Subgroups — #25253297

Order 11008.

676 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583331	C₂	1	2	yes	maximal, complemented
2	#19146317	C₂	1	2	yes	maximal, complemented
3	#19136848	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136686	C₂	1	2	yes	maximal, complemented
5	#19136641	C₂	1	2	yes	maximal, complemented
6	#19136537	C₂	1	4	yes	maximal, complemented
7	C₄×D₆₈₈	C₂	1	1	yes	maximal, characteristic, complemented
8	D₄×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13063347	C₂²	1	4	yes	—
10	#14583554	—	2	2	no	—
11	#14583331	—	2	8	no	—
12	C₂×Dic₃₄₄	C₂²	1	2	yes	—
13	C₂×Dic₃₄₄	C₂²	1	2	yes	—
14	C₂×#13063270	C₂²	1	4	yes	—
15	C₂×D₆₈₈	C₂²	1	1	yes	characteristic
16	#14583279	C₂²	1	2	yes	—
17	#14583140	C₂²	1	4	yes	—
18	#14583105	—	2	2	no	—
19	#14583063	—	2	4	no	—
20	C₄×D₃₄₄	C₂²	1	2	yes	—
21	#14582725	C₂²	1	1	yes	characteristic
22	#14582704	C₂²	1	4	yes	—
23	#14582701	C₂²	1	2	yes	—
24	C₂²×C₁₆×C₄₃	C₂²	1	2	yes	—
25	C₈×D₄×C₄₃	C₂²	1	1	yes	characteristic
26	C₄₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
27	C₄₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
29	C₂×#10621898	—	2	4	no	—
30	#13063347	—	2	8	no	—
31	#13063347	—	4	4	no	—
32	C₂×Dic₁₇₂	—	2	2	no	—
33	C₂×Dic₁₇₂	C₂³	1	4	yes	—
34	C₂×#10621873	—	2	4	no	—
35	C₂×D₃₄₄	C₂³	1	2	yes	—
36	#13063325	—	2	2	no	—
37	#13063307	—	4	4	no	—
38	#13063304	—	2	4	no	—
39	#13063296	—	4	2	no	—
40	C₄×D₁₇₂	—	2	2	no	—
41	C₄⋊C₄×D₄₃	—	4	2	no	—
42	Dic₃₄₄	—	2	4	no	—
43	Dic₃₄₄	—	4	2	no	—
44	#13063270	—	2	8	no	—
45	D₆₈₈	D₄	1	4	yes	complemented
46	#13063259	C₂³	1	2	yes	—
47	#13063250	—	2	4	no	—
48	#13063247	—	2	2	no	—
49	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
50	C₄×D₄×C₄₃	D₄	1	1	yes	characteristic
51	C₂×C₁₆×C₄₃	—	2	4	no	—
52	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
53	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
54	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₄₃	D₄	1	1	yes	characteristic
56	C₂²⋊C₈×C₄₃	D₄	1	2	yes	—
57	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
58	C₂×D₄×C₄₃	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
60	C₂×C₂²⋊D₄₃	—	4	4	no	—
61	#10621898	—	4	8	no	—
62	#10621898	—	8	4	no	—
63	C₂×D₁₇₂	—	2	2	no	—
64	C₂×C₄×D₄₃	—	4	2	no	—
65	C₂×C₄×D₄₃	—	4	4	no	—
66	C₂×Dic₈₆	—	2	4	no	—
67	C₂×C₈×C₄₃	—	2	2	no	—
68	C₂×C₈×C₄₃	—	2	2	no	—
69	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
72	C₄⋊C₄×C₄₃	D₈	1	1	yes	characteristic
73	C₂²⋊C₄×C₄₃	D₈	1	2	yes	—
74	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
75	C₂²⋊Dic₄₃	—	4	4	no	—
76	(C₂×D₄₃)⋊C₄	—	4	2	no	—
77	C₄⋊Dic₄₃	—	2	2	no	—
78	C₄⋊Dic₄₃	—	4	2	no	—
79	C₄⋊Dic₄₃	—	4	2	no	—
80	Dic₄₃⋊C₄	—	4	4	no	—
81	Dic₁₇₂	—	2	4	no	—
82	Dic₁₇₂	—	2	4	no	—
83	Dic₁₇₂	—	8	2	no	—
84	D₃₄₄	—	2	2	no	—
85	D₃₄₄	C₂×D₄	1	4	yes	—
86	#10621873	—	4	8	no	—
87	C₁₆×C₄₃	—	2	2	no	—
88	C₁₆×C₄₃	—	2	1	no	—
89	C₁₆×C₄₃	C₂×D₄	1	2	yes	—
