Subgroups — #25015068

Order 11008.

494 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583279	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₄₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582702	C₂	1	4	yes	maximal
4	C₄×#107×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₇₂	—	2	1	no	—
6	C₂²×Dic₁₇₂	C₂²	1	2	yes	—
7	C₂×#13063325	—	2	1	no	—
8	C₂²×C₄₃⋊C₁₆	C₂²	1	2	yes	—
9	#14583279	—	2	2	no	—
10	#14583279	C₄	1	4	yes	complemented
11	C₄⋊(C₄₃⋊C₁₆)	—	2	2	no	—
12	#14582702	—	2	8	no	—
13	C₂×#107×C₄₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₄₃	—	2	1	no	—
16	C₄₃×#340	—	2	1	no	—
17	C₂²×Dic₈₆	—	2	2	no	—
18	C₂×Dic₁₇₂	—	2	4	no	—
19	C₂×Dic₁₇₂	—	2	4	no	—
20	C₂×Dic₁₇₂	—	2	4	no	—
21	C₂×Dic₁₇₂	—	2	2	no	—
22	C₂×Dic₁₇₂	—	4	4	no	—
23	C₂×Dic₁₇₂	C₂×C₄	1	4	yes	—
24	#13063325	—	2	4	no	—
25	#13063325	—	4	2	no	—
26	C₂×C₄⋊Dic₄₃	—	4	1	no	—
27	C₂×C₄⋊Dic₄₃	—	4	1	no	—
28	C₂×C₄₃⋊C₁₆	—	2	4	no	—
29	C₂×C₄₃⋊C₁₆	—	2	2	no	—
30	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
33	C₂×C₄²×C₄₃	—	4	1	no	—
34	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
35	#107×C₄₃	—	2	2	no	—
36	#107×C₄₃	D₄	1	4	yes	—
37	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₄₃	D₄	1	2	yes	—
39	#95×C₄₃	—	4	1	no	—
40	C₂²×C₄×C₄₃	—	4	2	no	—
41	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₄₃	—	4	2	no	—
45	C₂×Dic₈₆	—	2	4	no	—
46	C₂×Dic₈₆	—	4	4	no	—
47	C₂×Dic₈₆	—	4	4	no	—
48	C₂×Dic₈₆	—	4	2	no	—
49	C₂×C₈×C₄₃	—	2	2	no	—
50	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
53	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₄₃	#41	1	2	yes	—
55	C₄⋊C₄×C₄₃	—	2	4	no	—
56	C₄⋊C₄×C₄₃	—	4	2	no	—
57	C₄²×C₄₃	—	8	2	no	—
58	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
59	C₄²×C₄₃	SD₁₆	1	2	yes	—
60	C₄⋊Dic₄₃	—	4	4	no	—
61	C₄⋊Dic₄₃	—	4	4	no	—
62	Dic₁₇₂	—	2	8	no	—
63	Dic₁₇₂	—	4	8	no	—
64	Dic₁₇₂	—	4	2	no	—
65	Dic₁₇₂	—	4	2	no	—
66	C₄₃⋊C₁₆	—	4	2	no	—
67	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₄₃	—	2	2	no	—
69	C₂×C₄×C₄₃	—	4	4	no	—
70	C₂×C₄×C₄₃	—	8	2	no	—
71	C₂×C₄×C₄₃	—	8	2	no	—
72	C₂×C₄×C₄₃	#102	1	2	yes	—
73	C₂×C₄×C₄₃	#102	1	2	yes	—
74	C₂×C₄×C₄₃	SD₃₂	1	2	yes	—
75	C₂×C₄×C₄₃	Q₃₂	1	2	yes	—
76	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₄₃	#121	1	2	yes	—
78	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×Dic₄₃	—	4	4	no	—
81	C₂×Dic₄₃	—	8	4	no	—
82	Dic₈₆	—	4	4	no	—
83	Dic₈₆	—	4	4	no	—
84	Dic₈₆	—	8	4	no	—
85	C₈×C₄₃	—	2	2	no	—
86	C₈×C₄₃	C₄×D₄	1	2	yes	—
87	C₈×C₄₃	#124	1	1	yes	characteristic
88	C₈×C₄₃	#127	1	1	yes	characteristic
89	#13253	—	43	1	no	maximal, Sylow
90	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
91	C₂²×C₄₃	#459	1	2	yes	—
92	C₂²×C₄₃	C₂×SD₃₂	1	1	yes	characteristic
93	C₂²×C₄₃	C₂×Q₃₂	1	1	yes	characteristic
94	C₂²×C₄₃	#510	1	2	yes	—
95	C₄×C₄₃	—	2	2	no	—
96	C₄×C₄₃	—	8	4	no	—
97	C₄×C₄₃	#358	1	4	yes	complemented
98	C₄×C₄₃	#390	1	1	yes	characteristic
99	C₄×C₄₃	#418	1	1	yes	characteristic
100	C₄×C₄₃	#460	1	2	yes	—
101	Dic₄₃	—	8	4	no	—
102	C₂×#494	—	43	1	no	—
103	C₂×C₄⋊C₁₆	—	43	1	no	—
104	C₂×#358	—	43	4	no	—
105	C₄×#107	D₄₃	1	1	yes	characteristic
106	C₂×C₄₃	#1982	1	1	yes	characteristic
107	C₂×C₄₃	C₂×#358	1	1	yes	characteristic
108	C₂×C₄₃	#2141	1	1	yes	characteristic
109	C₂×C₄₃	#2208	1	2	yes	—
110	C₂×C₄₃	#2209	1	2	yes	—
111	C₂²×Q₁₆	—	43	2	no	—
112	C₂²×Q₁₆	—	86	1	no	—
113	C₂×C₄⋊Q₈	—	86	1	no	—
114	C₂²×C₁₆	—	43	2	no	—
115	#494	—	43	4	no	—
116	#494	—	86	2	no	—
117	C₂×#107	C₂×D₄₃	1	2	yes	—
118	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
