Subgroups — #24995874

Order 11008.

442 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583304	C₂	1	1	yes	maximal, characteristic
2	#19135542	C₂	1	2	yes	maximal
3	#19134462	C₂	1	1	yes	maximal, characteristic
4	C₂×#14582647	C₂	1	2	yes	maximal
5	C₂×C₄₃×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063325	C₂²	1	2	yes	—
7	#14583304	—	2	2	no	—
8	#14583304	C₄	1	4	yes	complemented
9	C₂×#13063259	—	2	1	no	—
10	C₂×#13063259	—	2	1	no	—
11	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×#13063235	—	2	1	no	—
13	C₄×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
14	#14582662	—	2	4	no	—
15	#14582653	—	2	1	no	—
16	#14582647	—	2	4	no	—
17	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
19	C₄₃×#334	—	2	2	no	—
20	C₂²×Dic₈₆	—	2	2	no	—
21	C₂²×C₄₃⋊C₈	—	2	1	no	—
22	#13063325	—	2	2	no	—
23	#13063325	—	2	2	no	—
24	#13063325	—	2	2	no	—
25	#13063325	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊Dic₄₃	—	2	2	no	—
27	C₂×C₄⋊Dic₄₃	—	2	2	no	—
28	C₂×C₄⋊Dic₄₃	D₄	1	2	yes	—
29	C₂×C₄×Dic₄₃	—	2	1	no	—
30	#13063259	—	2	4	no	—
31	#13063259	—	2	4	no	—
32	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
33	C₄⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
34	#13063235	—	4	2	no	—
35	C₄₃⋊#95	—	2	1	no	—
36	C₂²×C₈×C₄₃	—	2	2	no	—
37	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₄₃	—	2	2	no	—
41	C₄⋊C₈×C₄₃	—	2	2	no	—
42	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₄₃	D₄	1	2	yes	—
44	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
47	C₂²×Dic₄₃	—	2	2	no	—
48	C₂²×Dic₄₃	—	4	2	no	—
49	C₂×Dic₈₆	—	2	4	no	—
50	C₂×Dic₈₆	—	4	2	no	—
51	C₂×Dic₈₆	—	8	4	no	—
52	C₂×C₈×C₄₃	—	2	1	no	—
53	C₂×C₈×C₄₃	—	2	1	no	—
54	C₂×C₈×C₄₃	—	4	2	no	—
55	C₂×C₈×C₄₃	—	4	4	no	—
56	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₄₃	SD₁₆	1	2	yes	—
59	C₂×C₈×C₄₃	Q₁₆	1	2	yes	—
60	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₄₃	#41	1	1	yes	characteristic
64	C₄⋊Dic₄₃	—	2	4	no	—
65	C₄⋊Dic₄₃	—	2	4	no	—
66	C₄⋊Dic₄₃	—	2	4	no	—
67	C₄⋊Dic₄₃	—	4	2	no	—
68	C₄⋊Dic₄₃	—	4	2	no	—
69	C₄×Dic₄₃	—	4	1	no	—
70	C₄×Dic₄₃	—	4	1	no	—
71	C₂×C₄₃⋊C₈	—	2	2	no	—
72	C₂×C₄₃⋊C₈	—	4	1	no	—
73	C₂×C₄₃⋊C₈	—	4	1	no	—
74	C₂³×C₄₃	#121	1	1	yes	characteristic
75	C₂×C₄×C₄₃	—	2	2	no	—
76	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
78	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
80	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
81	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
82	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
83	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
84	C₂×C₄×C₄₃	#137	1	4	yes	—
85	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
86	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
87	C₂×Dic₄₃	—	4	4	no	—
88	C₂×Dic₄₃	—	4	2	no	—
89	C₂×Dic₄₃	—	4	2	no	—
90	C₂×Dic₄₃	—	4	2	no	—
91	C₂×Dic₄₃	—	8	4	no	—
92	Dic₈₆	—	8	4	no	—
93	Dic₈₆	—	8	4	no	—
94	C₈×C₄₃	—	2	2	no	—
95	C₈×C₄₃	—	8	2	no	—
96	C₈×C₄₃	#103	1	4	yes	—
97	C₄₃⋊C₈	—	4	2	no	—
98	#9739	—	43	1	no	maximal, Sylow
99	C₂²×C₄₃	#390	1	1	yes	characteristic
100	C₂²×C₄₃	#461	1	1	yes	characteristic
101	C₂²×C₄₃	#462	1	1	yes	characteristic
102	C₂²×C₄₃	#470	1	2	yes	—
103	C₂²×C₄₃	#472	1	2	yes	—
104	C₄×C₄₃	—	2	1	no	—
105	C₄×C₄₃	—	2	1	no	—
106	C₄×C₄₃	C₂×#103	1	1	yes	characteristic
107	C₄×C₄₃	#419	1	1	yes	characteristic
108	C₄×C₄₃	#441	1	2	yes	—
109	C₄×C₄₃	#461	1	1	yes	characteristic
110	C₄×C₄₃	#462	1	1	yes	characteristic
111	C₄×C₄₃	#497	1	2	yes	—
112	Dic₄₃	—	8	2	no	—
113	Dic₄₃	—	8	4	no	—
114	C₂×#500	—	43	1	no	—
115	#1987	—	43	2	no	—
116	#1770	—	43	1	no	—
117	C₂×#334	D₄₃	1	1	yes	characteristic
118	C₂×#333	—	43	2	no	—
119	C₂×C₄₃	#1695	1	2	yes	—
120	C₂×C₄₃	#1696	1	2	yes	—
121	C₂×C₄₃	#1971	1	1	yes	characteristic
122	C₂×C₄₃	#1978	1	1	yes	characteristic
123	C₂×C₄₃	#1981	1	1	yes	characteristic
124	C₂×C₄⋊Q₈	—	43	2	no	—
125	#500	—	43	4	no	—
126	#500	—	86	2	no	—
127	C₂×#107	—	86	1	no	—
128	C₂×#107	—	86	1	no	—
