Subgroups — #24995644

Order 11008.

882 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19142990	C₂	1	1	yes	maximal, characteristic
2	#19134747	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14582642	C₂	1	4	yes	maximal, complemented
4	C₄×C₄⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13063322	—	2	2	no	—
6	C₂×#13063322	—	2	1	no	—
7	C₂×#13063322	—	2	1	no	—
8	C₂×#13063322	C₄	1	4	yes	—
9	C₂×#13063322	C₂²	1	2	yes	—
10	C₂×C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
11	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
12	#14582642	—	2	8	no	—
13	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₃	—	2	1	no	—
15	C₄³×C₄₃	C₂²	1	1	yes	characteristic
16	C₄₃×#336	—	2	1	no	—
17	C₂²×D₁₇₂	—	2	4	no	—
18	C₂²×D₁₇₂	—	2	4	no	—
19	C₂²×D₁₇₂	—	2	2	no	—
20	C₂²×D₁₇₂	—	2	2	no	—
21	C₂²×D₁₇₂	C₂×C₄	1	4	yes	—
22	C₂²×C₄₃⋊C₈	—	2	2	no	—
23	#13063322	—	2	4	no	—
24	#13063322	—	2	4	no	—
25	#13063322	—	4	8	no	—
26	#13063322	—	4	4	no	—
27	#13063322	—	4	4	no	—
28	#13063322	—	4	2	no	—
29	#13063322	—	4	4	no	—
30	#13063322	—	4	2	no	—
31	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
32	C₄⋊(C₄₃⋊C₈)	—	4	4	no	—
33	C₂²×C₈×C₄₃	—	2	2	no	—
34	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₄₃	D₄	1	2	yes	—
36	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₄₃	—	2	2	no	—
40	C₄⋊C₈×C₄₃	D₄	1	4	yes	complemented
41	C₄×C₈×C₄₃	—	4	2	no	—
42	C₂³×D₄₃	—	2	4	no	—
43	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
48	C₂×D₁₇₂	—	2	8	no	—
49	C₂×D₁₇₂	—	4	8	no	—
50	C₂×D₁₇₂	—	4	8	no	—
51	C₂×D₁₇₂	—	4	8	no	—
52	C₂×D₁₇₂	—	4	8	no	—
53	C₂×D₁₇₂	—	4	8	no	—
54	C₂×D₁₇₂	—	4	8	no	—
55	C₂×D₁₇₂	—	4	4	no	—
56	C₂×D₁₇₂	—	4	2	no	—
57	C₂×D₁₇₂	—	4	2	no	—
58	C₂×D₁₇₂	—	4	2	no	—
59	C₂×D₁₇₂	—	4	4	no	—
60	C₂×D₁₇₂	—	4	2	no	—
61	C₂×D₁₇₂	—	4	4	no	—
62	C₂×D₁₇₂	—	4	4	no	—
63	C₂×D₁₇₂	—	4	8	no	—
64	C₂×C₈×C₄₃	—	2	4	no	—
65	C₂×C₈×C₄₃	—	4	2	no	—
66	C₂×C₈×C₄₃	—	4	2	no	—
67	C₄²×C₄₃	—	2	4	no	—
68	C₄²×C₄₃	—	2	2	no	—
69	C₄²×C₄₃	D₈	1	4	yes	—
70	C₄²×C₄₃	D₈	1	2	yes	—
71	C₄²×C₄₃	SD₁₆	1	4	yes	—
72	C₄²×C₄₃	SD₁₆	1	2	yes	—
73	C₄²×C₄₃	C₂×D₄	1	2	yes	—
74	C₄²×C₄₃	#41	1	2	yes	—
75	C₂×C₄₃⋊C₈	—	4	2	no	—
76	C₂×C₄₃⋊C₈	—	4	4	no	—
77	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×D₄₃	—	4	4	no	—
79	C₂²×D₄₃	—	4	8	no	—
80	C₂²×D₄₃	—	4	4	no	—
81	C₂²×D₄₃	—	4	8	no	—
82	C₂²×D₄₃	—	4	4	no	—
83	C₂×C₄×C₄₃	—	2	2	no	—
84	C₂×C₄×C₄₃	—	2	2	no	—
85	C₂×C₄×C₄₃	—	2	2	no	—
86	C₂×C₄×C₄₃	#100	1	4	yes	—
87	C₂×C₄×C₄₃	#102	1	4	yes	—
88	C₂×C₄×C₄₃	#102	1	4	yes	—
89	C₂×C₄×C₄₃	#102	1	4	yes	—
90	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
91	C₂×C₄×C₄₃	#121	1	2	yes	—
92	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
93	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
94	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
