Subgroups — #15686979

Order 2944.

1068 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₄₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#13093143	C₂	1	2	yes	maximal, complemented
3	#13093127	C₂	1	3	yes	maximal, complemented
4	#13093123	C₂	1	6	yes	maximal, complemented
5	#13093082	C₂	1	3	yes	maximal, complemented
6	#13092973	C₂	1	6	yes	maximal, complemented
7	#13092972	C₂	1	3	yes	maximal, complemented
8	C₄×#67292	C₂	1	1	yes	maximal, characteristic
9	C₄×#67292	C₂	1	2	yes	maximal
10	C₄×Q₈×D₂₃	C₂	1	3	yes	maximal, complemented
11	C₂₃×#557	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#67292	C₂²	1	1	yes	characteristic
13	C₂×Q₈×D₂₃	C₂²	1	3	yes	—
14	C₂₃×C₄⋊Q₈	C₂²	1	3	yes	—
15	C₂₃×#125	C₂²	1	6	yes	—
16	C₄×Q₈×C₂₃	C₂²	1	3	yes	—
17	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
18	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
19	#10625117	C₂²	1	6	yes	—
20	Q₈×Dic₂₃	C₂²	1	3	yes	—
21	Q₈×Dic₂₃	C₂²	1	1	yes	characteristic
22	Q₈×Dic₂₃	C₂²	1	2	yes	—
23	#10625068	C₂²	1	6	yes	—
24	#10625067	C₂²	1	12	yes	—
25	#10625067	C₂²	1	12	yes	—
26	#10625064	C₂²	1	3	yes	—
27	#10625064	C₂²	1	6	yes	—
28	#10625063	C₂²	1	6	yes	—
29	#10625063	C₂²	1	3	yes	—
30	C₄⋊C₄×D₂₃	C₂²	1	6	yes	—
31	C₄⋊C₄×D₂₃	C₂²	1	3	yes	—
32	#10625061	C₂²	1	6	yes	—
33	#10625059	C₂²	1	6	yes	—
34	#10625059	C₂²	1	6	yes	—
35	#10625058	C₂²	1	1	yes	characteristic
36	#10625058	C₂²	1	6	yes	—
37	#10625058	C₂²	1	6	yes	—
38	C₄×D₉₂	C₂²	1	6	yes	—
39	C₄×D₉₂	C₂²	1	3	yes	—
40	C₄²×D₂₃	C₂²	1	3	yes	—
41	C₄²×D₂₃	C₂²	1	6	yes	—
42	#10625041	C₂²	1	6	yes	—
43	#10625040	C₂²	1	3	yes	—
44	C₄×Dic₄₆	C₂²	1	6	yes	—
45	C₄×Dic₄₆	C₂²	1	3	yes	—
46	C₂×Q₈×C₂₃	C₂³	1	3	yes	—
47	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
48	#67292	Q₈	1	8	yes	complemented
49	Q₈×D₂₃	—	2	12	no	—
50	C₂×D₉₂	C₂³	1	3	yes	—
51	C₂×C₄×D₂₃	C₂³	1	3	yes	—
52	C₂×C₄×D₂₃	C₂³	1	6	yes	—
53	C₂×C₄×D₂₃	C₂³	1	6	yes	—
54	C₂×C₄×D₂₃	C₂³	1	3	yes	—
55	C₂×C₄×D₂₃	C₂³	1	3	yes	—
56	C₂×Dic₄₆	C₂³	1	6	yes	—
57	C₂×Dic₄₆	C₂³	1	6	yes	—
58	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
59	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₂₃	C₂³	1	6	yes	—
61	C₄⋊C₄×C₂₃	C₂³	1	6	yes	—
62	C₄⋊C₄×C₂₃	C₂³	1	3	yes	—
63	C₄⋊C₄×C₂₃	C₂³	1	6	yes	—
64	C₄²×C₂₃	C₂³	1	3	yes	—
65	C₄²×C₂₃	C₂³	1	6	yes	—
66	(C₂×D₂₃)⋊C₄	C₂³	1	6	yes	—
67	(C₂×D₂₃)⋊C₄	C₂³	1	12	yes	—
68	C₄⋊Dic₂₃	C₂³	1	3	yes	—
69	C₄⋊Dic₂₃	C₂³	1	6	yes	—
70	C₄⋊Dic₂₃	C₂³	1	6	yes	—
71	C₄⋊Dic₂₃	C₂³	1	3	yes	—
72	Dic₂₃⋊C₄	C₂³	1	2	yes	—
73	Dic₂₃⋊C₄	C₂³	1	6	yes	—
74	Dic₂₃⋊C₄	C₂³	1	6	yes	—
75	Dic₂₃⋊C₄	C₂³	1	12	yes	—
76	C₄×Dic₂₃	C₂³	1	6	yes	—
77	C₄×Dic₂₃	C₂³	1	3	yes	—
78	C₄×Dic₂₃	C₂³	1	2	yes	—
79	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
