Subgroups — #25007898

Order 11008.

310 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145109	C₂	1	1	yes	maximal, characteristic
2	#19136351	C₂	1	1	yes	maximal, characteristic
3	#19132654	C₂	1	2	yes	maximal, complemented
4	#19132651	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₃×#335	C₂	1	1	yes	maximal, characteristic
6	#14583456	—	2	1	no	—
7	C₂×#13063265	—	2	1	no	—
8	#14583304	C₂²	1	1	yes	characteristic
9	#14583304	C₂²	1	1	yes	characteristic
10	#14583301	—	2	1	no	—
11	#14583301	C₄	1	2	yes	—
12	#14583297	—	2	1	no	—
13	#14582937	—	2	1	no	—
14	#14582937	C₄	1	2	yes	—
15	C₄⋊(C₄₃⋊C₁₆)	—	2	1	no	—
16	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₄₃	—	2	1	no	—
18	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
19	C₄₃×#335	—	2	1	no	—
20	C₄₃×#335	C₂²	1	2	yes	—
21	#13063325	—	2	1	no	—
22	#13063325	—	2	1	no	—
23	#13063317	—	2	2	no	—
24	#13063304	—	4	1	no	—
25	#13063304	—	4	1	no	—
26	C₄×Dic₈₆	—	4	1	no	—
27	C₄×Dic₈₆	—	4	1	no	—
28	#13063284	—	2	2	no	—
29	#13063265	—	4	2	no	—
30	C₂×C₄₃⋊C₁₆	—	2	2	no	—
31	#13063259	—	2	1	no	—
32	#13063259	—	2	1	no	—
33	#13063259	C₂×C₄	1	2	yes	—
34	#13063259	C₂×C₄	1	2	yes	—
35	#13063257	—	2	2	no	—
36	#13063257	—	2	2	no	—
37	C₂²×C₈×C₄₃	—	4	1	no	—
38	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
41	C₄⋊C₈×C₄₃	—	2	2	no	—
42	C₄⋊C₈×C₄₃	—	4	1	no	—
43	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
45	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
50	C₂×Dic₈₆	—	4	1	no	—
51	C₂×Dic₈₆	—	4	1	no	—
52	C₂×C₈×C₄₃	—	2	2	no	—
53	C₂×C₈×C₄₃	—	4	2	no	—
54	C₂×C₈×C₄₃	—	8	2	no	—
55	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₄₃	SD₁₆	1	2	yes	—
58	C₂×C₈×C₄₃	Q₁₆	1	2	yes	—
59	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₄₃	D₈	1	1	yes	characteristic
63	C₄²×C₄₃	SD₁₆	1	1	yes	characteristic
64	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂²⋊Dic₄₃	—	4	2	no	—
66	C₄⋊Dic₄₃	—	2	2	no	—
67	C₄⋊Dic₄₃	—	2	2	no	—
68	C₄⋊Dic₄₃	—	4	1	no	—
69	C₄⋊Dic₄₃	—	4	1	no	—
70	Dic₄₃⋊C₄	—	4	2	no	—
71	Dic₄₃⋊C₄	—	4	2	no	—
72	C₄×Dic₄₃	—	4	2	no	—
73	C₄₃⋊C₁₆	—	4	2	no	—
74	C₂³×C₄₃	#102	1	1	yes	characteristic
75	C₂×C₄×C₄₃	—	2	1	no	—
76	C₂×C₄×C₄₃	—	2	1	no	—
77	C₂×C₄×C₄₃	—	4	1	no	—
78	C₂×C₄×C₄₃	#101	1	2	yes	—
79	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
80	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
81	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
82	C₂×C₄×C₄₃	#103	1	2	yes	—
83	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×Dic₄₃	—	4	2	no	—
87	C₂×Dic₄₃	—	4	2	no	—
88	Dic₈₆	—	8	2	no	—
89	Dic₈₆	—	8	1	no	—
90	Dic₈₆	—	8	1	no	—
91	C₈×C₄₃	—	2	1	no	—
92	C₈×C₄₃	—	2	1	no	—
93	C₈×C₄₃	—	8	2	no	—
94	C₈×C₄₃	#103	1	2	yes	—
95	C₈×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
96	C₈×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
97	#11942	—	43	1	no	maximal, Sylow
98	C₂²×C₄₃	—	2	1	no	—
99	C₂²×C₄₃	—	2	1	no	—
100	C₂²×C₄₃	#332	1	2	yes	—
101	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
102	C₄×C₄₃	—	2	1	no	—
103	C₄×C₄₃	—	2	1	no	—
104	C₄×C₄₃	—	2	1	no	—
105	C₄×C₄₃	#332	1	2	yes	—
106	C₄×C₄₃	C₂×#101	1	1	yes	characteristic
107	C₄×C₄₃	C₂×#103	1	1	yes	characteristic
108	C₄×C₄₃	#509	1	1	yes	characteristic
109	C₄×C₄₃	#510	1	1	yes	characteristic
110	Dic₄₃	—	8	2	no	—
111	Dic₄₃	—	8	2	no	—
112	#3160	—	43	1	no	—
113	#2150	—	43	1	no	—
114	C₂×#335	D₄₃	1	1	yes	characteristic
115	#1365	—	43	2	no	—
116	#1364	—	43	2	no	—
117	C₂×C₄₃	—	2	2	no	—
118	C₂×C₄₃	C₂×#332	1	1	yes	characteristic
119	C₂×C₄₃	#2139	1	1	yes	characteristic
120	C₂×C₄₃	#2148	1	1	yes	characteristic
121	#533	—	86	1	no	—
122	C₂×M₃₂	—	86	1	no	—
123	#500	—	43	1	no	—
124	#500	—	43	1	no	—
125	#499	—	43	2	no	—
126	#499	—	86	1	no	—
127	#498	—	86	1	no	—
128	#427	—	43	2	no	—
129	#427	—	86	1	no	—
130	C₂×C₄⋊C₈	—	2	1	no	—
