Subgroups — #25966149

Order 11008.

3114 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583716	C₂	1	4	yes	maximal, complemented
2	#19148380	C₂	1	2	yes	maximal, complemented
3	#19145089	C₂	1	12	yes	maximal, complemented
4	#19145054	C₂	1	6	yes	maximal, complemented
5	#19145023	C₂	1	3	yes	maximal, complemented
6	C₂×#14583279	C₂	1	3	yes	maximal, complemented
7	C₄×#131×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063372	C₂²	1	4	yes	—
9	#14583716	—	2	32	no	—
10	C₂×#13063358	C₂²	1	24	yes	—
11	C₂×#13063347	C₂²	1	12	yes	—
12	C₂²×Dic₁₇₂	C₂²	1	12	yes	—
13	Q₈×D₁₇₂	—	2	4	no	—
14	#14583553	—	2	12	no	—
15	#14583475	C₂²	1	12	yes	—
16	#14583449	C₂²	1	6	yes	—
17	C₂×#13063325	C₂²	1	6	yes	—
18	#14583293	C₂²	1	24	yes	—
19	#14583283	C₂²	1	12	yes	—
20	#14583279	C₂²	1	3	yes	—
21	#14583279	C₂²	1	6	yes	—
22	#14583279	C₂²	1	12	yes	—
23	#14583276	C₂²	1	1	yes	characteristic
24	#14583272	C₂²	1	6	yes	—
25	C₂×#131×C₄₃	C₂²	1	2	yes	—
26	C₄×#41×C₄₃	C₂²	1	1	yes	characteristic
27	C₄₃×#405	C₂²	1	6	yes	—
28	C₄×M₁₆×C₄₃	C₂²	1	3	yes	—
29	C₂×C₄×C₈×C₄₃	C₂²	1	3	yes	—
30	#13063372	—	2	32	no	—
31	C₂×#10621898	C₂³	1	12	yes	—
32	C₂×#10621900	—	2	12	no	—
33	C₂²×Dic₈₆	C₂³	1	12	yes	—
34	C₂×Q₈×D₄₃	—	2	4	no	—
35	#13063358	—	2	96	no	—
36	#13063347	—	2	48	no	—
37	C₂×Dic₁₇₂	—	2	48	no	—
38	C₂×Dic₁₇₂	C₂³	1	12	yes	—
39	C₂×Dic₁₇₂	C₂³	1	24	yes	—
40	C₂×#10621873	C₂³	1	24	yes	—
41	C₂×D₃₄₄	C₂³	1	4	yes	—
42	#13063325	—	2	12	no	—
43	#13063325	C₂³	1	6	yes	—
44	#13063325	C₂³	1	12	yes	—
45	#13063322	C₂³	1	2	yes	—
46	#13063313	C₂³	1	12	yes	—
47	#13063309	—	2	24	no	—
48	#13063305	—	2	24	no	—
49	#13063299	—	2	12	no	—
50	C₄×D₁₇₂	—	2	12	no	—
51	C₂×C₄⋊Dic₄₃	—	2	12	no	—
52	C₂×#41×C₄₃	C₂³	1	1	yes	characteristic
53	#131×C₄₃	D₄	1	16	yes	complemented
54	C₂×M₁₆×C₄₃	C₂³	1	6	yes	—
55	C₂²×C₈×C₄₃	C₂³	1	6	yes	—
56	C₄×Q₈×C₄₃	D₄	1	2	yes	—
57	C₄×D₄×C₄₃	D₄	1	6	yes	—
58	#117×C₄₃	C₂³	1	3	yes	—
59	C₂×C₄²×C₄₃	C₂³	1	3	yes	—
60	#97×C₄₃	C₂³	1	6	yes	—
61	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₄₃	C₂³	1	3	yes	—
63	C₄×C₈×C₄₃	C₂³	1	6	yes	—
