Subgroups — C₂×#13093107 #15686909

Order 2944.

1124 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093107	C₂	1	16	yes	maximal, direct product factor
2	C₂×#10625050	C₂	1	2	yes	maximal, complemented
3	C₂×#10625041	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#10625060	C₂	1	2	yes	maximal, complemented
5	C₂×#10625106	C₂	1	2	yes	maximal
6	C₂×#10625095	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#10625095	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×Dic₄₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#10625097	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10625048	C₂	1	2	yes	maximal, complemented
11	C₂²×C₄⋊Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
13	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂³×Dic₂₃	C₂²	1	2	yes	—
17	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₂₃	C₂²	1	8	yes	—
19	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊Dic₂₃	C₂²	1	2	yes	—
24	C₂×C₂²⋊Dic₂₃	C₂²	1	2	yes	—
25	C₂×C₂²⋊Dic₂₃	C₂²	1	2	yes	—
26	#10625106	C₂²	1	16	yes	—
27	#10625097	C₂²	1	8	yes	complemented
28	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
31	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊Dic₂₃	C₂²	1	8	yes	complemented
33	#10625095	C₂²	1	8	yes	complemented
34	#10625095	C₂²	1	8	yes	complemented
35	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
36	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
37	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
38	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
39	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
40	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
41	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
42	#10625060	C₂²	1	16	yes	complemented
43	#10625050	C₂²	1	16	yes	complemented
44	#10625048	C₂²	1	16	yes	complemented
45	#10625041	C₂²	1	8	yes	complemented
46	C₄×Dic₄₆	C₂²	1	8	yes	complemented
47	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
48	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
49	C₂×D₄×C₂₃	—	2	4	no	—
50	C₂×D₄×C₂₃	C₂³	1	4	yes	—
51	C₂²×C₄×C₂₃	—	2	4	no	—
52	C₂²×C₄×C₂₃	—	2	4	no	—
53	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
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	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
58	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
59	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
60	C₂²×Dic₂₃	—	2	8	no	—
61	C₂²×Dic₂₃	C₂³	1	8	yes	—
62	C₂²×Dic₂₃	C₂³	1	2	yes	—
63	C₂²×Dic₂₃	C₂³	1	2	yes	—
64	C₂²×Dic₂₃	C₂³	1	2	yes	—
65	C₂²×Dic₂₃	C₂³	1	2	yes	—
66	C₂×Dic₄₆	—	2	4	no	—
67	C₂×Dic₄₆	C₂³	1	4	yes	—
68	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
69	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
70	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
71	C₄²×C₂₃	C₂³	1	4	yes	—
72	C₂²⋊Dic₂₃	C₂³	1	8	yes	—
73	C₂²⋊Dic₂₃	C₂³	1	8	yes	—
74	C₂²⋊Dic₂₃	C₂³	1	8	yes	—
75	C₄⋊Dic₂₃	C₂³	1	4	yes	—
76	C₄⋊Dic₂₃	C₂³	1	4	yes	—
77	C₄⋊Dic₂₃	C₂³	1	4	yes	—
78	C₄⋊Dic₂₃	C₂³	1	8	yes	—
79	Dic₂₃⋊C₄	C₂³	1	8	yes	—
80	Dic₂₃⋊C₄	C₂³	1	8	yes	—
81	Dic₂₃⋊C₄	C₂³	1	8	yes	—
82	Dic₂₃⋊C₄	C₂³	1	8	yes	—
83	Dic₂₃⋊C₄	C₂³	1	8	yes	—
