Subgroups — #24987230

Order 11008.

542 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19137072	C₂	1	1	yes	maximal, characteristic
2	#19134448	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582631	C₂	1	2	yes	maximal
4	#19132435	C₂	1	2	yes	maximal
5	C₄²×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
8	C₂×#13063236	—	2	1	no	—
9	#14582805	C₄	1	2	yes	—
10	#14582797	C₄	1	2	yes	—
11	#14582783	—	2	2	no	—
12	C₄×Dic₄₃⋊C₄	—	2	1	no	—
13	C₄×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
14	#14582745	—	2	1	no	—
15	#14582654	—	2	1	no	—
16	#14582631	C₄	1	8	yes	complemented
17	C₂×C₄×C₈×C₄₃	—	2	2	no	—
18	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
19	C₄³×C₄₃	C₂²	1	1	yes	characteristic
20	C₂²×Dic₈₆	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄₃⋊C₈	—	2	2	no	—
22	C₄×Dic₈₆	—	2	2	no	—
23	C₄×Dic₈₆	C₂×C₄	1	4	yes	—
24	C₂×C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
25	C₂×C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
26	C₂×Dic₄₃⋊C₄	—	2	1	no	—
27	C₂×Dic₄₃⋊C₄	—	2	2	no	—
28	C₂×C₄×Dic₄₃	—	2	1	no	—
29	C₂×C₄×Dic₄₃	—	2	1	no	—
30	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
31	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	4	yes	—
32	#13063236	—	4	2	no	—
33	C₄₃⋊#95	—	2	2	no	—
34	C₄₃⋊#95	—	2	2	no	—
35	C₄₃⋊#95	—	2	2	no	—
36	C₄₃⋊#95	—	2	1	no	—
37	C₂²×C₈×C₄₃	—	2	2	no	—
38	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₃	D₄	1	2	yes	—
42	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
43	C₄×C₈×C₄₃	—	2	8	no	—
44	C₄×C₈×C₄₃	—	2	2	no	—
45	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
46	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₄₃	—	2	1	no	—
52	C₂²×Dic₄₃	—	2	1	no	—
53	C₂²×Dic₄₃	—	2	2	no	—
54	C₂×Dic₈₆	—	2	4	no	—
55	C₂×Dic₈₆	—	2	1	no	—
56	C₂×Dic₈₆	C₄²	1	2	yes	—
57	C₂×C₈×C₄₃	—	2	8	no	—
58	C₂×C₈×C₄₃	—	2	4	no	—
59	C₄²×C₄₃	—	2	4	no	—
60	C₄²×C₄₃	—	2	4	no	—
61	C₄²×C₄₃	—	2	2	no	—
62	C₄²×C₄₃	SD₁₆	1	2	yes	—
63	C₄²×C₄₃	Q₁₆	1	2	yes	—
64	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
65	C₄²×C₄₃	#41	1	2	yes	—
66	C₄⋊Dic₄₃	—	2	1	no	—
67	C₄⋊Dic₄₃	—	2	4	no	—
68	C₄⋊Dic₄₃	C₄²	1	2	yes	—
69	Dic₄₃⋊C₄	—	2	4	no	—
70	Dic₄₃⋊C₄	—	4	4	no	—
71	C₄×Dic₄₃	—	2	4	no	—
72	C₄×Dic₄₃	—	4	2	no	—
73	C₂×C₄₃⋊C₈	—	2	4	no	—
74	C₂×C₄₃⋊C₈	—	4	2	no	—
75	C₂×C₄₃⋊C₈	—	4	2	no	—
76	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₄₃	—	2	2	no	—
78	C₂×C₄×C₄₃	—	2	2	no	—
79	C₂×C₄×C₄₃	—	2	2	no	—
80	C₂×C₄×C₄₃	—	2	2	no	—
81	C₂×C₄×C₄₃	—	2	2	no	—
82	C₂×C₄×C₄₃	#103	1	4	yes	—
83	C₂×C₄×C₄₃	#104	1	4	yes	—
84	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