64	#41×C₄₃	C₂×D₄	1	8	yes	—
65	C₂×Q₈×C₄₃	C₂×D₄	1	1	yes	characteristic
66	C₂×D₄×C₄₃	C₂×D₄	1	3	yes	—
67	C₂²×C₄×C₄₃	C₂×D₄	1	6	yes	—
68	C₂²×C₄×C₄₃	C₂⁴	1	3	yes	—
69	C₂×C₂²⋊D₄₃	—	2	12	no	—
70	C₂²×Dic₄₃	—	2	12	no	—
71	Q₈×D₄₃	—	4	16	no	—
72	#10621900	—	4	48	no	—
73	#10621898	—	2	48	no	—
74	C₂×D₁₇₂	—	2	4	no	—
75	C₂×D₁₇₂	C₂⁴	1	4	yes	—
76	C₂×C₄×D₄₃	—	2	12	no	—
77	C₂×Dic₈₆	—	2	48	no	—
78	C₂×Dic₈₆	—	2	12	no	—
79	C₂×Dic₈₆	C₂⁴	1	12	yes	—
80	M₁₆×C₄₃	C₂×D₄	1	24	yes	—
81	C₂×C₈×C₄₃	C₂×D₄	1	24	yes	—
82	C₂×C₈×C₄₃	C₂⁴	1	2	yes	—
83	C₂×C₈×C₄₃	C₂⁴	1	6	yes	—
84	C₄⋊C₄×C₄₃	C₂×D₄	1	6	yes	—
85	C₂²⋊C₄×C₄₃	C₂×D₄	1	6	yes	—
86	C₄²×C₄₃	C₂×D₄	1	6	yes	—
87	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
88	C₄²×C₄₃	C₂⁴	1	3	yes	—
89	(C₂×D₄₃)⋊C₄	—	2	24	no	—
90	C₄⋊Dic₄₃	—	2	12	no	—
91	C₄⋊Dic₄₃	—	2	24	no	—
92	Dic₁₇₂	—	2	24	no	—
93	Dic₁₇₂	—	2	48	no	—
94	D₃₄₄	—	2	8	no	—
95	#10621873	—	2	48	no	—
96	C₂³×C₄₃	C₂²×D₄	1	3	yes	—
97	C₂²×D₄₃	—	2	4	no	—
98	Q₈×C₄₃	#127	1	4	yes	—
99	D₄×C₄₃	#127	1	12	yes	—
100	C₂×C₄×C₄₃	—	2	12	no	—
101	C₂×C₄×C₄₃	C₂²×D₄	1	6	yes	—
102	C₂×C₄×C₄₃	C₂²×D₄	1	3	yes	—
103	C₂×C₄×C₄₃	C₂²×D₄	1	12	yes	—
104	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
105	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
106	C₂²⋊D₄₃	—	4	24	no	—
107	C₂×Dic₄₃	—	2	12	no	—
108	C₂×Dic₄₃	—	4	24	no	—
109	D₁₇₂	—	2	8	no	—
110	D₁₇₂	—	4	8	no	—
111	C₄×D₄₃	—	4	24	no	—
112	Dic₈₆	—	2	24	no	—
113	Dic₈₆	—	4	24	no	—
114	C₈×C₄₃	C₂²×D₄	1	12	yes	—
115	C₈×C₄₃	C₂²×D₄	1	4	yes	—
116	#60648	—	43	1	no	maximal, Sylow
117	C₂²×C₄₃	—	2	6	no	—
118	C₂²×C₄₃	C₂×#127	1	6	yes	—
119	C₂²×C₄₃	C₂³×D₄	1	1	yes	characteristic
120	C₂×D₄₃	—	4	4	no	—
121	C₂×D₄₃	—	4	8	no	—
122	C₄×C₄₃	—	2	6	no	—
123	C₄×C₄₃	C₂×#127	1	6	yes	—
124	C₄×C₄₃	#578	1	4	yes	complemented
125	C₄×C₄₃	C₂³×D₄	1	2	yes	—
126	Dic₄₃	—	4	12	no	—
127	C₂×#578	—	43	4	no	—
128	#3499	—	43	2	no	—
129	#3156	—	43	12	no	—
130	#3152	—	43	6	no	—
131	#3147	—	43	3	no	—
132	C₂×#494	—	43	3	no	—
133	C₄×#131	D₄₃	1	1	yes	characteristic, complemented
134	C₂×C₄₃	—	2	6	no	—
135	C₂×C₄₃	C₂²×#127	1	1	yes	characteristic
