Subgroups — #25007872

Order 11008.

310 conjugacy classes of subgroups.

The subgroup lattice has 224 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	#19132648	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₃×#334	C₂	1	1	yes	maximal, characteristic
5	#14583456	—	2	1	no	—
6	C₂×#13063265	—	2	1	no	—
7	#14583304	C₄	1	1	yes	characteristic
8	#14583304	C₄	1	1	yes	characteristic
9	#14583301	—	2	1	no	—
10	#14583301	C₂²	1	2	yes	—
11	#14583297	—	2	1	no	—
12	#14582937	—	2	1	no	—
13	#14582937	C₄	1	2	yes	—
14	C₄⋊(C₄₃⋊C₁₆)	—	2	1	no	—
15	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₄₃	—	2	1	no	—
17	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
18	C₄₃×#334	—	2	1	no	—
19	C₄₃×#334	C₂²	1	2	yes	—
20	#13063325	—	2	1	no	—
21	#13063325	—	2	1	no	—
22	#13063317	—	2	2	no	—
23	#13063304	—	4	1	no	—
24	#13063304	—	4	1	no	—
25	C₄×Dic₈₆	—	4	1	no	—
26	C₄×Dic₈₆	—	4	1	no	—
27	#13063284	—	2	2	no	—
28	#13063265	—	4	2	no	—
29	C₂×C₄₃⋊C₁₆	—	2	2	no	—
30	#13063259	—	2	1	no	—
31	#13063259	—	2	1	no	—
32	#13063259	C₂×C₄	1	2	yes	—
33	#13063259	C₂×C₄	1	2	yes	—
34	#13063257	—	2	2	no	—
35	#13063257	—	2	2	no	—
36	C₂²×C₈×C₄₃	—	4	1	no	—
37	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
40	C₄⋊C₈×C₄₃	—	2	2	no	—
41	C₄⋊C₈×C₄₃	—	4	1	no	—
42	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
45	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
49	C₂×Dic₈₆	—	4	1	no	—
50	C₂×Dic₈₆	—	4	1	no	—
51	C₂×C₈×C₄₃	—	2	2	no	—
52	C₂×C₈×C₄₃	—	4	2	no	—
53	C₂×C₈×C₄₃	—	8	2	no	—
54	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₄₃	SD₁₆	1	2	yes	—
57	C₂×C₈×C₄₃	Q₁₆	1	2	yes	—
58	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₄₃	D₈	1	1	yes	characteristic
62	C₄²×C₄₃	SD₁₆	1	1	yes	characteristic
63	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
64	C₂²⋊Dic₄₃	—	4	2	no	—
65	C₄⋊Dic₄₃	—	2	2	no	—
66	C₄⋊Dic₄₃	—	2	2	no	—
67	C₄⋊Dic₄₃	—	4	1	no	—
68	C₄⋊Dic₄₃	—	4	1	no	—
69	Dic₄₃⋊C₄	—	4	2	no	—
70	Dic₄₃⋊C₄	—	4	2	no	—
71	C₄×Dic₄₃	—	4	2	no	—
72	C₄₃⋊C₁₆	—	4	2	no	—
73	C₂³×C₄₃	#102	1	1	yes	characteristic
74	C₂×C₄×C₄₃	—	2	2	no	—
75	C₂×C₄×C₄₃	—	4	1	no	—
76	C₂×C₄×C₄₃	#101	1	2	yes	—
77	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
78	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
79	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
80	C₂×C₄×C₄₃	#103	1	2	yes	—
81	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
84	C₂×Dic₄₃	—	4	2	no	—
85	C₂×Dic₄₃	—	4	2	no	—
86	Dic₈₆	—	8	2	no	—
87	Dic₈₆	—	8	2	no	—
88	C₈×C₄₃	—	2	1	no	—
89	C₈×C₄₃	—	2	1	no	—
90	C₈×C₄₃	—	8	2	no	—
91	C₈×C₄₃	#103	1	2	yes	—
92	C₈×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
93	C₈×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
94	#11936	—	43	1	no	maximal, Sylow
95	C₂²×C₄₃	—	2	1	no	—
96	C₂²×C₄₃	—	2	1	no	—
97	C₂²×C₄₃	#332	1	2	yes	—
98	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
99	C₄×C₄₃	—	2	1	no	—
100	C₄×C₄₃	—	2	1	no	—
101	C₄×C₄₃	—	2	1	no	—
102	C₄×C₄₃	#332	1	2	yes	—
103	C₄×C₄₃	#360	1	2	yes	—
104	C₄×C₄₃	C₂×#101	1	1	yes	characteristic
105	C₄×C₄₃	C₂×#103	1	1	yes	characteristic
106	Dic₄₃	—	8	2	no	—
107	Dic₄₃	—	8	2	no	—
108	#3160	—	43	1	no	—
109	#2150	—	43	1	no	—
110	C₂×#334	D₄₃	1	1	yes	characteristic
111	#1363	—	43	4	no	—
112	C₂×C₄₃	—	2	2	no	—
113	C₂×C₄₃	C₂×#332	1	1	yes	characteristic
114	C₂×C₄₃	#2143	1	1	yes	characteristic
115	C₂×C₄₃	C₂×#360	1	1	yes	characteristic
116	#533	—	86	1	no	—
117	C₂×M₃₂	—	86	1	no	—
118	#500	—	43	1	no	—
119	#500	—	43	1	no	—
120	#499	—	43	2	no	—
121	#499	—	86	1	no	—
122	#498	—	86	1	no	—
123	#427	—	43	2	no	—
124	#427	—	86	1	no	—
125	C₂×C₄⋊C₈	—	2	1	no	—
