Subgroups — #25196485

Order 11008.

726 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19142973	C₂	1	1	yes	maximal, characteristic
2	#19135736	C₂	1	4	yes	maximal
3	#19135416	C₂	1	2	yes	maximal
4	#19134730	C₂	1	2	yes	maximal
5	#19134610	C₂	1	4	yes	maximal
6	#19134448	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13063325	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
10	C₂×#13063249	C₂²	1	8	yes	—
11	C₂×#13063236	C₂²	1	2	yes	—
12	#14582805	—	2	2	no	—
13	#14582805	—	2	2	no	—
14	#14582765	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×Dic₈₆	—	2	2	no	—
26	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
27	#13063325	—	4	4	no	—
28	C₂×C₄⋊Dic₄₃	—	2	2	no	—
29	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
30	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
31	#13063249	—	4	8	no	—
32	#13063236	—	2	4	no	—
33	C₄₃⋊#95	—	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	complemented
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	C₂×Dic₈₆	—	4	4	no	—
64	C₂×Dic₈₆	—	4	4	no	—
65	C₂×Dic₈₆	—	4	4	no	—
66	C₄⋊C₄×C₄₃	—	2	8	no	—
67	C₄⋊C₄×C₄₃	—	2	4	no	—
68	C₄⋊C₄×C₄₃	—	4	4	no	—
69	C₄⋊C₄×C₄₃	—	4	4	no	—
70	C₄²×C₄₃	—	2	4	no	—
71	C₄²×C₄₃	—	2	2	no	—
72	C₄²×C₄₃	—	4	2	no	—
73	C₄²×C₄₃	—	4	4	no	—
74	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
75	C₄²×C₄₃	#41	1	2	yes	—
76	C₄²×C₄₃	#41	1	4	yes	—
77	C₄⋊Dic₄₃	—	2	8	no	—
78	C₄⋊Dic₄₃	—	4	4	no	—
79	C₂×C₄₃⋊C₈	—	2	4	no	—
80	C₂×C₄₃⋊C₈	—	2	4	no	—
81	C₂×C₄₃⋊C₈	—	2	4	no	—
82	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
83	Q₈×C₄₃	—	2	8	no	—
84	Q₈×C₄₃	—	4	4	no	—
85	C₂×C₄×C₄₃	—	2	2	no	—
86	C₂×C₄×C₄₃	—	2	2	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	—	2	2	no	—
91	C₂×C₄×C₄₃	—	2	4	no	—
92	C₂×C₄×C₄₃	—	4	2	no	—
93	C₂×C₄×C₄₃	—	4	2	no	—
94	C₂×C₄×C₄₃	—	4	2	no	—
95	C₂×C₄×C₄₃	—	4	4	no	—
96	C₂×C₄×C₄₃	C₂²⋊Q₈	1	4	yes	—
97	C₂×C₄×C₄₃	#124	1	2	yes	—
98	C₂×C₄×C₄₃	#124	1	2	yes	—
99	C₂×C₄×C₄₃	#137	1	4	yes	—
100	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
102	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
103	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
104	C₂×Dic₄₃	—	4	4	no	—
105	C₂×Dic₄₃	—	4	4	no	—
106	C₂×Dic₄₃	—	4	4	no	—
107	Dic₈₆	—	8	8	no	—
108	C₄₃⋊C₈	—	4	4	no	—
109	#29147	—	43	1	no	maximal, Sylow
110	C₂²×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
111	C₂²×C₄₃	C₂×#124	1	1	yes	characteristic
112	C₂²×C₄₃	#535	1	1	yes	characteristic
113	C₂²×C₄₃	C₂×#137	1	2	yes	—
114	C₂²×C₄₃	#575	1	2	yes	—
115	C₄×C₄₃	—	2	4	no	—
116	C₄×C₄₃	—	2	2	no	—
117	C₄×C₄₃	—	2	2	no	—
118	C₄×C₄₃	—	2	4	no	—
119	C₄×C₄₃	—	4	4	no	—
120	C₄×C₄₃	—	8	2	no	—
121	C₄×C₄₃	#554	1	2	yes	—
122	C₄×C₄₃	#555	1	2	yes	—
123	Dic₄₃	—	8	4	no	—
124	#2862	—	43	1	no	—
125	C₄×C₄⋊Q₈	D₄₃	1	1	yes	characteristic
126	#2024	—	43	4	no	—
127	#1960	—	43	2	no	—
128	#1836	—	43	2	no	—
129	#1805	—	43	4	no	—
130	#1767	—	43	1	no	—
131	C₂×C₄₃	#2563	1	1	yes	characteristic
132	C₂×C₄₃	#3078	1	2	yes	—
133	C₂×C₄₃	#3135	1	1	yes	characteristic
134	C₂×C₄₃	#3139	1	1	yes	characteristic
135	C₂×C₄₃	#3215	1	2	yes	—
136	C₂×C₄⋊Q₈	—	43	2	no	—
137	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
138	C₂×C₄×Q₈	C₂×D₄₃	1	2	yes	—
139	C₂×C₄⋊C₈	—	43	2	no	—
140	C₂×#103	—	43	8	no	—
141	C₂×#97	—	43	2	no	—
142	#391	—	86	2	no	—
143	#391	—	86	2	no	—
144	#391	C₂×D₄₃	1	2	yes	—
145	#389	—	2	2	no	—
146	#384	C₂×D₄₃	1	1	yes	characteristic
