Subgroups — #25196462

Order 11008.

722 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	#19142958	C₂	1	1	yes	maximal, characteristic
2	#19135746	C₂	1	4	yes	maximal
3	#19135398	C₂	1	2	yes	maximal
4	#19134722	C₂	1	2	yes	maximal
5	#19134610	C₂	1	4	yes	maximal
6	C₄×C₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13063317	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
10	C₂×#13063249	C₂²	1	8	yes	—
11	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
12	#14582797	—	2	2	no	—
13	#14582797	—	2	2	no	—
14	#14582769	C₂²	1	2	yes	—
15	#14582765	C₂²	1	2	yes	—
16	#14582661	C₂²	1	8	yes	—
17	#14582654	C₂²	1	2	yes	—
18	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₄₃	C₂²	1	2	yes	—
20	C₄₃×#391	C₂²	1	2	yes	—
21	C₄₃×#389	—	2	2	no	—
22	C₄₃×#384	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₄₃	—	2	2	no	—
24	C₄³×C₄₃	C₂²	1	1	yes	characteristic
25	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
26	#13063317	—	4	4	no	—
27	C₂×C₄⋊Dic₄₃	—	2	2	no	—
28	C₂×C₄⋊Dic₄₃	—	2	2	no	—
29	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
30	C₂×Dic₄₃⋊C₄	—	2	4	no	—
31	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
32	#13063249	—	4	8	no	—
33	C₄×C₄₃⋊C₈	—	2	4	no	—
34	C₄₃⋊#95	—	2	2	no	—
35	C₄₃⋊#95	—	2	2	no	—
36	C₂²×Q₈×C₄₃	C₂³	1	2	yes	—
37	C₄⋊Q₈×C₄₃	—	2	2	no	—
38	C₄⋊Q₈×C₄₃	Q₈	1	4	yes	—
39	C₄×Q₈×C₄₃	—	2	8	no	—
40	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
41	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
42	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
44	C₂×C₄²×C₄₃	—	2	2	no	—
45	C₂×C₄²×C₄₃	D₄	1	2	yes	—
46	C₂×C₄²×C₄₃	D₄	1	2	yes	—
47	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
50	#95×C₄₃	—	2	4	no	—
51	C₂×Q₈×C₄₃	—	2	2	no	—
52	C₂×Q₈×C₄₃	—	2	4	no	—
53	C₂×Q₈×C₄₃	—	2	4	no	—
54	C₂×Q₈×C₄₃	C₂×Q₈	1	4	yes	—
55	C₂²×C₄×C₄₃	—	2	2	no	—
56	C₂²×C₄×C₄₃	—	2	2	no	—
57	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
62	C₂²×Dic₄₃	—	2	4	no	—
63	C₄⋊C₄×C₄₃	—	2	8	no	—
64	C₄⋊C₄×C₄₃	—	2	4	no	—
65	C₄⋊C₄×C₄₃	—	4	4	no	—
66	C₄⋊C₄×C₄₃	—	4	4	no	—
67	C₄²×C₄₃	—	2	4	no	—
68	C₄²×C₄₃	—	2	4	no	—
69	C₄²×C₄₃	—	2	2	no	—
70	C₄²×C₄₃	—	4	4	no	—
71	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
72	C₄²×C₄₃	#41	1	2	yes	—
73	C₄²×C₄₃	#41	1	4	yes	—
74	C₄⋊Dic₄₃	—	2	8	no	—
75	C₄⋊Dic₄₃	—	4	4	no	—
76	C₄⋊Dic₄₃	—	4	4	no	—
77	Dic₄₃⋊C₄	—	4	8	no	—
78	C₂×C₄₃⋊C₈	—	2	4	no	—
79	C₂×C₄₃⋊C₈	—	2	4	no	—
80	C₂×C₄₃⋊C₈	—	2	4	no	—
81	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
82	Q₈×C₄₃	—	2	8	no	—
83	Q₈×C₄₃	—	4	4	no	—
84	C₂×C₄×C₄₃	—	2	2	no	—
85	C₂×C₄×C₄₃	—	2	2	no	—
86	C₂×C₄×C₄₃	—	2	4	no	—