90	C₂³×C₄₃	C₂×D₈	1	2	yes	—
91	C₂²×D₄₃	—	4	2	no	—
92	D₄×C₄₃	—	2	2	no	—
93	C₂×C₄×C₄₃	—	2	2	no	—
94	C₂×C₄×C₄₃	—	2	2	no	—
95	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
96	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
99	C₂²⋊D₄₃	—	8	8	no	—
100	C₂×Dic₄₃	—	4	2	no	—
101	C₂×Dic₄₃	—	4	4	no	—
102	D₁₇₂	—	2	4	no	—
103	D₁₇₂	—	4	2	no	—
104	C₄×D₄₃	—	8	8	no	—
105	C₄×D₄₃	—	8	4	no	—
106	Dic₈₆	—	4	4	no	—
107	Dic₈₆	—	4	4	no	—
108	C₈×C₄₃	—	2	2	no	—
109	C₈×C₄₃	—	2	1	no	—
110	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
111	C₈×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
112	#33849	—	43	1	no	maximal, Sylow
113	C₂²×C₄₃	—	2	2	no	—
114	C₂²×C₄₃	—	2	2	no	—
115	C₂²×C₄₃	—	2	2	no	—
116	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
117	C₂×D₄₃	—	4	4	no	—
118	C₂×D₄₃	—	8	2	no	—
119	C₂×D₄₃	—	8	2	no	—
120	C₄×C₄₃	—	2	1	no	—
121	C₄×C₄₃	—	2	2	no	—
122	C₄×C₄₃	#508	1	2	yes	complemented
123	C₄×C₄₃	D₄²	1	1	yes	characteristic
124	C₄×C₄₃	#576	1	1	yes	characteristic
125	Dic₄₃	—	8	2	no	—
126	Dic₄₃	—	8	4	no	—
127	C₂×#508	—	43	2	no	—
128	#3283	—	43	2	no	—
129	#2247	—	43	1	no	—
130	#2216	—	43	2	no	—
131	#2208	—	43	2	no	—
132	#2189	—	43	4	no	—
133	C₄×D₁₆	—	43	1	no	—
134	D₄×C₁₆	D₄₃	1	1	yes	characteristic, complemented
135	C₂×C₄₃	—	4	2	no	—
136	C₂×C₄₃	D₄×D₈	1	1	yes	characteristic
137	C₂×C₄₃	C₂×#508	1	1	yes	characteristic
138	C₂×C₄₃	#3417	1	1	yes	characteristic
139	D₄₃	—	8	4	no	—
140	C₂×#135	—	43	4	no	—
141	#547	—	86	2	no	—
142	#508	—	86	8	no	—
143	C₂×Q₃₂	—	43	2	no	—
144	C₂×Q₃₂	—	43	2	no	—
145	C₂×SD₃₂	—	43	4	no	—
146	C₂×D₁₆	—	43	1	no	—
147	C₂²×C₁₆	C₂×D₄₃	1	2	yes	—
148	#494	—	43	2	no	—
149	#467	—	43	4	no	—
150	#461	—	86	2	no	—
151	#452	—	86	4	no	—
152	C₄×D₈	—	43	2	no	—
153	C₈×D₄	C₂×D₄₃	1	1	yes	characteristic
154	#366	—	43	1	no	—
155	C₄⋊C₁₆	C₂×D₄₃	1	1	yes	characteristic
156	#358	—	43	4	no	—
157	#357	—	43	2	no	—
158	C₂²⋊C₁₆	C₂×D₄₃	1	2	yes	—
159	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
160	C₄₃	#33849	1	1	yes	characteristic, complemented, Sylow
161	C₂×#41	—	86	4	no	—
162	#135	—	86	8	no	—
163	#135	—	172	4	no	—
164	C₂×Q₁₆	—	43	4	no	—
165	C₂×Q₁₆	—	86	2	no	—
166	C₂×SD₁₆	—	86	4	no	—
167	C₂×D₈	—	43	2	no	—
168	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
169	C₄⋊Q₈	—	86	2	no	—
170	#123	—	172	4	no	—
171	C₂²⋊Q₈	—	86	4	no	—
172	#121	—	172	2	no	—
173	C₄×D₄	—	86	2	no	—
174	C₄×D₄	D₁₇₂	1	1	yes	characteristic
175	C₂×C₄⋊C₄	—	172	2	no	—
176	Q₃₂	—	86	4	no	—
177	Q₃₂	—	172	2	no	—
178	SD₃₂	—	86	8	no	—
179	D₁₆	—	43	4	no	—
180	C₂×C₁₆	—	2	4	no	—
181	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
182	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
183	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
184	#107	—	43	2	no	—
185	C₄⋊C₈	D₁₇₂	1	1	yes	characteristic
186	#103	—	86	4	no	—
187	#102	—	86	2	no	—
188	C₂²⋊C₈	D₁₇₂	1	2	yes	—
