	—
41	#10625064	C₂²	1	8	yes	—
42	C₄⋊C₄×D₂₃	C₂²	1	16	yes	complemented
43	#10625061	C₂²	1	8	yes	complemented
44	#10625059	C₂²	1	8	yes	complemented
45	C₄×D₉₂	C₂²	1	8	yes	—
46	C₄×Dic₄₆	C₂²	1	8	yes	complemented
47	C₂³×D₂₃	C₂³	1	2	yes	—
48	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
50	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
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	1	yes	characteristic
56	C₂²×Dic₂₃	C₂³	1	2	yes	—
57	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
58	C₂×D₉₂	Q₈	1	8	yes	complemented
59	C₂×D₉₂	C₂³	1	4	yes	—
60	C₂×C₄×D₂₃	—	2	8	no	—
61	C₂×C₄×D₂₃	—	4	4	no	—
62	C₂×C₄×D₂₃	—	4	4	no	—
63	C₂×C₄×D₂₃	C₂³	1	8	yes	—
64	C₂×C₄×D₂₃	C₂³	1	8	yes	—
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	C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
73	C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
74	C₄²×C₂₃	C₂³	1	4	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	—
79	C₄⋊Dic₂₃	C₂³	1	4	yes	—
80	C₄⋊Dic₂₃	C₂³	1	8	yes	—
81	Dic₂₃⋊C₄	C₂³	1	4	yes	—
82	Dic₂₃⋊C₄	C₂³	1	4	yes	—
83	Dic₂₃⋊C₄	C₂³	1	8	yes	—
84	Dic₂₃⋊C₄	C₂³	1	8	yes	—
85	C₄×Dic₂₃	C₂³	1	4	yes	—
86	C₄×Dic₂₃	C₂³	1	4	yes	—
87	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
88	C₂²×D₂₃	—	2	4	no	—
89	C₂²×D₂₃	—	2	4	no	—
90	C₂²×D₂₃	C₂×Q₈	1	8	yes	—
91	C₂²×D₂₃	C₂⁴	1	4	yes	—
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	C₂×Q₈	1	4	yes	—
97	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
98	C₂×C₄×C₂₃	C₂⁴	1	4	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	2	no	—
103	C₂×Dic₂₃	—	2	2	no	—
104	C₂×Dic₂₃	—	2	4	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	C₂×Q₈	1	16	yes	complemented
113	C₄×D₂₃	—	2	16	no	—
114	C₄×D₂₃	—	4	8	no	—
115	C₄×D₂₃	—	4	8	no	—
116	Dic₄₆	—	2	8	no	—
117	Dic₄₆	—	4	8	no	—
118	C₂×#554	—	23	1	no	maximal, Sylow
119	C₂²×C₂₃	C₂²×Q₈	1	2	yes	—
120	C₂²×C₂₃	C₂×#41	1	2	yes	—
121	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
122	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
123	C₂×D₂₃	—	2	16	no	—
124	C₂×D₂₃	—	2	4	no	—
125	C₂×D₂₃	—	2	4	no	—
126	C₂×D₂₃	C₂²×Q₈	1	8	yes	—
127	C₄×C₂₃	—	2	2	no	—
128	C₄×C₂₃	—	2	4	no	—
129	C₄×C₂₃	—	2	4	no	—
130	C₄×C₂₃	—	2	2	no	—
131	C₄×C₂₃	C₂²×Q₈	1	4	yes	—
132	Dic₂₃	—	2	2	no	—
133	Dic₂₃	—	2	2	no	—
134	Dic₂₃	—	2	4	no	—
135	Dic₂₃	—	2	2	no	—
136	Dic₂₃	C₂×#41	1	4	yes	—
137	#554	—	23	16	no	—
138	C₂×C₄⋊Q₈	—	23	1	no	—
139	C₂×#125	—	23	1	no	—
140	C₂×#125	D₂₃	1	1	yes	characteristic, complemented
141	C₂×C₂²⋊Q₈	—	23	2	no	—
142	C₂×C₂²⋊Q₈	—	23	2	no	—
143	C₂×C₂²⋊Q₈	—	23	2	no	—
144	C₂×C₄×Q₈	—	23	1	no	—