80	C₄×Dic₂₃	C₂³	1	3	yes	—
81	C₄×Dic₂₃	C₂³	1	6	yes	—
82	C₂²×D₂₃	C₂⁴	1	3	yes	—
83	Q₈×C₂₃	—	2	6	no	—
84	Q₈×C₂₃	C₂×Q₈	1	4	yes	complemented
85	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
86	C₂×C₄×C₂₃	C₂⁴	1	6	yes	—
87	C₂×C₄×C₂₃	C₂⁴	1	3	yes	—
88	C₂×C₄×C₂₃	C₂⁴	1	3	yes	—
89	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
90	C₂×Dic₂₃	C₂⁴	1	1	yes	characteristic
91	C₂×Dic₂₃	C₂⁴	1	3	yes	—
92	C₂×Dic₂₃	C₂⁴	1	6	yes	—
93	C₂×Dic₂₃	C₂⁴	1	1	yes	characteristic
94	C₂×Dic₂₃	C₂⁴	1	2	yes	—
95	D₉₂	C₂×Q₈	1	12	yes	complemented
96	C₄×D₂₃	—	2	6	no	—
97	C₄×D₂₃	—	2	12	no	—
98	C₄×D₂₃	—	4	6	no	—
99	C₄×D₂₃	—	4	3	no	—
100	C₄×D₂₃	C₂×Q₈	1	12	yes	complemented
101	Dic₄₆	—	2	12	no	—
102	Dic₄₆	—	4	6	no	—
103	Dic₄₆	#41	1	4	yes	complemented
104	Q₈×#41	—	23	1	no	maximal, Sylow
105	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
106	C₂×D₂₃	—	2	3	no	—
107	C₂×D₂₃	—	2	3	no	—
108	C₂×D₂₃	C₂²×Q₈	1	6	yes	—
109	C₄×C₂₃	—	2	6	no	—
110	C₄×C₂₃	—	2	3	no	—
111	C₄×C₂₃	—	2	2	no	—
112	C₄×C₂₃	C₂²×Q₈	1	6	yes	—
113	C₄×C₂₃	C₂×#41	1	2	yes	—
114	Dic₂₃	—	2	3	no	—
115	Dic₂₃	—	2	6	no	—
116	Dic₂₃	—	2	1	no	—
117	Dic₂₃	C₂²×Q₈	1	2	yes	—
118	Dic₂₃	C₂×#41	1	4	yes	—
119	Q₈²	—	23	1	no	—
120	#557	—	23	2	no	—
121	#557	D₂₃	1	1	yes	characteristic, complemented
122	#554	—	23	6	no	—
123	#554	—	23	3	no	—
124	D₄×Q₈	—	23	3	no	—
125	#533	—	23	3	no	—
126	#533	—	23	6	no	—
127	C₄×#41	—	23	2	no	—
128	C₄×#41	—	23	1	no	—
129	C₂×C₄×Q₈	—	23	3	no	—
130	C₂×C₂₃	C₂³×Q₈	1	1	yes	characteristic
131	C₂×C₂₃	C₂²×#41	1	1	yes	characteristic
132	C₂×C₂₃	#585	1	1	yes	characteristic
133	D₂₃	—	2	6	no	—
134	C₂×#41	—	23	1	no	—
135	C₂²×Q₈	—	23	3	no	—
136	C₄⋊Q₈	—	23	6	no	—
137	C₄⋊Q₈	—	23	6	no	—
138	C₄⋊Q₈	—	23	3	no	—
139	C₄⋊Q₈	C₂×D₂₃	1	3	yes	—
140	#125	—	23	6	no	—
141	#125	—	23	6	no	—
142	#125	C₂×D₂₃	1	6	yes	—
143	C₂²⋊Q₈	—	23	6	no	—
144	C₂²⋊Q₈	—	23	12	no	—
145	C₂²⋊Q₈	—	23	12	no	—
146	C₂²⋊Q₈	—	23	6	no	—
147	C₄×Q₈	—	23	3	no	—
148	C₄×Q₈	—	23	1	no	—
149	C₄×Q₈	—	23	2	no	—
150	C₄×Q₈	—	23	1	no	—
151	C₄×Q₈	—	23	6	no	—
152	C₄×Q₈	—	23	6	no	—
153	C₄×Q₈	—	23	3	no	—
154	C₄×Q₈	—	23	6	no	—
155	C₄×Q₈	C₂×D₂₃	1	3	yes	—
156	C₄×Q₈	C₂×D₂₃	1	2	yes	—
157	C₄×Q₈	C₂×D₂₃	1	1	yes	characteristic
158	C₄×D₄	—	23	6	no	—
159	C₄×D₄	—	23	3	no	—
160	C₄×D₄	—	23	3	no	—
161	C₄×D₄	—	23	6	no	—
162	#117	—	23	6	no	—
163	#117	—	23	3	no	—
164	C₂×C₄⋊C₄	—	23	6	no	—
165	C₂×C₄⋊C₄	—	23	3	no	—
166	C₂×C₄²	—	23	3	no	—
167	C₂×C₄²	—	23	6	no	—
168	C₂₃	Q₈×#41	1	1	yes	characteristic, complemented, Sylow
169	#41	—	23	8	no	—
170	C₂×Q₈	—	23	1	no	—
171	C₂×Q₈	—	23	6	no	—
172	C₂×Q₈	—	23	6	no	—
173	C₂×Q₈	—	46	12	no	—
174	C₂×Q₈	C₂²×D₂₃	1	3	yes	—
175	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
