Subgroups — #24995499

Order 11008.

482 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×C₄×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic
2	#19135531	C₂	1	2	yes	maximal
3	C₂×#14582650	C₂	1	1	yes	maximal, characteristic
4	C₂×#14582631	C₂	1	2	yes	maximal
5	C₂×C₄₃×#322	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₂	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₈₆	C₂²	1	2	yes	—
8	C₄×Dic₁₇₂	—	2	2	no	—
9	C₄×Dic₁₇₂	—	2	2	no	—
10	C₂×#13063259	C₄	1	1	yes	characteristic
11	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
12	C₂×#13063250	C₄	1	2	yes	—
13	#14582653	—	2	4	no	—
14	#14582650	—	2	2	no	—
15	#14582631	—	2	4	no	—
16	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
19	C₄₃×#322	—	2	2	no	—
20	C₂²×Dic₈₆	C₂×C₄	1	2	yes	—
21	C₂×Dic₁₇₂	—	2	4	no	—
22	C₂×Dic₁₇₂	—	2	2	no	—
23	C₂²×C₄₃⋊C₈	—	2	2	no	—
24	C₄×Dic₈₆	—	2	2	no	—
25	C₄×Dic₈₆	—	2	2	no	—
26	C₄×Dic₈₆	—	2	2	no	—
27	C₄×Dic₈₆	—	2	2	no	—
28	C₂×C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
29	C₂×Dic₄₃⋊C₄	—	2	2	no	—
30	C₂×C₄×Dic₄₃	—	2	2	no	—
31	#13063259	—	2	2	no	—
32	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
33	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
34	#13063250	—	2	4	no	—
35	C₂²×C₈×C₄₃	—	2	1	no	—
36	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₄₃	D₄	1	2	yes	—
39	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₄₃	—	2	1	no	—
41	C₄×C₈×C₄₃	—	2	1	no	—
42	C₄×C₈×C₄₃	—	2	1	no	—
43	C₄×C₈×C₄₃	D₄	1	2	yes	—
44	C₄×C₈×C₄₃	D₄	1	2	yes	—
45	C₄×C₈×C₄₃	D₄	1	2	yes	—
46	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
49	C₂²×Dic₄₃	—	2	2	no	—
50	C₂²×Dic₄₃	—	2	2	no	—
51	C₂×Dic₈₆	—	2	2	no	—
52	C₂×Dic₈₆	—	2	2	no	—
53	C₂×Dic₈₆	—	2	2	no	—
54	C₂×Dic₈₆	—	2	2	no	—
55	C₂×Dic₈₆	—	2	2	no	—
56	C₂×Dic₈₆	M₁₆	1	4	yes	—
57	C₂×C₈×C₄₃	—	2	4	no	—
58	C₂×C₈×C₄₃	—	2	2	no	—
59	C₂×C₈×C₄₃	—	2	2	no	—
60	C₂×C₈×C₄₃	—	2	2	no	—
61	C₂×C₈×C₄₃	—	2	2	no	—
62	C₂×C₈×C₄₃	—	2	4	no	—
63	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₄₃	#41	1	1	yes	characteristic
69	C₄⋊Dic₄₃	—	2	2	no	—
70	C₄⋊Dic₄₃	—	2	2	no	—
71	Dic₄₃⋊C₄	—	4	2	no	—
72	Dic₄₃⋊C₄	—	4	2	no	—
73	C₄×Dic₄₃	—	4	2	no	—
74	C₄×Dic₄₃	—	4	2	no	—
75	C₂×C₄₃⋊C₈	—	4	2	no	—
76	C₂×C₄₃⋊C₈	—	4	4	no	—
77	Dic₁₇₂	—	2	8	no	—
78	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₄₃	—	2	2	no	—
80	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
83	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
84	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
85	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
86	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
87	C₂×C₄×C₄₃	#131	1	2	yes	—
88	C₂×C₄×C₄₃	#137	1	4	yes	—
89	C₂×Dic₄₃	—	2	4	no	—
90	C₂×Dic₄₃	—	4	2	no	—
91	C₂×Dic₄₃	—	4	2	no	—
92	C₂×Dic₄₃	—	4	2	no	—
93	C₂×Dic₄₃	—	4	4	no	—
94	Dic₈₆	—	2	4	no	—
95	Dic₈₆	—	2	4	no	—
96	Dic₈₆	—	4	2	no	—
97	Dic₈₆	—	4	2	no	—
98	C₈×C₄₃	—	2	4	no	—
99	C₈×C₄₃	—	2	2	no	—
100	C₈×C₄₃	—	2	2	no	—
101	C₈×C₄₃	—	4	2	no	—
102	C₄₃⋊C₈	—	8	2	no	—
103	#9664	—	43	1	no	maximal, Sylow
104	C₂²×C₄₃	#390	1	1	yes	characteristic
105	C₂²×C₄₃	#435	1	2	yes	—
106	C₂²×C₄₃	#441	1	2	yes	—
107	C₂²×C₄₃	#443	1	2	yes	—
108	C₄×C₄₃	—	2	1	no	—
109	C₄×C₄₃	—	2	1	no	—
110	C₄×C₄₃	#407	1	1	yes	characteristic
111	C₄×C₄₃	#408	1	1	yes	characteristic
112	C₄×C₄₃	#435	1	2	yes	—
113	C₄×C₄₃	#470	1	2	yes	—
114	C₄×C₄₃	#497	1	2	yes	—
115	Dic₄₃	—	4	4	no	—
116	Dic₄₃	—	8	2	no	—
117	C₂×C₄×Q₁₆	—	43	1	no	—
118	#1984	—	43	2	no	—
119	C₂×#334	—	43	1	no	—
120	C₂×Q₈⋊C₈	—	43	2	no	—
121	C₂×#322	D₄₃	1	1	yes	characteristic
122	C₂×C₄₃	#1587	1	4	yes	—
123	C₂×C₄₃	#1955	1	1	yes	characteristic
124	C₂×C₄₃	#1971	1	1	yes	characteristic
125	C₂×C₄₃	#1974	1	1	yes	characteristic
126	C₂²×Q₁₆	—	43	1	no	—
127	C₂×C₄×Q₈	—	43	2	no	—
128	C₄×Q₁₆	—	86	2	no	—
129	C₄×Q₁₆	—	86	2	no	—
