Subgroups — #15686787

Order 2944.

1040 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093038	C₂	1	4	yes	maximal, complemented
2	#13093029	C₂	1	2	yes	maximal, complemented
3	#13093022	C₂	1	2	yes	maximal, complemented
4	#13093014	C₂	1	4	yes	maximal, complemented
5	#13093005	C₂	1	4	yes	maximal, complemented
6	#13092974	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092967	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092945	C₂	1	8	yes	maximal, complemented
9	#13092928	C₂	1	2	yes	maximal, complemented
10	C₄×#67288	C₂	1	2	yes	maximal, complemented
11	C₂×C₂₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×#67288	C₂²	1	2	yes	—
14	C₂₃×#117	C₂²	1	4	yes	—
15	C₂₃×#117	C₂²	1	2	yes	—
16	C₂₃×#117	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
21	#10625124	C₂²	1	4	yes	—
22	#10625101	C₂²	1	4	yes	—
23	#10625101	C₂²	1	4	yes	—
24	#10625100	C₂²	1	4	yes	—
25	C₄×C₂²⋊D₂₃	C₂²	1	8	yes	—
26	C₄×C₂²⋊D₂₃	C₂²	1	8	yes	—
27	#10625097	C₂²	1	2	yes	—
28	#10625095	C₂²	1	4	yes	—
29	#10625095	C₂²	1	4	yes	—
30	#10625069	C₂²	1	8	yes	complemented
31	#10625067	C₂²	1	8	yes	complemented
32	#10625065	C₂²	1	8	yes	complemented
33	#10625060	C₂²	1	8	yes	complemented
34	#10625056	C₂²	1	8	yes	—
35	#10625055	C₂²	1	8	yes	—
36	#10625054	C₂²	1	8	yes	—
37	#10625050	C₂²	1	8	yes	—
38	#10625046	C₂²	1	2	yes	complemented
39	#10625045	C₂²	1	1	yes	characteristic, complemented
40	C₄×D₉₂	C₂²	1	4	yes	complemented
41	C₄×D₉₂	C₂²	1	4	yes	complemented
42	#10625043	C₂²	1	4	yes	complemented
43	C₄²×D₂₃	C₂²	1	4	yes	complemented
44	#10625041	C₂²	1	2	yes	complemented
45	#10625040	C₂²	1	1	yes	characteristic, complemented
46	#10625040	C₂²	1	2	yes	complemented
47	C₄×Dic₄₆	C₂²	1	4	yes	complemented
48	C₄×Dic₄₆	C₂²	1	4	yes	complemented
49	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	—	2	4	no	—
51	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
52	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
53	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
55	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
56	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
57	C₂×C₂²⋊D₂₃	C₂³	1	4	yes	—
58	C₂×C₂²⋊D₂₃	C₂³	1	8	yes	—
59	#67288	—	2	8	no	—
60	C₂×D₉₂	C₂³	1	2	yes	—
61	C₂×D₉₂	C₂³	1	4	yes	—
62	C₂×C₄×D₂₃	C₂³	1	4	yes	—
63	C₂×C₄×D₂₃	C₂³	1	8	yes	—
64	C₂×Dic₄₆	C₂³	1	2	yes	—
65	C₂×Dic₄₆	C₂³	1	4	yes	—
66	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
67	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
68	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
69	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
70	C₄²×C₂₃	C₂³	1	2	yes	—
71	C₄²×C₂₃	C₂³	1	2	yes	—
72	C₄²×C₂₃	C₂³	1	1	yes	characteristic
73	C₄²×C₂₃	C₂³	1	1	yes	characteristic
74	C₄²×C₂₃	C₂³	1	2	yes	—
75	C₂²⋊Dic₂₃	C₂³	1	8	yes	—
76	C₂²⋊Dic₂₃	C₂³	1	4	yes	—
77	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
78	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
79	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
80	C₄⋊Dic₂₃	C₂³	1	4	yes	—
81	C₄⋊Dic₂₃	C₂³	1	4	yes	—
82	C₄⋊Dic₂₃	C₂³	1	4	yes	—
83	Dic₂₃⋊C₄	C₂³	1	8	yes	—
84	Dic₂₃⋊C₄	C₂³	1	8	yes	—
