Subgroups — C₂×#13092532 #15682285

Order 2944.

472 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625121	C₂	1	1	yes	maximal, characteristic
2	C₂×#10625072	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092532	C₂	1	8	yes	maximal, direct product factor
4	C₂×#10625004	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10625019	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10625007	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×#67288	C₂²	1	1	yes	characteristic
11	C₂₃×#131	—	2	2	no	—
12	C₂₃×#131	C₂²	1	4	yes	complemented
13	C₂×M₁₆×C₂₃	—	2	1	no	—
14	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₂₃	—	2	1	no	—
16	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
17	#10625121	C₄	1	4	yes	complemented
18	#10625121	C₂²	1	4	yes	—
19	C₂×#67275	C₄	1	1	yes	characteristic, complemented
20	C₂×#67274	C₄	1	1	yes	characteristic, complemented
21	C₂×#67273	C₄	1	1	yes	characteristic, complemented
22	C₂×#67272	C₄	1	1	yes	characteristic, complemented
23	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
24	C₂×#67267	C₂²	1	1	yes	characteristic
25	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
26	#10625072	—	2	2	no	—
27	#10625072	C₂²	1	4	yes	complemented
28	C₂×#67262	—	2	1	no	—
29	C₂×C₈×D₂₃	—	2	1	no	—
30	#10625019	C₂²	1	4	yes	complemented
31	#10625007	C₂²	1	4	yes	complemented
32	#10625004	C₂²	1	4	yes	complemented
33	C₈×Dic₂₃	C₂²	1	4	yes	complemented
34	#41×C₂₃	—	2	1	no	—
35	#41×C₂₃	—	2	1	no	—
36	#41×C₂₃	C₂×C₄	1	2	yes	—
37	#41×C₂₃	C₂³	1	2	yes	—
38	C₂×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
39	C₂×D₄×C₂₃	—	2	1	no	—
40	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₂₃	—	2	1	no	—
42	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
43	C₂×C₂²⋊D₂₃	—	2	1	no	—
44	C₂²×Dic₂₃	—	2	1	no	—
45	#67288	—	2	1	no	—
46	#67288	—	2	1	no	—
47	#67288	C₂×C₄	1	2	yes	—
48	#67288	C₂³	1	2	yes	—
49	C₂×D₉₂	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₂₃	—	2	1	no	—
51	C₂×Dic₄₆	C₂×C₄	1	1	yes	characteristic
52	M₁₆×C₂₃	—	2	1	no	—
53	M₁₆×C₂₃	—	2	4	no	—
54	M₁₆×C₂₃	C₂³	1	2	yes	—
55	C₂×C₈×C₂₃	—	2	1	no	—
56	C₂×C₈×C₂₃	—	2	4	no	—
57	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
58	C₂×C₈×C₂₃	D₄	1	2	yes	—
59	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
60	C₂×C₈×C₂₃	C₂³	1	2	yes	—
61	#67275	C₂×C₄	1	4	yes	complemented
62	#67274	C₂×C₄	1	4	yes	complemented
63	#67273	C₂×C₄	1	4	yes	complemented
64	#67272	C₂×C₄	1	4	yes	complemented
65	C₄×Dic₂₃	—	2	1	no	—
66	C₄×Dic₂₃	C₂³	1	2	yes	—
67	#67267	—	2	1	no	—
68	#67267	C₂³	1	2	yes	—
69	C₂×C₂₃⋊C₈	—	2	1	no	—
70	C₂×C₂₃⋊C₈	C₂×C₄	1	1	yes	characteristic
71	C₂×C₂₃⋊C₈	C₂×C₄	1	1	yes	characteristic
72	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
73	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
74	#67262	—	2	4	no	—
75	C₈×D₂₃	—	2	4	no	—
76	C₂³×C₂₃	—	2	1	no	—
77	C₂³×C₂₃	#41	1	1	yes	characteristic
78	C₂²×D₂₃	—	2	1	no	—
79	Q₈×C₂₃	—	2	1	no	—
80	Q₈×C₂₃	C₂²×C₄	1	2	yes	—
81	D₄×C₂₃	—	2	2	no	—
82	D₄×C₂₃	—	2	2	no	—
83	D₄×C₂₃	—	2	1	no	—
84	D₄×C₂₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₂₃	—	2	1	no	—
