Subgroups — #25015187

Order 11008.

366 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19136351	C₂	1	1	yes	maximal, characteristic
2	C₄×#13063261	C₂	1	1	yes	maximal, characteristic
3	#19132703	C₂	1	4	yes	maximal
4	C₂×C₄₃×#500	C₂	1	1	yes	maximal, characteristic
5	C₂×#13063265	C₂²	1	2	yes	—
6	C₂×#13063261	C₂²	1	2	yes	—
7	C₄×#10621877	—	2	1	no	—
8	C₄⋊(C₄₃⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₄₃⋊C₁₆)	—	2	1	no	—
10	#14582664	—	2	1	no	—
11	C₂×C₄⋊Q₈×C₄₃	—	2	1	no	—
12	C₄₃×#500	—	2	1	no	—
13	C₄₃×#500	—	2	1	no	—
14	C₄₃×#500	C₄	1	2	yes	—
15	C₄₃×#500	C₄	1	2	yes	—
16	C₂×#107×C₄₃	—	2	1	no	—
17	C₂×#107×C₄₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
19	C₂×#10621877	—	2	2	no	—
20	#13063265	—	2	4	no	—
21	C₂×C₄₃⋊C₁₆	—	2	2	no	—
22	#13063261	—	2	2	no	—
23	#13063261	Q₈	1	4	yes	complemented
24	#13063236	—	4	1	no	—
25	C₄×C₄₃⋊C₈	—	4	1	no	—
26	C₂²×Q₈×C₄₃	—	4	1	no	—
27	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
29	C₄⋊Q₈×C₄₃	—	2	2	no	—
30	C₄⋊Q₈×C₄₃	—	2	2	no	—
31	C₄⋊Q₈×C₄₃	—	4	1	no	—
32	C₄⋊Q₈×C₄₃	—	4	1	no	—
33	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₄₃	—	4	1	no	—
35	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
36	#107×C₄₃	—	2	2	no	—
37	#107×C₄₃	—	2	2	no	—
38	#107×C₄₃	—	2	2	no	—
39	#107×C₄₃	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
44	C₂×Q₈×C₄₃	—	4	2	no	—
45	C₂×Q₈×C₄₃	—	4	2	no	—
46	C₂×Q₈×C₄₃	—	8	4	no	—
47	C₂²×C₄×C₄₃	—	4	2	no	—
48	C₂²×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₄₃	—	2	2	no	—
52	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
57	C₂×C₈×C₄₃	#41	1	2	yes	—
58	C₄⋊C₄×C₄₃	—	2	4	no	—
59	C₄⋊C₄×C₄₃	—	4	2	no	—
60	C₄⋊C₄×C₄₃	—	4	2	no	—
61	C₄⋊C₄×C₄₃	—	4	2	no	—
62	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₄₃	D₈	1	1	yes	characteristic
65	C₄²×C₄₃	D₈	1	1	yes	characteristic
66	#10621877	—	2	4	no	—
67	#10621877	—	4	2	no	—
68	C₂×C₄₃⋊C₈	—	4	2	no	—
69	C₄₃⋊C₁₆	—	4	2	no	—
70	C₂³×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
71	Q₈×C₄₃	—	8	4	no	—
72	Q₈×C₄₃	—	8	4	no	—
73	C₂×C₄×C₄₃	—	2	2	no	—
74	C₂×C₄×C₄₃	—	2	2	no	—
75	C₂×C₄×C₄₃	—	4	4	no	—
76	C₂×C₄×C₄₃	—	8	4	no	—
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	#102	1	1	yes	characteristic
81	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
83	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₄₃	#137	1	2	yes	—
85	C₈×C₄₃	—	2	1	no	—
86	C₈×C₄₃	—	2	1	no	—
87	C₈×C₄₃	C₄×Q₈	1	2	yes	—
88	C₈×C₄₃	#124	1	1	yes	characteristic
89	C₈×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
90	C₄₃⋊C₈	—	4	4	no	—
91	#13279	—	43	1	no	maximal, Sylow
92	C₂²×C₄₃	—	2	1	no	—
93	C₂²×C₄₃	—	2	1	no	—
94	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
95	C₂²×C₄₃	#475	1	2	yes	—
96	C₄×C₄₃	—	2	1	no	—
97	C₄×C₄₃	—	2	1	no	—
98	C₄×C₄₃	—	8	4	no	—
99	C₄×C₄₃	#361	1	2	yes	—
100	C₄×C₄₃	#362	1	2	yes	—
101	C₄×C₄₃	#391	1	1	yes	characteristic
102	C₄×C₄₃	#419	1	1	yes	characteristic
103	C₄×C₄₃	#474	1	2	yes	—
104	C₂×#500	D₄₃	1	1	yes	characteristic
105	#2150	—	43	1	no	—
106	C₄×#108	—	43	1	no	—
107	#1387	—	43	4	no	—
108	C₂×C₄₃	—	2	2	no	—
109	C₂×C₄₃	#1988	1	1	yes	characteristic
110	C₂×C₄₃	C₂×#361	1	1	yes	characteristic
111	C₂×C₄₃	C₂×#362	1	1	yes	characteristic
112	C₂×C₄⋊Q₈	—	2	1	no	—
113	C₂×M₃₂	—	43	2	no	—
114	#500	—	2	1	no	—
115	#500	—	2	1	no	—
116	#500	Dic₄₃	1	2	yes	—
117	#500	Dic₄₃	1	2	yes	—
118	C₂×#108	—	43	2	no	—
119	C₂×#107	—	2	1	no	—