85	Dic₂₃⋊C₄	C₂³	1	8	yes	—
86	C₄×Dic₂₃	C₂³	1	4	yes	—
87	C₄×Dic₂₃	C₂³	1	8	yes	—
88	C₂³×C₂₃	—	2	2	no	—
89	C₂³×C₂₃	—	4	2	no	—
90	C₂³×C₂₃	C₂⁴	1	2	yes	—
91	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
92	C₂²×D₂₃	C₂⁴	1	4	yes	—
93	C₂×C₄×C₂₃	—	2	2	no	—
94	C₂×C₄×C₂₃	—	2	4	no	—
95	C₂×C₄×C₂₃	—	2	4	no	—
96	C₂×C₄×C₂₃	—	4	2	no	—
97	C₂×C₄×C₂₃	—	4	2	no	—
98	C₂×C₄×C₂₃	#41	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×C₄×C₂₃	C₂⁴	1	2	yes	—
103	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
105	C₂²⋊D₂₃	—	2	8	no	—
106	C₂²⋊D₂₃	—	4	8	no	—
107	C₂×Dic₂₃	C₂⁴	1	4	yes	—
108	C₂×Dic₂₃	C₂⁴	1	8	yes	—
109	D₉₂	—	2	4	no	—
110	D₉₂	—	4	4	no	—
111	C₄×D₂₃	—	2	8	no	—
112	C₄×D₂₃	—	4	8	no	—
113	Dic₄₆	—	2	4	no	—
114	Dic₄₆	—	4	4	no	—
115	#3455	—	23	1	no	maximal, Sylow
116	C₂²×C₂₃	—	2	2	no	—
117	C₂²×C₂₃	—	2	2	no	—
118	C₂²×C₂₃	—	2	4	no	—
119	C₂²×C₂₃	—	4	4	no	—
120	C₂²×C₂₃	C₂×#41	1	2	yes	—
121	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
122	C₂×D₂₃	—	2	4	no	—
123	C₂×D₂₃	—	2	8	no	—
124	C₄×C₂₃	—	2	2	no	—
125	C₄×C₂₃	—	2	2	no	—
126	C₄×C₂₃	—	2	4	no	—
127	C₄×C₂₃	—	2	2	no	—
128	C₄×C₂₃	C₂×#41	1	2	yes	—
129	C₄×C₂₃	C₂×#41	1	2	yes	—
130	Dic₂₃	—	2	4	no	—
131	Dic₂₃	—	2	8	no	—
132	#542	—	23	4	no	—
133	#541	—	23	2	no	—
134	#540	—	23	2	no	—
135	#539	—	23	4	no	—
136	#538	—	23	4	no	—
137	#533	—	23	1	no	—
138	#532	—	23	1	no	—
139	#529	—	23	8	no	—
140	#525	—	23	2	no	—
141	C₄×#41	—	23	2	no	—
142	C₂×#117	D₂₃	1	1	yes	characteristic, complemented
143	C₂×C₂₃	—	2	2	no	—
144	C₂×C₂₃	—	4	2	no	—
145	C₂×C₂₃	C₂²×#41	1	1	yes	characteristic
146	C₂×C₂₃	#585	1	2	yes	—
147	D₂₃	—	4	4	no	—
148	C₂×#41	—	23	2	no	—
149	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
150	C₄⋊Q₈	—	23	1	no	—
151	C₄⋊Q₈	—	23	2	no	—
152	#127	—	23	1	no	—
153	#126	—	23	8	no	—
154	#126	—	23	8	no	—
155	#125	—	23	8	no	—
156	#125	—	23	2	no	—
157	#124	—	23	8	no	—
158	#124	—	23	2	no	—
159	#123	—	23	4	no	—
160	#123	—	23	8	no	—
161	#123	—	23	8	no	—
162	C₂²⋊Q₈	—	23	4	no	—
163	C₂²⋊Q₈	—	23	4	no	—
164	C₂²⋊Q₈	—	23	8	no	—
165	#121	—	23	4	no	—
166	#121	—	23	4	no	—
167	#121	—	23	8	no	—
168	#120	—	23	4	no	—
169	C₄×Q₈	—	23	4	no	—
170	C₄×Q₈	—	23	4	no	—
171	C₄×D₄	—	23	8	no	—
172	C₄×D₄	—	23	8	no	—
173	C₄×D₄	—	23	4	no	—
174	C₄×D₄	—	23	4	no	—
175	#117	—	23	2	no	—
176	#117	—	23	4	no	—
177	#117	C₂×D₂₃	1	4	yes	—
178	#117	C₂×D₂₃	1	2	yes	—
179	#117	C₂×D₂₃	1	2	yes	—
180	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
181	C₂×C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
182	C₂×C₄²	—	23	4	no	—
183	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
184	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
185	C₂₃	#3455	1	1	yes	characteristic, complemented, Sylow
186	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
187	#41	—	46	8	no	—
188	C₂×Q₈	—	23	2	no	—
189	C₂×Q₈	—	23	4	no	—
190	C₂×D₄	—	23	4	no	—
191	C₂×D₄	—	23	8	no	—
192	C₂×D₄	—	23	2	no	—
193	C₂×D₄	—	23	4	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	23	4	no	—
196	C₂²×C₄	—	23	8	no	—
197	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
198	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