136	C₂×C₄₃	C₂×#578	1	2	yes	—
137	D₄₃	—	8	4	no	—
138	C₂×2₋¹⁺⁴	—	43	4	no	—
139	#578	—	86	32	no	—
140	C₂×#137	—	43	24	no	—
141	C₂×#135	—	43	12	no	—
142	C₂²×Q₁₆	—	43	12	no	—
143	C₂×#131	C₂×D₄₃	1	2	yes	—
144	D₄×Q₈	—	86	4	no	—
145	#547	—	86	12	no	—
146	#536	—	43	12	no	—
147	#532	—	43	6	no	—
148	C₂×C₄⋊Q₈	—	43	6	no	—
149	C₄×#41	C₂×D₄₃	1	1	yes	characteristic
150	#497	—	43	24	no	—
151	#495	—	43	12	no	—
152	#494	—	43	3	no	—
153	#494	—	43	6	no	—
154	#494	—	43	12	no	—
155	#493	—	43	1	no	—
156	#492	—	43	6	no	—
157	#405	C₂×D₄₃	1	6	yes	—
158	C₄×M₁₆	C₂×D₄₃	1	3	yes	—
159	C₂×C₄×C₈	C₂×D₄₃	1	3	yes	—
160	C₄₃	#60648	1	1	yes	characteristic, complemented, Sylow
161	2₋¹⁺⁴	—	86	32	no	—
162	C₂×#41	—	43	12	no	—
163	C₂×#41	—	86	12	no	—
164	C₂×#41	C₂²×D₄₃	1	1	yes	characteristic
165	C₂²×Q₈	—	43	12	no	—
166	C₂²×Q₈	—	86	4	no	—
167	#137	—	86	96	no	—
168	#135	—	86	48	no	—
169	C₂×Q₁₆	—	43	12	no	—
170	C₂×Q₁₆	—	43	24	no	—
171	C₂×Q₁₆	—	86	48	no	—
172	C₂×SD₁₆	—	43	24	no	—
173	C₂×D₈	—	43	4	no	—
174	#131	D₁₇₂	1	16	yes	complemented
175	C₂×M₁₆	C₂²×D₄₃	1	6	yes	—
176	C₂²×C₈	C₂²×D₄₃	1	6	yes	—
177	C₄⋊Q₈	—	43	6	no	—
178	C₄⋊Q₈	—	43	12	no	—
179	C₄⋊Q₈	—	86	12	no	—
180	#127	—	43	2	no	—
181	#124	—	43	12	no	—
182	#123	—	86	24	no	—
183	C₂²⋊Q₈	—	86	24	no	—
184	#121	—	86	12	no	—
185	C₄×Q₈	D₁₇₂	1	2	yes	—
186	C₄×D₄	—	86	12	no	—
187	C₄×D₄	D₁₇₂	1	6	yes	—
188	#117	C₂²×D₄₃	1	3	yes	—
189	C₂×C₄⋊C₄	—	86	12	no	—
190	C₂×C₄²	C₂²×D₄₃	1	3	yes	—
191	#97	C₂²×D₄₃	1	6	yes	—
192	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₄₃	1	3	yes	—
194	C₄×C₈	C₂²×D₄₃	1	6	yes	—
195	#41	—	86	48	no	—
196	#41	—	172	48	no	—
197	#41	C₂×D₁₇₂	1	8	yes	—
198	C₂×Q₈	—	43	12	no	—
199	C₂×Q₈	—	86	12	no	—
200	C₂×Q₈	—	86	48	no	—
201	C₂×Q₈	—	172	16	no	—
202	C₂×Q₈	C₂×D₁₇₂	1	1	yes	characteristic
203	C₂×D₄	—	43	4	no	—
204	C₂×D₄	—	86	12	no	—
205	C₂×D₄	—	86	4	no	—
206	C₂×D₄	C₂×D₁₇₂	1	3	yes	—
207	C₂²×C₄	—	86	12	no	—
208	C₂²×C₄	—	86	12	no	—
209	C₂²×C₄	C₂×D₁₇₂	1	6	yes	—
210	C₂²×C₄	C₂³×D₄₃	1	3	yes	—
211	Q₁₆	—	86	24	no	—
212	Q₁₆	—	86	48	no	—
213	SD₁₆	—	86	48	no	—
214	D₈	—	86	8	no	—