130	C₂×#107	—	43	1	no	—
131	C₂×C₄⋊C₈	—	43	2	no	—
132	C₂×#103	—	43	2	no	—
133	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
134	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
135	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
136	#336	—	86	4	no	—
137	#334	—	86	2	no	—
138	Q₈⋊C₈	—	86	4	no	—
139	#322	—	2	2	no	—
140	C₄₃	#9664	1	1	yes	characteristic, complemented, Sylow
141	C₂²×Q₈	—	43	2	no	—
142	C₂×Q₁₆	—	86	4	no	—
143	C₂×Q₁₆	—	86	2	no	—
144	C₂²×C₈	—	2	1	no	—
145	C₂²×C₈	—	86	2	no	—
146	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
147	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
148	C₂²×C₈	D₁₇₂	1	2	yes	—
149	C₄×Q₈	—	86	2	no	—
150	C₄×Q₈	—	86	2	no	—
151	C₄×Q₈	—	86	2	no	—
152	C₄×Q₈	—	86	2	no	—
153	C₂×C₄⋊C₄	—	43	2	no	—
154	C₂×C₄⋊C₄	—	86	2	no	—
155	C₂×C₄²	—	86	2	no	—
156	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
157	#107	—	86	2	no	—
158	C₄⋊C₈	—	86	2	no	—
159	C₄⋊C₈	—	86	2	no	—
160	#103	—	86	4	no	—
161	C₄×C₈	—	2	1	no	—
162	C₄×C₈	—	2	1	no	—
163	C₄×C₈	—	2	1	no	—
164	C₄×C₈	D₁₇₂	1	2	yes	—
165	C₄×C₈	D₁₇₂	1	2	yes	—
166	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
167	C₂×Q₈	—	43	4	no	—
168	C₂×Q₈	—	86	2	no	—
169	C₂×Q₈	—	86	2	no	—
170	C₂×Q₈	—	86	2	no	—
171	C₂×Q₈	—	86	2	no	—
172	C₂×Q₈	—	86	2	no	—
173	C₂²×C₄	—	86	2	no	—
174	C₂²×C₄	—	86	2	no	—
175	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
176	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
177	C₂²×C₄	#10621898	1	1	yes	characteristic
178	Q₁₆	—	86	8	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	172	2	no	—
186	C₂×C₈	—	172	4	no	—
187	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
188	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
189	C₄⋊C₄	—	86	2	no	—
190	C₄⋊C₄	—	86	2	no	—
191	C₄⋊C₄	—	172	2	no	—
192	C₄⋊C₄	—	172	2	no	—
193	C₄²	—	172	2	no	—
194	C₄²	—	172	2	no	—
195	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
196	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
197	C₄²	#10621898	1	1	yes	characteristic
198	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
199	C₂³	C₄×D₁₇₂	1	1	yes	characteristic
200	Q₈	—	86	4	no	—
201	Q₈	—	86	4	no	—
202	Q₈	—	172	2	no	—
203	Q₈	—	172	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	86	4	no	—
206	C₂×C₄	—	172	2	no	—
207	C₂×C₄	—	172	4	no	—
208	C₂×C₄	—	172	2	no	—
209	C₂×C₄	—	172	2	no	—
210	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
211	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
212	C₂×C₄	#13063299	1	1	yes	characteristic
213	C₂×C₄	#13063299	1	1	yes	characteristic
214	C₂×C₄	#13063313	1	1	yes	characteristic
215	C₂×C₄	#13063322	1	1	yes	characteristic
216	C₂×C₄	M₁₆×D₄₃	1	2	yes	—
217	C₂×C₄	#13063333	1	2	yes	—
218	C₂×C₄	#13063358	1	4	yes	—
219	C₈	—	2	4	no	—
220	C₈	—	2	2	no	—
221	C₈	—	2	2	no	—
222	C₈	—	4	2	no	—
223	C₈	—	344	2	no	—
224	C₂²	#14582801	1	1	yes	characteristic
225	C₂²	#14582972	1	2	yes	—
226	C₂²	#14583007	1	2	yes	—
227	C₂²	#14583017	1	2	yes	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	344	2	no	—
231	C₄	—	172	4	no	—
232	C₄	#14582862	1	1	yes	characteristic
233	C₄	#14582867	1	1	yes	characteristic
234	C₄	#14582972	1	2	yes	—
235	C₄	#14583157	1	2	yes	—
236	C₄	#14583293	1	2	yes	—
237	C₂	#19133468	1	4	yes	—
238	C₂	#19135393	1	1	yes	characteristic
239	C₂	#19135466	1	1	yes	characteristic
240	C₂	#19135483	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₇₂
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₂×C₄₃×#322	C₂
socle, prime radical	C₂³×C₄₃	C₄×D₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₂×#322	D₄₃
hypocenter	C₄₃	#9664
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
140. C₄₃	#9664	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₁₇₂	2
#19135531	2
C₂×#14582650	2
C₂×#14582631	2
C₂×C₄₃×#322	2
#9664	43