Subgroups — #25110386

Order 11008.

886 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×#14583279	C₂	1	1	yes	maximal, characteristic
2	C₄×#13063325	C₂	1	2	yes	maximal
3	#19135397	C₂	1	2	yes	maximal
4	#19134935	C₂	1	8	yes	maximal
5	#19134723	C₂	1	1	yes	maximal, characteristic
6	C₄₃×#1744	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₇₂	C₂²	1	4	yes	—
8	C₂×#13063325	C₂²	1	2	yes	—
9	C₂×C₄×Dic₈₆	C₂²	1	4	yes	—
10	#14583279	C₄	1	8	yes	complemented
11	C₂×#13063250	C₂²	1	8	yes	—
12	#14582805	C₂²	1	4	yes	—
13	#14582797	—	2	2	no	—
14	#14582797	—	2	2	no	—
15	#14582769	C₂²	1	2	yes	—
16	C₄×C₄⋊Dic₄₃	—	2	2	no	—
17	C₄×C₄⋊Dic₄₃	—	2	2	no	—
18	#14582662	C₂²	1	4	yes	—
19	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
21	C₄³×C₄₃	C₂²	1	1	yes	characteristic
22	C₄₃×#336	C₂²	1	4	yes	—
23	C₂²×Dic₈₆	C₂³	1	4	yes	—
24	C₂×Dic₁₇₂	—	4	8	no	—
25	C₂×Dic₁₇₂	C₂×C₄	1	16	yes	—
26	#13063325	—	2	4	no	—
27	#13063325	C₂×C₄	1	8	yes	—
28	C₄×Dic₈₆	—	4	8	no	—
29	C₂×C₄⋊Dic₄₃	—	2	2	no	—
30	C₂×C₄⋊Dic₄₃	—	2	2	no	—
31	C₂×C₄⋊Dic₄₃	—	2	2	no	—
32	C₂×C₄⋊Dic₄₃	—	2	2	no	—
33	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
34	C₂×Dic₄₃⋊C₄	—	2	4	no	—
35	C₂×C₄×Dic₄₃	—	2	4	no	—
36	#13063250	—	4	8	no	—
37	C₄₃⋊#95	—	2	4	no	—
38	C₄₃⋊#95	—	2	4	no	—
39	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
40	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
41	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₃	D₄	1	2	yes	—
43	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
45	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₄₃	—	2	2	no	—
47	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
48	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
53	C₂²×Dic₄₃	—	2	4	no	—
54	C₂²×Dic₄₃	—	2	4	no	—
55	C₂×Dic₈₆	—	2	8	no	—
56	C₂×Dic₈₆	—	2	8	no	—
57	C₂×Dic₈₆	—	2	4	no	—
58	C₂×Dic₈₆	C₂²×C₄	1	8	yes	—
59	C₂×C₈×C₄₃	—	2	4	no	—
60	C₂×C₈×C₄₃	—	2	2	no	—
61	C₂×C₈×C₄₃	—	2	4	no	—
62	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
63	C₄²×C₄₃	—	2	2	no	—
64	C₄²×C₄₃	—	2	4	no	—
65	C₄²×C₄₃	—	2	4	no	—
66	C₄²×C₄₃	Q₁₆	1	4	yes	complemented
67	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
68	C₄²×C₄₃	#41	1	2	yes	—
69	C₄⋊Dic₄₃	—	2	8	no	—
70	C₄⋊Dic₄₃	—	2	8	no	—
71	C₄⋊Dic₄₃	—	2	8	no	—
72	C₄⋊Dic₄₃	—	4	4	no	—
73	C₄⋊Dic₄₃	—	4	4	no	—
74	Dic₄₃⋊C₄	—	4	8	no	—
75	C₄×Dic₄₃	—	4	8	no	—
76	Dic₁₇₂	—	4	16	no	—
77	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
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	—	2	2	no	—
83	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₄₃	C₂×Q₁₆	1	4	yes	—
86	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
87	C₂×C₄×C₄₃	#135	1	2	yes	—
88	C₂×C₄×C₄₃	#137	1	4	yes	—
89	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
90	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
91	C₂×C₄×C₄₃	H(5,𝔽₂)	1	4	yes	—
92	C₂×Dic₄₃	—	2	8	no	—
93	C₂×Dic₄₃	—	4	4	no	—
94	C₂×Dic₄₃	—	4	4	no	—
95	C₂×Dic₄₃	—	4	4	no	—
96	C₂×Dic₄₃	—	4	4	no	—
97	C₂×Dic₄₃	—	4	4	no	—
98	Dic₈₆	—	2	16	no	—
99	Dic₈₆	—	4	8	no	—
100	C₈×C₄₃	—	2	4	no	—
101	C₈×C₄₃	—	4	2	no	—
102	#22270	—	43	1	no	maximal, Sylow
103	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
104	C₂²×C₄₃	#535	1	2	yes	—
105	C₂²×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
106	C₂²×C₄₃	C₂×#135	1	1	yes	characteristic
107	C₂²×C₄₃	C₂×#137	1	2	yes	—
108	C₄×C₄₃	—	2	4	no	—
109	C₄×C₄₃	—	4	2	no	—
110	C₄×C₄₃	C₄×Q₁₆	1	4	yes	—
111	C₄×C₄₃	#441	1	4	yes	—
112	C₄×C₄₃	#518	1	2	yes	—
113	C₄×C₄₃	#540	1	2	yes	—
114	Dic₄₃	—	4	8	no	—
115	Dic₄₃	—	8	4	no	—
116	C₂×#494	—	43	1	no	—
117	C₄×C₄⋊Q₈	—	43	2	no	—
118	#1956	—	43	2	no	—
119	#1878	—	43	8	no	—
120	#1834	—	43	1	no	—
121	#1744	D₄₃	1	1	yes	characteristic
122	C₂×C₄₃	#2328	1	1	yes	characteristic
123	C₂×C₄₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
124	C₂×C₄₃	C₂×#441	1	1	yes	characteristic
125	C₂×C₄₃	#3209	1	2	yes	—
126	C₂×C₄₃	#3210	1	2	yes	—
127	C₂²×Q₁₆	—	43	4	no	—
128	C₂×C₄⋊Q₈	—	43	2	no	—
