Subgroups — #25015165

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	C₂×C₄⋊(C₄₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#19132692	C₂	1	4	yes	maximal
4	C₄×#108×C₄₃	C₂	1	1	yes	maximal, characteristic
5	#14583456	—	2	1	no	—
6	C₂²×C₄₃⋊C₁₆	C₂²	1	2	yes	—
7	#14583304	C₄	1	1	yes	characteristic
8	#14583304	C₄	1	1	yes	characteristic
9	#14583301	—	2	1	no	—
10	#14583301	C₄	1	2	yes	—
11	#14583297	—	2	1	no	—
12	#14582937	—	2	1	no	—
13	#14582937	C₂²	1	2	yes	—
14	C₄⋊(C₄₃⋊C₁₆)	—	2	1	no	—
15	C₄⋊(C₄₃⋊C₁₆)	—	2	1	no	—
16	C₂×#108×C₄₃	C₂²	1	2	yes	—
17	C₄×M₁₆×C₄₃	—	2	1	no	—
18	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
19	C₄₃×#341	—	2	1	no	—
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	C₂×C₄₃⋊C₁₆	—	2	2	no	—
29	C₂×C₄₃⋊C₁₆	—	2	4	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	C₂×M₁₆×C₄₃	—	2	2	no	—
37	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
40	#108×C₄₃	—	2	2	no	—
41	#108×C₄₃	Q₈	1	4	yes	complemented
42	#97×C₄₃	—	4	1	no	—
43	C₄×C₈×C₄₃	—	4	1	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	M₁₆×C₄₃	—	2	4	no	—
54	M₁₆×C₄₃	—	4	2	no	—
55	C₂×C₈×C₄₃	—	2	2	no	—
56	C₂×C₈×C₄₃	—	4	2	no	—
57	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
62	C₂×C₈×C₄₃	#41	1	2	yes	—
63	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₄²×C₄₃	D₈	1	1	yes	characteristic
66	C₄²×C₄₃	D₈	1	1	yes	characteristic
67	C₂²⋊Dic₄₃	—	4	2	no	—
68	C₄⋊Dic₄₃	—	2	2	no	—
69	C₄⋊Dic₄₃	—	2	2	no	—
70	C₄⋊Dic₄₃	—	4	1	no	—
71	C₄⋊Dic₄₃	—	4	1	no	—
72	Dic₄₃⋊C₄	—	4	2	no	—
73	Dic₄₃⋊C₄	—	4	2	no	—
74	C₄×Dic₄₃	—	4	2	no	—
75	C₄₃⋊C₁₆	—	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	—	4	4	no	—
94	C₈×C₄₃	C₄×Q₈	1	2	yes	—
95	C₈×C₄₃	#124	1	1	yes	characteristic
96	C₈×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
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	—	43	1	no	—
113	C₄×#108	D₄₃	1	1	yes	characteristic
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	—	43	2	no	—
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	C₂×D₄₃	1	2	yes	—
127	#427	—	43	2	no	—
128	#427	—	86	1	no	—
129	C₄×M₁₆	—	2	1	no	—
130	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
131	C₄⋊C₁₆	—	86	1	no	—
132	C₄⋊C₁₆	—	86	1	no	—
133	#341	—	2	1	no	—
134	C₄₃	#13274	1	1	yes	characteristic, complemented, Sylow
135	C₂×M₁₆	—	2	2	no	—
136	C₂²×C₈	C₂²⋊D₄₃	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	C₂²⋊D₄₃	1	1	yes	characteristic
147	C₂×C₁₆	—	86	2	no	—
148	C₂×C₁₆	—	86	4	no	—
149	#108	—	2	2	no	—
150	#108	Dic₈₆	1	4	yes	complemented
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	—	4	1	no	—
158	C₄×C₈	—	4	1	no	—
159	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
160	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
161	C₄×C₈	D₁₇₂	1	1	yes	characteristic
162	C₄×C₈	D₁₇₂	1	1	yes	characteristic
163	C₂×Q₈	—	172	1	no	—
164	C₂×Q₈	—	172	1	no	—
165	C₂²×C₄	#10621884	1	1	yes	characteristic
166	C₂²×C₄	#10621884	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
168	M₁₆	—	2	4	no	—
169	M₁₆	—	4	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	4	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×Dic₈₆	1	2	yes	—
175	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
176	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
177	C₂×C₈	#10621898	1	2	yes	—
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	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
187	C₄²	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
188	C₄²	#10621882	1	1	yes	characteristic
189	C₄²	#10621882	1	1	yes	characteristic
190	C₁₆	—	172	2	no	—
191	C₂³	C₂×#10621884	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	#13063245	1	1	yes	characteristic
199	C₂×C₄	#13063245	1	1	yes	characteristic
200	C₂×C₄	#13063245	1	1	yes	characteristic
201	C₂×C₄	#13063245	1	1	yes	characteristic
202	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
203	C₂×C₄	#13063304	1	2	yes	—
204	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
205	C₂×C₄	#13063355	1	2	yes	—
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	4	4	no	—
209	C₈	C₄×Dic₈₆	1	2	yes	—
210	C₈	#13063313	1	1	yes	characteristic
211	C₈	#13063325	1	1	yes	characteristic
212	C₂²	—	2	1	no	—
213	C₂²	—	2	1	no	—
214	C₂²	C₂×#13063245	1	1	yes	characteristic
215	C₂²	#14583184	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	#14582705	1	2	yes	—
222	C₄	#14582805	1	1	yes	characteristic
223	C₄	#14582907	1	1	yes	characteristic
224	C₄	#14583179	1	2	yes	—
225	C₂	—	2	2	no	—
226	C₂	#19135547	1	1	yes	characteristic
227	C₂	C₂×#14582705	1	1	yes	characteristic
228	C₂	#19136322	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#10621882
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄×#108×C₄₃	C₂
socle	C₂²×C₄₃	C₂×#102
prime radical	C₂³×C₄₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₄×#108	D₄₃
hypocenter	C₄₃	#13274
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
41. #108×C₄₃	Q₈	semidirect
134. C₄₃	#13274	semidirect
150. #108	Dic₈₆	semidirect

Maximal subgroups

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

subgroup	index
#19145109	2
C₂×C₄⋊(C₄₃⋊C₁₆)	2
#19132692	2
C₄×#108×C₄₃	2
#13274	43