Subgroups — #25105052

Order 11008.

1398 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583276	C₂	1	1	yes	maximal, characteristic
2	C₂×#14583012	C₂	1	4	yes	maximal, complemented
3	C₄×#13063322	C₂	1	2	yes	maximal, complemented
4	#19135471	C₂	1	4	yes	maximal, complemented
5	#19135401	C₂	1	2	yes	maximal, complemented
6	#19134752	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#97×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₃₄₄	C₂²	1	4	yes	—
9	C₂×#13063322	C₂²	1	2	yes	—
10	C₂×C₄×D₁₇₂	—	2	4	no	—
11	C₂×C₄×D₁₇₂	C₂²	1	4	yes	—
12	#14583276	C₄	1	8	yes	complemented
13	#14583012	—	2	16	no	—
14	C₂×#13063257	C₂²	1	4	yes	—
15	C₂×#13063247	C₂²	1	8	yes	—
16	#14582801	—	2	4	no	—
17	#14582801	C₂²	1	4	yes	—
18	#14582769	C₂²	1	2	yes	—
19	C₂×#97×C₄₃	C₂²	1	2	yes	—
20	C₂×#97×C₄₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₄³×C₄₃	C₂²	1	1	yes	characteristic
24	C₂²×D₁₇₂	—	2	4	no	—
25	C₂²×D₁₇₂	C₂³	1	4	yes	—
26	C₂²×C₄×D₄₃	—	2	4	no	—
27	C₂×D₃₄₄	—	2	8	no	—
28	C₂×D₃₄₄	—	2	8	no	—
29	C₂×D₃₄₄	C₂×C₄	1	16	yes	complemented
30	#13063322	—	2	4	no	—
31	#13063322	C₂×C₄	1	8	yes	complemented
32	C₄×D₁₇₂	—	2	16	no	—
33	C₄×D₁₇₂	—	4	16	no	—
34	C₂×C₄⋊Dic₄₃	—	2	4	no	—
35	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
36	C₂×(C₂×D₄₃)⋊C₄	—	2	4	no	—
37	C₂×(C₂×D₄₃)⋊C₄	—	2	8	no	—
38	#13063257	—	2	8	no	—
39	#13063247	—	2	16	no	—
40	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
41	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
42	C₂×C₄²×C₄₃	D₄	1	4	yes	—
43	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
46	#97×C₄₃	—	2	4	no	—
47	#97×C₄₃	D₄	1	8	yes	complemented
48	C₄×C₈×C₄₃	—	2	2	no	—
49	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	complemented
50	C₂³×D₄₃	—	2	4	no	—
51	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
55	C₂²×Dic₄₃	—	2	4	no	—
56	C₂×D₁₇₂	—	2	16	no	—
57	C₂×D₁₇₂	—	2	4	no	—
58	C₂×D₁₇₂	—	2	8	no	—
59	C₂×D₁₇₂	—	2	8	no	—
60	C₂×D₁₇₂	—	4	8	no	—
61	C₂×D₁₇₂	—	4	8	no	—
62	C₂×D₁₇₂	C₂²×C₄	1	8	yes	—
63	C₂×C₄×D₄₃	—	4	16	no	—
64	C₂×C₄×D₄₃	—	4	8	no	—
65	C₂×C₈×C₄₃	—	2	2	no	—
66	C₂×C₈×C₄₃	—	2	2	no	—
67	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
68	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
69	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
70	C₂×C₈×C₄₃	#41	1	4	yes	—
71	C₄²×C₄₃	—	2	8	no	—
72	C₄²×C₄₃	—	2	2	no	—
73	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
74	C₄²×C₄₃	C₂×D₄	1	4	yes	—
75	C₄²×C₄₃	#41	1	2	yes	—
76	(C₂×D₄₃)⋊C₄	—	4	16	no	—
77	(C₂×D₄₃)⋊C₄	—	4	8	no	—
78	C₄⋊Dic₄₃	—	2	8	no	—
79	C₄⋊Dic₄₃	—	4	8	no	—
80	D₃₄₄	—	2	32	no	—
81	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
82	C₂²×D₄₃	—	2	8	no	—
83	C₂²×D₄₃	—	4	4	no	—
84	C₂²×D₄₃	—	4	4	no	—
85	C₂²×D₄₃	—	4	8	no	—
86	C₂²×D₄₃	—	4	8	no	—
87	C₂×C₄×C₄₃	—	2	4	no	—
88	C₂×C₄×C₄₃	—	2	2	no	—
89	C₂×C₄×C₄₃	—	2	4	no	—
90	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
91	C₂×C₄×C₄₃	#127	1	4	yes	—