86	C₂×C₄×C₂₃	—	2	2	no	—
87	C₂×C₄×C₂₃	—	2	2	no	—
88	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
94	C₂²⋊D₂₃	—	2	2	no	—
95	C₂²⋊D₂₃	—	2	2	no	—
96	C₂×Dic₂₃	—	2	4	no	—
97	C₂×Dic₂₃	—	2	1	no	—
98	C₂×Dic₂₃	—	2	2	no	—
99	D₉₂	—	2	1	no	—
100	D₉₂	C₂²×C₄	1	2	yes	—
101	C₄×D₂₃	—	2	2	no	—
102	C₄×D₂₃	—	2	2	no	—
103	Dic₄₆	—	2	1	no	—
104	Dic₄₆	C₂²×C₄	1	2	yes	—
105	C₈×C₂₃	—	2	2	no	—
106	C₈×C₂₃	C₂×D₄	1	2	yes	—
107	C₈×C₂₃	C₂×D₄	1	2	yes	—
108	C₂₃⋊C₈	—	2	2	no	—
109	C₂₃⋊C₈	C₂²×C₄	1	2	yes	—
110	C₂₃⋊C₈	C₂²×C₄	1	2	yes	—
111	C₂×#443	—	23	1	no	maximal, Sylow
112	C₂²×C₂₃	—	2	2	no	—
113	C₂²×C₂₃	—	2	2	no	—
114	C₂²×C₂₃	—	4	2	no	—
115	C₂²×C₂₃	C₄×D₄	1	1	yes	characteristic
116	C₂²×C₂₃	C₄×D₄	1	1	yes	characteristic
117	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
118	C₂×D₂₃	—	2	2	no	—
119	C₂×D₂₃	—	4	2	no	—
120	C₄×C₂₃	—	2	2	no	—
121	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
122	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
123	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
124	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
125	Dic₂₃	—	2	4	no	—
126	Dic₂₃	—	2	2	no	—
127	C₂×#135	—	23	1	no	—
128	C₂×#131	—	23	1	no	—
129	C₂×#131	D₂₃	1	1	yes	characteristic, complemented
130	#443	—	23	8	no	—
131	C₂×#108	—	23	1	no	—
132	C₂×#104	—	23	1	no	—
133	C₂×#104	—	23	1	no	—
134	C₂×C₄×C₈	—	23	1	no	—
135	C₂×C₂₃	—	2	1	no	—
136	C₂×C₂₃	—	2	1	no	—
137	C₂×C₂₃	—	4	2	no	—
138	C₂×C₂₃	#443	1	2	yes	complemented
139	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
140	D₂₃	—	4	2	no	—
141	C₂×#41	—	23	1	no	—
142	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
143	#135	—	23	4	no	—
144	#135	—	23	4	no	—
145	C₂×Q₁₆	—	23	1	no	—
146	C₂×SD₁₆	—	23	1	no	—
147	C₂×SD₁₆	—	23	1	no	—
148	C₂×D₈	—	23	1	no	—
149	#131	—	2	2	no	—
150	#131	—	23	4	no	—
151	#131	—	46	2	no	—
152	#131	C₂×D₂₃	1	4	yes	complemented
153	C₂×M₁₆	—	2	1	no	—
154	C₂×M₁₆	—	23	1	no	—
155	C₂×M₁₆	—	46	1	no	—
156	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
157	C₂²×C₈	—	2	1	no	—
158	C₂²×C₈	—	23	1	no	—
159	C₂²×C₈	—	46	1	no	—
160	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
161	C₂×C₄²	—	23	1	no	—
162	#108	—	23	4	no	—
163	#104	—	23	4	no	—
164	#104	—	23	4	no	—
165	C₄×C₈	—	23	4	no	—
166	C₂₃	C₂×#443	1	1	yes	characteristic, complemented, Sylow
167	#41	—	2	1	no	—
168	#41	—	2	1	no	—
169	#41	—	23	2	no	—
170	#41	—	23	2	no	—
171	#41	—	46	1	no	—
172	#41	—	46	1	no	—
173	#41	C₄×D₂₃	1	2	yes	—
174	#41	C₂²×D₂₃	1	2	yes	—
175	C₂×Q₈	—	23	1	no	—
176	C₂×Q₈	C₄×D₂₃	1	1	yes	characteristic
177	C₂×D₄	—	2	1	no	—
178	C₂×D₄	—	23	1	no	—
179	C₂×D₄	—	46	1	no	—
180	C₂×D₄	C₄×D₂₃	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	46	1	no	—
183	C₂²×C₄	—	46	1	no	—
184	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
185	Q₁₆	—	23	4	no	—
186	SD₁₆	—	23	4	no	—
187	SD₁₆	—	23	4	no	—
188	D₈	—	23	4	no	—
189	M₁₆	—	2	1	no	—
190	M₁₆	—	2	4	no	—
191	M₁₆	—	23	2	no	—
192	M₁₆	—	46	1	no	—
193	M₁₆	—	46	4	no	—
194	M₁₆	C₂²×D₂₃	1	2	yes	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	23	1	no	—
198	C₂×C₈	—	23	1	no	—
