Subgroups — #25105476

Order 11008.

670 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145119	C₂	1	1	yes	maximal, characteristic
2	#19143535	C₂	1	2	yes	maximal
3	#19135572	C₂	1	2	yes	maximal
4	#19135348	C₂	1	2	yes	maximal
5	#19135343	C₂	1	2	yes	maximal
6	#19134770	C₂	1	1	yes	maximal, characteristic
7	#19134479	C₂	1	2	yes	maximal
8	#19134474	C₂	1	2	yes	maximal
9	C₄×#97×C₄₃	C₂	1	1	yes	maximal, characteristic
10	#14583456	C₂²	1	2	yes	—
11	#14583308	C₄	1	8	yes	complemented
12	#14583027	—	2	8	no	—
13	C₂×#13063261	C₂²	1	2	yes	—
14	C₂×#13063261	C₂²	1	2	yes	—
15	#14582937	C₂²	1	2	yes	—
16	#14582937	C₂²	1	2	yes	—
17	#14582925	C₂²	1	4	yes	—
18	#14582923	C₂²	1	2	yes	—
19	#14582854	C₂²	1	4	yes	—
20	#14582659	C₂²	1	4	yes	—
21	#14582659	C₂²	1	4	yes	—
22	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
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	C₂×#10621877	C₂³	1	4	yes	—
30	#13063325	C₂×C₄	1	4	yes	—
31	#13063317	C₂×C₄	1	4	yes	—
32	#13063304	—	2	4	no	—
33	C₄×Dic₈₆	—	2	4	no	—
34	#13063284	C₂³	1	4	yes	—
35	#13063261	—	2	4	no	—
36	#13063261	—	2	4	no	—
37	#13063259	C₂×C₄	1	4	yes	—
38	#13063259	C₂×C₄	1	4	yes	—
39	#13063257	C₂×C₄	1	4	yes	—
40	#13063257	C₂×C₄	1	4	yes	—
41	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
42	Dic₄₃⋊C₈	—	2	8	no	—
43	#13063235	—	2	4	no	—
44	C₈×Dic₄₃	—	2	4	no	—
45	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₄₃	—	2	2	no	—
50	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
53	#97×C₄₃	—	2	2	no	—
54	#97×C₄₃	C₂×C₄	1	4	yes	—
55	#97×C₄₃	D₄	1	4	yes	—
56	#97×C₄₃	Q₈	1	4	yes	complemented
57	C₄×C₈×C₄₃	—	2	2	no	—
58	C₄×C₈×C₄₃	Q₈	1	4	yes	complemented
59	C₂²×C₄×C₄₃	—	2	1	no	—
60	C₂²×C₄×C₄₃	—	2	1	no	—
61	C₂²×C₄×C₄₃	#41	1	2	yes	—
62	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
63	C₂×Dic₈₆	—	2	4	no	—
64	C₂×C₈×C₄₃	—	2	1	no	—
65	C₂×C₈×C₄₃	—	2	1	no	—
66	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
71	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
72	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
73	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
74	C₂×C₈×C₄₃	#41	1	2	yes	—
75	C₂×C₈×C₄₃	#41	1	2	yes	—
76	C₄²×C₄₃	—	2	1	no	—
77	C₄²×C₄₃	—	2	1	no	—
78	C₄²×C₄₃	—	4	4	no	—
79	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
80	C₄²×C₄₃	C₂×D₄	1	2	yes	—
81	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
82	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
83	C₂²⋊Dic₄₃	—	2	4	no	—
84	C₄⋊Dic₄₃	—	2	4	no	—
85	C₄⋊Dic₄₃	C₂²×C₄	1	8	yes	—
86	Dic₄₃⋊C₄	—	2	8	no	—
87	C₄×Dic₄₃	—	2	4	no	—
88	#10621877	—	2	4	no	—
89	#10621877	—	2	4	no	—
90	C₂×C₄₃⋊C₈	—	2	4	no	—
91	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
92	C₂×C₄×C₄₃	—	2	4	no	—
93	C₂×C₄×C₄₃	—	2	2	no	—
94	C₂×C₄×C₄₃	—	2	2	no	—
95	C₂×C₄×C₄₃	—	4	2	no	—
96	C₂×C₄×C₄₃	—	4	2	no	—
97	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
98	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
103	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
104	C₂×Dic₄₃	—	2	8	no	—
105	Dic₈₆	—	8	4	no	—
106	C₈×C₄₃	—	2	2	no	—
107	C₈×C₄₃	C₄×Q₈	1	4	yes	—
108	C₈×C₄₃	C₄×Q₈	1	4	yes	—
109	C₈×C₄₃	C₄⋊Q₈	1	4	yes	—
110	C₄₃⋊C₈	—	4	4	no	—
111	#21736	—	43	1	no	maximal, Sylow
112	C₂²×C₄₃	—	2	2	no	—
113	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
114	C₂²×C₄₃	#533	1	1	yes	characteristic
115	C₂²×C₄₃	#533	1	1	yes	characteristic
116	C₄×C₄₃	—	2	4	no	—
117	C₄×C₄₃	—	2	4	no	—
118	C₄×C₄₃	—	4	2	no	—
119	C₄×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
120	C₄×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
121	C₄×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
122	C₄×C₄₃	#532	1	1	yes	characteristic
123	Dic₄₃	—	4	8	no	—
124	#3162	—	43	1	no	—
125	#2961	—	43	2	no	—
126	#1994	—	43	2	no	—
127	#1946	—	43	2	no	—
128	#1945	—	43	2	no	—
129	#1849	—	43	1	no	—
130	#1773	—	43	2	no	—
131	#1772	—	43	2	no	—
132	C₄×#97	D₄₃	1	1	yes	characteristic
133	C₂×C₄₃	—	2	1	no	—
134	C₂×C₄₃	—	2	1	no	—
135	C₂×C₄₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
136	C₂×C₄₃	#2956	1	2	yes	—
137	#533	—	43	2	no	—
138	#501	—	43	8	no	—
139	#446	—	86	8	no	—
140	C₂×#108	—	43	2	no	—
141	C₂×#108	—	43	2	no	—
142	#427	—	43	2	no	—
143	#427	—	43	2	no	—
144	#424	—	43	4	no	—
145	#423	—	43	2	no	—
