Subgroups — #25015164

Order 11008.

338 conjugacy classes of subgroups.

The subgroup lattice has 230 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	C₄×#13063261	C₂	1	1	yes	maximal, characteristic
3	#19132691	C₂	1	4	yes	maximal
4	C₂×C₄₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	#14583456	—	2	1	no	—
6	#14583304	C₄	1	1	yes	characteristic
7	#14583304	C₄	1	1	yes	characteristic
8	#14583301	—	2	1	no	—
9	#14583301	C₄	1	2	yes	—
10	#14583297	—	2	1	no	—
11	C₂×#13063261	C₂²	1	2	yes	—
12	#14582937	—	2	1	no	—
13	#14582937	C₂²	1	2	yes	—
14	C₄×#10621877	—	2	1	no	—
15	#14582664	—	2	1	no	—
16	C₂²×C₁₆×C₄₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
18	C₄₃×C₄⋊C₁₆	—	2	1	no	—
19	C₄₃×C₄⋊C₁₆	—	2	1	no	—
20	C₂×#10621877	—	2	2	no	—
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	#13063261	—	2	2	no	—
30	#13063261	Q₈	1	4	yes	complemented
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	#13063236	—	4	1	no	—
38	C₄×C₄₃⋊C₈	—	4	1	no	—
39	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
42	C₂×C₁₆×C₄₃	—	2	2	no	—
43	C₂×C₁₆×C₄₃	—	2	4	no	—
44	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
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×Dic₈₆	—	4	1	no	—
52	C₂×Dic₈₆	—	4	1	no	—
53	C₂×C₈×C₄₃	—	2	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	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
59	C₂×C₈×C₄₃	#41	1	2	yes	—
60	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₄₃	D₈	1	1	yes	characteristic
63	C₄²×C₄₃	D₈	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	#10621877	—	2	4	no	—
73	#10621877	—	4	2	no	—
74	C₂×C₄₃⋊C₈	—	4	2	no	—
75	C₁₆×C₄₃	—	4	2	no	—
76	C₂³×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₄₃	—	2	2	no	—
78	C₂×C₄×C₄₃	—	4	1	no	—
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	#102	1	1	yes	characteristic
83	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
85	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₄₃	#137	1	2	yes	—
87	C₂×Dic₄₃	—	4	2	no	—
88	C₂×Dic₄₃	—	4	2	no	—
89	Dic₈₆	—	8	2	no	—
90	Dic₈₆	—	8	2	no	—
91	C₈×C₄₃	—	2	1	no	—
92	C₈×C₄₃	—	2	1	no	—
93	C₈×C₄₃	C₄×Q₈	1	2	yes	—
94	C₈×C₄₃	#124	1	1	yes	characteristic
95	C₈×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
96	C₄₃⋊C₈	—	4	4	no	—
97	#13274	—	43	1	no	maximal, Sylow
98	C₂²×C₄₃	—	2	1	no	—
99	C₂²×C₄₃	—	2	1	no	—
100	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
101	C₂²×C₄₃	#475	1	2	yes	—
102	C₄×C₄₃	—	2	1	no	—
103	C₄×C₄₃	—	2	1	no	—
104	C₄×C₄₃	—	2	1	no	—
105	C₄×C₄₃	#359	1	2	yes	—
106	C₄×C₄₃	#391	1	1	yes	characteristic
107	C₄×C₄₃	#419	1	1	yes	characteristic
108	C₄×C₄₃	#474	1	2	yes	—
109	Dic₄₃	—	8	2	no	—
110	Dic₄₃	—	8	2	no	—
111	#3160	—	43	1	no	—
112	C₂×C₄⋊C₁₆	D₄₃	1	1	yes	characteristic
113	C₄×#108	—	43	1	no	—
114	#1382	—	43	4	no	—
115	C₂×C₄₃	—	2	2	no	—
116	C₂×C₄₃	#1988	1	1	yes	characteristic
117	C₂×C₄₃	C₂×#359	1	1	yes	characteristic
118	C₂×C₄₃	#2143	1	1	yes	characteristic
119	#533	—	86	1	no	—
120	C₂²×C₁₆	C₂×D₄₃	1	2	yes	—
121	#500	—	43	1	no	—
122	#500	—	43	1	no	—
123	#499	—	43	2	no	—
124	#499	—	86	1	no	—
125	#498	—	86	1	no	—
126	C₂×#108	—	43	2	no	—
127	#427	—	43	2	no	—
128	#427	—	86	1	no	—
129	C₄×M₁₆	—	86	1	no	—
130	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
131	C₄⋊C₁₆	—	2	1	no	—
132	C₄⋊C₁₆	—	2	1	no	—
133	#341	—	86	1	no	—
134	C₄₃	#13274	1	1	yes	characteristic, complemented, Sylow