215	M₁₆	C₂×D₁₇₂	1	24	yes	—
216	C₂×C₈	C₂×D₁₇₂	1	24	yes	—
217	C₂×C₈	C₂³×D₄₃	1	2	yes	—
218	C₂×C₈	C₂³×D₄₃	1	6	yes	—
219	C₄⋊C₄	—	86	12	no	—
220	C₄⋊C₄	—	86	24	no	—
221	C₄⋊C₄	C₂×D₁₇₂	1	6	yes	—
222	C₂²⋊C₄	—	86	24	no	—
223	C₂²⋊C₄	C₂×D₁₇₂	1	6	yes	—
224	C₄²	C₂×D₁₇₂	1	6	yes	—
225	C₄²	C₂³×D₄₃	1	1	yes	characteristic
226	C₄²	C₂³×D₄₃	1	3	yes	—
227	C₂³	—	86	4	no	—
228	C₂³	C₂²×D₁₇₂	1	3	yes	—
229	Q₈	—	86	24	no	—
230	Q₈	—	172	24	no	—
231	Q₈	#13063322	1	4	yes	—
232	D₄	—	86	8	no	—
233	D₄	—	172	24	no	—
234	D₄	—	172	8	no	—
235	D₄	#13063322	1	12	yes	—
236	C₂×C₄	—	2	12	no	—
237	C₂×C₄	—	86	12	no	—
238	C₂×C₄	—	172	24	no	—
239	C₂×C₄	—	172	24	no	—
240	C₂×C₄	C₂²×D₁₇₂	1	6	yes	—
241	C₂×C₄	C₂²×D₁₇₂	1	3	yes	—
242	C₂×C₄	C₂²×D₁₇₂	1	12	yes	—
243	C₂×C₄	C₂²×D₁₇₂	1	2	yes	—
244	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
245	C₈	C₂²×D₁₇₂	1	4	yes	—
246	C₈	C₂²×D₁₇₂	1	12	yes	—
247	C₂²	—	2	6	no	—
248	C₂²	—	172	8	no	—
249	C₂²	—	172	4	no	—
250	C₂²	C₂×#13063322	1	6	yes	—
251	C₂²	C₂³×D₁₇₂	1	1	yes	characteristic
252	C₄	—	2	6	no	—
253	C₄	—	172	12	no	—
254	C₄	C₂×#13063322	1	6	yes	—
255	C₄	#14583716	1	4	yes	complemented
256	C₄	C₂³×D₁₇₂	1	2	yes	—
257	C₂	—	2	6	no	—
258	C₂	—	344	4	no	—
259	C₂	C₂²×#13063322	1	1	yes	characteristic
260	C₂	C₂×#14583716	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂³×D₁₇₂
commutator, cosocle	C₂×C₄×C₄₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₄×#131×C₄₃	C₂
socle	C₂²×C₄₃	C₂³×D₄
hypercenter	C₄×#131	D₄₃
hypocenter	C₄₃	#60648
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
1. C₂×#14583716	C₂	semidirect
2. #19148380	C₂	semidirect
3. #19145089	C₂	semidirect
4. #19145054	C₂	semidirect
5. #19145023	C₂	semidirect
6. C₂×#14583279	C₂	semidirect
7. C₄×#131×C₄₃	C₂	semidirect
53. #131×C₄₃	D₄	semidirect
124. C₄×C₄₃	#578	semidirect
133. C₄×#131	D₄₃	semidirect
160. C₄₃	#60648	semidirect
174. #131	D₁₇₂	semidirect
255. C₄	#14583716	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583716	2
#19148380	2
#19145089	2
#19145054	2
#19145023	2
C₂×#14583279	2
C₄×#131×C₄₃	2
#60648	43