146	#405	—	43	4	no	—
147	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
148	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
149	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
150	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
151	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
152	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
153	C₄³	C₂×D₄₃	1	1	yes	characteristic
154	#339	—	43	4	no	—
155	#339	—	43	4	no	—
156	C₄₃	#21736	1	1	yes	characteristic, complemented, Sylow
157	C₂×M₁₆	—	43	4	no	—
158	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
159	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
160	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
161	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
162	C₄⋊Q₈	—	43	4	no	—
163	#125	—	43	4	no	—
164	C₂²⋊Q₈	—	86	4	no	—
165	C₄×Q₈	—	86	4	no	—
166	#117	—	43	4	no	—
167	C₂×C₄²	—	2	2	no	—
168	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
169	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
170	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
171	#108	—	86	4	no	—
172	#108	—	86	4	no	—
173	#107	—	43	4	no	—
174	#107	—	43	4	no	—
175	#106	—	43	4	no	—
176	#106	—	43	4	no	—
177	C₄⋊C₈	—	86	8	no	—
178	C₄⋊C₈	—	86	4	no	—
179	#97	—	2	2	no	—
180	#97	—	86	4	no	—
181	#97	Dic₈₆	1	4	yes	complemented
182	#97	C₄×D₄₃	1	4	yes	—
183	#97	D₁₇₂	1	4	yes	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	—	86	4	no	—
186	C₄×C₈	Dic₈₆	1	4	yes	complemented
187	C₂×Q₈	—	86	4	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	#10621898	1	2	yes	—
191	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
192	M₁₆	—	86	4	no	—
193	M₁₆	—	86	4	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	86	4	no	—
197	C₂×C₈	C₂×Dic₈₆	1	2	yes	—
198	C₂×C₈	C₂×Dic₈₆	1	2	yes	—
199	C₂×C₈	C₂×Dic₈₆	1	2	yes	—
200	C₂×C₈	C₂×Dic₈₆	1	2	yes	—
201	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
202	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
203	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
204	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
205	C₂×C₈	#10621898	1	2	yes	—
206	C₂×C₈	#10621898	1	2	yes	—
207	C₄⋊C₄	—	43	8	no	—
208	C₄⋊C₄	—	86	8	no	—
209	C₄⋊C₄	—	86	4	no	—
210	C₂²⋊C₄	—	86	4	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	4	4	no	—
214	C₄²	—	86	4	no	—
215	C₄²	C₂×Dic₈₆	1	2	yes	—
216	C₄²	C₂×Dic₈₆	1	2	yes	—
217	C₄²	C₂×C₄×D₄₃	1	2	yes	—
218	C₄²	C₂×D₁₇₂	1	2	yes	—
219	C₂³	C₂×#10621898	1	1	yes	characteristic
220	Q₈	—	344	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	86	8	no	—
227	C₂×C₄	C₄×D₁₇₂	1	4	yes	—
228	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
229	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
230	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
231	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
232	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
233	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
234	C₈	—	2	2	no	—
235	C₈	—	172	4	no	—
236	C₈	C₄×Dic₈₆	1	4	yes	—
237	C₈	C₄×Dic₈₆	1	4	yes	—
238	C₈	#13063325	1	4	yes	—
239	C₂²	—	2	2	no	—
240	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
241	C₂²	#14583456	1	1	yes	characteristic
242	C₂²	#14583456	1	1	yes	characteristic
243	C₄	—	2	4	no	—
244	C₄	—	2	4	no	—
245	C₄	—	4	2	no	—
246	C₄	—	172	8	no	—
247	C₄	C₂×C₄×Dic₈₆	1	1	yes	characteristic
248	C₄	C₂×C₄×Dic₈₆	1	1	yes	characteristic
249	C₄	C₂×#13063325	1	1	yes	characteristic
250	C₄	#14583449	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	1	no	—
253	C₂	C₄×#13063325	1	1	yes	characteristic
254	C₂	#19143506	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄×#97×C₄₃	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂³×C₄₃	C₂×#41
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄×#97	D₄₃
hypocenter	C₄₃	#21736
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
11. #14583308	C₄	semidirect
56. #97×C₄₃	Q₈	semidirect
58. C₄×C₈×C₄₃	Q₈	semidirect
156. C₄₃	#21736	semidirect
181. #97	Dic₈₆	semidirect
186. C₄×C₈	Dic₈₆	semidirect

Maximal subgroups

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

subgroup	index
#19145119	2
#19143535	2
#19135572	2
#19135348	2
#19135343	2
#19134770	2
#19134479	2
#19134474	2
C₄×#97×C₄₃	2
#21736	43