Subgroups — #25011812

Order 11008.

526 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₃₄₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582701	C₂	1	2	yes	maximal
3	C₄×#13063259	C₂	1	1	yes	maximal, characteristic, complemented
4	#19132687	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₁₇₂	—	2	1	no	—
8	C₄×D₃₄₄	—	2	2	no	—
9	C₄×D₃₄₄	C₄	1	4	yes	complemented
10	C₂×#13063259	C₂²	1	1	yes	characteristic
11	C₂×#13063259	C₂²	1	2	yes	—
12	C₂×#13063247	—	2	1	no	—
13	C₄×C₄⋊Dic₄₃	—	2	1	no	—
14	#14582701	C₄	1	8	yes	complemented
15	#14582662	—	2	1	no	—
16	C₂²×C₁₆×C₄₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
18	C₄₃×#346	—	2	2	no	—
19	C₂²×D₁₇₂	—	2	1	no	—
20	C₂²×C₄×D₄₃	—	4	1	no	—
21	C₂×D₃₄₄	—	2	1	no	—
22	C₂×D₃₄₄	—	2	2	no	—
23	C₂×D₃₄₄	C₂×C₄	1	4	yes	—
24	C₂×D₃₄₄	C₂×C₄	1	2	yes	—
25	C₄×D₁₇₂	—	2	4	no	—
26	C₄×D₁₇₂	—	4	2	no	—
27	C₂×C₄⋊Dic₄₃	—	2	1	no	—
28	C₂×C₄⋊Dic₄₃	—	2	2	no	—
29	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
30	C₂×C₄×Dic₄₃	—	4	1	no	—
31	#13063259	—	2	1	no	—
32	#13063259	—	2	2	no	—
33	#13063259	C₂×C₄	1	2	yes	—
34	#13063259	C₂×C₄	1	4	yes	complemented
35	#13063247	—	2	4	no	—
36	C₄₃⋊#95	—	4	1	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	C₂×C₁₆×C₄₃	—	2	2	no	—
41	C₂×C₁₆×C₄₃	—	2	2	no	—
42	C₂×C₁₆×C₄₃	C₂×C₄	1	4	yes	complemented
43	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₄₃	D₄	1	2	yes	—
45	C₂³×D₄₃	—	4	1	no	—
46	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₄₃	—	4	2	no	—
50	C₂²×Dic₄₃	—	4	1	no	—
51	C₂×D₁₇₂	—	2	2	no	—
52	C₂×D₁₇₂	—	2	4	no	—
53	C₂×D₁₇₂	—	4	1	no	—
54	C₂×D₁₇₂	—	4	2	no	—
55	C₂×C₄×D₄₃	—	4	4	no	—
56	C₂×C₄×D₄₃	—	8	4	no	—
57	C₂×C₈×C₄₃	—	2	2	no	—
58	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
59	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₄₃	#41	1	2	yes	—
63	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₄₃	SD₁₆	1	2	yes	—
65	(C₂×D₄₃)⋊C₄	—	4	4	no	—
66	C₄⋊Dic₄₃	—	2	2	no	—
67	C₄⋊Dic₄₃	—	2	4	no	—
68	C₄⋊Dic₄₃	—	4	1	no	—
69	C₄⋊Dic₄₃	—	4	2	no	—
70	C₄×Dic₄₃	—	8	2	no	—
71	D₃₄₄	—	2	2	no	—
72	D₃₄₄	—	2	4	no	—
73	D₃₄₄	C₄²	1	4	yes	complemented
74	C₁₆×C₄₃	—	2	4	no	—
75	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
76	C₂²×D₄₃	—	4	2	no	—
77	C₂²×D₄₃	—	4	4	no	—
78	C₂²×D₄₃	—	8	2	no	—
79	C₂²×D₄₃	—	8	1	no	—
80	C₂²×D₄₃	—	8	1	no	—
81	C₂×C₄×C₄₃	—	2	1	no	—
82	C₂×C₄×C₄₃	—	2	2	no	—
83	C₂×C₄×C₄₃	—	2	1	no	—
84	C₂×C₄×C₄₃	#102	1	2	yes	—
85	C₂×C₄×C₄₃	#102	1	2	yes	—
86	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
89	C₂×C₄×C₄₃	#135	1	2	yes	—
90	C₂×Dic₄₃	—	4	2	no	—
91	C₂×Dic₄₃	—	4	4	no	—
92	C₂×Dic₄₃	—	8	2	no	—
93	C₂×Dic₄₃	—	8	2	no	—
94	C₂×Dic₄₃	—	8	2	no	—
95	D₁₇₂	—	2	4	no	—
96	D₁₇₂	—	4	2	no	—
97	D₁₇₂	—	4	4	no	—
98	C₄×D₄₃	—	8	8	no	—
