Subgroups — #15683179

Order 2944.

388 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#67288	C₂	1	1	yes	maximal, characteristic
2	#13092721	C₂	1	2	yes	maximal, complemented
3	#13092714	C₂	1	2	yes	maximal, complemented
4	#13092709	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092704	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092699	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092694	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092453	C₂	1	2	yes	maximal, complemented
9	C₂×C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
10	#13092411	C₂	1	2	yes	maximal
11	C₂₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#67288	C₂²	1	1	yes	characteristic
13	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂₃×#125	C₂²	1	2	yes	—
16	C₂₃×C₂²⋊Q₈	—	2	1	no	—
17	C₂₃×C₂²⋊Q₈	—	2	1	no	—
18	C₄×Q₈×C₂₃	—	2	1	no	—
19	C₄×Q₈×C₂₃	—	2	1	no	—
20	C₂₃×#117	C₂²	1	2	yes	—
21	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊D₂₃	—	2	1	no	—
23	C₄×C₂²⋊D₂₃	—	2	1	no	—
24	#10625097	C₂²	1	1	yes	characteristic
25	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
26	C₄×D₉₂	C₂²	1	1	yes	characteristic
27	C₄×D₉₂	C₂²	1	1	yes	characteristic
28	#10625043	—	2	1	no	—
29	C₄²×D₂₃	—	2	1	no	—
30	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
31	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
32	#10624992	C₂²	1	2	yes	—
33	#10624992	C₂²	1	2	yes	—
34	#10624991	C₂²	1	2	yes	—
35	#10624991	C₂²	1	2	yes	—
36	#10624990	C₂²	1	2	yes	—
37	#10624990	C₂²	1	2	yes	—
38	#10624989	C₂²	1	2	yes	—
39	#10624989	C₂²	1	2	yes	—
40	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
41	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
42	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
43	C₂×Q₈×C₂₃	—	2	1	no	—
44	C₂×Q₈×C₂₃	—	2	1	no	—
45	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊D₂₃	—	2	1	no	—
49	#67288	—	2	2	no	—
50	#67288	Q₈	1	4	yes	complemented
51	C₂×D₉₂	C₂³	1	1	yes	characteristic
52	C₂×C₄×D₂₃	—	2	1	no	—
53	C₂×C₄×D₂₃	—	2	1	no	—
54	C₂×C₄×D₂₃	—	2	1	no	—
55	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
56	C₄⋊C₄×C₂₃	—	2	2	no	—
57	C₄⋊C₄×C₂₃	—	2	2	no	—
58	C₄⋊C₄×C₂₃	—	2	1	no	—
59	C₄⋊C₄×C₂₃	—	2	1	no	—
60	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
61	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
62	C₂²⋊C₄×C₂₃	—	2	2	no	—
63	C₄²×C₂₃	—	2	2	no	—
64	C₄²×C₂₃	D₄	1	1	yes	characteristic
65	C₄²×C₂₃	D₄	1	1	yes	characteristic
66	C₄²×C₂₃	C₂³	1	1	yes	characteristic
67	C₄²×C₂₃	C₂³	1	1	yes	characteristic
68	C₂²⋊Dic₂₃	—	2	1	no	—
69	(C₂×D₂₃)⋊C₄	—	2	1	no	—
70	(C₂×D₂₃)⋊C₄	—	2	1	no	—
71	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
73	Dic₂₃⋊C₄	—	2	1	no	—
74	Dic₂₃⋊C₄	—	2	1	no	—
75	C₄×Dic₂₃	—	2	1	no	—
76	C₄×Dic₂₃	—	2	1	no	—
77	C₄×Dic₂₃	—	2	1	no	—
78	C₂×C₂₃⋊C₈	—	2	4	no	—
79	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
80	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
81	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×D₂₃	—	2	1	no	—
