Subgroups — #25253335

Order 11008.

696 conjugacy classes of subgroups.

The subgroup lattice has 268 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	#19146355	C₂	1	2	yes	maximal, complemented
3	#19136845	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136681	C₂	1	2	yes	maximal, complemented
5	#19136636	C₂	1	2	yes	maximal, complemented
6	#19136520	C₂	1	4	yes	maximal, complemented
7	C₄×Dic₃₄₄	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	#14583570	—	2	2	no	—
11	#14583331	—	2	8	no	—
12	C₂×Dic₃₄₄	C₂²	1	1	yes	characteristic
13	C₂×#13063270	C₂²	1	4	yes	—
14	C₂×D₆₈₈	C₂²	1	2	yes	—
15	C₂×D₆₈₈	C₂²	1	2	yes	—
16	#14583276	C₂²	1	2	yes	—
17	#14583135	C₂²	1	4	yes	—
18	#14583100	—	2	2	no	—
19	#14583046	—	2	4	no	—
20	C₄×Dic₁₇₂	C₂²	1	2	yes	—
21	#14582725	C₂²	1	1	yes	characteristic
22	#14582704	C₂²	1	2	yes	—
23	#14582701	C₂²	1	4	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	C₂³	1	2	yes	—
33	C₂×#10621873	—	2	4	no	—
34	C₂×D₃₄₄	—	2	2	no	—
35	C₂×D₃₄₄	C₂³	1	4	yes	—
36	#13063322	—	2	2	no	—
37	#13063321	—	4	2	no	—
38	#13063299	—	2	4	no	—
39	#13063298	—	4	4	no	—
40	C₄×Dic₈₆	—	2	2	no	—
41	#13063287	—	4	2	no	—
42	Dic₃₄₄	D₄	1	4	yes	complemented
43	#13063270	—	2	8	no	—
44	D₆₈₈	—	2	4	no	—
45	D₆₈₈	—	4	2	no	—
46	#13063259	C₂³	1	2	yes	—
47	#13063250	—	2	2	no	—
48	#13063247	—	2	4	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	C₂×C₂²⋊D₄₃	—	4	4	no	—
62	#10621898	—	4	8	no	—
63	#10621898	—	8	4	no	—
64	C₂×D₁₇₂	—	2	4	no	—
65	C₂×D₁₇₂	—	4	2	no	—
66	C₂×D₁₇₂	—	4	2	no	—
67	C₂×C₄×D₄₃	—	4	4	no	—
68	C₂×Dic₈₆	—	2	2	no	—
69	C₂×C₈×C₄₃	—	2	2	no	—
70	C₂×C₈×C₄₃	—	2	2	no	—
71	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
74	C₄⋊C₄×C₄₃	D₈	1	1	yes	characteristic
75	C₂²⋊C₄×C₄₃	D₈	1	2	yes	—
76	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
77	(C₂×D₄₃)⋊C₄	—	4	4	no	—
78	C₄⋊Dic₄₃	—	2	2	no	—
79	Dic₄₃⋊C₄	—	4	2	no	—
80	C₄×Dic₄₃	—	4	2	no	—
81	Dic₁₇₂	—	2	2	no	—
82	Dic₁₇₂	C₂×D₄	1	4	yes	—
83	D₃₄₄	—	2	4	no	—
84	D₃₄₄	—	2	4	no	—
85	D₃₄₄	—	8	2	no	—
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	4	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	1	yes	characteristic
96	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
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	C₂²⋊D₄₃	—	16	4	no	—
101	C₂×Dic₄₃	—	4	2	no	—
102	C₂×Dic₄₃	—	4	2	no	—
103	D₁₇₂	—	4	4	no	—
104	D₁₇₂	—	4	4	no	—
105	D₁₇₂	—	8	4	no	—
106	D₁₇₂	—	16	2	no	—
107	C₄×D₄₃	—	8	8	no	—
108	Dic₈₆	—	2	4	no	—
109	Dic₈₆	—	4	2	no	—
110	C₈×C₄₃	—	2	2	no	—
111	C₈×C₄₃	—	2	1	no	—
112	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
113	C₈×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
114	#33853	—	43	1	no	maximal, Sylow
115	C₂²×C₄₃	—	2	2	no	—
116	C₂²×C₄₃	—	2	2	no	—
117	C₂²×C₄₃	—	2	2	no	—
118	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
119	C₂×D₄₃	—	8	4	no	—
120	C₂×D₄₃	—	8	4	no	—
121	C₂×D₄₃	—	8	4	no	—
122	C₄×C₄₃	—	2	1	no	—
123	C₄×C₄₃	—	2	2	no	—
124	C₄×C₄₃	#508	1	2	yes	complemented
125	C₄×C₄₃	D₄²	1	1	yes	characteristic
126	C₄×C₄₃	#576	1	1	yes	characteristic
127	Dic₄₃	—	4	4	no	—
128	Dic₄₃	—	8	2	no	—
129	C₂×#508	—	43	2	no	—
130	#3287	—	43	2	no	—
131	#2246	—	43	1	no	—
132	#2215	—	43	2	no	—
133	#2207	—	43	2	no	—
134	#2186	—	43	4	no	—
135	C₄×Q₃₂	—	43	1	no	—
136	D₄×C₁₆	D₄₃	1	1	yes	characteristic, complemented
137	C₂×C₄₃	—	4	2	no	—
138	C₂×C₄₃	D₄×D₈	1	1	yes	characteristic
139	C₂×C₄₃	C₂×#508	1	1	yes	characteristic
140	C₂×C₄₃	#3415	1	1	yes	characteristic
141	D₄₃	—	16	4	no	—
142	C₂×#135	—	43	4	no	—
143	#550	—	86	2	no	—
144	#508	—	86	8	no	—
145	C₂×Q₃₂	—	43	1	no	—
146	C₂×SD₃₂	—	43	4	no	—
147	C₂×D₁₆	—	43	2	no	—
148	C₂×D₁₆	—	43	2	no	—
149	C₂²×C₁₆	C₂×D₄₃	1	2	yes	—
150	#493	—	43	2	no	—
151	#466	—	43	4	no	—
152	#460	—	86	2	no	—
153	#449	—	86	4	no	—
154	C₄×Q₁₆	—	43	2	no	—
