Subgroups — #25097224

Order 11008.

718 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143233	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582855	C₂	1	1	yes	maximal, characteristic
3	#19134769	C₂	1	2	yes	maximal, complemented
4	#19134732	C₂	1	2	yes	maximal, complemented
5	#19134635	C₂	1	2	yes	maximal
6	#19134631	C₂	1	2	yes	maximal
7	#19134502	C₂	1	2	yes	maximal, complemented
8	C₄×#13063260	C₂	1	2	yes	maximal, complemented
9	C₄₃×#2921	C₂	1	1	yes	maximal, characteristic
10	C₂²×#10621877	C₂²	1	1	yes	characteristic
11	C₂³×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
12	#14582960	—	2	4	no	—
13	C₂×#13063260	C₂²	1	4	yes	—
14	C₂×#13063260	C₂²	1	4	yes	—
15	#14582923	C₂²	1	2	yes	—
16	#14582863	C₂²	1	2	yes	—
17	#14582855	C₄	1	8	yes	—
18	C₄×#10621877	C₂²	1	2	yes	—
19	C₂×#13063236	C₂²	1	1	yes	characteristic
20	C₂×#13063236	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
23	#14582663	C₂²	1	4	yes	—
24	#14582663	C₂²	1	4	yes	—
25	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
26	C₄₃×#433	—	2	4	no	—
27	C₄₃×#423	C₂²	1	2	yes	—
28	C₄₃×#407	C₂²	1	2	yes	—
29	C₄×M₁₆×C₄₃	C₂²	1	2	yes	—
30	C₂×#10621877	—	2	4	no	—
31	C₂×#10621877	C₂×C₄	1	4	yes	—
32	C₂×#10621877	C₂³	1	4	yes	—
33	C₂²×C₄₃⋊C₈	—	2	4	no	—
34	C₂²×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
35	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
36	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
37	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
38	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
39	#13063260	—	2	8	no	—
40	#13063260	—	2	8	no	—
41	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
42	C₂²⋊(C₄₃⋊C₈)	—	2	4	no	—
43	#13063236	—	2	2	no	—
44	#13063236	C₂×C₄	1	4	yes	—
45	#13063236	C₂×C₄	1	4	yes	—
46	C₄×C₄₃⋊C₈	—	2	2	no	—
47	C₄×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
48	C₄×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
49	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
50	C₂×M₁₆×C₄₃	C₂³	1	4	yes	—
51	#117×C₄₃	C₂×C₄	1	4	yes	—
52	#117×C₄₃	D₄	1	4	yes	—
53	C₂×C₄⋊C₄×C₄₃	Q₈	1	2	yes	—
54	C₂×C₂²⋊C₄×C₄₃	Q₈	1	2	yes	—
55	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₄₃	—	2	4	no	—
58	C₂²⋊C₈×C₄₃	—	2	4	no	—
59	#97×C₄₃	—	2	2	no	—
60	C₄×C₈×C₄₃	—	2	2	no	—
61	C₂⁴×C₄₃	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	—	2	2	no	—
63	C₂²×C₄×C₄₃	—	2	2	no	—
64	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
70	M₁₆×C₄₃	—	2	4	no	—
71	M₁₆×C₄₃	—	2	4	no	—
72	C₂×C₈×C₄₃	—	2	4	no	—
73	C₄⋊C₄×C₄₃	C₄⋊C₄	1	8	yes	—
74	C₂²⋊C₄×C₄₃	C₄⋊C₄	1	8	yes	—
75	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
76	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
77	C₄²×C₄₃	C₂×D₄	1	2	yes	—
78	C₄²×C₄₃	C₂×D₄	1	2	yes	—
79	#10621877	—	2	8	no	—
80	#10621877	—	2	4	no	—
81	#10621877	—	2	4	no	—
82	C₂×C₄₃⋊C₈	—	2	2	no	—
83	C₂×C₄₃⋊C₈	—	2	2	no	—
84	C₂×C₄₃⋊C₈	—	2	8	no	—
85	C₂×C₄₃⋊C₈	—	2	4	no	—
