Subgroups — #25012695

Order 11008.

430 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583017	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582720	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14582708	C₂	1	2	yes	maximal
4	#19132693	C₂	1	2	yes	maximal, complemented
5	C₄₃×#1844	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13063347	C₂²	1	1	yes	characteristic
7	C₂×#13063331	—	2	1	no	—
8	C₂×#13063265	C₂²	1	2	yes	—
9	#14583017	—	2	2	no	—
10	#14583017	C₄	1	4	yes	complemented
11	C₂×#13063261	C₂²	1	1	yes	characteristic
12	C₂×#13063253	—	2	1	no	—
13	#14582720	—	2	2	no	—
14	#14582708	—	2	4	no	—
15	C₂×#106×C₄₃	—	2	1	no	—
16	C₂×#106×C₄₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
18	C₄₃×#384	—	2	1	no	—
19	C₂×#10621898	—	2	1	no	—
20	#13063347	—	2	1	no	—
21	#13063347	—	2	1	no	—
22	#13063347	C₂×C₄	1	2	yes	—
23	#13063347	D₄	1	2	yes	—
24	C₂×Dic₁₇₂	Q₈	1	1	yes	characteristic
25	C₂×#10621873	—	2	1	no	—
26	C₂×D₃₄₄	Q₈	1	1	yes	characteristic
27	#13063331	—	2	4	no	—
28	#13063331	—	4	2	no	—
29	C₂×#10621877	—	2	1	no	—
30	C₂×#10621872	—	4	1	no	—
31	C₂×C₈×D₄₃	—	4	1	no	—
32	#13063265	—	2	2	no	—
33	#13063265	—	2	2	no	—
34	C₂×C₄₃⋊C₁₆	—	2	2	no	—
35	#13063261	—	2	1	no	—
36	#13063261	C₂×C₄	1	2	yes	—
37	#13063253	—	2	4	no	—
38	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
41	C₂×C₄⋊C₄×C₄₃	—	4	2	no	—
42	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
43	#106×C₄₃	—	2	2	no	—
44	#106×C₄₃	—	2	2	no	—
45	#106×C₄₃	—	4	2	no	—
46	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₄₃	D₄	1	2	yes	—
48	C₂²×C₄×C₄₃	—	2	1	no	—
49	C₂²×C₄×C₄₃	—	4	2	no	—
50	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₂²⋊D₄₃	—	4	1	no	—
52	#10621898	—	2	2	no	—
53	#10621898	—	2	2	no	—
54	#10621898	—	4	1	no	—
55	#10621898	—	4	1	no	—
56	C₂×D₁₇₂	—	2	1	no	—
57	C₂×C₄×D₄₃	—	4	1	no	—
58	C₂×Dic₈₆	—	2	1	no	—
59	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₄₃	D₈	1	2	yes	—
61	C₂×C₈×C₄₃	SD₁₆	1	1	yes	characteristic
62	C₂×C₈×C₄₃	SD₁₆	1	1	yes	characteristic
63	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
68	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
69	C₄⋊C₄×C₄₃	—	4	2	no	—
70	C₄⋊C₄×C₄₃	—	4	2	no	—
71	C₄⋊C₄×C₄₃	—	8	4	no	—
72	C₄²×C₄₃	—	2	1	no	—
73	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
74	#10621877	—	2	2	no	—
75	#10621877	—	4	1	no	—
76	C₂×C₄₃⋊C₈	—	4	1	no	—
77	Dic₁₇₂	—	2	1	no	—
78	Dic₁₇₂	C₄⋊C₄	1	2	yes	complemented
79	D₃₄₄	—	2	1	no	—
80	D₃₄₄	C₄⋊C₄	1	2	yes	complemented
81	#10621873	—	2	2	no	—
82	#10621873	—	2	2	no	—
83	#10621872	—	4	4	no	—
84	C₈×D₄₃	—	4	4	no	—
85	C₄₃⋊C₁₆	—	4	2	no	—
86	C₂³×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
87	C₂²×D₄₃	—	4	1	no	—
88	C₂×C₄×C₄₃	—	2	4	no	—
89	C₂×C₄×C₄₃	—	2	2	no	—
90	C₂×C₄×C₄₃	—	8	4	no	—
91	C₂×C₄×C₄₃	—	8	2	no	—
92	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