155	C₈×D₄	C₂×D₄₃	1	1	yes	characteristic
156	#366	—	43	1	no	—
157	C₄⋊C₁₆	C₂×D₄₃	1	1	yes	characteristic
158	#358	—	43	2	no	—
159	#357	—	43	4	no	—
160	C₂²⋊C₁₆	C₂×D₄₃	1	2	yes	—
161	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
162	C₄₃	#33853	1	1	yes	characteristic, complemented, Sylow
163	C₂×#41	—	86	4	no	—
164	#135	—	86	8	no	—
165	#135	—	172	4	no	—
166	C₂×Q₁₆	—	43	2	no	—
167	C₂×SD₁₆	—	86	4	no	—
168	C₂×D₈	—	43	4	no	—
169	C₂×D₈	—	86	2	no	—
170	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
171	#127	—	86	2	no	—
172	#127	—	172	2	no	—
173	#121	—	86	4	no	—
174	#121	—	172	4	no	—
175	C₄×Q₈	—	86	2	no	—
176	C₄×D₄	—	172	2	no	—
177	C₄×D₄	D₁₇₂	1	1	yes	characteristic
178	Q₃₂	—	43	4	no	—
179	SD₃₂	—	86	8	no	—
180	D₁₆	—	86	4	no	—
181	D₁₆	—	172	2	no	—
182	C₂×C₁₆	—	2	4	no	—
183	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
184	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
185	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
186	#107	—	43	2	no	—
187	C₄⋊C₈	D₁₇₂	1	1	yes	characteristic
188	#103	—	86	2	no	—
189	#102	—	86	4	no	—
190	C₂²⋊C₈	D₁₇₂	1	2	yes	—
191	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
192	#41	—	344	4	no	—
193	#41	—	172	8	no	—
194	C₂×Q₈	—	86	2	no	—
195	C₂×D₄	—	86	4	no	—
196	C₂×D₄	—	172	4	no	—
197	C₂×D₄	—	172	4	no	—
198	C₂×D₄	—	172	2	no	—
199	C₂×D₄	—	172	2	no	—
200	C₂×D₄	D₃₄₄	1	1	yes	characteristic
201	C₂²×C₄	—	172	4	no	—
202	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
203	Q₁₆	—	43	4	no	—
204	Q₁₆	—	86	2	no	—
205	SD₁₆	—	172	8	no	—
206	D₈	—	86	4	no	—
207	D₈	—	86	4	no	—
208	D₈	—	344	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
212	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
213	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
214	C₄⋊C₄	—	86	2	no	—
215	C₄⋊C₄	—	172	2	no	—
216	C₄⋊C₄	D₃₄₄	1	1	yes	characteristic
217	C₂²⋊C₄	—	172	4	no	—
218	C₂²⋊C₄	D₃₄₄	1	2	yes	—
219	C₄²	—	172	2	no	—
220	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
221	C₁₆	—	2	2	no	—
222	C₁₆	—	2	1	no	—
223	C₁₆	D₄×D₄₃	1	2	yes	—
224	C₂³	—	172	4	no	—
225	C₂³	C₂×D₃₄₄	1	2	yes	—
226	Q₈	—	86	4	no	—
227	Q₈	—	172	2	no	—
228	D₄	—	2	2	no	—
229	D₄	—	344	8	no	—
230	D₄	—	344	4	no	—
231	D₄	—	172	4	no	—
232	D₄	—	172	4	no	—
233	D₄	—	688	2	no	—
234	D₄	—	688	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	344	8	no	—
238	C₂×C₄	—	172	2	no	—
239	C₂×C₄	—	172	2	no	—
240	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
241	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
242	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
243	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	2	no	—
246	C₈	C₂×D₄×D₄₃	1	1	yes	characteristic
247	C₈	#13063371	1	1	yes	characteristic
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	344	4	no	—
252	C₂²	—	344	4	no	—
253	C₂²	—	344	4	no	—
254	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	1	no	—
257	C₄	—	344	2	no	—
258	C₄	—	172	4	no	—
259	C₄	#14583331	1	2	yes	complemented
260	C₄	D₄×D₁₇₂	1	1	yes	characteristic
261	C₄	#14583704	1	1	yes	characteristic
262	C₂	—	4	2	no	—
263	C₂	—	688	4	no	—
264	C₂	D₄×D₃₄₄	1	1	yes	characteristic
265	C₂	C₂×#14583331	1	1	yes	characteristic
266	C₂	#19147736	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	#33853
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. #19146355	C₂	semidirect
3. #19136845	C₂	semidirect
4. #19136681	C₂	semidirect
5. #19136636	C₂	semidirect
6. #19136520	C₂	semidirect
7. C₄×Dic₃₄₄	C₂	semidirect
8. D₄×C₁₆×C₄₃	C₂	semidirect
42. Dic₃₄₄	D₄	semidirect
124. C₄×C₄₃	#508	semidirect
136. D₄×C₁₆	D₄₃	semidirect
162. C₄₃	#33853	semidirect
259. 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
#19146355	2
#19136845	2
#19136681	2
#19136636	2
#19136520	2
C₄×Dic₃₄₄	2
D₄×C₁₆×C₄₃	2
#33853	43