84	C₄×Dic₂₃	C₂³	1	8	yes	—
85	C₄×Dic₂₃	C₂³	1	8	yes	—
86	C₂³×C₂₃	—	2	2	no	—
87	C₂³×C₂₃	—	2	2	no	—
88	C₂³×C₂₃	—	2	2	no	—
89	C₂³×C₂₃	—	2	2	no	—
90	C₂³×C₂₃	—	2	2	no	—
91	C₂³×C₂₃	#41	1	4	yes	—
92	C₂³×C₂₃	C₂⁴	1	2	yes	—
93	C₂³×C₂₃	C₂⁴	1	2	yes	—
94	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
95	D₄×C₂₃	—	2	8	no	—
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	8	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×Dic₂₃	—	2	4	no	—
109	C₂×Dic₂₃	—	2	16	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	C₂×Dic₂₃	C₂⁴	1	4	yes	—
117	Dic₄₆	—	2	8	no	—
118	C₂×#552	—	23	1	no	maximal, Sylow
119	C₂²×C₂₃	—	2	8	no	—
120	C₂²×C₂₃	—	2	2	no	—
121	C₂²×C₂₃	—	2	2	no	—
122	C₂²×C₂₃	—	2	2	no	—
123	C₂²×C₂₃	—	2	2	no	—
124	C₂²×C₂₃	—	2	2	no	—
125	C₂²×C₂₃	—	4	4	no	—
126	C₂²×C₂₃	C₂×#41	1	4	yes	—
127	C₂²×C₂₃	C₂×#41	1	2	yes	—
128	C₂²×C₂₃	C₂×#41	1	2	yes	—
129	C₂²×C₂₃	2₋¹⁺⁴	1	2	yes	—
130	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
131	C₄×C₂₃	—	2	2	no	—
132	C₄×C₂₃	—	2	2	no	—
133	C₄×C₂₃	—	2	2	no	—
134	C₄×C₂₃	—	2	2	no	—
135	C₄×C₂₃	C₂×#41	1	4	yes	—
136	Dic₂₃	—	2	4	no	—
137	Dic₂₃	—	2	4	no	—
138	Dic₂₃	—	2	4	no	—
139	Dic₂₃	—	2	4	no	—
140	#552	—	23	16	no	—
141	C₂×#126	—	23	2	no	—
142	C₂×#125	—	23	2	no	—
143	C₂×#125	—	23	1	no	—
144	C₂×#123	—	23	2	no	—
145	C₂×C₂²⋊Q₈	—	23	1	no	—
146	C₂×C₂²⋊Q₈	—	23	1	no	—
147	C₂×C₄×Q₈	—	23	1	no	—
148	C₂×C₄×D₄	D₂₃	1	1	yes	characteristic
149	C₂×#117	—	23	1	no	—
150	C₂×#117	—	23	2	no	—
151	C₂²×C₄⋊C₄	—	23	1	no	—
152	C₂×C₂₃	—	2	4	no	—
153	C₂×C₂₃	—	4	2	no	—
154	C₂×C₂₃	#552	1	4	yes	complemented
155	C₂×C₂₃	C₂²×#41	1	1	yes	characteristic
156	C₂×C₂₃	C₂²×#41	1	1	yes	characteristic
157	C₂×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
158	C₂²×Q₈	—	23	1	no	—
159	C₂²×D₄	C₂×D₂₃	1	1	yes	characteristic
160	C₂³×C₄	—	23	2	no	—
161	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
162	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
163	#126	—	23	16	no	—
164	#125	—	23	16	no	—
165	#125	—	23	8	no	—
166	#123	—	23	16	no	—
167	C₂²⋊Q₈	—	23	8	no	—
168	C₂²⋊Q₈	—	23	8	no	—
169	C₄×Q₈	—	23	8	no	—
170	C₄×D₄	C₂×D₂₃	1	8	yes	—
171	#117	—	23	8	no	—
172	#117	—	23	16	no	—
173	C₂×C₄⋊C₄	—	23	1	no	—
174	C₂×C₄⋊C₄	—	23	1	no	—
175	C₂×C₄⋊C₄	—	23	2	no	—
176	C₂×C₄⋊C₄	—	23	8	no	—
177	C₂×C₄⋊C₄	—	23	2	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	—	23	2	no	—
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	C₂×D₂₃	1	1	yes	characteristic
188	C₂×C₂²⋊C₄	C₂×D₂₃	1	1	yes	characteristic
189	C₂×C₄²	—	23	2	no	—
190	C₂×C₄²	—	23	2	no	—
191	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
192	C₂₃	C₂×#552	1	1	yes	characteristic, complemented, Sylow
193	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
194	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
195	C₂×Q₈	—	23	4	no	—
196	C₂×Q₈	—	46	4	no	—
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	C₂²×D₂₃	1	4	yes	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	23	8	no	—
202	C₂²×C₄	—	23	2	no	—
203	C₂²×C₄	—	23	2	no	—
204	C₂²×C₄	—	23	2	no	—
205	C₂²×C₄	—	23	2	no	—
206	C₂²×C₄	—	46	8	no	—
207	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
208	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
