Subgroups — #15683117

Order 2944.

452 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	#13093215	C₂	1	2	yes	maximal, complemented
2	C₂×#10625075	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092692	C₂	1	2	yes	maximal, complemented
4	#13092687	C₂	1	2	yes	maximal, complemented
5	#13092680	C₂	1	4	yes	maximal, complemented
6	C₂×#10625001	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092399	C₂	1	2	yes	maximal, 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	2	yes	—
11	C₂₃×#131	—	2	1	no	—
12	C₂₃×#131	—	2	1	no	—
13	C₂₃×#131	C₂²	1	2	yes	—
14	C₂₃×#131	C₂²	1	2	yes	—
15	C₂×M₁₆×C₂₃	—	2	1	no	—
16	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₂₃	—	2	1	no	—
18	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	#10625122	C₂²	1	4	yes	—
20	#10625121	—	2	4	no	—
21	#10625120	C₂²	1	4	yes	—
22	#10625113	—	2	2	no	—
23	#10625103	—	2	2	no	—
24	C₂×#67267	C₂²	1	2	yes	—
25	C₂×Dic₉₂	C₂²	1	1	yes	characteristic, complemented
26	#10625075	—	2	1	no	—
27	#10625075	—	2	1	no	—
28	#10625075	—	2	1	no	—
29	#10625075	—	2	1	no	—
30	C₂×D₁₈₄	C₂²	1	1	yes	characteristic, complemented
31	C₂×#67263	C₂²	1	2	yes	complemented
32	#10625006	C₂²	1	4	yes	complemented
33	#10625005	C₂²	1	4	yes	complemented
34	#10625001	C₂²	1	2	yes	complemented
35	#10625001	C₂²	1	2	yes	complemented
36	#41×C₂₃	—	2	1	no	—
37	#41×C₂₃	—	2	1	no	—
38	#41×C₂₃	D₄	1	2	yes	—
39	#41×C₂₃	C₂³	1	2	yes	—
40	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
41	C₂×D₄×C₂₃	—	2	1	no	—
42	C₂×D₄×C₂₃	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	—	2	1	no	—
44	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
45	C₂×C₂²⋊D₂₃	—	2	2	no	—
46	#67288	—	2	2	no	—
47	#67288	—	2	2	no	—
48	#67288	—	2	2	no	—
49	#67288	—	2	2	no	—
50	C₂×D₉₂	C₂³	1	2	yes	—
51	C₂×C₄×D₂₃	—	2	2	no	—
52	C₂×Dic₄₆	C₂³	1	2	yes	—
53	M₁₆×C₂₃	—	2	1	no	—
54	M₁₆×C₂₃	—	2	2	no	—
55	M₁₆×C₂₃	—	2	2	no	—
56	M₁₆×C₂₃	C₂³	1	2	yes	—
57	C₂×C₈×C₂₃	—	2	1	no	—
58	C₂×C₈×C₂₃	—	2	2	no	—
59	C₂×C₈×C₂₃	—	2	2	no	—
60	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
61	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
62	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
63	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
64	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
66	#67275	—	2	4	no	—
67	#67274	—	2	4	no	—
68	#67273	—	2	4	no	—
69	#67272	—	2	4	no	—
70	#67267	—	2	2	no	—
71	#67267	C₂³	1	4	yes	—
72	C₂×C₂₃⋊C₈	—	2	2	no	—
73	Dic₉₂	—	2	1	no	—
74	Dic₉₂	—	2	1	no	—
75	D₁₈₄	—	2	1	no	—
76	D₁₈₄	—	2	1	no	—
77	#67263	—	2	2	no	—
78	#67263	—	2	2	no	—
79	C₂³×C₂₃	—	2	1	no	—
80	C₂³×C₂₃	#41	1	1	yes	characteristic
81	C₂²×D₂₃	—	2	2	no	—
82	Q₈×C₂₃	—	2	1	no	—
83	Q₈×C₂₃	C₂×D₄	1	2	yes	—
84	D₄×C₂₃	—	2	2	no	—
85	D₄×C₂₃	—	2	2	no	—
86	D₄×C₂₃	—	2	1	no	—
87	D₄×C₂₃	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₂₃	—	2	1	no	—
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	—	2	2	no	—
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×D₄	1	1	yes	characteristic
