Subgroups — #25249194

Order 11008.

894 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583308	C₂	1	4	yes	maximal
2	#19142993	C₂	1	1	yes	maximal, characteristic
3	#19142958	C₂	1	1	yes	maximal, characteristic
4	#19134763	C₂	1	4	yes	maximal
5	#19134755	C₂	1	2	yes	maximal
6	#19134752	C₂	1	2	yes	maximal
7	C₄×#97×C₄₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13063325	—	2	2	no	—
9	C₂×#13063325	C₂²	1	2	yes	—
10	C₂×#13063317	C₂²	1	2	yes	—
11	#14583308	—	2	16	no	—
12	C₂×#13063259	C₂²	1	8	yes	—
13	C₂×#13063257	C₂²	1	8	yes	—
14	#14582797	—	2	2	no	—
15	#14582797	—	2	2	no	—
16	#14582769	—	2	2	no	—
17	#14582769	C₂²	1	1	yes	characteristic
18	#14582769	C₂²	1	2	yes	—
19	#14582769	C₂²	1	1	yes	characteristic
20	#14582769	C₂²	1	2	yes	—
21	#14582765	C₂²	1	2	yes	—
22	C₂×#97×C₄₃	C₂²	1	2	yes	—
23	C₂×#97×C₄₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
25	C₄³×C₄₃	C₂²	1	1	yes	characteristic
26	C₂²×Dic₈₆	—	2	2	no	—
27	#13063325	—	2	8	no	—
28	#13063325	—	4	8	no	—
29	#13063317	—	2	8	no	—
30	C₂×C₄⋊Dic₄₃	—	2	2	no	—
31	C₂×C₄⋊Dic₄₃	—	2	2	no	—
32	C₂×C₄⋊Dic₄₃	—	2	4	no	—
33	C₂×C₄⋊Dic₄₃	—	2	4	no	—
34	C₂×C₄⋊Dic₄₃	—	2	2	no	—
35	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
36	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
37	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
38	C₂×Dic₄₃⋊C₄	—	2	2	no	—
39	C₂×Dic₄₃⋊C₄	—	2	2	no	—
40	#13063259	—	2	16	no	—
41	#13063257	—	2	16	no	—
42	C₄₃⋊#95	—	2	4	no	—
43	C₂²×C₈×C₄₃	C₂³	1	4	yes	—
44	C₂×C₄²×C₄₃	D₄	1	2	yes	—
45	C₂×C₄²×C₄₃	D₄	1	2	yes	—
46	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
49	#97×C₄₃	D₄	1	8	yes	—
50	#97×C₄₃	Q₈	1	8	yes	complemented
51	C₄×C₈×C₄₃	Q₈	1	8	yes	complemented
52	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
57	C₂²×Dic₄₃	—	2	2	no	—
58	C₂²×Dic₄₃	—	2	2	no	—
59	C₂²×Dic₄₃	—	2	4	no	—
60	C₂×Dic₈₆	—	4	8	no	—
61	C₂×Dic₈₆	—	4	4	no	—
62	C₂×C₈×C₄₃	C₂×D₄	1	8	yes	—
63	C₂×C₈×C₄₃	C₂×Q₈	1	8	yes	—
64	C₂×C₈×C₄₃	C₂×Q₈	1	8	yes	—
65	C₄²×C₄₃	—	2	4	no	—
66	C₄²×C₄₃	—	2	4	no	—
67	C₄²×C₄₃	C₂×D₄	1	4	yes	—
68	C₄²×C₄₃	C₂×Q₈	1	4	yes	—
69	C₄²×C₄₃	C₂×Q₈	1	4	yes	—
70	C₄⋊Dic₄₃	—	2	4	no	—
71	C₄⋊Dic₄₃	—	2	8	no	—
72	C₄⋊Dic₄₃	—	2	4	no	—
73	C₄⋊Dic₄₃	—	4	8	no	—
74	C₄⋊Dic₄₃	—	4	4	no	—
75	C₄⋊Dic₄₃	—	4	4	no	—
76	C₄⋊Dic₄₃	—	4	8	no	—
77	C₄⋊Dic₄₃	—	4	4	no	—
78	Dic₄₃⋊C₄	—	4	4	no	—
79	Dic₄₃⋊C₄	—	4	4	no	—
80	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
81	C₂×C₄×C₄₃	—	2	2	no	—
82	C₂×C₄×C₄₃	—	2	2	no	—
83	C₂×C₄×C₄₃	—	2	2	no	—
84	C₂×C₄×C₄₃	—	2	2	no	—
85	C₂×C₄×C₄₃	#127	1	2	yes	—
86	C₂×C₄×C₄₃	#127	1	2	yes	—
87	C₂×C₄×C₄₃	C₄⋊Q₈	1	4	yes	—
88	C₂×C₄×C₄₃	C₄⋊Q₈	1	4	yes	—