199	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
200	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
201	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
202	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
203	C₄⋊C₄	—	23	4	no	—
204	C₄⋊C₄	—	23	8	no	—
205	C₄⋊C₄	—	23	4	no	—
206	C₄⋊C₄	—	23	8	no	—
207	C₄⋊C₄	—	23	8	no	—
208	C₄⋊C₄	—	23	4	no	—
209	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
210	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
211	C₂²⋊C₄	—	23	4	no	—
212	C₂²⋊C₄	—	23	8	no	—
213	C₂²⋊C₄	—	23	8	no	—
214	C₂²⋊C₄	—	23	8	no	—
215	C₂²⋊C₄	—	23	8	no	—
216	C₂²⋊C₄	C₂²×D₂₃	1	4	yes	—
217	C₂²⋊C₄	C₂²×D₂₃	1	4	yes	—
218	C₄²	—	23	8	no	—
219	C₄²	—	23	4	no	—
220	C₄²	C₂²×D₂₃	1	2	yes	—
221	C₄²	C₂²×D₂₃	1	1	yes	characteristic
222	C₄²	C₂²×D₂₃	1	1	yes	characteristic
223	C₄²	C₂²×D₂₃	1	2	yes	—
224	C₄²	C₂²×D₂₃	1	2	yes	—
225	C₂³	—	2	2	no	—
226	C₂³	—	4	2	no	—
227	C₂³	—	23	4	no	—
228	C₂³	C₂³×D₂₃	1	2	yes	—
229	C₂³	C₂³×D₂₃	1	1	yes	characteristic
230	Q₈	—	46	4	no	—
231	Q₈	—	92	4	no	—
232	D₄	—	46	8	no	—
233	D₄	—	46	4	no	—
234	D₄	—	92	4	no	—
235	D₄	—	92	8	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	23	4	no	—
242	C₂×C₄	—	23	8	no	—
243	C₂×C₄	—	46	8	no	—
244	C₂×C₄	—	92	8	no	—
245	C₂×C₄	#67288	1	4	yes	—
246	C₂×C₄	C₂³×D₂₃	1	4	yes	—
247	C₂×C₄	C₂³×D₂₃	1	2	yes	—
248	C₂×C₄	C₂³×D₂₃	1	2	yes	—
249	C₂×C₄	C₂³×D₂₃	1	2	yes	—
250	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
251	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	46	4	no	—
257	C₂²	—	46	8	no	—
258	C₂²	C₂×#67288	1	2	yes	—
259	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	4	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	46	4	no	—
265	C₄	—	46	8	no	—
266	C₄	C₂×#67288	1	2	yes	—
267	C₄	C₂×#67288	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	4	2	no	—
270	C₂	—	92	4	no	—
271	C₂	C₂²×#67288	1	1	yes	characteristic
272	C₂	#13093253	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁵
Frattini	C₂²	C₂⁴×D₂₃
Fitting	C₂×C₂₃×#117	C₂
hypercenter	C₂×#117	D₂₃
hypocenter	C₂₃	#3455
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #13093038	C₂	semidirect
2. #13093029	C₂	semidirect
3. #13093022	C₂	semidirect
4. #13093014	C₂	semidirect
5. #13093005	C₂	semidirect
6. #13092974	C₂	semidirect
7. #13092967	C₂	semidirect
8. #13092945	C₂	semidirect
9. #13092928	C₂	semidirect
10. C₄×#67288	C₂	semidirect
11. C₂×C₂₃×#117	C₂	semidirect
30. #10625069	C₂²	semidirect
31. #10625067	C₂²	semidirect
32. #10625065	C₂²	semidirect
33. #10625060	C₂²	semidirect
38. #10625046	C₂²	semidirect
39. #10625045	C₂²	semidirect
40. C₄×D₉₂	C₂²	semidirect
41. C₄×D₉₂	C₂²	semidirect
42. #10625043	C₂²	semidirect
43. C₄²×D₂₃	C₂²	semidirect
44. #10625041	C₂²	semidirect
45. #10625040	C₂²	semidirect
46. #10625040	C₂²	semidirect
47. C₄×Dic₄₆	C₂²	semidirect
48. C₄×Dic₄₆	C₂²	semidirect
142. C₂×#117	D₂₃	semidirect
185. C₂₃	#3455	semidirect

Maximal subgroups

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

subgroup	index
#13093038	2
#13093029	2
#13093022	2
#13093014	2
#13093005	2
#13092974	2
#13092967	2
#13092945	2
#13092928	2
C₄×#67288	2
C₂×C₂₃×#117	2
#3455	23