126	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
127	C₄⋊C₁₆	—	43	2	no	—
128	C₄⋊C₁₆	—	86	1	no	—
129	#334	—	2	1	no	—
130	#334	C₂×D₄₃	1	2	yes	—
131	C₄₃	#11936	1	1	yes	characteristic, complemented, Sylow
132	C₂²×C₈	—	4	1	no	—
133	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
134	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
135	C₄⋊Q₈	—	86	1	no	—
136	C₄⋊Q₈	—	86	1	no	—
137	#125	—	86	2	no	—
138	C₂²⋊Q₈	—	172	1	no	—
139	C₂²⋊Q₈	—	172	1	no	—
140	C₄×Q₈	—	172	1	no	—
141	C₄×Q₈	—	172	1	no	—
142	#117	—	86	2	no	—
143	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
144	M₃₂	—	172	2	no	—
145	C₂×C₁₆	—	86	2	no	—
146	#107	—	43	2	no	—
147	#107	—	43	2	no	—
148	#107	—	86	1	no	—
149	#107	—	86	1	no	—
150	#106	—	86	2	no	—
151	#106	—	86	2	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	4	1	no	—
154	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
155	C₄×C₈	D₁₇₂	1	1	yes	characteristic
156	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
157	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
158	C₂×Q₈	—	172	1	no	—
159	C₂×Q₈	—	172	1	no	—
160	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
161	C₂²×C₄	#10621882	1	1	yes	characteristic
162	C₂²×C₄	#10621884	1	1	yes	characteristic
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	4	2	no	—
165	C₂×C₈	—	8	2	no	—
166	C₂×C₈	#10621873	1	2	yes	—
167	C₂×C₈	Dic₁₇₂	1	2	yes	—
168	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
169	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
170	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
171	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
172	C₄⋊C₄	—	86	2	no	—
173	C₄⋊C₄	—	86	2	no	—
174	C₄⋊C₄	—	172	1	no	—
175	C₄⋊C₄	—	172	1	no	—
176	C₄⋊C₄	—	172	2	no	—
177	C₄⋊C₄	—	172	2	no	—
178	C₂²⋊C₄	—	172	2	no	—
179	C₄²	—	172	2	no	—
180	C₄²	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
181	C₄²	#10621882	1	1	yes	characteristic
182	C₄²	#10621884	1	1	yes	characteristic
183	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
184	C₁₆	—	172	2	no	—
185	C₂³	#13063245	1	1	yes	characteristic
186	Q₈	—	344	2	no	—
187	Q₈	—	344	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	4	1	no	—
190	C₂×C₄	—	172	2	no	—
191	C₂×C₄	—	172	2	no	—
192	C₂×C₄	#13063243	1	2	yes	—
193	C₂×C₄	#13063245	1	1	yes	characteristic
194	C₂×C₄	#13063245	1	1	yes	characteristic
195	C₂×C₄	#13063245	1	1	yes	characteristic
196	C₂×C₄	#13063250	1	2	yes	—
197	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
198	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
199	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
200	C₈	—	2	1	no	—
201	C₈	—	2	1	no	—
202	C₈	—	8	2	no	—
203	C₈	#13063250	1	2	yes	—
204	C₈	C₂×#10621873	1	1	yes	characteristic
205	C₈	C₂×Dic₁₇₂	1	1	yes	characteristic
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#14582646	1	2	yes	—
209	C₂²	C₂×#13063245	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	344	2	no	—
214	C₄	—	344	2	no	—
215	C₄	#14582646	1	2	yes	—
216	C₄	#14582708	1	2	yes	—
217	C₄	C₂×#13063243	1	1	yes	characteristic
218	C₄	C₂×#13063250	1	1	yes	characteristic
219	C₂	—	2	2	no	—
220	C₂	C₂×#14582646	1	1	yes	characteristic
221	C₂	#19136322	1	1	yes	characteristic
222	C₂	C₂×#14582708	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×#334	C₂
socle	C₂²×C₄₃	C₂×#102
prime radical	C₂³×C₄₃	#102
cosocle	C₄×C₈×C₄₃	C₂³
hypercenter	C₂×#334	D₄₃
hypocenter	C₄₃	#11936
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
3. #19132648	C₂	semidirect
131. C₄₃	#11936	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
#19132648	2
C₂×C₄₃×#334	2
#11936	43