145	C₂×C₄×D₄	—	23	1	no	—
146	C₂×C₄×D₄	—	23	1	no	—
147	C₂×C₄×D₄	—	23	1	no	—
148	C₂²×C₄⋊C₄	—	23	2	no	—
149	C₂×C₂₃	#554	1	4	yes	complemented
150	C₂×C₂₃	C₂³×Q₈	1	1	yes	characteristic
151	C₂×C₂₃	C₂²×#41	1	1	yes	characteristic
152	C₂×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
153	D₂₃	—	2	8	no	—
154	C₂²×Q₈	—	23	1	no	—
155	C₂²×Q₈	—	23	2	no	—
156	C₂²×D₄	—	23	1	no	—
157	C₂³×C₄	—	23	2	no	—
158	C₂³×C₄	—	23	2	no	—
159	C₂³×C₄	—	23	2	no	—
160	C₄⋊Q₈	—	23	8	no	—
161	#125	—	23	8	no	—
162	#125	C₂×D₂₃	1	8	yes	complemented
163	C₂²⋊Q₈	—	23	16	no	—
164	C₂²⋊Q₈	—	23	16	no	—
165	C₂²⋊Q₈	—	23	16	no	—
166	C₄×Q₈	—	23	8	no	—
167	C₄×D₄	—	23	8	no	—
168	C₄×D₄	—	23	8	no	—
169	C₄×D₄	—	23	8	no	—
170	C₂×C₄⋊C₄	—	23	1	no	—
171	C₂×C₄⋊C₄	—	23	2	no	—
172	C₂×C₄⋊C₄	—	23	2	no	—
173	C₂×C₄⋊C₄	—	23	1	no	—
174	C₂×C₄⋊C₄	—	23	1	no	—
175	C₂×C₄⋊C₄	—	23	1	no	—
176	C₂×C₄⋊C₄	—	23	2	no	—
177	C₂×C₄⋊C₄	—	23	16	no	—
178	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
179	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
180	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
181	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
182	C₂×C₂²⋊C₄	—	23	2	no	—
183	C₂×C₂²⋊C₄	—	23	2	no	—
184	C₂×C₂²⋊C₄	—	23	2	no	—
185	C₂×C₄²	—	23	1	no	—
186	C₂×C₄²	—	23	1	no	—
187	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
188	C₂₃	C₂×#554	1	1	yes	characteristic, complemented, Sylow
189	C₂⁴	—	23	2	no	—
190	C₂×Q₈	—	23	4	no	—
191	C₂×Q₈	—	23	8	no	—
192	C₂×Q₈	—	46	4	no	—
193	C₂×Q₈	—	92	4	no	—
194	C₂×D₄	—	23	8	no	—
195	C₂×D₄	—	23	4	no	—
196	C₂²×C₄	—	23	1	no	—
197	C₂²×C₄	—	23	1	no	—
198	C₂²×C₄	—	23	1	no	—
199	C₂²×C₄	—	23	1	no	—
200	C₂²×C₄	—	23	2	no	—
201	C₂²×C₄	—	23	8	no	—
202	C₂²×C₄	—	23	8	no	—
203	C₂²×C₄	—	23	8	no	—
204	C₂²×C₄	—	46	8	no	—
205	C₂²×C₄	—	92	4	no	—
206	C₂²×C₄	—	92	4	no	—
207	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
208	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
209	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
210	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
211	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
212	C₄⋊C₄	—	23	4	no	—
213	C₄⋊C₄	—	23	8	no	—
214	C₄⋊C₄	—	23	8	no	—
215	C₄⋊C₄	—	23	4	no	—
216	C₄⋊C₄	—	23	4	no	—
217	C₄⋊C₄	—	23	4	no	—
218	C₄⋊C₄	—	23	8	no	—
219	C₄⋊C₄	C₂²×D₂₃	1	8	yes	—
220	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
221	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
222	C₄⋊C₄	C₂²×D₂₃	1	8	yes	—
223	C₂²⋊C₄	—	23	8	no	—
224	C₂²⋊C₄	—	23	8	no	—
225	C₂²⋊C₄	—	23	8	no	—
226	C₄²	—	23	4	no	—
227	C₄²	—	23	4	no	—
228	C₄²	C₂²×D₂₃	1	4	yes	—