147	#382	—	43	2	no	—
148	#382	—	43	2	no	—
149	C₄×C₄⋊C₄	—	2	2	no	—
150	C₄³	C₂×D₄₃	1	1	yes	characteristic
151	#340	—	43	8	no	—
152	#336	—	43	2	no	—
153	C₄₃	#29147	1	1	yes	characteristic, complemented, Sylow
154	C₂²×Q₈	—	86	2	no	—
155	C₂²×Q₈	C₂²×D₄₃	1	2	yes	—
156	C₂²×C₈	—	43	4	no	—
157	C₄⋊Q₈	—	2	2	no	—
158	C₄⋊Q₈	—	172	4	no	—
159	C₄⋊Q₈	Dic₈₆	1	4	yes	complemented
160	C₄×Q₈	—	2	8	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	43	4	no	—
165	C₂×C₄⋊C₄	—	86	2	no	—
166	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
167	C₂×C₄²	—	2	2	no	—
168	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
169	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
170	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
171	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
172	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
173	C₄⋊C₈	—	86	4	no	—
174	#103	—	172	8	no	—
175	#97	—	86	4	no	—
176	#95	—	2	4	no	—
177	#95	—	86	4	no	—
178	#95	—	86	2	no	—
179	#95	—	86	2	no	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	—	2	4	no	—
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	172	4	no	—
184	C₂×Q₈	—	172	4	no	—
185	C₂×Q₈	—	172	4	no	—
186	C₂×Q₈	C₂×Dic₈₆	1	4	yes	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	86	4	no	—
190	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
191	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
194	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
195	C₂×C₈	—	86	4	no	—
196	C₂×C₈	—	86	4	no	—
197	C₂×C₈	—	86	4	no	—
198	C₄⋊C₄	—	2	8	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	4	4	no	—
201	C₄⋊C₄	—	4	4	no	—
202	C₄⋊C₄	—	86	8	no	—
203	C₄⋊C₄	—	172	4	no	—
204	C₄²	—	2	4	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	4	2	no	—
207	C₄²	—	4	4	no	—
208	C₄²	C₂×Dic₈₆	1	2	yes	—
209	C₄²	#10621898	1	2	yes	—
210	C₄²	#10621900	1	4	yes	—
211	C₂³	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
212	Q₈	—	2	8	no	—
213	Q₈	—	4	4	no	—
214	Q₈	—	344	8	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	172	4	no	—
227	C₂×C₄	—	172	4	no	—
228	C₂×C₄	—	172	4	no	—
229	C₂×C₄	#13063304	1	4	yes	—
230	C₂×C₄	#13063312	1	2	yes	—
231	C₂×C₄	#13063312	1	2	yes	—
232	C₂×C₄	#13063356	1	4	yes	—
233	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
234	C₂×C₄	C₂×#10621900	1	2	yes	—
235	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
236	C₂×C₄	#13063371	1	2	yes	—
237	C₈	—	172	4	no	—
238	C₂²	C₂×#13063304	1	1	yes	characteristic
239	C₂²	C₂×#13063312	1	1	yes	characteristic
240	C₂²	#14583463	1	1	yes	characteristic
241	C₂²	C₂×#13063356	1	2	yes	—
242	C₂²	#14583697	1	2	yes	—
243	C₄	—	2	2	no	—
244	C₄	—	2	4	no	—
245	C₄	—	2	4	no	—
246	C₄	—	2	2	no	—
247	C₄	—	4	4	no	—
248	C₄	—	8	2	no	—
249	C₄	—	344	4	no	—
250	C₄	#14583606	1	2	yes	—
251	C₄	#14583614	1	2	yes	—
252	C₂	#19139768	1	1	yes	characteristic
253	C₂	#19144400	1	2	yes	—
254	C₂	#19144940	1	1	yes	characteristic
255	C₂	#19144975	1	1	yes	characteristic
256	C₂	#19145594	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	#29147
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
38. C₄⋊Q₈×C₄₃	Q₈	semidirect
153. C₄₃	#29147	semidirect
159. C₄⋊Q₈	Dic₈₆	semidirect

Maximal subgroups

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

subgroup	index
#19142973	2
#19135736	2
#19135416	2
#19134730	2
#19134610	2
#19134448	2
C₄×C₄⋊Q₈×C₄₃	2
#29147	43