89	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
90	C₂×C₄×C₄₃	#137	1	4	yes	—
91	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
92	C₂×C₄×C₄₃	C₂²×Q₈	1	2	yes	—
93	C₂×C₄×C₄₃	C₂²×Q₈	1	2	yes	—
94	C₂×Dic₄₃	—	4	2	no	—
95	C₂×Dic₄₃	—	4	2	no	—
96	C₂×Dic₄₃	—	4	4	no	—
97	C₂×Dic₄₃	—	4	8	no	—
98	C₂×Dic₄₃	—	4	4	no	—
99	C₂×Dic₄₃	—	4	4	no	—
100	Dic₈₆	—	8	8	no	—
101	C₈×C₄₃	C₄⋊Q₈	1	16	yes	complemented
102	#33316	—	43	1	no	maximal, Sylow
103	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
104	C₂²×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
105	C₂²×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
106	C₂²×C₄₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
107	C₂²×C₄₃	C₂×#137	1	2	yes	—
108	C₄×C₄₃	—	2	2	no	—
109	C₄×C₄₃	—	2	2	no	—
110	C₄×C₄₃	—	2	4	no	—
111	C₄×C₄₃	#501	1	8	yes	complemented
112	C₄×C₄₃	C₂×C₄⋊Q₈	1	4	yes	—
113	Dic₄₃	—	8	4	no	—
114	Dic₄₃	—	8	2	no	—
115	Dic₄₃	—	8	2	no	—
116	C₂×#501	—	43	4	no	—
117	#2868	—	43	1	no	—
118	#2859	—	43	1	no	—
119	#1847	—	43	4	no	—
120	#1845	—	43	2	no	—
121	#1844	—	43	2	no	—
122	C₄×#97	D₄₃	1	1	yes	characteristic
123	C₂×C₄₃	#2868	1	1	yes	characteristic
124	C₂×C₄₃	#3151	1	1	yes	characteristic
125	C₂×C₄₃	#3152	1	1	yes	characteristic
126	C₂×C₄₃	C₂×#501	1	2	yes	—
127	C₂×C₄₃	#3165	1	2	yes	—
128	C₂×C₄⋊Q₈	—	43	2	no	—
129	C₂×C₄⋊Q₈	—	86	2	no	—
130	C₂×#125	—	43	2	no	—
131	#501	—	86	16	no	—
132	C₂×#107	—	43	8	no	—
133	C₂×#106	—	43	8	no	—
134	C₂×#97	C₂×D₄₃	1	2	yes	—
135	C₂×#97	C₂×D₄₃	1	2	yes	—
136	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
137	#389	—	86	2	no	—
138	#389	—	86	2	no	—
139	#384	—	43	1	no	—
140	#384	—	43	1	no	—
141	#384	—	43	2	no	—
142	#384	—	43	2	no	—
143	#384	—	86	2	no	—
144	#382	—	43	2	no	—
145	C₄³	C₂×D₄₃	1	1	yes	characteristic
146	C₄₃	#33316	1	1	yes	characteristic, complemented, Sylow
147	C₂²×Q₈	—	86	2	no	—
148	C₂²×C₈	C₂²×D₄₃	1	4	yes	—
149	C₄⋊Q₈	—	86	8	no	—
150	C₄⋊Q₈	—	172	8	no	—
151	#125	—	86	8	no	—
152	C₂×C₄⋊C₄	—	43	2	no	—
153	C₂×C₄⋊C₄	—	43	2	no	—
154	C₂×C₄⋊C₄	—	43	4	no	—
155	C₂×C₄⋊C₄	—	86	2	no	—
156	C₂×C₄⋊C₄	—	86	2	no	—
157	C₂×C₄⋊C₄	—	86	2	no	—
158	C₂×C₄⋊C₄	—	86	4	no	—
159	C₂×C₄⋊C₄	—	86	4	no	—
160	C₂×C₄⋊C₄	—	86	2	no	—
161	C₂×C₄⋊C₄	—	86	2	no	—
162	C₂×C₄²	D₁₇₂	1	2	yes	—
163	C₂×C₄²	D₁₇₂	1	2	yes	—
164	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
165	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
166	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
167	#107	—	86	16	no	—
168	#106	—	86	16	no	—
169	#97	Dic₈₆	1	8	yes	complemented
170	#97	D₁₇₂	1	8	yes	—
171	C₄×C₈	Dic₈₆	1	8	yes	complemented
172	#95	—	86	4	no	—