96	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
98	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
99	D₁₇₂	—	8	8	no	—
100	D₁₇₂	—	8	16	no	—
101	D₁₇₂	—	8	16	no	—
102	D₁₇₂	—	8	4	no	—
103	D₁₇₂	—	8	4	no	—
104	D₁₇₂	—	8	4	no	—
105	D₁₇₂	—	8	2	no	—
106	D₁₇₂	—	8	2	no	—
107	C₈×C₄₃	—	4	4	no	—
108	C₄₃⋊C₈	—	8	2	no	—
109	#9695	—	43	1	no	maximal, Sylow
110	C₂²×C₄₃	#390	1	1	yes	characteristic
111	C₂²×C₄₃	C₂×#100	1	2	yes	—
112	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
113	C₂²×C₄₃	C₂×#102	1	2	yes	—
114	C₂²×C₄₃	#418	1	1	yes	characteristic
115	C₂×D₄₃	—	8	8	no	—
116	C₂×D₄₃	—	8	8	no	—
117	C₂×D₄₃	—	8	4	no	—
118	C₂×D₄₃	—	8	4	no	—
119	C₂×D₄₃	—	8	4	no	—
120	C₄×C₄₃	—	2	4	no	—
121	C₄×C₄₃	—	2	2	no	—
122	C₄×C₄₃	#331	1	8	yes	complemented
123	C₄×C₄₃	#459	1	2	yes	—
124	C₄×C₄₃	#460	1	2	yes	—
125	C₄×C₄₃	#486	1	2	yes	—
126	C₄×C₄₃	#492	1	1	yes	characteristic
127	C₄×C₄₃	#493	1	1	yes	characteristic
128	#2867	—	43	1	no	—
129	#1842	—	43	1	no	—
130	C₄×C₄⋊C₈	D₄₃	1	1	yes	characteristic, complemented
131	C₂×#331	—	43	4	no	—
132	C₂×C₄₃	C₂×#331	1	2	yes	—
133	C₂×C₄₃	#1545	1	2	yes	—
134	C₂×C₄₃	#1957	1	1	yes	characteristic
135	C₂×C₄₃	#1965	1	1	yes	characteristic
136	C₂×C₄₃	#1982	1	1	yes	characteristic
137	D₄₃	—	16	4	no	—
138	C₂×#127	—	43	2	no	—
139	C₂×#127	—	43	4	no	—
140	C₂×#127	—	86	2	no	—
141	C₂×#127	—	86	1	no	—
142	C₂×#127	—	86	1	no	—
143	C₂×C₄⋊C₈	—	43	2	no	—
144	C₂×C₄⋊C₈	—	86	2	no	—
145	C₂×C₄⋊C₈	C₂×D₄₃	1	2	yes	—
146	C₂×C₄×C₈	—	2	1	no	—
147	C₄³	C₂×D₄₃	1	1	yes	characteristic
148	#336	—	2	1	no	—
149	#331	—	86	8	no	—
150	C₄₃	#9695	1	1	yes	characteristic, complemented, Sylow
151	C₂²×D₄	—	43	4	no	—
152	C₂²×D₄	—	86	4	no	—
153	C₂²×D₄	—	86	4	no	—
154	C₂²×D₄	—	86	2	no	—
155	C₂²×D₄	—	86	2	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	86	2	no	—
158	#127	—	86	4	no	—
159	#127	—	86	4	no	—
160	#127	—	172	8	no	—
161	#127	—	172	4	no	—
162	#127	—	172	4	no	—
163	#127	—	172	4	no	—
164	#127	—	172	2	no	—
165	#127	—	172	2	no	—
166	C₂×C₄²	C₄×D₄₃	1	2	yes	—
167	C₂×C₄²	D₁₇₂	1	2	yes	—
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	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	86	4	no	—
173	C₄⋊C₈	—	172	4	no	—
174	C₄⋊C₈	D₁₇₂	1	4	yes	complemented
175	C₄×C₈	—	4	2	no	—
176	C₂⁴	—	86	4	no	—
177	C₂×D₄	—	86	8	no	—
178	C₂×D₄	—	172	8	no	—
179	C₂×D₄	—	172	8	no	—
180	C₂×D₄	—	172	8	no	—
181	C₂×D₄	—	172	8	no	—
182	C₂×D₄	—	172	8	no	—
183	C₂×D₄	—	172	8	no	—
184	C₂×D₄	—	172	8	no	—
185	C₂×D₄	—	172	4	no	—
186	C₂×D₄	—	172	2	no	—
187	C₂×D₄	—	172	2	no	—
188	C₂×D₄	—	172	2	no	—
189	C₂×D₄	—	172	4	no	—
190	C₂×D₄	—	172	2	no	—
191	C₂×D₄	—	172	4	no	—
192	C₂×D₄	—	172	4	no	—
193	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
194	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