119	C₄×C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₁₆	—	86	2	no	—
121	#358	—	86	8	no	—
122	#340	—	2	1	no	—
123	C₄₃	#13253	1	1	yes	characteristic, complemented, Sylow
124	C₂²×Q₈	—	86	2	no	—
125	C₂×Q₁₆	—	43	4	no	—
126	C₂×Q₁₆	—	86	4	no	—
127	C₂×Q₁₆	—	86	4	no	—
128	C₂×Q₁₆	—	86	2	no	—
129	C₂×Q₁₆	—	86	4	no	—
130	C₂×Q₁₆	—	172	4	no	—
131	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
132	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
133	C₄⋊Q₈	—	86	4	no	—
134	C₄⋊Q₈	—	172	2	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	172	1	no	—
137	C₂×C₄⋊C₄	—	172	1	no	—
138	C₂×C₄²	—	4	1	no	—
139	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
140	C₂×C₁₆	—	86	4	no	—
141	C₂×C₁₆	—	86	2	no	—
142	#107	—	2	2	no	—
143	#107	D₁₇₂	1	4	yes	—
144	C₄×C₈	C₄×D₄₃	1	2	yes	—
145	C₄×C₈	D₁₇₂	1	2	yes	—
146	#95	—	4	1	no	—
147	C₂×Q₈	—	86	4	no	—
148	C₂×Q₈	—	172	4	no	—
149	C₂×Q₈	—	172	4	no	—
150	C₂×Q₈	—	172	2	no	—
151	C₂²×C₄	—	4	2	no	—
152	C₂²×C₄	—	172	2	no	—
153	C₂²×C₄	#10621882	1	1	yes	characteristic
154	C₂²×C₄	#10621882	1	1	yes	characteristic
155	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
156	Q₁₆	—	86	8	no	—
157	Q₁₆	—	172	8	no	—
158	Q₁₆	—	172	2	no	—
159	Q₁₆	—	172	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
162	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
163	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
164	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
165	C₂×C₈	#10621898	1	2	yes	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄⋊C₄	—	172	4	no	—
169	C₄⋊C₄	—	172	4	no	—
170	C₄²	—	8	2	no	—
171	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
172	C₄²	#10621884	1	2	yes	—
173	C₁₆	—	172	2	no	—
174	C₂³	C₂×#10621882	1	1	yes	characteristic
175	Q₈	—	344	4	no	—
176	Q₈	—	172	4	no	—
177	Q₈	—	172	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	8	2	no	—
181	C₂×C₄	—	8	2	no	—
182	C₂×C₄	—	344	4	no	—
183	C₂×C₄	—	172	4	no	—
184	C₂×C₄	#13063245	1	2	yes	—
185	C₂×C₄	#13063245	1	2	yes	—
186	C₂×C₄	#13063269	1	2	yes	—
187	C₂×C₄	#13063271	1	2	yes	—
188	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
189	C₂×C₄	#13063299	1	2	yes	—
190	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
191	C₂×C₄	#13063350	1	2	yes	—
192	C₈	—	2	2	no	—
193	C₈	C₄×D₁₇₂	1	2	yes	—
194	C₈	#13063313	1	1	yes	characteristic
195	C₈	#13063322	1	1	yes	characteristic
196	C₂²	C₂×#13063245	1	1	yes	characteristic
197	C₂²	#14583094	1	2	yes	—
198	C₂²	C₂×#13063269	1	1	yes	characteristic
199	C₂²	C₂×#13063271	1	1	yes	characteristic
200	C₂²	#14583339	1	2	yes	—
201	C₄	—	2	2	no	—
202	C₄	—	8	4	no	—
203	C₄	—	344	4	no	—
204	C₄	#14582702	1	4	yes	complemented
205	C₄	#14582801	1	1	yes	characteristic
206	C₄	#14582903	1	1	yes	characteristic
207	C₄	#14583099	1	2	yes	—
208	C₂	#19135522	1	1	yes	characteristic
209	C₂	C₂×#14582702	1	1	yes	characteristic
210	C₂	#19136314	1	1	yes	characteristic
211	C₂	#19136640	1	2	yes	—
212	C₂	#19136645	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#10621882
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄×#107×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂×D₈
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₄×#107	D₄₃
hypocenter	C₄₃	#13253
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
10. #14583279	C₄	semidirect
97. C₄×C₄₃	#358	semidirect
123. C₄₃	#13253	semidirect
204. C₄	#14582702	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583279	2
C₂×C₄⋊(C₄₃⋊C₁₆)	2
C₂×#14582702	2
C₄×#107×C₄₃	2
#13253	43