173	C₂×Q₈	—	172	8	no	—
174	C₂×Q₈	—	172	4	no	—
175	C₂²×C₄	—	86	2	no	—
176	C₂²×C₄	—	86	2	no	—
177	C₂²×C₄	—	86	4	no	—
178	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
181	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
182	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
183	C₂×C₈	C₂×Dic₈₆	1	8	yes	—
184	C₂×C₈	C₂×Dic₈₆	1	8	yes	—
185	C₂×C₈	C₂×D₁₇₂	1	8	yes	—
186	C₄⋊C₄	—	86	4	no	—
187	C₄⋊C₄	—	86	8	no	—
188	C₄⋊C₄	—	86	4	no	—
189	C₄⋊C₄	—	172	8	no	—
190	C₄⋊C₄	—	172	4	no	—
191	C₄⋊C₄	—	172	4	no	—
192	C₄⋊C₄	—	172	4	no	—
193	C₄⋊C₄	—	172	8	no	—
194	C₄⋊C₄	—	172	4	no	—
195	C₄⋊C₄	—	172	4	no	—
196	C₄²	—	2	4	no	—
197	C₄²	—	2	4	no	—
198	C₄²	C₂×Dic₈₆	1	4	yes	—
199	C₄²	C₂×Dic₈₆	1	4	yes	—
200	C₄²	C₂×D₁₇₂	1	4	yes	—
201	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
202	Q₈	—	344	8	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	172	8	no	—
208	C₂×C₄	—	172	4	no	—
209	C₂×C₄	—	172	4	no	—
210	C₂×C₄	—	172	2	no	—
211	C₂×C₄	—	172	2	no	—
212	C₂×C₄	—	172	4	no	—
213	C₂×C₄	#13063322	1	2	yes	—
214	C₂×C₄	#13063322	1	2	yes	—
215	C₂×C₄	#13063325	1	4	yes	—
216	C₂×C₄	#13063325	1	4	yes	—
217	C₂×C₄	#13063353	1	4	yes	—
218	C₂×C₄	#13063358	1	4	yes	—
219	C₂×C₄	C₂²×D₁₇₂	1	2	yes	—
220	C₂×C₄	C₂²×Dic₈₆	1	2	yes	—
221	C₂×C₄	C₂²×Dic₈₆	1	2	yes	—
222	C₈	#13063325	1	16	yes	complemented
223	C₂²	C₂×#13063322	1	1	yes	characteristic
224	C₂²	C₂×#13063325	1	1	yes	characteristic
225	C₂²	C₂×#13063325	1	1	yes	characteristic
226	C₂²	C₂×#13063353	1	2	yes	—
227	C₂²	C₂×#13063358	1	2	yes	—
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	4	no	—
231	C₄	—	344	4	no	—
232	C₄	—	344	2	no	—
233	C₄	—	344	2	no	—
234	C₄	#14583308	1	8	yes	complemented
235	C₄	C₂×#13063325	1	4	yes	—
236	C₂	#19142993	1	1	yes	characteristic
237	C₂	#19145047	1	1	yes	characteristic
238	C₂	#19145054	1	1	yes	characteristic
239	C₂	C₂×#14583308	1	2	yes	—
240	C₂	#19145141	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₇₂
commutator, cosocle	C₂²×C₄×C₄₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄×#97×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
hypercenter	C₄×#97	D₄₃
hypocenter	C₄₃	#33316
LCS	G ▷ C₂²×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
50. #97×C₄₃	Q₈	semidirect
51. C₄×C₈×C₄₃	Q₈	semidirect
101. C₈×C₄₃	C₄⋊Q₈	semidirect
111. C₄×C₄₃	#501	semidirect
146. C₄₃	#33316	semidirect
169. #97	Dic₈₆	semidirect
171. C₄×C₈	Dic₈₆	semidirect
222. C₈	#13063325	semidirect
234. C₄	#14583308	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583308	2
#19142993	2
#19142958	2
#19134763	2
#19134755	2
#19134752	2
C₄×#97×C₄₃	2
#33316	43