85	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
86	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
87	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₄₃	#131	1	4	yes	—
89	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
91	C₂×C₄×C₄₃	#135	1	2	yes	—
92	C₂×Dic₄₃	—	2	2	no	—
93	C₂×Dic₄₃	—	2	2	no	—
94	C₂×Dic₄₃	—	4	2	no	—
95	C₂×Dic₄₃	—	4	2	no	—
96	C₂×Dic₄₃	—	4	2	no	—
97	C₂×Dic₄₃	—	4	2	no	—
98	C₂×Dic₄₃	—	4	2	no	—
99	Dic₈₆	—	2	4	no	—
100	Dic₈₆	—	4	2	no	—
101	C₈×C₄₃	—	2	8	no	—
102	C₄₃⋊C₈	—	4	4	no	—
103	#8176	—	43	1	no	maximal, Sylow
104	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
105	C₂²×C₄₃	#408	1	2	yes	—
106	C₂²×C₄₃	C₂×#103	1	1	yes	characteristic
107	C₂²×C₄₃	#416	1	1	yes	characteristic
108	C₂²×C₄₃	C₂×#104	1	2	yes	—
109	C₄×C₄₃	—	2	4	no	—
110	C₄×C₄₃	—	2	4	no	—
111	C₄×C₄₃	—	2	2	no	—
112	C₄×C₄₃	#405	1	2	yes	—
113	C₄×C₄₃	#433	1	2	yes	—
114	C₄×C₄₃	C₄×SD₁₆	1	2	yes	—
115	C₄×C₄₃	C₄×Q₁₆	1	2	yes	—
116	Dic₄₃	—	4	2	no	—
117	Dic₄₃	—	4	2	no	—
118	Dic₄₃	—	8	2	no	—
119	#2295	—	43	1	no	—
120	#1767	—	43	1	no	—
121	C₄²×C₈	D₄₃	1	1	yes	characteristic
122	C₂×Q₈⋊C₈	—	43	2	no	—
123	#1279	—	43	2	no	—
124	C₂×C₄₃	#1493	1	2	yes	—
125	C₂×C₄₃	#1494	1	2	yes	—
126	C₂×C₄₃	#1750	1	1	yes	characteristic
127	C₂×C₄₃	C₄×#104	1	1	yes	characteristic
128	C₂×C₄₃	C₄×#103	1	1	yes	characteristic
129	C₂×C₄×Q₈	—	43	1	no	—
130	C₂×C₄⋊C₈	—	43	2	no	—
131	C₂×#97	—	86	1	no	—
132	C₂×C₄×C₈	—	2	2	no	—
133	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
134	#391	—	43	2	no	—
135	#389	—	43	2	no	—
136	#387	—	86	2	no	—
137	C₄×C₄⋊C₄	—	43	1	no	—
138	C₄×C₄⋊C₄	—	86	1	no	—
139	#376	—	86	1	no	—
140	C₄³	C₂×D₄₃	1	1	yes	characteristic
141	#336	—	86	1	no	—
142	Q₈⋊C₈	—	43	8	no	—
143	C₄₃	#8176	1	1	yes	characteristic, complemented, Sylow
144	C₂²×Q₈	—	43	1	no	—
145	C₂²×C₈	—	2	2	no	—
146	C₂²×C₈	—	86	2	no	—
147	C₄×Q₈	—	43	4	no	—
148	C₄×Q₈	—	86	2	no	—
149	C₂×C₄⋊C₄	—	43	1	no	—
150	C₂×C₄⋊C₄	—	43	2	no	—
151	C₂×C₄⋊C₄	—	86	1	no	—
152	C₂×C₄⋊C₄	—	86	2	no	—
153	C₂×C₄²	—	86	1	no	—
154	C₂×C₄²	—	86	1	no	—
155	C₂×C₄²	C₄×D₄₃	1	2	yes	—
156	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
157	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
158	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
159	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
160	C₄⋊C₈	—	43	4	no	—
161	C₄⋊C₈	—	86	2	no	—
162	#97	—	172	2	no	—
163	C₄×C₈	—	2	8	no	—
164	C₄×C₈	—	2	2	no	—
165	C₄×C₈	C₄×D₄₃	1	4	yes	—
166	#95	—	86	2	no	—
167	#95	—	86	2	no	—
168	#95	—	86	2	no	—
169	#95	—	86	1	no	—
170	C₂×Q₈	—	43	2	no	—
171	C₂×Q₈	—	86	1	no	—