86	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
87	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
88	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
89	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
90	C₂×C₄₃⋊C₈	#41	1	4	yes	—
91	C₂×C₄₃⋊C₈	#41	1	4	yes	—
92	C₂³×C₄₃	—	2	2	no	—
93	C₂³×C₄₃	—	4	2	no	—
94	C₂³×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
95	C₂³×C₄₃	C₂²×Q₈	1	1	yes	characteristic
96	C₂×C₄×C₄₃	—	2	2	no	—
97	C₂×C₄×C₄₃	—	2	2	no	—
98	C₂×C₄×C₄₃	—	2	2	no	—
99	C₂×C₄×C₄₃	—	2	4	no	—
100	C₂×C₄×C₄₃	—	4	2	no	—
101	C₂×C₄×C₄₃	—	4	2	no	—
102	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
103	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
108	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
109	C₈×C₄₃	—	4	4	no	—
110	C₄₃⋊C₈	—	2	2	no	—
111	C₄₃⋊C₈	—	2	2	no	—
112	C₄₃⋊C₈	—	4	4	no	—
113	C₄₃⋊C₈	#117	1	4	yes	—
114	C₄₃⋊C₈	#117	1	4	yes	—
115	#20829	—	43	1	no	maximal, Sylow
116	C₂²×C₄₃	—	2	2	no	—
117	C₂²×C₄₃	—	2	4	no	—
118	C₂²×C₄₃	—	2	2	no	—
119	C₂²×C₄₃	—	4	4	no	—
120	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₄₃	#533	1	2	yes	—
122	C₄×C₄₃	—	2	4	no	—
123	C₄×C₄₃	—	2	4	no	—
124	C₄×C₄₃	—	2	2	no	—
125	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
126	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
127	C₄×C₄₃	#532	1	2	yes	—
128	#2921	—	43	1	no	—
129	#2921	D₄₃	1	1	yes	characteristic
130	C₂×#405	—	43	1	no	—
131	#1849	—	43	2	no	—
132	#1837	—	43	2	no	—
133	#1811	—	43	2	no	—
134	#1810	—	43	2	no	—
135	#1778	—	43	2	no	—
136	C₄×#108	—	43	2	no	—
137	C₂×C₄₃	—	2	2	no	—
138	C₂×C₄₃	—	4	2	no	—
139	C₂×C₄₃	#2285	1	1	yes	characteristic
140	C₂×C₄₃	#2916	1	2	yes	—
141	C₂²×M₁₆	—	43	1	no	—
142	C₂³×C₈	—	43	1	no	—
143	C₂×#117	C₂×D₄₃	1	1	yes	characteristic
144	#433	—	2	4	no	—
145	#433	—	86	4	no	—
146	C₂×#108	—	43	4	no	—
147	C₂×#108	—	43	4	no	—
148	#423	—	43	2	no	—
149	#423	C₂×D₄₃	1	2	yes	—
150	#407	—	43	2	no	—
151	#407	C₂×D₄₃	1	2	yes	—
152	#405	—	43	8	no	—
153	C₄×M₁₆	—	43	2	no	—
154	C₄×M₁₆	C₂×D₄₃	1	2	yes	—
155	C₂×#97	—	43	1	no	—
156	C₂×#97	—	43	1	no	—
157	C₂×C₄×C₈	—	43	1	no	—
158	C₂×C₄×C₈	—	43	1	no	—
159	#341	—	43	4	no	—
160	#341	—	43	4	no	—
161	C₄₃	#20829	1	1	yes	characteristic, complemented, Sylow
162	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
163	C₂×M₁₆	—	43	4	no	—
164	C₂×M₁₆	—	43	4	no	—
165	C₂×M₁₆	—	86	4	no	—
166	C₂×M₁₆	C₂²×D₄₃	1	4	yes	—
167	C₂²×C₈	—	43	1	no	—
168	C₂²×C₈	—	43	1	no	—
169	C₂²×C₈	—	43	4	no	—
170	C₂²×C₈	—	43	1	no	—
171	C₂²×C₈	—	43	1	no	—
172	C₂²×C₈	—	86	4	no	—
173	#117	C₄×D₄₃	1	4	yes	—
174	#117	C₂²⋊D₄₃	1	4	yes	—
175	C₂×C₄⋊C₄	Dic₈₆	1	2	yes	—
176	C₂×C₂²⋊C₄	Dic₈₆	1	2	yes	—
177	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
178	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
179	#108	—	86	8	no	—
180	#108	—	86	8	no	—
181	C₄⋊C₈	—	2	4	no	—
182	C₄⋊C₈	—	86	4	no	—
183	C₂²⋊C₈	—	2	4	no	—
184	C₂²⋊C₈	—	86	4	no	—