120	C₂×#107	C₂×D₄₃	1	2	yes	—
121	C₄×M₁₆	—	86	1	no	—
122	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
123	C₄⋊C₁₆	—	86	1	no	—
124	C₄⋊C₁₆	—	86	1	no	—
125	#341	—	86	1	no	—
126	C₄₃	#13279	1	1	yes	characteristic, complemented, Sylow
127	C₂²×Q₈	—	4	1	no	—
128	C₂×M₁₆	—	86	2	no	—
129	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	—	4	1	no	—
134	C₄⋊Q₈	—	4	1	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	4	1	no	—
137	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
138	M₃₂	—	86	4	no	—
139	C₂×C₁₆	—	86	2	no	—
140	#108	—	43	4	no	—
141	#108	—	86	2	no	—
142	#107	—	2	2	no	—
143	#107	—	2	2	no	—
144	#107	—	2	2	no	—
145	#107	C₂×Dic₄₃	1	4	yes	—
146	#97	—	172	1	no	—
147	C₄×C₈	—	172	1	no	—
148	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
149	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
150	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
151	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
152	C₂×Q₈	—	4	2	no	—
153	C₂×Q₈	—	4	2	no	—
154	C₂×Q₈	—	8	4	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	#10621883	1	1	yes	characteristic
157	C₂²×C₄	#10621883	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
159	M₁₆	—	86	4	no	—
160	M₁₆	—	172	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	172	2	no	—
163	C₂×C₈	C₂²⋊Dic₄₃	1	1	yes	characteristic
164	C₂×C₈	C₂²⋊Dic₄₃	1	1	yes	characteristic
165	C₂×C₈	Q₈×D₄₃	1	2	yes	—
166	C₂×C₈	#10621902	1	2	yes	—
167	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
168	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄²	#10621882	1	1	yes	characteristic
174	C₄²	#10621882	1	1	yes	characteristic
175	C₄²	C₂²⋊Dic₄₃	1	1	yes	characteristic
176	C₄²	C₂²⋊Dic₄₃	1	1	yes	characteristic
177	C₁₆	—	172	2	no	—
178	C₂³	C₂×#10621883	1	1	yes	characteristic
179	Q₈	—	8	4	no	—
180	Q₈	—	8	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	8	4	no	—
185	C₂×C₄	#13063246	1	1	yes	characteristic
186	C₂×C₄	#13063246	1	1	yes	characteristic
187	C₂×C₄	#13063246	1	1	yes	characteristic
188	C₂×C₄	#13063246	1	1	yes	characteristic
189	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
190	C₂×C₄	#13063301	1	2	yes	—
191	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
192	C₂×C₄	#13063356	1	2	yes	—
193	C₈	—	2	1	no	—
194	C₈	—	2	1	no	—
195	C₈	—	172	4	no	—
196	C₈	Q₈×Dic₄₃	1	2	yes	—
197	C₈	#13063314	1	1	yes	characteristic
198	C₈	#13063323	1	1	yes	characteristic
199	C₂²	—	2	1	no	—
200	C₂²	—	2	1	no	—
201	C₂²	C₂×#13063246	1	1	yes	characteristic
202	C₂²	#14583181	1	2	yes	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	8	4	no	—
206	C₄	#14582712	1	2	yes	—
207	C₄	#14582715	1	2	yes	—
208	C₄	#14582806	1	1	yes	characteristic
209	C₄	#14582908	1	1	yes	characteristic
210	C₄	#14583176	1	2	yes	—
211	C₂	—	2	2	no	—
212	C₂	#19135548	1	1	yes	characteristic
213	C₂	C₂×#14582712	1	1	yes	characteristic
214	C₂	C₂×#14582715	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13063246
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂×C₄₃×#500	C₂
socle	C₂²×C₄₃	C₂×#102
prime radical	C₂³×C₄₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×#500	D₄₃
hypocenter	C₄₃	#13279
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
23. #13063261	Q₈	semidirect
126. C₄₃	#13279	semidirect

Maximal subgroups

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

subgroup	index
#19136351	2
C₄×#13063261	2
#19132703	2
C₂×C₄₃×#500	2
#13279	43