95	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
96	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₂₃	—	4	2	no	—
98	C₂²⋊D₂₃	—	4	2	no	—
99	C₂×Dic₂₃	—	2	2	no	—
100	D₉₂	—	2	2	no	—
101	D₉₂	—	2	2	no	—
102	C₄×D₂₃	—	4	2	no	—
103	C₄×D₂₃	—	4	2	no	—
104	Dic₄₆	—	2	2	no	—
105	Dic₄₆	—	2	2	no	—
106	C₈×C₂₃	—	2	2	no	—
107	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
108	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
109	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
110	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
111	C₂₃⋊C₈	—	2	4	no	—
112	#3067	—	23	1	no	maximal, Sylow
113	C₂²×C₂₃	—	2	2	no	—
114	C₂²×C₂₃	—	4	1	no	—
115	C₂²×C₂₃	—	4	2	no	—
116	C₂²×C₂₃	#121	1	1	yes	characteristic
117	C₂²×C₂₃	#121	1	1	yes	characteristic
118	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
119	C₂×D₂₃	—	4	2	no	—
120	C₂×D₂₃	—	8	2	no	—
121	C₄×C₂₃	—	2	2	no	—
122	C₄×C₂₃	#121	1	1	yes	characteristic
123	C₄×C₂₃	#121	1	1	yes	characteristic
124	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
125	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
126	Dic₂₃	—	4	2	no	—
127	#575	—	23	2	no	—
128	C₂×#135	—	23	1	no	—
129	C₂×#131	D₂₃	1	1	yes	characteristic, complemented
130	#473	—	23	2	no	—
131	#472	—	23	2	no	—
132	#471	—	23	4	no	—
133	C₂×#108	—	23	1	no	—
134	#413	—	23	2	no	—
135	C₂×C₂₃	—	2	1	no	—
136	C₂×C₂₃	—	2	1	no	—
137	C₂×C₂₃	—	2	1	no	—
138	C₂×C₂₃	—	4	2	no	—
139	C₂×C₂₃	C₂×#121	1	1	yes	characteristic
140	D₂₃	—	8	2	no	—
141	C₂×#41	—	23	2	no	—
142	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
143	#137	—	23	4	no	—
144	#137	—	46	2	no	—
145	AGL(1,ℤ/8)	—	23	4	no	—
146	AGL(1,ℤ/8)	—	46	2	no	—
147	#135	—	46	4	no	—
148	#135	—	46	1	no	—
149	#135	—	46	1	no	—
150	#135	—	46	1	no	—
151	#135	—	46	1	no	—
152	C₂×Q₁₆	—	23	1	no	—
153	C₂×SD₁₆	—	23	2	no	—
154	C₂×D₈	—	23	1	no	—
155	#131	—	2	1	no	—
156	#131	—	2	1	no	—
157	#131	C₂×D₂₃	1	2	yes	—
158	#131	C₂×D₂₃	1	2	yes	—
159	C₂×M₁₆	—	2	1	no	—
160	C₂×M₁₆	—	23	2	no	—
161	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
162	C₂²×C₈	—	2	1	no	—
163	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
164	#108	—	23	2	no	—
165	#108	—	23	2	no	—
166	#101	—	23	4	no	—
167	#100	—	23	4	no	—
168	C₂₃	#3067	1	1	yes	characteristic, complemented, Sylow
169	#41	—	2	1	no	—
170	#41	—	2	1	no	—
171	#41	—	46	2	no	—
172	#41	—	46	2	no	—
173	#41	—	46	2	no	—
174	#41	—	46	2	no	—
175	#41	D₉₂	1	2	yes	—
176	#41	C₂²×D₂₃	1	2	yes	—
177	C₂×Q₈	—	23	2	no	—
178	C₂×Q₈	D₉₂	1	1	yes	characteristic
179	C₂×D₄	—	2	1	no	—
180	C₂×D₄	—	23	2	no	—
181	C₂×D₄	—	46	2	no	—
182	C₂×D₄	D₉₂	1	1	yes	characteristic
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	46	2	no	—
185	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
186	Q₁₆	—	46	4	no	—
187	Q₁₆	—	46	1	no	—
188	Q₁₆	—	46	1	no	—
189	SD₁₆	—	46	4	no	—
190	SD₁₆	—	46	4	no	—
191	SD₁₆	—	46	2	no	—
192	SD₁₆	—	46	2	no	—
193	D₈	—	46	4	no	—
194	D₈	—	46	1	no	—
195	D₈	—	46	1	no	—
196	M₁₆	—	2	1	no	—