92	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	8	yes	—
93	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
95	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
96	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
97	C₂×Dic₄₃	—	4	4	no	—
98	C₂×Dic₄₃	—	4	8	no	—
99	D₁₇₂	—	2	16	no	—
100	D₁₇₂	—	4	8	no	—
101	D₁₇₂	—	4	32	no	—
102	C₄×D₄₃	—	8	16	no	—
103	C₈×C₄₃	—	2	4	no	—
104	C₈×C₄₃	C₄×D₄	1	8	yes	complemented
105	#21690	—	43	1	no	maximal, Sylow
106	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
107	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
108	C₂²×C₄₃	#532	1	1	yes	characteristic
109	C₂²×C₄₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
110	C₂×D₄₃	—	4	8	no	—
111	C₂×D₄₃	—	4	16	no	—
112	C₂×D₄₃	—	8	4	no	—
113	C₂×D₄₃	—	8	4	no	—
114	C₂×D₄₃	—	8	8	no	—
115	C₄×C₄₃	—	2	4	no	—
116	C₄×C₄₃	—	4	2	no	—
117	C₄×C₄₃	#442	1	8	yes	complemented
118	C₄×C₄₃	C₂×C₄×D₄	1	2	yes	—
119	C₄×C₄₃	#532	1	2	yes	—
120	Dic₄₃	—	8	4	no	—
121	C₂×#493	—	43	1	no	—
122	C₂×#442	—	43	4	no	—
123	C₄×#127	—	43	2	no	—
124	#1972	—	43	4	no	—
125	#1957	—	43	2	no	—
126	#1844	—	43	1	no	—
127	C₄×#97	D₄₃	1	1	yes	characteristic, complemented
128	C₂×C₄₃	C₄×#127	1	1	yes	characteristic
129	C₂×C₄₃	C₂×#442	1	2	yes	—
130	C₂×C₄₃	#3151	1	2	yes	—
131	C₂×C₄₃	#3184	1	2	yes	—
132	D₄₃	—	8	8	no	—
133	C₂²×D₈	—	43	4	no	—
134	C₂×#127	—	43	2	no	—
135	C₂×C₄×D₄	—	43	4	no	—
136	C₂×C₄×D₄	—	86	4	no	—
137	#493	—	43	8	no	—
138	#442	—	86	16	no	—
139	C₂×#106	—	43	4	no	—
140	C₂×#102	—	43	8	no	—
141	C₂×#97	C₂×D₄₃	1	2	yes	—
142	C₂×#97	C₂×D₄₃	1	2	yes	—
143	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
144	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
145	#390	—	43	4	no	—
146	#390	—	86	4	no	—
147	#384	—	43	2	no	—
148	C₄³	C₂×D₄₃	1	1	yes	characteristic
149	C₄₃	#21690	1	1	yes	characteristic, complemented, Sylow
150	C₂²×D₄	—	43	4	no	—
151	C₂²×D₄	—	86	4	no	—
152	C₂³×C₄	—	86	4	no	—
153	C₂×D₈	—	43	16	no	—
154	C₂×D₈	—	86	8	no	—
155	C₂×D₈	—	86	8	no	—
156	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
157	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
158	#127	—	43	8	no	—
159	#127	—	86	4	no	—
160	C₄×D₄	—	86	16	no	—
161	C₄×D₄	—	172	16	no	—
162	C₂×C₄⋊C₄	—	43	4	no	—
163	C₂×C₄⋊C₄	—	86	4	no	—
164	C₂×C₂²⋊C₄	—	86	4	no	—
165	C₂×C₂²⋊C₄	—	86	8	no	—
166	C₂×C₄²	D₁₇₂	1	4	yes	—
167	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
168	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
169	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
170	#106	—	86	8	no	—
171	#102	—	86	16	no	—
172	#97	—	2	4	no	—
173	#97	D₁₇₂	1	8	yes	complemented
174	C₄×C₈	—	2	2	no	—
175	C₄×C₈	C₄×D₄₃	1	4	yes	complemented
176	C₂⁴	—	86	4	no	—
177	C₂×D₄	—	43	8	no	—
178	C₂×D₄	—	86	16	no	—
179	C₂×D₄	—	86	8	no	—
180	C₂×D₄	—	86	8	no	—
181	C₂×D₄	—	86	4	no	—
182	C₂×D₄	—	172	8	no	—
183	C₂×D₄	—	172	8	no	—
184	C₂²×C₄	—	86	4	no	—