129	C₂×C₄⋊C₈	—	43	1	no	—
130	C₂×C₄⋊C₈	C₂×D₄₃	1	2	yes	—
131	C₂×#97	—	86	1	no	—
132	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
133	C₄×C₄⋊C₄	—	43	1	no	—
134	#340	—	86	4	no	—
135	#336	—	86	1	no	—
136	#334	—	2	2	no	—
137	#333	—	86	4	no	—
138	C₄₃	#9739	1	1	yes	characteristic, complemented, Sylow
139	C₂²×Q₈	—	86	2	no	—
140	C₂²×C₈	—	2	2	no	—
141	C₂²×C₈	—	86	1	no	—
142	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
143	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
144	C₄⋊Q₈	—	43	4	no	—
145	C₄⋊Q₈	—	86	2	no	—
146	C₄⋊Q₈	—	86	2	no	—
147	C₄⋊Q₈	—	86	2	no	—
148	C₂×C₄⋊C₄	—	43	2	no	—
149	C₂×C₄⋊C₄	—	86	2	no	—
150	C₂×C₄⋊C₄	—	86	2	no	—
151	C₂×C₄²	—	86	1	no	—
152	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
153	#107	—	86	4	no	—
154	#107	—	86	4	no	—
155	C₄⋊C₈	—	2	2	no	—
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	43	2	no	—
158	C₄⋊C₈	—	86	1	no	—
159	#97	—	172	2	no	—
160	C₄×C₈	C₄×D₄₃	1	2	yes	—
161	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
162	#95	—	86	1	no	—
163	C₂×Q₈	—	86	4	no	—
164	C₂×Q₈	—	344	4	no	—
165	C₂×Q₈	—	172	2	no	—
166	C₂²×C₄	—	86	2	no	—
167	C₂²×C₄	—	172	2	no	—
168	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
169	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
170	C₂²×C₄	#10621902	1	1	yes	characteristic
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	—	4	4	no	—
175	C₂×C₈	—	86	2	no	—
176	C₂×C₈	—	172	1	no	—
177	C₂×C₈	—	172	1	no	—
178	C₂×C₈	#10621873	1	2	yes	—
179	C₂×C₈	Dic₁₇₂	1	2	yes	—
180	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
181	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
182	C₄⋊C₄	—	86	4	no	—
183	C₄⋊C₄	—	86	4	no	—
184	C₄⋊C₄	—	86	4	no	—
185	C₄⋊C₄	—	172	2	no	—
186	C₄⋊C₄	—	172	2	no	—
187	C₄²	—	172	1	no	—
188	C₄²	—	172	1	no	—
189	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
190	C₄²	D₄×D₄₃	1	1	yes	characteristic
191	C₄²	#10621902	1	1	yes	characteristic
192	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
193	C₂³	#13063300	1	1	yes	characteristic
194	Q₈	—	344	4	no	—
195	Q₈	—	344	4	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	344	4	no	—
198	C₂×C₄	—	172	4	no	—
199	C₂×C₄	—	172	2	no	—
200	C₂×C₄	—	172	2	no	—
201	C₂×C₄	—	172	2	no	—
202	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
203	C₂×C₄	#13063287	1	1	yes	characteristic
204	C₂×C₄	#13063287	1	1	yes	characteristic
205	C₂×C₄	#13063300	1	1	yes	characteristic
206	C₂×C₄	#13063314	1	1	yes	characteristic
207	C₂×C₄	#13063321	1	1	yes	characteristic
208	C₂×C₄	C₂×#10621873	1	1	yes	characteristic
209	C₂×C₄	C₂×Dic₁₇₂	1	1	yes	characteristic
210	C₂×C₄	#13063354	1	4	yes	—
211	C₂×C₄	#13063358	1	1	yes	characteristic
212	C₂×C₄	#13063358	1	1	yes	characteristic
213	C₈	—	2	2	no	—
214	C₈	—	8	2	no	—
215	C₈	—	172	2	no	—
216	C₈	#13063250	1	4	yes	—
217	C₂²	#14582800	1	1	yes	characteristic
218	C₂²	#14583105	1	1	yes	characteristic
219	C₂²	#14583110	1	1	yes	characteristic
220	C₂²	#14583158	1	2	yes	—
221	C₂²	#14583170	1	2	yes	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	344	4	no	—
225	C₄	—	344	2	no	—
226	C₄	C₂×#13063250	1	1	yes	characteristic
227	C₄	#14582909	1	1	yes	characteristic
228	C₄	#14583005	1	2	yes	—
229	C₄	#14583105	1	1	yes	characteristic
230	C₄	#14583110	1	1	yes	characteristic
231	C₄	#14583290	1	2	yes	—
232	C₂	#19134119	1	2	yes	—
233	C₂	#19134124	1	2	yes	—
234	C₂	#19135464	1	1	yes	characteristic
235	C₂	#19135503	1	1	yes	characteristic
236	C₂	#19135517	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13063300
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₂×C₄₃×#334	C₂
socle, prime radical	C₂³×C₄₃	#121
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₂×#334	D₄₃
hypocenter	C₄₃	#9739
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
8. #14583304	C₄	semidirect
138. C₄₃	#9739	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583304	2
#19135542	2
#19134462	2
C₂×#14582647	2
C₂×C₄₃×#334	2
#9739	43