87	C₂×C₄×C₄₃	—	2	2	no	—
88	C₂×C₄×C₄₃	—	2	2	no	—
89	C₂×C₄×C₄₃	—	2	2	no	—
90	C₂×C₄×C₄₃	—	4	2	no	—
91	C₂×C₄×C₄₃	—	4	2	no	—
92	C₂×C₄×C₄₃	—	4	2	no	—
93	C₂×C₄×C₄₃	—	4	4	no	—
94	C₂×C₄×C₄₃	C₂²⋊Q₈	1	4	yes	—
95	C₂×C₄×C₄₃	#124	1	2	yes	—
96	C₂×C₄×C₄₃	#124	1	2	yes	—
97	C₂×C₄×C₄₃	#135	1	4	yes	—
98	C₂×C₄×C₄₃	#137	1	4	yes	—
99	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
101	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
102	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
103	C₂×Dic₄₃	—	4	4	no	—
104	C₂×Dic₄₃	—	4	4	no	—
105	C₂×Dic₄₃	—	4	4	no	—
106	C₄₃⋊C₈	—	4	4	no	—
107	#29144	—	43	1	no	maximal, Sylow
108	C₂²×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
109	C₂²×C₄₃	C₂×#124	1	1	yes	characteristic
110	C₂²×C₄₃	#535	1	1	yes	characteristic
111	C₂²×C₄₃	C₂×#135	1	2	yes	—
112	C₂²×C₄₃	C₂×#137	1	2	yes	—
113	C₄×C₄₃	—	2	2	no	—
114	C₄×C₄₃	—	2	2	no	—
115	C₄×C₄₃	—	4	4	no	—
116	C₄×C₄₃	—	4	2	no	—
117	C₄×C₄₃	—	8	2	no	—
118	C₄×C₄₃	#479	1	8	yes	—
119	C₄×C₄₃	#554	1	2	yes	—
120	C₄×C₄₃	#555	1	2	yes	—
121	Dic₄₃	—	8	4	no	—
122	#2859	—	43	1	no	—
123	C₄×C₄⋊Q₈	D₄₃	1	1	yes	characteristic
124	#2026	—	43	4	no	—
125	#1956	—	43	2	no	—
126	#1834	—	43	2	no	—
127	#1805	—	43	4	no	—
128	C₄×C₄⋊C₈	—	43	1	no	—
129	C₂×C₄₃	#2563	1	1	yes	characteristic
130	C₂×C₄₃	C₂×#479	1	2	yes	—
131	C₂×C₄₃	#3131	1	1	yes	characteristic
132	C₂×C₄₃	#3139	1	1	yes	characteristic
133	C₂×C₄₃	#3210	1	2	yes	—
134	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
135	C₂×#125	—	43	2	no	—
136	C₂×C₄×Q₈	C₂×D₄₃	1	2	yes	—
137	C₂×C₄⋊C₈	—	43	2	no	—
138	C₂×#103	—	43	8	no	—
139	C₂×C₄×C₈	—	43	2	no	—
140	#391	C₂×D₄₃	1	2	yes	—
141	#389	—	2	2	no	—
142	#389	—	86	2	no	—
143	#389	—	86	2	no	—
144	#384	—	43	2	no	—
145	#384	C₂×D₄₃	1	1	yes	characteristic
146	#382	—	43	2	no	—
147	C₄×C₄⋊C₄	—	2	2	no	—
148	C₄³	C₂×D₄₃	1	1	yes	characteristic
149	#340	—	43	8	no	—
150	#336	—	43	2	no	—
151	C₄₃	#29144	1	1	yes	characteristic, complemented, Sylow
152	C₂²×Q₈	C₂²×D₄₃	1	2	yes	—
153	C₂²×C₈	—	43	4	no	—
154	C₄⋊Q₈	—	2	2	no	—
155	C₄⋊Q₈	Dic₈₆	1	4	yes	—
156	#125	—	172	4	no	—
157	C₄×Q₈	—	2	8	no	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	43	4	no	—
162	C₂×C₄⋊C₄	—	86	4	no	—
163	C₂×C₄⋊C₄	—	86	2	no	—
164	C₂×C₄⋊C₄	—	86	2	no	—
165	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
166	C₂×C₄²	—	2	2	no	—
167	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
168	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
169	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
170	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
171	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