94	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
98	C₂²⋊D₄₃	—	4	2	no	—
99	C₂²⋊D₄₃	—	4	2	no	—
100	C₂×Dic₄₃	—	4	1	no	—
101	D₁₇₂	—	2	2	no	—
102	D₁₇₂	—	4	1	no	—
103	C₄×D₄₃	—	4	2	no	—
104	C₄×D₄₃	—	4	2	no	—
105	Dic₈₆	—	2	2	no	—
106	Dic₈₆	—	4	1	no	—
107	C₈×C₄₃	#102	1	2	yes	—
108	C₈×C₄₃	#102	1	2	yes	—
109	C₈×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₈×C₄₃	#117	1	1	yes	characteristic
111	C₈×C₄₃	#121	1	1	yes	characteristic
112	C₈×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
113	C₄₃⋊C₈	—	4	2	no	—
114	#12823	—	43	1	no	maximal, Sylow
115	C₂²×C₄₃	—	2	2	no	—
116	C₂²×C₄₃	#417	1	1	yes	characteristic
117	C₂²×C₄₃	#459	1	1	yes	characteristic
118	C₂²×C₄₃	#476	1	1	yes	characteristic
119	C₂×D₄₃	—	4	2	no	—
120	C₂×D₄₃	—	8	2	no	—
121	C₄×C₄₃	—	2	4	no	—
122	C₄×C₄₃	—	16	2	no	—
123	C₄×C₄₃	#380	1	1	yes	characteristic
124	C₄×C₄₃	C₂×#102	1	1	yes	characteristic
125	C₄×C₄₃	#461	1	1	yes	characteristic
126	C₄×C₄₃	#474	1	1	yes	characteristic
127	Dic₄₃	—	4	2	no	—
128	C₂×#443	—	43	1	no	—
129	C₂×#364	—	43	1	no	—
130	C₂×#360	—	43	2	no	—
131	#1844	D₄₃	1	1	yes	characteristic, complemented
132	#1383	—	43	2	no	—
133	C₂×C₄₃	—	2	1	no	—
134	C₂×C₄₃	—	2	1	no	—
135	C₂×C₄₃	#1802	1	1	yes	characteristic
136	C₂×C₄₃	#2230	1	2	yes	—
137	D₄₃	—	8	2	no	—
138	C₂×#135	—	43	1	no	—
139	C₂×#131	—	86	1	no	—
140	C₂×M₃₂	—	43	2	no	—
141	#443	—	43	4	no	—
142	#443	—	86	2	no	—
143	C₂×#108	—	43	1	no	—
144	C₂×#106	—	2	1	no	—
145	C₂×#106	C₂×D₄₃	1	2	yes	—
146	C₂×#104	—	86	1	no	—
147	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
148	#384	—	2	1	no	—
149	#364	—	86	2	no	—
150	#360	—	86	4	no	—
151	C₄₃	#12823	1	1	yes	characteristic, complemented, Sylow
152	C₂×#41	—	86	1	no	—
153	#135	—	43	2	no	—
154	#135	—	43	2	no	—
155	#135	—	86	1	no	—
156	#135	—	86	1	no	—
157	C₂×Q₁₆	—	43	1	no	—
158	C₂×SD₁₆	—	86	1	no	—
159	C₂×D₈	—	43	1	no	—
160	#131	—	86	4	no	—
161	#131	—	172	2	no	—
162	C₂×M₁₆	—	86	1	no	—
163	C₂×M₁₆	—	172	1	no	—
164	C₂²×C₈	—	172	1	no	—
165	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
166	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	—	4	2	no	—
169	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
170	M₃₂	—	86	2	no	—
171	M₃₂	—	86	2	no	—
172	C₂×C₁₆	—	86	2	no	—
173	#108	—	43	2	no	—
174	#108	—	86	1	no	—
175	#106	—	2	2	no	—
176	#106	—	2	2	no	—
177	#106	—	4	2	no	—
178	#104	—	86	4	no	—
179	C₄×C₈	C₄×D₄₃	1	2	yes	—
180	C₄×C₈	D₁₇₂	1	2	yes	—
181	#41	—	86	2	no	—
182	#41	—	86	2	no	—
183	#41	—	172	1	no	—
184	#41	—	172	1	no	—
185	C₂×Q₈	—	86	1	no	—
186	C₂×D₄	—	86	1	no	—
187	C₂×D₄	—	172	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	172	1	no	—
191	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
192	Q₁₆	—	43	2	no	—
193	Q₁₆	—	86	1	no	—
194	SD₁₆	—	86	2	no	—
195	SD₁₆	—	86	2	no	—
196	D₈	—	43	2	no	—
197	D₈	—	86	1	no	—
198	M₁₆	—	86	2	no	—
199	M₁₆	—	172	1	no	—
200	M₁₆	—	172	4	no	—
201	C₂×C₈	—	172	1	no	—