185	#97	—	2	2	no	—
186	#97	—	43	4	no	—
187	#97	—	43	4	no	—
188	#97	—	86	2	no	—
189	C₄×C₈	—	2	2	no	—
190	C₄×C₈	—	43	4	no	—
191	C₄×C₈	—	43	4	no	—
192	C₄×C₈	—	86	2	no	—
193	C₂⁴	C₂×Dic₈₆	1	1	yes	characteristic
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
197	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
198	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
199	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
200	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
201	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
202	M₁₆	—	2	4	no	—
203	M₁₆	—	2	4	no	—
204	M₁₆	—	86	8	no	—
205	M₁₆	—	86	4	no	—
206	M₁₆	—	86	4	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	43	2	no	—
209	C₂×C₈	—	43	2	no	—
210	C₂×C₈	—	43	4	no	—
211	C₂×C₈	—	43	4	no	—
212	C₂×C₈	—	43	2	no	—
213	C₂×C₈	—	43	2	no	—
214	C₂×C₈	—	86	8	no	—
215	C₂×C₈	—	86	2	no	—
216	C₂×C₈	—	86	2	no	—
217	C₂×C₈	—	86	4	no	—
218	C₄⋊C₄	Dic₄₃⋊C₄	1	8	yes	—
219	C₂²⋊C₄	Dic₄₃⋊C₄	1	8	yes	—
220	C₄²	C₂×C₄×D₄₃	1	2	yes	—
221	C₄²	C₂×C₄×D₄₃	1	2	yes	—
222	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
223	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
224	C₂³	—	2	2	no	—
225	C₂³	—	4	2	no	—
226	C₂³	C₂×Dic₄₃⋊C₄	1	2	yes	—
227	C₂³	C₂²×Dic₈₆	1	1	yes	characteristic
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	C₂×Dic₄₃⋊C₄	1	4	yes	—
235	C₂×C₄	C₂×Dic₄₃⋊C₄	1	4	yes	—
236	C₂×C₄	C₂×Dic₄₃⋊C₄	1	2	yes	—
237	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
238	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
239	C₂×C₄	#41×D₄₃	1	2	yes	—
240	C₂×C₄	#41×D₄₃	1	2	yes	—
241	C₈	—	4	4	no	—
242	C₈	—	43	4	no	—
243	C₈	—	43	4	no	—
244	C₈	—	86	2	no	—
245	C₈	—	86	2	no	—
246	C₈	—	172	4	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	4	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	4	4	no	—
251	C₂²	C₂²×Dic₄₃⋊C₄	1	1	yes	characteristic
252	C₂²	#14583452	1	2	yes	—
253	C₄	—	2	4	no	—
254	C₄	—	2	4	no	—
255	C₄	—	2	2	no	—
256	C₄	#117×D₄₃	1	1	yes	characteristic
257	C₄	#117×D₄₃	1	1	yes	characteristic
258	C₄	#14583445	1	2	yes	—
259	C₂	—	2	2	no	—
260	C₂	—	4	2	no	—
261	C₂	#19136998	1	1	yes	characteristic
262	C₂	#19143204	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#2921	C₂
socle	C₂²×C₄₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₄₃	C₂×Q₈
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#2921	D₄₃
hypocenter	C₄₃	#20829
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ #2921

Complements

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

N	H ≅ G/N	type
3. #19134769	C₂	semidirect
4. #19134732	C₂	semidirect
7. #19134502	C₂	semidirect
8. C₄×#13063260	C₂	semidirect
161. C₄₃	#20829	semidirect

Maximal subgroups

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

subgroup	index
#19143233	2
C₂×#14582855	2
#19134769	2
#19134732	2
#19134635	2
#19134631	2
#19134502	2
C₄×#13063260	2
C₄₃×#2921	2
#20829	43