172	C₄⋊C₈	—	86	4	no	—
173	#103	—	172	8	no	—
174	C₄×C₈	—	86	4	no	—
175	#95	—	2	4	no	—
176	#95	—	86	2	no	—
177	#95	—	86	2	no	—
178	C₂×Q₈	—	2	2	no	—
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	—	2	4	no	—
181	C₂×Q₈	C₂×Dic₈₆	1	4	yes	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	86	4	no	—
185	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
186	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
189	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
190	C₂×C₈	—	86	4	no	—
191	C₂×C₈	—	86	4	no	—
192	C₂×C₈	—	86	4	no	—
193	C₄⋊C₄	—	2	8	no	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	4	4	no	—
196	C₄⋊C₄	—	4	4	no	—
197	C₄⋊C₄	—	86	8	no	—
198	C₄⋊C₄	—	172	8	no	—
199	C₄⋊C₄	—	172	4	no	—
200	C₄⋊C₄	—	172	4	no	—
201	C₄²	—	2	4	no	—
202	C₄²	—	2	4	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	4	4	no	—
205	C₄²	C₂×Dic₈₆	1	2	yes	—
206	C₄²	#10621898	1	2	yes	—
207	C₄²	#10621900	1	4	yes	—
208	C₂³	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
209	Q₈	—	2	8	no	—
210	Q₈	—	4	4	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	172	4	no	—
222	C₂×C₄	—	172	4	no	—
223	C₂×C₄	—	172	4	no	—
224	C₂×C₄	#13063304	1	4	yes	—
225	C₂×C₄	#13063312	1	2	yes	—
226	C₂×C₄	#13063312	1	2	yes	—
227	C₂×C₄	#13063345	1	4	yes	—
228	C₂×C₄	#13063356	1	4	yes	—
229	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
230	C₂×C₄	C₂×#10621900	1	2	yes	—
231	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
232	C₂×C₄	#13063371	1	2	yes	—
233	C₈	—	172	4	no	—
234	C₂²	C₂×#13063304	1	1	yes	characteristic
235	C₂²	C₂×#13063312	1	1	yes	characteristic
236	C₂²	#14583463	1	1	yes	characteristic
237	C₂²	C₂×#13063345	1	2	yes	—
238	C₂²	C₂×#13063356	1	2	yes	—
239	C₄	—	2	2	no	—
240	C₄	—	2	2	no	—
241	C₄	—	4	2	no	—
242	C₄	—	4	4	no	—
243	C₄	—	8	2	no	—
244	C₄	—	344	4	no	—
245	C₄	#14583206	1	8	yes	—
246	C₄	#14583606	1	2	yes	—
247	C₄	#14583614	1	2	yes	—
248	C₂	#19139768	1	1	yes	characteristic
249	C₂	C₂×#14583206	1	2	yes	—
250	C₂	#19144917	1	1	yes	characteristic
251	C₂	#19144975	1	1	yes	characteristic
252	C₂	#19145547	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₃
commutator, cosocle	C₂²×C₄×C₄₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄×C₄⋊Q₈×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
hypercenter	C₄×C₄⋊Q₈	D₄₃
hypocenter	C₄₃	#29144
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
151. C₄₃	#29144	semidirect

Maximal subgroups

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

subgroup	index
#19142958	2
#19135746	2
#19135398	2
#19134722	2
#19134610	2
C₄×C₄⋊(C₄₃⋊C₈)	2
C₄×C₄⋊Q₈×C₄₃	2
#29144	43