199	C₂×C₈	—	23	2	no	—
200	C₂×C₈	—	23	2	no	—
201	C₂×C₈	—	46	1	no	—
202	C₂×C₈	—	46	4	no	—
203	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊D₂₃	1	2	yes	—
205	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
206	C₂×C₈	C₂²×D₂₃	1	2	yes	—
207	C₄²	—	23	2	no	—
208	C₄²	—	46	1	no	—
209	C₂³	—	2	1	no	—
210	C₂³	—	46	1	no	—
211	C₂³	#67288	1	1	yes	characteristic
212	Q₈	—	2	1	no	—
213	Q₈	—	23	2	no	—
214	Q₈	—	46	1	no	—
215	Q₈	C₂×C₄×D₂₃	1	2	yes	—
216	D₄	—	2	2	no	—
217	D₄	—	2	2	no	—
218	D₄	—	2	1	no	—
219	D₄	—	23	2	no	—
220	D₄	—	46	2	no	—
221	D₄	—	46	2	no	—
222	D₄	—	46	1	no	—
223	D₄	C₂×C₄×D₂₃	1	2	yes	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	46	4	no	—
228	C₂×C₄	—	46	1	no	—
229	C₂×C₄	—	46	2	no	—
230	C₂×C₄	—	46	2	no	—
231	C₂×C₄	—	46	2	no	—
232	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
234	C₂×C₄	#67288	1	1	yes	characteristic
235	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
237	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	23	2	no	—
240	C₈	—	23	2	no	—
241	C₈	—	46	2	no	—
242	C₈	C₂×C₂²⋊D₂₃	1	2	yes	—
243	C₈	C₂×C₂²⋊D₂₃	1	2	yes	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	46	2	no	—
248	C₂²	—	92	2	no	—
249	C₂²	C₄×C₂²⋊D₂₃	1	1	yes	characteristic
250	C₂²	C₄×C₂²⋊D₂₃	1	1	yes	characteristic
251	C₂²	C₂×#67288	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	46	4	no	—
254	C₄	—	46	2	no	—
255	C₄	C₄×C₂²⋊D₂₃	1	1	yes	characteristic
256	C₄	C₄×C₂²⋊D₂₃	1	1	yes	characteristic
257	C₄	C₂²×C₄×D₂₃	1	1	yes	characteristic
258	C₄	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	4	2	no	—
262	C₂	—	92	2	no	—
263	C₂	#13092532	1	2	yes	direct product factor
264	C₂	C₂×C₄×C₂²⋊D₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊D₂₃
commutator	C₄×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂×C₂₃×#131	C₂
socle	C₂²×C₂₃	C₄×D₄
prime radical	C₂×#10625121	C₂
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₂×#131	D₂₃
hypocenter	C₂₃	C₂×#443
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. C₂×#10625072	C₂	semidirect
3. #13092532	C₂	direct (N × H)
4. C₂×#10625004	C₂	semidirect
5. C₂×#10625019	C₂	semidirect
6. C₂×#10625007	C₂	semidirect
7. C₂×C₈×Dic₂₃	C₂	semidirect
8. C₂×C₂₃×#131	C₂	semidirect
12. C₂₃×#131	C₂²	semidirect
17. #10625121	C₄	semidirect
19. C₂×#67275	C₄	semidirect
20. C₂×#67274	C₄	semidirect
21. C₂×#67273	C₄	semidirect
22. C₂×#67272	C₄	semidirect
27. #10625072	C₂²	semidirect
30. #10625019	C₂²	semidirect
31. #10625007	C₂²	semidirect
32. #10625004	C₂²	semidirect
33. C₈×Dic₂₃	C₂²	semidirect
61. #67275	C₂×C₄	semidirect
62. #67274	C₂×C₄	semidirect
63. #67273	C₂×C₄	semidirect
64. #67272	C₂×C₄	semidirect
129. C₂×#131	D₂₃	semidirect
138. C₂×C₂₃	#443	semidirect
152. #131	C₂×D₂₃	semidirect
166. C₂₃	C₂×#443	semidirect
263. C₂	#13092532	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#10625121	2
C₂×#10625072	2
#13092532	2
C₂×#10625004	2
C₂×#10625019	2
C₂×#10625007	2
C₂×C₈×Dic₂₃	2
C₂×C₂₃×#131	2
C₂×#443	23

Subgroups — C2×#13092532 #15682285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13092532 #15682285