129	C₂×C₄×Q₈	—	43	4	no	—
130	#494	—	43	8	no	—
131	C₂×#103	—	43	8	no	—
132	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
133	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
134	#391	—	43	4	no	—
135	#389	—	86	2	no	—
136	#389	—	86	2	no	—
137	#384	—	43	2	no	—
138	C₄×C₄⋊C₄	—	86	2	no	—
139	C₄×C₄⋊C₄	—	86	2	no	—
140	C₄³	C₂×D₄₃	1	1	yes	characteristic
141	#340	—	43	4	no	—
142	#336	C₂×D₄₃	1	4	yes	—
143	C₄₃	#22270	1	1	yes	characteristic, complemented, Sylow
144	C₂²×Q₈	—	43	4	no	—
145	C₂×Q₁₆	—	43	16	no	—
146	C₂×Q₁₆	—	172	8	no	—
147	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
148	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
149	C₄⋊Q₈	—	43	8	no	—
150	C₄⋊Q₈	—	86	4	no	—
151	C₄×Q₈	—	172	8	no	—
152	C₂×C₄⋊C₄	—	43	4	no	—
153	C₂×C₄⋊C₄	—	86	2	no	—
154	C₂×C₄⋊C₄	—	86	2	no	—
155	C₂×C₄⋊C₄	—	86	4	no	—
156	C₂×C₄⋊C₄	—	86	2	no	—
157	C₂×C₄⋊C₄	—	86	2	no	—
158	C₂×C₄²	—	86	4	no	—
159	C₂×C₄²	D₁₇₂	1	2	yes	—
160	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
161	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
162	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
163	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
164	#103	—	172	8	no	—
165	C₄×C₈	—	2	2	no	—
166	C₄×C₈	C₄×D₄₃	1	4	yes	—
167	#95	—	86	4	no	—
168	#95	—	86	4	no	—
169	C₂×Q₈	—	43	8	no	—
170	C₂×Q₈	—	86	4	no	—
171	C₂×Q₈	—	86	8	no	—
172	C₂×Q₈	—	86	8	no	—
173	C₂²×C₄	—	86	4	no	—
174	C₂²×C₄	—	86	4	no	—
175	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
176	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
177	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
178	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
179	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
180	Q₁₆	—	172	16	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	C₂×C₄×D₄₃	1	4	yes	—
185	C₄⋊C₄	—	86	8	no	—
186	C₄⋊C₄	—	86	8	no	—
187	C₄⋊C₄	—	86	8	no	—
188	C₄⋊C₄	—	172	8	no	—
189	C₄⋊C₄	—	172	4	no	—
190	C₄⋊C₄	—	172	4	no	—
191	C₄²	—	2	2	no	—
192	C₄²	—	2	4	no	—
193	C₄²	—	2	4	no	—
194	C₄²	—	172	8	no	—
195	C₄²	Dic₁₇₂	1	4	yes	complemented
196	C₄²	C₂×C₄×D₄₃	1	2	yes	—
197	C₄²	#10621898	1	2	yes	—
198	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
199	Q₈	—	86	16	no	—
200	Q₈	—	172	8	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	86	8	no	—
207	C₂×C₄	—	172	4	no	—
208	C₂×C₄	—	172	4	no	—
209	C₂×C₄	—	172	4	no	—
210	C₂×C₄	—	172	4	no	—
211	C₂×C₄	—	172	4	no	—
212	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
213	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
214	C₂×C₄	C₂×Dic₁₇₂	1	4	yes	—
215	C₂×C₄	C₂×Dic₁₇₂	1	2	yes	—
216	C₂×C₄	#13063347	1	2	yes	—
217	C₂×C₄	#13063358	1	4	yes	—
218	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
219	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
220	C₂×C₄	#13063371	1	4	yes	—
221	C₈	—	2	4	no	—
222	C₈	—	4	2	no	—
223	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
224	C₂²	#14583468	1	2	yes	—
225	C₂²	C₂²×Dic₁₇₂	1	1	yes	characteristic
226	C₂²	C₂×#13063347	1	1	yes	characteristic
227	C₂²	C₂×#13063358	1	2	yes	—
228	C₄	—	2	4	no	—
229	C₄	—	4	2	no	—
230	C₄	—	344	4	no	—
231	C₄	—	172	8	no	—
232	C₄	C₄×Dic₁₇₂	1	4	yes	—
233	C₄	#14583007	1	4	yes	—
234	C₄	#14583373	1	2	yes	—
235	C₄	#14583504	1	2	yes	—
236	C₂	#19137320	1	1	yes	characteristic
237	C₂	C₂×C₄×Dic₁₇₂	1	1	yes	characteristic
238	C₂	C₂×#14583007	1	1	yes	characteristic
239	C₂	#19145545	1	2	yes	—
240	C₂	#19145553	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1744	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1744	D₄₃
hypocenter	C₄₃	#22270
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1744

Complements

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

N	H ≅ G/N	type
10. #14583279	C₄	semidirect
66. C₄²×C₄₃	Q₁₆	semidirect
143. C₄₃	#22270	semidirect
195. C₄²	Dic₁₇₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583279	2
C₄×#13063325	2
#19135397	2
#19134935	2
#19134723	2
C₄₃×#1744	2
#22270	43