131	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
132	C₄⋊C₁₆	—	43	2	no	—
133	C₄⋊C₁₆	—	86	1	no	—
134	#335	—	2	1	no	—
135	#335	C₂×D₄₃	1	2	yes	—
136	C₄₃	#11942	1	1	yes	characteristic, complemented, Sylow
137	C₂²×C₈	—	4	1	no	—
138	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
139	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
140	C₄⋊Q₈	—	86	1	no	—
141	C₄⋊Q₈	—	86	1	no	—
142	#125	—	86	2	no	—
143	C₂²⋊Q₈	—	172	1	no	—
144	C₂²⋊Q₈	—	172	1	no	—
145	C₄×Q₈	—	172	1	no	—
146	C₄×Q₈	—	172	1	no	—
147	#117	—	86	2	no	—
148	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
149	M₃₂	—	172	2	no	—
150	C₂×C₁₆	—	86	2	no	—
151	#107	—	43	2	no	—
152	#107	—	43	2	no	—
153	#107	—	86	1	no	—
154	#107	—	86	1	no	—
155	#106	—	86	2	no	—
156	#106	—	86	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	4	1	no	—
159	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
160	C₄×C₈	D₁₇₂	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
162	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
163	C₂×Q₈	—	172	1	no	—
164	C₂×Q₈	—	172	1	no	—
165	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
166	C₂²×C₄	#10621882	1	1	yes	characteristic
167	C₂²×C₄	#10621884	1	1	yes	characteristic
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	—	8	2	no	—
171	C₂×C₈	#10621873	1	2	yes	—
172	C₂×C₈	Dic₁₇₂	1	2	yes	—
173	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
174	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
175	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
176	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
177	C₄⋊C₄	—	86	2	no	—
178	C₄⋊C₄	—	86	2	no	—
179	C₄⋊C₄	—	172	2	no	—
180	C₄⋊C₄	—	172	2	no	—
181	C₄⋊C₄	—	172	1	no	—
182	C₄⋊C₄	—	172	1	no	—
183	C₂²⋊C₄	—	172	2	no	—
184	C₄²	—	172	2	no	—
185	C₄²	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
186	C₄²	#10621882	1	1	yes	characteristic
187	C₄²	#10621884	1	1	yes	characteristic
188	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
189	C₁₆	—	172	2	no	—
190	C₂³	#13063245	1	1	yes	characteristic
191	Q₈	—	344	2	no	—
192	Q₈	—	344	1	no	—
193	Q₈	—	344	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	172	2	no	—
198	C₂×C₄	—	172	2	no	—
199	C₂×C₄	#13063243	1	2	yes	—
200	C₂×C₄	#13063245	1	1	yes	characteristic
201	C₂×C₄	#13063245	1	1	yes	characteristic
202	C₂×C₄	#13063245	1	1	yes	characteristic
203	C₂×C₄	#13063250	1	2	yes	—
204	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
205	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
206	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
207	C₈	—	2	1	no	—
208	C₈	—	2	1	no	—
209	C₈	—	8	2	no	—
210	C₈	#13063250	1	2	yes	—
211	C₈	C₂×#10621873	1	1	yes	characteristic
212	C₈	C₂×Dic₁₇₂	1	1	yes	characteristic
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	#14582646	1	2	yes	—
216	C₂²	C₂×#13063245	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	344	2	no	—
221	C₄	—	344	2	no	—
222	C₄	#14582646	1	2	yes	—
223	C₄	C₂×#13063243	1	1	yes	characteristic
224	C₄	C₂×#13063250	1	1	yes	characteristic
225	C₄	#14583335	1	1	yes	characteristic
226	C₄	#14583340	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	C₂×#14582646	1	1	yes	characteristic
229	C₂	#19136305	1	1	yes	characteristic
230	C₂	#19136342	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13063245
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₂×C₄₃×#335	C₂
socle	C₂²×C₄₃	C₂×#102
prime radical	C₂³×C₄₃	#102
cosocle	C₄×C₈×C₄₃	C₂³
hypercenter	C₂×#335	D₄₃
hypocenter	C₄₃	#11942
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
3. #19132654	C₂	semidirect
4. #19132651	C₂	semidirect
136. C₄₃	#11942	semidirect

Maximal subgroups

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

subgroup	index
#19145109	2
#19136351	2
#19132654	2
#19132651	2
C₂×C₄₃×#335	2
#11942	43