209	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
210	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
211	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
212	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
213	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
214	C₄⋊C₄	—	23	4	no	—
215	C₄⋊C₄	—	23	4	no	—
216	C₄⋊C₄	—	23	8	no	—
217	C₄⋊C₄	—	23	8	no	—
218	C₄⋊C₄	—	23	4	no	—
219	C₄⋊C₄	—	23	8	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	C₂²⋊C₄	—	23	8	no	—
225	C₂²⋊C₄	—	23	8	no	—
226	C₂²⋊C₄	—	23	8	no	—
227	C₂²⋊C₄	C₂²×D₂₃	1	4	yes	—
228	C₂²⋊C₄	C₂²×D₂₃	1	4	yes	—
229	C₄²	—	23	8	no	—
230	C₄²	—	23	8	no	—
231	C₄²	C₂²×D₂₃	1	4	yes	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	2	2	no	—
237	C₂³	#67288	1	4	yes	—
238	C₂³	C₂³×D₂₃	1	2	yes	—
239	C₂³	C₂³×D₂₃	1	2	yes	—
240	C₂³	C₂³×D₂₃	1	1	yes	characteristic
241	Q₈	—	46	8	no	—
242	D₄	—	2	8	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	2	8	no	—
249	C₂×C₄	—	23	4	no	—
250	C₂×C₄	—	23	4	no	—
251	C₂×C₄	—	23	4	no	—
252	C₂×C₄	—	23	4	no	—
253	C₂×C₄	—	46	16	no	—
254	C₂×C₄	—	46	4	no	—
255	C₂×C₄	—	46	4	no	—
256	C₂×C₄	—	46	4	no	—
257	C₂×C₄	—	46	4	no	—
258	C₂×C₄	#67290	1	4	yes	—
259	C₂×C₄	C₂³×D₂₃	1	2	yes	—
260	C₂×C₄	C₂³×D₂₃	1	2	yes	—
261	C₂×C₄	C₂³×D₂₃	1	2	yes	—
262	C₂×C₄	C₂³×D₂₃	1	2	yes	—
263	C₂×C₄	C₂³×D₂₃	1	2	yes	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	8	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	4	4	no	—
271	C₂²	C₂×#67288	1	4	yes	—
272	C₂²	C₂×#67288	1	2	yes	—
273	C₂²	C₂×#67290	1	2	yes	—
274	C₂²	#10625161	1	2	yes	—
275	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	46	4	no	—
281	C₄	—	46	4	no	—
282	C₄	—	46	4	no	—
283	C₄	—	46	4	no	—
284	C₄	C₂×#67290	1	4	yes	—
285	C₂	—	2	4	no	—
286	C₂	—	4	2	no	—
287	C₂	#13093107	1	4	yes	direct product factor
288	C₂	C₂²×#67290	1	1	yes	characteristic
289	C₂	C₂²×#67288	1	1	yes	characteristic
290	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×C₄×D₄×C₂₃	C₂
socle	C₂³×C₂₃	C₂⁴
prime radical	C₂²×D₄×C₂₃	C₂²
hypercenter	C₂×C₄×D₄	D₂₃
hypocenter	C₂₃	C₂×#552
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #13093107	C₂	direct (N × H)
2. C₂×#10625050	C₂	semidirect
3. C₂×#10625041	C₂	semidirect
4. C₂×#10625060	C₂	semidirect
6. C₂×#10625095	C₂	semidirect
7. C₂×#10625095	C₂	semidirect
8. C₂×C₄×Dic₄₆	C₂	semidirect
9. C₂×#10625097	C₂	semidirect
10. C₂×#10625048	C₂	semidirect
11. C₂²×C₄⋊Dic₂₃	C₂	semidirect
27. #10625097	C₂²	semidirect
32. C₂×C₄⋊Dic₂₃	C₂²	semidirect
33. #10625095	C₂²	semidirect
34. #10625095	C₂²	semidirect
42. #10625060	C₂²	semidirect
43. #10625050	C₂²	semidirect
44. #10625048	C₂²	semidirect
45. #10625041	C₂²	semidirect
46. C₄×Dic₄₆	C₂²	semidirect
154. C₂×C₂₃	#552	semidirect
192. C₂₃	C₂×#552	semidirect
287. C₂	#13093107	direct (N × H)

Maximal subgroups

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

subgroup	index
#13093107	2
C₂×#10625050	2
C₂×#10625041	2
C₂×#10625060	2
C₂×#10625106	2
C₂×#10625095	2
C₂×C₄×Dic₄₆	2
C₂×#10625097	2
C₂×#10625048	2
C₂²×C₄⋊Dic₂₃	2
C₂×C₄×D₄×C₂₃	2
C₂×#552	23

Subgroups — C2×#13093107 #15686909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13093107 #15686909