Subgroups — #25100840

Order 11008.

798 conjugacy classes of subgroups.

The subgroup lattice has 270 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	#19137072	C₂	1	2	yes	maximal
3	#19135466	C₂	1	2	yes	maximal
4	#19135333	C₂	1	2	yes	maximal
5	#19135285	C₂	1	4	yes	maximal
6	C₄×#13063257	C₂	1	1	yes	maximal, characteristic
7	C₄×#13063250	C₂	1	2	yes	maximal
8	C₄×#97×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₈₆	C₂²	1	2	yes	—
11	C₄×Dic₁₇₂	C₄	1	8	yes	complemented
12	C₂×#13063259	C₂²	1	1	yes	characteristic
13	C₂×#13063257	C₂²	1	2	yes	—
14	C₂×#13063250	C₂²	1	4	yes	—
15	C₂×#13063250	C₂²	1	2	yes	—
16	#14582805	C₂²	1	4	yes	—
17	#14582797	C₂²	1	4	yes	—
18	#14582783	—	2	4	no	—
19	C₄×Dic₄₃⋊C₄	—	2	2	no	—
20	C₄×C₄⋊Dic₄₃	C₂²	1	2	yes	—
21	#14582745	—	2	2	no	—
22	#14582662	C₂²	1	2	yes	—
23	#14582662	C₂²	1	4	yes	—
24	C₂×#97×C₄₃	C₂²	1	2	yes	—
25	C₂×#97×C₄₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₄³×C₄₃	C₂²	1	1	yes	characteristic
29	C₂²×Dic₈₆	C₂³	1	2	yes	—
30	C₂×Dic₁₇₂	—	2	4	no	—
31	C₂×Dic₁₇₂	C₂×C₄	1	4	yes	—
32	C₄×Dic₈₆	—	2	4	no	—
33	C₄×Dic₈₆	C₂×C₄	1	8	yes	—
34	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
35	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
36	C₂×Dic₄₃⋊C₄	—	2	2	no	—
37	C₂×Dic₄₃⋊C₄	—	2	4	no	—
38	C₂×C₄×Dic₄₃	—	2	2	no	—
39	C₂×C₄×Dic₄₃	—	2	2	no	—
40	#13063259	C₂×C₄	1	4	yes	—
41	#13063257	—	2	4	no	—
42	#13063250	—	2	8	no	—
43	#13063250	C₂×C₄	1	8	yes	—
44	C₄₃⋊#95	—	2	4	no	—
45	C₄₃⋊#95	—	2	4	no	—
46	C₄₃⋊#95	—	2	4	no	—
47	C₄₃⋊#95	—	2	2	no	—
48	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
50	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₄₃	D₄	1	2	yes	—
52	C₂×C₄²×C₄₃	D₄	1	2	yes	—
53	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
54	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
55	#97×C₄₃	—	2	4	no	—
56	#97×C₄₃	—	2	4	no	—
57	C₄×C₈×C₄₃	—	2	2	no	—
58	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
59	C₂²×C₄×C₄₃	C₂×D₄	1	4	yes	—
60	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
63	C₂²×Dic₄₃	—	2	2	no	—
64	C₂²×Dic₄₃	—	2	2	no	—
65	C₂²×Dic₄₃	—	2	4	no	—
66	C₂×Dic₈₆	—	2	4	no	—
67	C₂×Dic₈₆	—	2	4	no	—
68	C₂×Dic₈₆	—	2	2	no	—
69	C₂×Dic₈₆	C₂²×C₄	1	4	yes	—
70	C₂×C₈×C₄₃	—	2	2	no	—
71	C₂×C₈×C₄₃	—	2	2	no	—
72	C₂×C₈×C₄₃	—	2	2	no	—
73	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₄₃	#41	1	4	yes	—
76	C₂×C₈×C₄₃	#41	1	4	yes	—
77	C₄²×C₄₃	—	2	4	no	—
78	C₄²×C₄₃	—	2	2	no	—
79	C₄²×C₄₃	—	2	2	no	—
80	C₄²×C₄₃	—	4	2	no	—
81	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
82	C₄²×C₄₃	#41	1	2	yes	—
83	C₄⋊Dic₄₃	—	2	2	no	—
84	C₄⋊Dic₄₃	—	2	4	no	—
85	C₄⋊Dic₄₃	—	2	4	no	—
86	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
87	Dic₄₃⋊C₄	—	2	8	no	—
88	Dic₄₃⋊C₄	—	4	8	no	—