189	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
190	#41	—	344	4	no	—
191	#41	—	172	8	no	—
192	C₂×Q₈	—	86	4	no	—
193	C₂×D₄	—	86	2	no	—
194	C₂×D₄	—	172	4	no	—
195	C₂×D₄	D₃₄₄	1	1	yes	characteristic
196	C₂²×C₄	—	172	2	no	—
197	C₂²×C₄	—	172	4	no	—
198	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
199	Q₁₆	—	86	4	no	—
200	Q₁₆	—	86	4	no	—
201	Q₁₆	—	344	2	no	—
202	SD₁₆	—	172	8	no	—
203	D₈	—	43	4	no	—
204	D₈	—	86	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
208	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
209	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
210	C₄⋊C₄	—	86	2	no	—
211	C₄⋊C₄	—	172	2	no	—
212	C₄⋊C₄	—	172	4	no	—
213	C₄⋊C₄	—	172	2	no	—
214	C₄⋊C₄	D₃₄₄	1	1	yes	characteristic
215	C₂²⋊C₄	—	172	4	no	—
216	C₂²⋊C₄	—	172	2	no	—
217	C₂²⋊C₄	D₃₄₄	1	2	yes	—
218	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
219	C₁₆	—	2	2	no	—
220	C₁₆	—	2	1	no	—
221	C₁₆	D₄×D₄₃	1	2	yes	—
222	C₂³	—	172	2	no	—
223	C₂³	C₂×D₃₄₄	1	2	yes	—
224	Q₈	—	172	4	no	—
225	Q₈	—	172	4	no	—
226	D₄	—	2	2	no	—
227	D₄	—	86	4	no	—
228	D₄	—	344	8	no	—
229	D₄	—	172	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	344	8	no	—
233	C₂×C₄	—	344	4	no	—
234	C₂×C₄	—	172	2	no	—
235	C₂×C₄	—	172	4	no	—
236	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
237	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
238	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
239	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
240	C₈	—	2	1	no	—
241	C₈	—	2	2	no	—
242	C₈	C₂×D₄×D₄₃	1	1	yes	characteristic
243	C₈	#13063371	1	1	yes	characteristic
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	344	2	no	—
248	C₂²	—	344	2	no	—
249	C₂²	—	172	4	no	—
250	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	2	1	no	—
253	C₄	—	344	4	no	—
254	C₄	—	344	2	no	—
255	C₄	#14583331	1	2	yes	complemented
256	C₄	D₄×D₁₇₂	1	1	yes	characteristic
257	C₄	#14583704	1	1	yes	characteristic
258	C₂	—	4	2	no	—
259	C₂	—	344	4	no	—
260	C₂	D₄×D₃₄₄	1	1	yes	characteristic
261	C₂	C₂×#14583331	1	1	yes	characteristic
262	C₂	#19147748	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₄₄
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	D₄×C₁₆×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₈
hypercenter	D₄×C₁₆	D₄₃
hypocenter	C₄₃	#33849
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#14583331	C₂	semidirect
2. #19146317	C₂	semidirect
3. #19136848	C₂	semidirect
4. #19136686	C₂	semidirect
5. #19136641	C₂	semidirect
6. #19136537	C₂	semidirect
7. C₄×D₆₈₈	C₂	semidirect
8. D₄×C₁₆×C₄₃	C₂	semidirect
45. D₆₈₈	D₄	semidirect
122. C₄×C₄₃	#508	semidirect
134. D₄×C₁₆	D₄₃	semidirect
160. C₄₃	#33849	semidirect
255. C₄	#14583331	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583331	2
#19146317	2
#19136848	2
#19136686	2
#19136641	2
#19136537	2
C₄×D₆₈₈	2
D₄×C₁₆×C₄₃	2
#33849	43