83	Q₈×C₂₃	—	4	2	no	—
84	Q₈×C₂₃	—	4	2	no	—
85	C₂×C₄×C₂₃	—	2	2	no	—
86	C₂×C₄×C₂₃	—	2	2	no	—
87	C₂×C₄×C₂₃	—	2	2	no	—
88	C₂×C₄×C₂₃	—	4	1	no	—
89	C₂×C₄×C₂₃	C₂×D₄	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	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
95	C₂×C₄×C₂₃	#41	1	2	yes	—
96	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₂₃	—	2	4	no	—
98	C₂×Dic₂₃	—	2	1	no	—
99	C₂×Dic₂₃	—	2	1	no	—
100	C₂×Dic₂₃	—	2	1	no	—
101	D₉₂	—	2	1	no	—
102	D₉₂	C₂×Q₈	1	2	yes	—
103	C₄×D₂₃	—	2	4	no	—
104	C₄×D₂₃	—	4	2	no	—
105	C₄×D₂₃	—	4	2	no	—
106	Dic₄₆	—	2	1	no	—
107	Dic₄₆	C₂×Q₈	1	2	yes	—
108	C₂₃⋊C₈	—	2	2	no	—
109	C₂₃⋊C₈	—	2	2	no	—
110	#3076	—	23	1	no	maximal, Sylow
111	C₂²×C₂₃	—	2	1	no	—
112	C₂²×C₂₃	—	2	1	no	—
113	C₂²×C₂₃	C₂²⋊Q₈	1	2	yes	—
114	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×D₂₃	—	2	2	no	—
116	C₂×D₂₃	—	4	1	no	—
117	C₂×D₂₃	—	4	1	no	—
118	C₄×C₂₃	—	2	1	no	—
119	C₄×C₂₃	—	2	1	no	—
120	C₄×C₂₃	—	2	1	no	—
121	C₄×C₂₃	—	4	2	no	—
122	C₄×C₂₃	—	4	2	no	—
123	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
124	C₄×C₂₃	#135	1	2	yes	—
125	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
126	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
127	Dic₂₃	—	2	2	no	—
128	Dic₂₃	—	4	1	no	—
129	Dic₂₃	—	4	1	no	—
130	#533	D₂₃	1	1	yes	characteristic, complemented
131	C₄×#41	—	23	1	no	—
132	#479	—	23	2	no	—
133	#478	—	23	2	no	—
134	#477	—	23	1	no	—
135	#476	—	23	1	no	—
136	#475	—	23	1	no	—
137	#474	—	23	1	no	—
138	#427	—	23	2	no	—
139	C₂×C₄⋊C₈	—	23	1	no	—
140	#416	—	23	2	no	—
141	C₂×C₂₃	—	2	2	no	—
142	C₂×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
143	C₂×C₂₃	C₂×#135	1	1	yes	characteristic
144	C₂×C₂₃	#575	1	1	yes	characteristic
145	D₂₃	—	4	2	no	—
146	C₂×#41	—	23	1	no	—
147	C₂²×C₈	—	23	2	no	—
148	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
149	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
150	#125	C₂×D₂₃	1	2	yes	—
151	C₂²⋊Q₈	—	2	1	no	—
152	C₂²⋊Q₈	—	2	1	no	—
153	C₄×Q₈	—	2	1	no	—
154	C₄×Q₈	—	2	1	no	—
155	C₄×Q₈	—	23	1	no	—
156	C₄×Q₈	—	23	1	no	—
157	C₄×D₄	—	23	1	no	—
158	C₄×D₄	—	23	1	no	—
159	C₄×D₄	—	46	1	no	—
160	C₄×D₄	—	46	1	no	—
161	#117	—	23	1	no	—
162	#117	—	46	1	no	—
163	#117	C₂×D₂₃	1	2	yes	—
164	C₂×C₄²	—	46	1	no	—
165	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
166	#107	—	23	2	no	—
167	#107	—	23	2	no	—
168	#106	—	23	2	no	—
169	#106	—	23	2	no	—
170	C₄⋊C₈	—	23	1	no	—
171	C₄⋊C₈	—	23	1	no	—
172	C₄⋊C₈	—	23	2	no	—
173	#103	—	23	2	no	—
174	#103	—	23	2	no	—
175	#102	—	23	2	no	—
176	#102	—	23	2	no	—
177	C₂₃	#3076	1	1	yes	characteristic, complemented, Sylow
178	#41	—	23	4	no	—
179	#41	—	46	2	no	—
180	C₂×Q₈	—	2	1	no	—
181	C₂×Q₈	—	2	1	no	—
182	C₂×Q₈	—	23	1	no	—
183	C₂×D₄	—	23	1	no	—
184	C₂×D₄	—	46	1	no	—
185	C₂²×C₄	—	46	1	no	—
186	C₂²×C₄	—	46	1	no	—
187	C₂²×C₄	—	46	1	no	—
188	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
189	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
190	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