172	C₂×Q₈	—	86	4	no	—
173	C₂²×C₄	—	86	1	no	—
174	C₂²×C₄	—	86	1	no	—
175	C₂²×C₄	—	86	2	no	—
176	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
177	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
178	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
181	C₂×C₈	—	2	8	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	86	4	no	—
184	C₂×C₈	—	172	2	no	—
185	C₂×C₈	—	172	2	no	—
186	C₄⋊C₄	—	43	2	no	—
187	C₄⋊C₄	—	86	1	no	—
188	C₄⋊C₄	—	86	4	no	—
189	C₄⋊C₄	—	86	4	no	—
190	C₄⋊C₄	—	172	4	no	—
191	C₄²	—	2	4	no	—
192	C₄²	—	2	4	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	86	4	no	—
195	C₄²	—	172	2	no	—
196	C₄²	#10621873	1	2	yes	—
197	C₄²	Dic₁₇₂	1	2	yes	—
198	C₄²	C₂×C₄×D₄₃	1	2	yes	—
199	C₄²	#10621898	1	2	yes	—
200	C₂³	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
201	Q₈	—	86	4	no	—
202	Q₈	—	172	2	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	—	2	2	no	—
208	C₂×C₄	—	86	2	no	—
209	C₂×C₄	—	86	2	no	—
210	C₂×C₄	—	172	2	no	—
211	C₂×C₄	—	172	2	no	—
212	C₂×C₄	—	172	2	no	—
213	C₂×C₄	—	172	2	no	—
214	C₂×C₄	—	172	2	no	—
215	C₂×C₄	#13063250	1	4	yes	—
216	C₂×C₄	#13063253	1	4	yes	—
217	C₂×C₄	C₄²×D₄₃	1	1	yes	characteristic
218	C₂×C₄	#13063282	1	1	yes	characteristic
219	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
220	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
221	C₂×C₄	#13063331	1	4	yes	—
222	C₂×C₄	C₂×#10621873	1	1	yes	characteristic
223	C₂×C₄	C₂×Dic₁₇₂	1	1	yes	characteristic
224	C₂×C₄	#13063347	1	2	yes	—
225	C₈	—	2	8	no	—
226	C₈	—	172	4	no	—
227	C₂²	C₄×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
228	C₂²	#14582865	1	2	yes	—
229	C₂²	C₂×#13063250	1	1	yes	characteristic
230	C₂²	#14582896	1	1	yes	characteristic
231	C₂²	C₂×#13063253	1	2	yes	—
232	C₄	—	2	4	no	—
233	C₄	—	2	4	no	—
234	C₄	—	2	2	no	—
235	C₄	—	344	2	no	—
236	C₄	—	172	2	no	—
237	C₄	—	172	2	no	—
238	C₄	#14582853	1	2	yes	—
239	C₄	#14582958	1	2	yes	—
240	C₄	C₄×#10621873	1	2	yes	—
241	C₄	C₄×Dic₁₇₂	1	2	yes	—
242	C₂	#19133051	1	2	yes	—
243	C₂	#19133056	1	2	yes	—
244	C₂	#19134362	1	1	yes	characteristic
245	C₂	C₄×#13063253	1	1	yes	characteristic
246	C₂	C₄×#13063250	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄²×C₈×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₄²×C₈	D₄₃
hypocenter	C₄₃	#8176
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
16. #14582631	C₄	semidirect
143. C₄₃	#8176	semidirect

Maximal subgroups

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

subgroup	index
#19137072	2
#19134448	2
C₂×#14582631	2
#19132435	2
C₄²×C₈×C₄₃	2
#8176	43