99	C₈×C₄₃	—	2	2	no	—
100	C₈×C₄₃	C₂×C₄²	1	1	yes	characteristic
101	C₈×C₄₃	#117	1	1	yes	characteristic
102	C₈×C₄₃	C₄×D₄	1	2	yes	—
103	#12637	—	43	1	no	maximal, Sylow
104	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
105	C₂²×C₄₃	C₄×D₈	1	2	yes	—
106	C₂²×C₄₃	#509	1	1	yes	characteristic
107	C₂²×C₄₃	#509	1	2	yes	—
108	C₂²×C₄₃	#510	1	1	yes	characteristic
109	C₂×D₄₃	—	4	4	no	—
110	C₂×D₄₃	—	8	2	no	—
111	C₂×D₄₃	—	8	2	no	—
112	C₂×D₄₃	—	8	4	no	—
113	C₂×D₄₃	—	8	4	no	—
114	C₄×C₄₃	—	2	2	no	—
115	C₄×C₄₃	—	2	2	no	—
116	C₄×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
117	C₄×C₄₃	#416	1	1	yes	characteristic
118	C₄×C₄₃	C₄×SD₁₆	1	2	yes	—
119	Dic₄₃	—	8	2	no	—
120	Dic₄₃	—	8	4	no	—
121	C₂×C₄×D₈	—	43	1	no	—
122	C₂×#357	—	43	2	no	—
123	C₂×#346	D₄₃	1	1	yes	characteristic, complemented
124	C₄×#107	—	43	1	no	—
125	#1380	—	43	2	no	—
126	C₂×C₄₃	C₄×#102	1	1	yes	characteristic
127	C₂×C₄₃	#2139	1	1	yes	characteristic
128	C₂×C₄₃	#2140	1	1	yes	characteristic
129	C₂×C₄₃	#2175	1	2	yes	—
130	C₂×C₄₃	#2177	1	2	yes	—
131	D₄₃	—	8	4	no	—
132	C₂²×D₈	—	43	1	no	—
133	C₂×C₄×D₄	—	86	1	no	—
134	C₂²×C₁₆	C₂×D₄₃	1	2	yes	—
135	C₄×D₈	—	43	4	no	—
136	C₄×D₈	—	86	2	no	—
137	C₂×#107	—	43	2	no	—
138	C₂×#107	—	43	1	no	—
139	C₂×#102	—	86	1	no	—
140	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
141	C₄×C₄⋊C₄	—	86	1	no	—
142	#357	—	43	8	no	—
143	#346	—	2	2	no	—
144	#340	—	86	1	no	—
145	C₄₃	#12637	1	1	yes	characteristic, complemented, Sylow
146	C₂²×D₄	—	86	1	no	—
147	C₂³×C₄	—	172	1	no	—
148	C₂×D₈	—	43	4	no	—
149	C₂×D₈	—	43	2	no	—
150	C₂×D₈	—	86	2	no	—
151	C₂×D₈	—	86	1	no	—
152	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
153	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
154	C₄×D₄	—	86	4	no	—
155	C₄×D₄	—	172	2	no	—
156	C₂×C₄⋊C₄	—	86	1	no	—
157	C₂×C₄⋊C₄	—	86	2	no	—
158	C₂×C₂²⋊C₄	—	172	1	no	—
159	C₂×C₄²	—	172	1	no	—
160	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
161	C₂×C₁₆	—	2	2	no	—
162	C₂×C₁₆	—	2	2	no	—
163	C₂×C₁₆	C₄×D₄₃	1	4	yes	complemented
164	#107	—	43	2	no	—
165	#107	—	43	4	no	—
166	#107	—	86	2	no	—
167	#107	—	86	1	no	—
168	#102	—	86	4	no	—
169	C₄×C₈	C₄×D₄₃	1	2	yes	—
170	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
171	#95	—	172	1	no	—
172	C₂⁴	—	172	1	no	—
173	C₂×D₄	—	86	2	no	—
174	C₂×D₄	—	86	4	no	—
175	C₂×D₄	—	172	1	no	—
176	C₂×D₄	—	172	2	no	—
177	C₂²×C₄	—	344	4	no	—
178	C₂²×C₄	—	172	2	no	—
179	C₂²×C₄	—	172	1	no	—
180	C₂²×C₄	—	172	4	no	—
181	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
182	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
183	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
184	D₈	—	43	4	no	—
185	D₈	—	86	2	no	—
186	D₈	—	86	4	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