229	C₂³	—	23	8	no	—
230	C₂³	—	23	4	no	—
231	C₂³	—	46	4	no	—
232	C₂³	—	46	4	no	—
233	C₂³	C₂³×D₂₃	1	1	yes	characteristic
234	Q₈	—	46	8	no	—
235	Q₈	—	92	8	no	—
236	D₄	—	23	16	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	23	4	no	—
242	C₂×C₄	—	23	2	no	—
243	C₂×C₄	—	23	2	no	—
244	C₂×C₄	—	23	2	no	—
245	C₂×C₄	—	23	2	no	—
246	C₂×C₄	—	23	4	no	—
247	C₂×C₄	—	46	16	no	—
248	C₂×C₄	—	46	2	no	—
249	C₂×C₄	—	46	2	no	—
250	C₂×C₄	—	46	4	no	—
251	C₂×C₄	—	46	2	no	—
252	C₂×C₄	—	92	8	no	—
253	C₂×C₄	—	92	8	no	—
254	C₂×C₄	Q₈×D₂₃	1	4	yes	—
255	C₂×C₄	C₂³×D₂₃	1	2	yes	—
256	C₂×C₄	C₂³×D₂₃	1	2	yes	—
257	C₂×C₄	C₂³×D₂₃	1	4	yes	—
258	C₂×C₄	C₂³×D₂₃	1	4	yes	—
259	C₂×C₄	C₂³×D₂₃	1	2	yes	—
260	C₂²	—	23	8	no	—
261	C₂²	—	46	16	no	—
262	C₂²	—	46	4	no	—
263	C₂²	—	46	4	no	—
264	C₂²	C₂×Q₈×D₂₃	1	2	yes	—
265	C₂²	#41×D₂₃	1	2	yes	—
266	C₂²	#10625160	1	2	yes	—
267	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
268	C₄	—	2	4	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	4	no	—
272	C₄	—	23	4	no	—
273	C₄	—	46	2	no	—
274	C₄	—	46	2	no	—
275	C₄	—	46	4	no	—
276	C₄	—	46	2	no	—
277	C₄	C₂×Q₈×D₂₃	1	4	yes	—
278	C₂	—	46	8	no	—
279	C₂	#13093123	1	4	yes	direct product factor
280	C₂	C₂²×Q₈×D₂₃	1	1	yes	characteristic
281	C₂	C₂×#41×D₂₃	1	1	yes	characteristic
282	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×#125	C₂
socle	C₂³×C₂₃	C₂⁴
prime radical	C₂²×D₉₂	C₂²
hypercenter	C₂×#125	D₂₃
hypocenter	C₂₃	C₂×#554
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. #13093123	C₂	direct (N × H)
2. C₂×#10625059	C₂	semidirect
3. C₂×#10625061	C₂	semidirect
4. C₂×#10625068	C₂	semidirect
5. C₂×#10625067	C₂	semidirect
6. C₂×#10625067	C₂	semidirect
7. C₂×C₄×Dic₄₆	C₂	semidirect
11. C₂×C₄⋊C₄×D₂₃	C₂	semidirect
12. C₂×C₂₃×#125	C₂	semidirect
19. C₂₃×#125	C₂²	semidirect
37. #10625068	C₂²	semidirect
38. #10625067	C₂²	semidirect
39. #10625067	C₂²	semidirect
42. C₄⋊C₄×D₂₃	C₂²	semidirect
43. #10625061	C₂²	semidirect
44. #10625059	C₂²	semidirect
46. C₄×Dic₄₆	C₂²	semidirect
58. C₂×D₉₂	Q₈	semidirect
112. D₉₂	C₂×Q₈	semidirect
140. C₂×#125	D₂₃	semidirect
149. C₂×C₂₃	#554	semidirect
162. #125	C₂×D₂₃	semidirect
188. C₂₃	C₂×#554	semidirect
279. C₂	#13093123	direct (N × H)

Maximal subgroups

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

subgroup	index
#13093123	2
C₂×#10625059	2
C₂×#10625061	2
C₂×#10625068	2
C₂×#10625067	2
C₂×C₄×Dic₄₆	2
C₂×C₄×D₉₂	2
C₂×#10625064	2
C₂×C₄⋊C₄×D₂₃	2
C₂×C₂₃×#125	2
C₂×#554	23

Subgroups — C2×#13093123 #15686927

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13093123 #15686927