135	C₂×M₁₆	—	86	2	no	—
136	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
137	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
138	C₄⋊Q₈	—	86	1	no	—
139	C₄⋊Q₈	—	86	1	no	—
140	#125	—	86	2	no	—
141	C₂²⋊Q₈	—	172	1	no	—
142	C₂²⋊Q₈	—	172	1	no	—
143	C₄×Q₈	—	172	1	no	—
144	C₄×Q₈	—	172	1	no	—
145	#117	—	86	2	no	—
146	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
147	C₂×C₁₆	—	2	4	no	—
148	C₂×C₁₆	—	2	2	no	—
149	#108	—	43	4	no	—
150	#108	—	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	#97	—	172	1	no	—
158	C₄×C₈	—	172	1	no	—
159	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
160	C₄×C₈	C₄×D₄₃	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	#10621873	1	1	yes	characteristic
166	C₂²×C₄	#10621873	1	1	yes	characteristic
167	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
168	M₁₆	—	86	4	no	—
169	M₁₆	—	172	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	172	2	no	—
172	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
173	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
174	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
175	C₂×C₈	#10621900	1	2	yes	—
176	C₂×C₈	Q₈×D₄₃	1	2	yes	—
177	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
178	C₄⋊C₄	—	86	2	no	—
179	C₄⋊C₄	—	86	2	no	—
180	C₄⋊C₄	—	172	2	no	—
181	C₄⋊C₄	—	172	2	no	—
182	C₄⋊C₄	—	172	1	no	—
183	C₄⋊C₄	—	172	1	no	—
184	C₂²⋊C₄	—	172	2	no	—
185	C₄²	—	172	2	no	—
186	C₄²	D₃₄₄	1	1	yes	characteristic
187	C₄²	D₃₄₄	1	1	yes	characteristic
188	C₄²	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
189	C₄²	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
190	C₁₆	—	4	2	no	—
191	C₂³	C₂×#10621873	1	1	yes	characteristic
192	Q₈	—	344	2	no	—
193	Q₈	—	344	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	172	2	no	—
197	C₂×C₄	—	172	2	no	—
198	C₂×C₄	#13063247	1	1	yes	characteristic
199	C₂×C₄	#13063247	1	1	yes	characteristic
200	C₂×C₄	#13063247	1	1	yes	characteristic
201	C₂×C₄	#13063247	1	1	yes	characteristic
202	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
203	C₂×C₄	#13063305	1	2	yes	—
204	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
205	C₂×C₄	#13063358	1	2	yes	—
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	172	4	no	—
209	C₈	#13063290	1	2	yes	—
210	C₈	#13063312	1	1	yes	characteristic
211	C₈	#13063323	1	1	yes	characteristic
212	C₂²	—	2	1	no	—
213	C₂²	—	2	1	no	—
214	C₂²	C₂×#13063247	1	1	yes	characteristic
215	C₂²	#14583185	1	2	yes	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	344	2	no	—
220	C₄	—	344	2	no	—
221	C₄	#14582707	1	2	yes	—
222	C₄	#14582804	1	1	yes	characteristic
223	C₄	#14582909	1	1	yes	characteristic
224	C₄	#14583180	1	2	yes	—
225	C₂	—	2	2	no	—
226	C₂	#19135546	1	1	yes	characteristic
227	C₂	C₂×#14582707	1	1	yes	characteristic
228	C₂	#19136324	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃₄₄
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂×C₄₃×C₄⋊C₁₆	C₂
socle	C₂²×C₄₃	C₂×#102
prime radical	C₂³×C₄₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₄₃
hypocenter	C₄₃	#13274
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
30. #13063261	Q₈	semidirect
134. C₄₃	#13274	semidirect

Maximal subgroups

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

subgroup	index
#19145109	2
C₄×#13063261	2
#19132691	2
C₂×C₄₃×C₄⋊C₁₆	2
#13274	43