176	C₂×D₄	—	23	3	no	—
177	C₂²×C₄	—	23	3	no	—
178	C₂²×C₄	—	23	6	no	—
179	C₂²×C₄	—	23	3	no	—
180	C₂²×C₄	—	23	3	no	—
181	C₂²×C₄	—	23	6	no	—
182	C₄⋊C₄	—	23	2	no	—
183	C₄⋊C₄	—	23	6	no	—
184	C₄⋊C₄	—	23	6	no	—
185	C₄⋊C₄	—	23	6	no	—
186	C₄⋊C₄	—	23	3	no	—
187	C₄⋊C₄	—	23	3	no	—
188	C₄⋊C₄	—	23	6	no	—
189	C₄⋊C₄	—	23	12	no	—
190	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
191	C₄⋊C₄	C₂²×D₂₃	1	6	yes	—
192	C₄⋊C₄	C₂²×D₂₃	1	6	yes	—
193	C₄⋊C₄	C₂²×D₂₃	1	3	yes	—
194	C₄⋊C₄	C₂²×D₂₃	1	6	yes	—
195	C₂²⋊C₄	—	23	12	no	—
196	C₂²⋊C₄	—	23	6	no	—
197	C₄²	—	23	2	no	—
198	C₄²	—	23	1	no	—
199	C₄²	—	23	6	no	—
200	C₄²	—	23	3	no	—
201	C₄²	—	23	3	no	—
202	C₄²	—	23	6	no	—
203	C₄²	C₂²×D₂₃	1	6	yes	—
204	C₄²	C₂²×D₂₃	1	3	yes	—
205	C₂³	—	23	3	no	—
206	Q₈	—	2	6	no	—
207	Q₈	—	23	4	no	—
208	Q₈	—	46	12	no	—
209	Q₈	—	92	6	no	—
210	Q₈	Q₈×D₂₃	1	4	yes	complemented
211	D₄	—	23	12	no	—
212	C₂×C₄	—	23	1	no	—
213	C₂×C₄	—	23	2	no	—
214	C₂×C₄	—	23	12	no	—
215	C₂×C₄	—	23	1	no	—
216	C₂×C₄	—	23	3	no	—
217	C₂×C₄	—	23	6	no	—
218	C₂×C₄	—	46	12	no	—
219	C₂×C₄	—	46	6	no	—
220	C₂×C₄	—	92	6	no	—
221	C₂×C₄	—	92	3	no	—
222	C₂×C₄	C₂³×D₂₃	1	2	yes	—
223	C₂×C₄	C₂³×D₂₃	1	6	yes	—
224	C₂×C₄	C₂³×D₂₃	1	3	yes	—
225	C₂×C₄	C₂³×D₂₃	1	3	yes	—
226	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
227	C₂²	—	23	6	no	—
228	C₂²	—	46	3	no	—
229	C₂²	—	46	3	no	—
230	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
231	C₄	—	2	6	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	3	no	—
234	C₄	—	23	2	no	—
235	C₄	—	23	4	no	—
236	C₄	—	46	3	no	—
237	C₄	—	46	1	no	—
238	C₄	—	46	6	no	—
239	C₄	C₂×Q₈×D₂₃	1	6	yes	—
240	C₄	#41×D₂₃	1	2	yes	—
241	C₂	—	46	6	no	—
242	C₂	C₂²×Q₈×D₂₃	1	1	yes	characteristic
243	C₂	C₂×#41×D₂₃	1	1	yes	characteristic
244	C₂	#13093254	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁵
Fitting	C₂₃×#557	C₂
prime radical	C₂×#67292	C₂²
hypercenter	#557	D₂₃
hypocenter	C₂₃	Q₈×#41
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #557

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₄₆	C₂	semidirect
2. #13093143	C₂	semidirect
3. #13093127	C₂	semidirect
4. #13093123	C₂	semidirect
5. #13093082	C₂	semidirect
6. #13092973	C₂	semidirect
7. #13092972	C₂	semidirect
10. C₄×Q₈×D₂₃	C₂	semidirect
11. C₂₃×#557	C₂	semidirect
48. #67292	Q₈	semidirect
84. Q₈×C₂₃	C₂×Q₈	semidirect
95. D₉₂	C₂×Q₈	semidirect
100. C₄×D₂₃	C₂×Q₈	semidirect
103. Dic₄₆	#41	semidirect
121. #557	D₂₃	semidirect
168. C₂₃	Q₈×#41	semidirect
210. Q₈	Q₈×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₄₆	2
#13093143	2
#13093127	2
#13093123	2
#13093082	2
#13092973	2
#13092972	2
C₄×#67292	2
C₄×Q₈×D₂₃	2
C₂₃×#557	2
Q₈×#41	23