89	C₄×Dic₄₃	—	2	8	no	—
90	C₄×Dic₄₃	—	4	4	no	—
91	Dic₁₇₂	—	2	8	no	—
92	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₄₃	—	2	2	no	—
94	C₂×C₄×C₄₃	—	2	4	no	—
95	C₂×C₄×C₄₃	—	2	4	no	—
96	C₂×C₄×C₄₃	—	2	4	no	—
97	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₄₃	#137	1	4	yes	—
100	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
102	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
103	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
104	C₂×Dic₄₃	—	2	4	no	—
105	C₂×Dic₄₃	—	2	4	no	—
106	C₂×Dic₄₃	—	4	4	no	—
107	C₂×Dic₄₃	—	4	4	no	—
108	C₂×Dic₄₃	—	4	4	no	—
109	C₂×Dic₄₃	—	4	4	no	—
110	C₂×Dic₄₃	—	4	4	no	—
111	Dic₈₆	—	2	4	no	—
112	Dic₈₆	—	2	4	no	—
113	Dic₈₆	—	4	4	no	—
114	C₈×C₄₃	—	2	2	no	—
115	C₈×C₄₃	—	2	4	no	—
116	C₈×C₄₃	#117	1	4	yes	—
117	#21278	—	43	1	no	maximal, Sylow
118	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
119	C₂²×C₄₃	#532	1	2	yes	—
120	C₂²×C₄₃	C₂×#137	1	2	yes	—
121	C₂²×C₄₃	#575	1	2	yes	—
122	C₄×C₄₃	—	2	2	no	—
123	C₄×C₄₃	—	4	4	no	—
124	C₄×C₄₃	#441	1	4	yes	—
125	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
126	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
127	C₄×C₄₃	#529	1	2	yes	—
128	Dic₄₃	—	4	4	no	—
129	Dic₄₃	—	4	4	no	—
130	Dic₄₃	—	8	4	no	—
131	C₂×C₄×Q₁₆	—	43	1	no	—
132	#2295	—	43	2	no	—
133	#1971	—	43	2	no	—
134	#1943	—	43	2	no	—
135	#1935	—	43	4	no	—
136	C₄×#106	—	43	1	no	—
137	C₄×#103	—	43	2	no	—
138	C₄×#97	D₄₃	1	1	yes	characteristic
139	C₂×C₄₃	#2293	1	1	yes	characteristic
140	C₂×C₄₃	C₂×#441	1	1	yes	characteristic
141	C₂×C₄₃	#2943	1	1	yes	characteristic
142	C₂×C₄₃	#3172	1	4	yes	—
143	C₂²×Q₁₆	—	43	1	no	—
144	C₂×C₄×Q₈	—	43	2	no	—
145	C₄×Q₁₆	—	43	8	no	—
146	C₂×#107	—	43	1	no	—
147	C₂×#106	—	43	2	no	—
148	C₂×#103	—	43	2	no	—
149	C₂×#103	—	43	4	no	—
150	C₂×#97	C₂×D₄₃	1	2	yes	—
151	C₂×#97	C₂×D₄₃	1	2	yes	—
152	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
154	#391	—	43	4	no	—
155	#389	—	43	4	no	—
156	#387	—	86	4	no	—
157	C₄×C₄⋊C₄	—	43	2	no	—
158	C₄×C₄⋊C₄	—	86	2	no	—
159	#376	—	86	2	no	—
160	C₄³	C₂×D₄₃	1	1	yes	characteristic
161	#340	—	43	2	no	—
162	#340	—	43	4	no	—
163	C₄₃	#21278	1	1	yes	characteristic, complemented, Sylow
164	C₂²×Q₈	—	43	2	no	—
165	C₂×Q₁₆	—	43	4	no	—
166	C₂×Q₁₆	—	86	4	no	—
167	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
169	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
170	C₄×Q₈	—	43	8	no	—
171	C₄×Q₈	—	86	4	no	—
172	C₂×C₄⋊C₄	—	43	2	no	—
173	C₂×C₄⋊C₄	—	43	4	no	—
174	C₂×C₄⋊C₄	—	86	2	no	—
175	C₂×C₄⋊C₄	—	86	4	no	—
176	C₂×C₄²	—	86	2	no	—
177	C₂×C₄²	—	86	2	no	—
178	C₂×C₄²	D₁₇₂	1	2	yes	—
179	C₂×C₄²	D₁₇₂	1	2	yes	—
180	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
181	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