202	C₂×C₈	—	172	4	no	—
203	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
204	C₂×C₈	#10621882	1	2	yes	—
205	C₂×C₈	#10621884	1	1	yes	characteristic
206	C₂×C₈	#10621884	1	1	yes	characteristic
207	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
208	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
209	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
210	C₂×C₈	#10621900	1	1	yes	characteristic
211	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
212	C₂×C₈	#10621902	1	1	yes	characteristic
213	C₄⋊C₄	—	4	2	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	8	4	no	—
216	C₄²	—	2	1	no	—
217	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
218	C₁₆	—	172	2	no	—
219	C₂³	—	172	1	no	—
220	C₂³	#13063351	1	1	yes	characteristic
221	Q₈	—	86	2	no	—
222	Q₈	—	172	1	no	—
223	D₄	—	86	2	no	—
224	D₄	—	172	2	no	—
225	D₄	—	172	2	no	—
226	D₄	—	172	1	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	8	4	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	—	172	1	no	—
232	C₂×C₄	—	172	2	no	—
233	C₂×C₄	—	172	2	no	—
234	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
235	C₂×C₄	#13063296	1	1	yes	characteristic
236	C₂×C₄	#13063301	1	1	yes	characteristic
237	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
238	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
239	C₂×C₄	#13063356	1	1	yes	characteristic
240	C₈	—	172	2	no	—
241	C₈	#13063245	1	2	yes	—
242	C₈	#13063245	1	2	yes	—
243	C₈	C₄⋊C₄×D₄₃	1	1	yes	characteristic
244	C₈	#13063283	1	1	yes	characteristic
245	C₈	#13063300	1	1	yes	characteristic
246	C₈	#13063305	1	1	yes	characteristic
247	C₂²	—	2	2	no	—
248	C₂²	—	344	2	no	—
249	C₂²	—	172	2	no	—
250	C₂²	#14582898	1	1	yes	characteristic
251	C₂²	#14583091	1	1	yes	characteristic
252	C₂²	#14583186	1	1	yes	characteristic
253	C₄	—	2	4	no	—
254	C₄	—	16	2	no	—
255	C₄	—	172	2	no	—
256	C₄	#14582756	1	1	yes	characteristic
257	C₄	C₂×#13063245	1	1	yes	characteristic
258	C₄	#14583101	1	1	yes	characteristic
259	C₄	#14583176	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	344	2	no	—
263	C₂	#19134592	1	1	yes	characteristic
264	C₂	#19136763	1	2	yes	—

	subgroup	G/H
center	C₂²	#14583186
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1844	C₂
socle	C₂²×C₄₃	#476
prime radical	C₂×#13063347	C₂²
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1844	D₄₃
hypocenter	C₄₃	#12823
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1844

Complements

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

N	H ≅ G/N	type
2. C₂×#14582720	C₂	semidirect
4. #19132693	C₂	semidirect
5. C₄₃×#1844	C₂	semidirect
10. #14583017	C₄	semidirect
78. Dic₁₇₂	C₄⋊C₄	semidirect
80. D₃₄₄	C₄⋊C₄	semidirect
131. #1844	D₄₃	semidirect
151. C₄₃	#12823	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583017	2
C₂×#14582720	2
C₂×#14582708	2
#19132693	2
C₄₃×#1844	2
#12823	43