191	C₂×C₈	—	23	2	no	—
192	C₂×C₈	—	23	2	no	—
193	C₂×C₈	—	46	4	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	2	1	no	—
197	C₄⋊C₄	—	2	1	no	—
198	C₄⋊C₄	—	23	1	no	—
199	C₄⋊C₄	—	23	1	no	—
200	C₄⋊C₄	—	46	1	no	—
201	C₄⋊C₄	—	46	1	no	—
202	C₄⋊C₄	C₂²×D₂₃	1	2	yes	—
203	C₄⋊C₄	C₂²×D₂₃	1	2	yes	—
204	C₂²⋊C₄	—	2	2	no	—
205	C₂²⋊C₄	—	46	1	no	—
206	C₂²⋊C₄	—	46	1	no	—
207	C₂²⋊C₄	—	46	1	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	46	1	no	—
210	C₄²	—	46	1	no	—
211	C₄²	—	46	1	no	—
212	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
213	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
214	C₄²	C₂²×D₂₃	1	1	yes	characteristic
215	C₄²	C₂²×D₂₃	1	1	yes	characteristic
216	C₂³	—	46	1	no	—
217	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
218	Q₈	—	4	2	no	—
219	Q₈	—	4	2	no	—
220	Q₈	—	23	2	no	—
221	Q₈	—	46	1	no	—
222	D₄	—	23	2	no	—
223	D₄	—	46	4	no	—
224	D₄	—	46	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	46	1	no	—
230	C₂×C₄	—	46	4	no	—
231	C₂×C₄	—	46	1	no	—
232	C₂×C₄	—	46	1	no	—
233	C₂×C₄	—	92	2	no	—
234	C₂×C₄	—	92	2	no	—
235	C₂×C₄	Q₈×D₂₃	1	2	yes	—
236	C₂×C₄	#67292	1	2	yes	—
237	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
238	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
239	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
240	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
241	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
242	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
243	C₈	—	46	2	no	—
244	C₈	—	46	2	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	46	2	no	—
248	C₂²	—	92	1	no	—
249	C₂²	—	92	1	no	—
250	C₂²	#10625117	1	2	yes	—
251	C₂²	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	4	2	no	—
256	C₄	—	4	2	no	—
257	C₄	—	46	2	no	—
258	C₄	—	92	1	no	—
259	C₄	—	92	1	no	—
260	C₄	#10625117	1	2	yes	—
261	C₄	#10625121	1	2	yes	—
262	C₄	C₂×Q₈×D₂₃	1	1	yes	characteristic
263	C₄	C₂×#67292	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	92	2	no	—
266	C₂	C₂×#10625117	1	1	yes	characteristic
267	C₂	C₂×#10625121	1	1	yes	characteristic
268	C₂	#13093215	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#533	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂×#67288	C₂²
hypercenter	#533	D₂₃
hypocenter	C₂₃	#3076
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

Complements

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

N	H ≅ G/N	type
2. #13092721	C₂	semidirect
3. #13092714	C₂	semidirect
4. #13092709	C₂	semidirect
5. #13092704	C₂	semidirect
6. #13092699	C₂	semidirect
7. #13092694	C₂	semidirect
8. #13092453	C₂	semidirect
9. C₂×C₄⋊(C₂₃⋊C₈)	C₂	semidirect
11. C₂₃×#533	C₂	semidirect
50. #67288	Q₈	semidirect
130. #533	D₂₃	semidirect
177. C₂₃	#3076	semidirect

Maximal subgroups

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

subgroup	index
C₄×#67288	2
#13092721	2
#13092714	2
#13092709	2
#13092704	2
#13092699	2
#13092694	2
#13092453	2
C₂×C₄⋊(C₂₃⋊C₈)	2
#13092411	2
C₂₃×#533	2
#3076	23