182	#107	—	43	4	no	—
183	#106	—	86	4	no	—
184	#103	—	43	8	no	—
185	#103	—	86	8	no	—
186	#97	—	2	4	no	—
187	#97	—	2	4	no	—
188	C₄×C₈	—	2	2	no	—
189	C₄×C₈	C₄×D₄₃	1	4	yes	—
190	#95	—	86	4	no	—
191	#95	—	86	4	no	—
192	#95	—	86	2	no	—
193	#95	—	86	4	no	—
194	C₂×Q₈	—	43	4	no	—
195	C₂×Q₈	—	86	2	no	—
196	C₂×Q₈	—	86	4	no	—
197	C₂×Q₈	—	86	4	no	—
198	C₂²×C₄	—	86	2	no	—
199	C₂²×C₄	—	86	2	no	—
200	C₂²×C₄	—	86	4	no	—
201	C₂²×C₄	C₂×D₁₇₂	1	4	yes	—
202	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
204	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
205	Q₁₆	—	86	8	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
210	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
211	C₂×C₈	#10621898	1	4	yes	—
212	C₂×C₈	#10621898	1	4	yes	—
213	C₄⋊C₄	—	43	4	no	—
214	C₄⋊C₄	—	86	8	no	—
215	C₄⋊C₄	—	86	2	no	—
216	C₄⋊C₄	—	86	4	no	—
217	C₄⋊C₄	—	86	4	no	—
218	C₄⋊C₄	—	172	8	no	—
219	C₄²	—	2	4	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	4	2	no	—
223	C₄²	—	86	8	no	—
224	C₄²	—	172	4	no	—
225	C₄²	C₂×C₄×D₄₃	1	2	yes	—
226	C₄²	#10621898	1	2	yes	—
227	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
228	Q₈	—	86	4	no	—
229	Q₈	—	86	4	no	—
230	Q₈	—	172	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	86	4	no	—
236	C₂×C₄	—	86	4	no	—
237	C₂×C₄	—	172	4	no	—
238	C₂×C₄	—	172	4	no	—
239	C₂×C₄	—	172	4	no	—
240	C₂×C₄	—	172	4	no	—
241	C₂×C₄	—	172	4	no	—
242	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
243	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
244	C₂×C₄	#13063358	1	4	yes	—
245	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
246	C₂×C₄	C₂×#10621898	1	2	yes	—
247	C₂×C₄	C₂×#10621898	1	2	yes	—
248	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	2	4	no	—
251	C₈	#13063282	1	4	yes	—
252	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
253	C₂²	#14583449	1	2	yes	—
254	C₂²	C₂×#13063358	1	2	yes	—
255	C₂²	#14583699	1	2	yes	—
256	C₄	—	2	2	no	—
257	C₄	—	4	4	no	—
258	C₄	—	344	4	no	—
259	C₄	—	172	4	no	—
260	C₄	—	172	4	no	—
261	C₄	#14583007	1	4	yes	—
262	C₄	C₂×#13063282	1	1	yes	characteristic
263	C₄	C₄×#10621898	1	1	yes	characteristic
264	C₄	#14583433	1	2	yes	—
265	C₂	#19137059	1	1	yes	characteristic
266	C₂	C₂×#14583007	1	1	yes	characteristic
267	C₂	#19143397	1	1	yes	characteristic
268	C₂	#19145201	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄×#97×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄×#97	D₄₃
hypocenter	C₄₃	#21278
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
11. C₄×Dic₁₇₂	C₄	semidirect
163. C₄₃	#21278	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
#19137072	2
#19135466	2
#19135333	2
#19135285	2
C₄×#13063257	2
C₄×#13063250	2
C₄×#97×C₄₃	2
#21278	43