189	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
190	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
191	C₂×C₈	#10621898	1	2	yes	—
192	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
193	C₄⋊C₄	—	86	2	no	—
194	C₄⋊C₄	—	86	4	no	—
195	C₄⋊C₄	—	172	1	no	—
196	C₄⋊C₄	—	172	2	no	—
197	C₂²⋊C₄	—	172	4	no	—
198	C₄²	—	344	2	no	—
199	C₄²	#10621873	1	2	yes	—
200	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
201	C₁₆	—	2	4	no	—
202	C₂³	—	344	2	no	—
203	C₂³	—	344	1	no	—
204	C₂³	—	344	1	no	—
205	C₂³	—	172	2	no	—
206	C₂³	—	172	4	no	—
207	C₂³	C₂×D₃₄₄	1	1	yes	characteristic
208	D₄	—	86	4	no	—
209	D₄	—	172	2	no	—
210	D₄	—	172	4	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	344	8	no	—
215	C₂×C₄	—	344	2	no	—
216	C₂×C₄	—	344	2	no	—
217	C₂×C₄	—	344	2	no	—
218	C₂×C₄	—	172	2	no	—
219	C₂×C₄	—	172	4	no	—
220	C₂×C₄	#13063247	1	2	yes	—
221	C₂×C₄	#13063247	1	2	yes	—
222	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
223	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
224	C₂×C₄	C₂×#10621873	1	1	yes	characteristic
225	C₂×C₄	#13063347	1	2	yes	—
226	C₈	—	2	2	no	—
227	C₈	C₄²×D₄₃	1	1	yes	characteristic
228	C₈	#13063282	1	1	yes	characteristic
229	C₈	C₄×C₂²⋊D₄₃	1	2	yes	—
230	C₂²	—	344	2	no	—
231	C₂²	—	344	2	no	—
232	C₂²	—	344	4	no	—
233	C₂²	—	344	4	no	—
234	C₂²	—	172	4	no	—
235	C₂²	C₂×#13063247	1	1	yes	characteristic
236	C₂²	C₄×D₃₄₄	1	2	yes	—
237	C₂²	#14583337	1	1	yes	characteristic
238	C₂²	#14583337	1	2	yes	—
239	C₂²	#14583342	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	344	2	no	—
243	C₄	—	344	4	no	—
244	C₄	C₄×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
245	C₄	#14582896	1	1	yes	characteristic
246	C₄	C₄×#10621873	1	2	yes	—
247	C₂	—	344	4	no	—
248	C₂	C₄×#13063247	1	1	yes	characteristic
249	C₂	#19136308	1	1	yes	characteristic
250	C₂	#19136312	1	1	yes	characteristic
251	C₂	#19136461	1	2	yes	—
252	C₂	#19136472	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₃₄₄
commutator	C₈×C₄₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂×C₄₃×#346	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂²×D₃₄₄	C₂²
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×#346	D₄₃
hypocenter	C₄₃	#12637
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
3. C₄×#13063259	C₂	semidirect
4. #19132687	C₂	semidirect
5. C₂×C₄₃×#346	C₂	semidirect
9. C₄×D₃₄₄	C₄	semidirect
14. #14582701	C₄	semidirect
34. #13063259	C₂×C₄	semidirect
42. C₂×C₁₆×C₄₃	C₂×C₄	semidirect
73. D₃₄₄	C₄²	semidirect
123. C₂×#346	D₄₃	semidirect
145. C₄₃	#12637	semidirect
163. C₂×C₁₆	C₄×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₃₄₄	2
C₂×#14582701	2
C₄×#13063259	2
#19132687	2
C₂×C₄₃×#346	2
#12637	43