195	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	172	2	no	—
202	C₂×C₈	—	172	4	no	—
203	C₄²	—	2	4	no	—
204	C₄²	—	2	2	no	—
205	C₄²	#10621873	1	4	yes	—
206	C₄²	D₃₄₄	1	4	yes	—
207	C₄²	#10621882	1	2	yes	—
208	C₄²	#10621884	1	2	yes	—
209	C₄²	C₂×D₁₇₂	1	2	yes	—
210	C₄²	#10621898	1	2	yes	—
211	C₂³	—	172	4	no	—
212	C₂³	—	172	8	no	—
213	C₂³	—	172	4	no	—
214	C₂³	—	172	8	no	—
215	C₂³	—	172	4	no	—
216	C₂³	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
217	D₄	—	344	8	no	—
218	D₄	—	344	16	no	—
219	D₄	—	344	4	no	—
220	D₄	—	344	4	no	—
221	D₄	—	344	16	no	—
222	D₄	—	344	4	no	—
223	D₄	—	344	2	no	—
224	D₄	—	344	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	#13063241	1	4	yes	—
229	C₂×C₄	#13063245	1	4	yes	—
230	C₂×C₄	#13063247	1	4	yes	—
231	C₂×C₄	#13063247	1	4	yes	—
232	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
233	C₂×C₄	#13063299	1	2	yes	—
234	C₂×C₄	#13063313	1	1	yes	characteristic
235	C₂×C₄	#13063322	1	1	yes	characteristic
236	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
237	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
238	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
239	C₂×C₄	C₂×#10621873	1	2	yes	—
240	C₂×C₄	#13063350	1	2	yes	—
241	C₈	—	4	4	no	—
242	C₈	—	344	2	no	—
243	C₂²	—	344	4	no	—
244	C₂²	—	344	8	no	—
245	C₂²	—	344	8	no	—
246	C₂²	—	344	4	no	—
247	C₂²	—	344	4	no	—
248	C₂²	#14582801	1	1	yes	characteristic
249	C₂²	C₂×#13063241	1	2	yes	—
250	C₂²	C₂×#13063245	1	1	yes	characteristic
251	C₂²	C₂×#13063247	1	2	yes	—
252	C₂²	#14582903	1	1	yes	characteristic
253	C₄	—	2	4	no	—
254	C₄	—	2	2	no	—
255	C₄	#14582642	1	8	yes	complemented
256	C₄	#14583094	1	2	yes	—
257	C₄	#14583099	1	2	yes	—
258	C₄	#14583246	1	2	yes	—
259	C₄	#14583272	1	1	yes	characteristic
260	C₄	#14583276	1	1	yes	characteristic
261	C₂	—	688	4	no	—
262	C₂	C₂×#14582642	1	2	yes	—
263	C₂	#19133262	1	2	yes	—
264	C₂	#19135401	1	1	yes	characteristic
265	C₂	#19135435	1	1	yes	characteristic
266	C₂	#19135522	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₃)⋊C₄
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄×C₄⋊C₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
prime radical	#19142990	C₂
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₄×C₄⋊C₈	D₄₃
hypocenter	C₄₃	#9695
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #19134747	C₂	semidirect
3. C₂×#14582642	C₂	semidirect
4. C₄×C₄⋊C₈×C₄₃	C₂	semidirect
40. C₄⋊C₈×C₄₃	D₄	semidirect
122. C₄×C₄₃	#331	semidirect
130. C₄×C₄⋊C₈	D₄₃	semidirect
150. C₄₃	#9695	semidirect
174. C₄⋊C₈	D₁₇₂	semidirect
255. C₄	#14582642	semidirect

Maximal subgroups

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

subgroup	index
#19142990	2
#19134747	2
C₂×#14582642	2
C₄×C₄⋊C₈×C₄₃	2
#9695	43