185	C₂²×C₄	—	172	16	no	—
186	C₂²×C₄	—	172	8	no	—
187	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
188	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
189	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
190	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
191	D₈	—	86	32	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	C₂×C₄×D₄₃	1	4	yes	—
195	C₂×C₈	C₂×D₁₇₂	1	4	yes	—
196	C₂×C₈	C₂×D₁₇₂	1	4	yes	—
197	C₂×C₈	#10621898	1	4	yes	—
198	C₄⋊C₄	—	86	8	no	—
199	C₄⋊C₄	—	172	8	no	—
200	C₂²⋊C₄	—	172	16	no	—
201	C₂²⋊C₄	—	172	8	no	—
202	C₄²	—	2	8	no	—
203	C₄²	—	2	2	no	—
204	C₄²	C₂×C₄×D₄₃	1	2	yes	—
205	C₄²	C₂×D₁₇₂	1	4	yes	—
206	C₄²	#10621898	1	2	yes	—
207	C₂³	—	86	8	no	—
208	C₂³	—	172	8	no	—
209	C₂³	—	172	8	no	—
210	C₂³	—	172	4	no	—
211	C₂³	—	172	4	no	—
212	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
213	D₄	—	86	16	no	—
214	D₄	—	172	8	no	—
215	D₄	—	172	32	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	344	16	no	—
220	C₂×C₄	—	172	8	no	—
221	C₂×C₄	—	172	4	no	—
222	C₂×C₄	C₄×D₁₇₂	1	4	yes	—
223	C₂×C₄	#13063322	1	4	yes	—
224	C₂×C₄	#13063353	1	8	yes	—
225	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
226	C₂×C₄	C₂²×D₁₇₂	1	2	yes	—
227	C₂×C₄	C₂×#10621898	1	2	yes	—
228	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
229	C₈	—	2	4	no	—
230	C₈	C₄×D₁₇₂	1	8	yes	complemented
231	C₂²	—	344	4	no	—
232	C₂²	—	344	4	no	—
233	C₂²	—	344	8	no	—
234	C₂²	—	172	8	no	—
235	C₂²	—	172	16	no	—
236	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
237	C₂²	C₂×#13063322	1	1	yes	characteristic
238	C₂²	#14583449	1	1	yes	characteristic
239	C₂²	C₂×#13063353	1	4	yes	—
240	C₄	—	2	4	no	—
241	C₄	—	4	2	no	—
242	C₄	—	344	4	no	—
243	C₄	#14583012	1	8	yes	complemented
244	C₄	C₂×C₄×D₁₇₂	1	2	yes	—
245	C₄	#14583449	1	2	yes	—
246	C₂	—	344	8	no	—
247	C₂	C₄×#13063322	1	1	yes	characteristic
248	C₂	C₂×#14583012	1	2	yes	—
249	C₂	#19145047	1	2	yes	—
250	C₂	#19145333	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄×#97×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂×#14583276	C₂
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄×#97	D₄₃
hypocenter	C₄₃	#21690
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
2. C₂×#14583012	C₂	semidirect
3. C₄×#13063322	C₂	semidirect
4. #19135471	C₂	semidirect
5. #19135401	C₂	semidirect
6. #19134752	C₂	semidirect
7. C₄×#97×C₄₃	C₂	semidirect
12. #14583276	C₄	semidirect
29. C₂×D₃₄₄	C₂×C₄	semidirect
31. #13063322	C₂×C₄	semidirect
47. #97×C₄₃	D₄	semidirect
49. C₄×C₈×C₄₃	C₂×C₄	semidirect
104. C₈×C₄₃	C₄×D₄	semidirect
117. C₄×C₄₃	#442	semidirect
127. C₄×#97	D₄₃	semidirect
149. C₄₃	#21690	semidirect
173. #97	D₁₇₂	semidirect
175. C₄×C₈	C₄×D₄₃	semidirect
230. C₈	C₄×D₁₇₂	semidirect
243. C₄	#14583012	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583276	2
C₂×#14583012	2
C₄×#13063322	2
#19135471	2
#19135401	2
#19134752	2
C₄×#97×C₄₃	2
#21690	43