197	M₁₆	—	2	2	no	—
198	M₁₆	—	2	2	no	—
199	M₁₆	—	23	4	no	—
200	M₁₆	—	46	2	no	—
201	M₁₆	C₂²×D₂₃	1	2	yes	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	46	2	no	—
206	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
207	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
208	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
209	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
210	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
211	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
212	C₂³	—	2	1	no	—
213	C₂³	—	46	2	no	—
214	C₂³	#67288	1	1	yes	characteristic
215	Q₈	—	2	1	no	—
216	Q₈	—	46	2	no	—
217	Q₈	—	46	2	no	—
218	Q₈	C₂×D₉₂	1	2	yes	—
219	D₄	—	2	2	no	—
220	D₄	—	2	2	no	—
221	D₄	—	2	1	no	—
222	D₄	—	46	2	no	—
223	D₄	—	46	2	no	—
224	D₄	—	92	2	no	—
225	D₄	—	92	2	no	—
226	D₄	C₂×D₉₂	1	2	yes	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	46	2	no	—
231	C₂×C₄	—	92	2	no	—
232	C₂×C₄	—	92	2	no	—
233	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
234	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
235	C₂×C₄	#67288	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
237	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
238	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	—	46	4	no	—
241	C₈	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
242	C₈	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
243	C₈	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
244	C₈	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
245	C₂²	—	2	2	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	92	2	no	—
249	C₂²	—	184	2	no	—
250	C₂²	#10625101	1	1	yes	characteristic
251	C₂²	#10625101	1	1	yes	characteristic
252	C₂²	C₂×#67288	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	92	2	no	—
255	C₄	#10625101	1	1	yes	characteristic
256	C₄	#10625101	1	1	yes	characteristic
257	C₄	C₂²×D₉₂	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	—	2	1	no	—
262	C₂	—	4	2	no	—
263	C₂	—	184	2	no	—
264	C₂	C₂×#10625101	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#10625101
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂×C₂₃×#131	C₂
socle	C₂×C₂₃	C₂×#121
hypercenter	C₂×#131	D₂₃
hypocenter	C₂₃	#3067
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. #13093215	C₂	semidirect
2. C₂×#10625075	C₂	semidirect
3. #13092692	C₂	semidirect
4. #13092687	C₂	semidirect
5. #13092680	C₂	semidirect
6. C₂×#10625001	C₂	semidirect
7. #13092399	C₂	semidirect
8. C₂×C₂₃×#131	C₂	semidirect
25. C₂×Dic₉₂	C₂²	semidirect
30. C₂×D₁₈₄	C₂²	semidirect
31. C₂×#67263	C₂²	semidirect
32. #10625006	C₂²	semidirect
33. #10625005	C₂²	semidirect
34. #10625001	C₂²	semidirect
35. #10625001	C₂²	semidirect
129. C₂×#131	D₂₃	semidirect
168. C₂₃	#3067	semidirect

Maximal subgroups

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

subgroup	index
#13093215	2
C₂×#10625075	2
#13092692	2
#13092687	2
#13092680	2
C₂×#10625001	2
#13092399	2
C₂×C₂₃×#131	2
#3067	23