Subgroups — #25001773

Order 11008.

998 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#10621877	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582745	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582658	C₂	1	4	yes	maximal
4	C₄×#117×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10621877	C₂²	1	2	yes	—
6	C₂²×C₄×Dic₄₃	C₂²	1	2	yes	—
7	C₄×#10621877	—	2	2	no	—
8	C₄×#10621877	C₄	1	4	yes	complemented
9	C₂×#13063236	—	2	1	no	—
10	C₂×C₄×C₄₃⋊C₈	—	2	1	no	—
11	#14582745	—	2	2	no	—
12	#14582745	C₄	1	4	yes	complemented
13	C₂×C₄₃⋊#95	—	2	2	no	—
14	#14582658	C₄	1	16	yes	complemented
15	C₂×#117×C₄₃	C₂²	1	2	yes	—
16	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₄₃	C₄	1	4	yes	complemented
18	C₄×C₂²⋊C₄×C₄₃	—	2	2	no	—
19	C₄₃×#376	—	2	1	no	—
20	C₄³×C₄₃	—	2	1	no	—
21	C₂³×Dic₄₃	—	2	2	no	—
22	C₂×#10621877	—	2	2	no	—
23	C₂×#10621877	—	2	4	no	—
24	C₂×#10621877	C₂×C₄	1	4	yes	—
25	C₂×#10621877	C₂×C₄	1	8	yes	—
26	C₂²×C₄₃⋊C₈	—	2	2	no	—
27	C₂×C₄×Dic₄₃	—	2	2	no	—
28	C₂×C₄×Dic₄₃	—	2	4	no	—
29	C₂×C₄×Dic₄₃	C₂×C₄	1	8	yes	—
30	C₂×C₄×Dic₄₃	C₂×C₄	1	4	yes	—
31	#13063236	—	2	4	no	—
32	C₄×C₄₃⋊C₈	—	2	4	no	—
33	C₄₃⋊#95	—	2	8	no	—
34	C₂³×C₄×C₄₃	D₄	1	2	yes	—
35	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
36	#117×C₄₃	—	2	4	no	—
37	#117×C₄₃	C₂×C₄	1	8	yes	—
38	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
39	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
40	C₂×C₄²×C₄₃	—	2	2	no	—
41	C₂×C₄²×C₄₃	—	2	4	no	—
42	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₄₃	D₄	1	2	yes	—
46	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₃	Q₈	1	2	yes	—
49	#95×C₄₃	—	2	4	no	—
50	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₃	—	2	4	no	—
52	C₂²×C₄×C₄₃	—	4	4	no	—
53	C₂²×C₄×C₄₃	C₂²⋊C₄	1	4	yes	—
54	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
57	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	#41	1	4	yes	—
65	C₂²×Dic₄₃	—	2	4	no	—
66	C₂²×Dic₄₃	—	2	8	no	—
67	C₂²×Dic₄₃	—	4	8	no	—
68	C₄⋊C₄×C₄₃	—	2	4	no	—
69	C₄⋊C₄×C₄₃	C₄²	1	8	yes	—
70	C₂²⋊C₄×C₄₃	—	2	8	no	—
71	C₄²×C₄₃	—	2	8	no	—
72	C₄²×C₄₃	—	4	8	no	—
73	C₄²×C₄₃	C₄²	1	4	yes	—
74	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
76	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
77	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
78	C₄²×C₄₃	C₄⋊C₄	1	4	yes	—
79	C₄×Dic₄₃	—	2	4	no	—
80	C₄×Dic₄₃	—	2	8	no	—
81	C₄×Dic₄₃	C₄²	1	8	yes	complemented
82	#10621877	—	2	4	no	—
83	#10621877	—	2	8	no	—
84	#10621877	C₄²	1	8	yes	complemented
85	C₂×C₄₃⋊C₈	—	2	4	no	—
86	C₂×C₄₃⋊C₈	—	2	8	no	—
87	C₂³×C₄₃	—	2	4	no	—
88	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂³×C₄₃	C₄×D₄	1	4	yes	—
92	C₂×C₄×C₄₃	—	2	4	no	—
93	C₂×C₄×C₄₃	—	2	4	no	—
94	C₂×C₄×C₄₃	—	2	8	no	—
95	C₂×C₄×C₄₃	—	4	4	no	—
96	C₂×C₄×C₄₃	—	4	4	no	—
97	C₂×C₄×C₄₃	—	4	8	no	—
98	C₂×C₄×C₄₃	#95	1	4	yes	—
99	C₂×C₄×C₄₃	#95	1	4	yes	—
100	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
101	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
102	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
104	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₃	#117	1	2	yes	—
108	C₂×C₄×C₄₃	#117	1	2	yes	—
109	C₂×C₄×C₄₃	#117	1	2	yes	—
110	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
111	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
112	C₂×C₄×C₄₃	C₄×Q₈	1	4	yes	—
113	C₂×Dic₄₃	—	2	8	no	—
114	C₂×Dic₄₃	—	4	8	no	—
115	C₂×Dic₄₃	—	4	16	no	—
116	C₄₃⋊C₈	—	2	8	no	—
117	#10701	—	43	1	no	maximal, Sylow
118	C₂²×C₄₃	—	2	2	no	—
119	C₂²×C₄₃	—	2	2	no	—
120	C₂²×C₄₃	—	2	8	no	—
121	C₂²×C₄₃	C₂×#95	1	1	yes	characteristic
122	C₂²×C₄₃	C₄×C₂²⋊C₄	1	2	yes	—
123	C₂²×C₄₃	C₄×C₂²⋊C₄	1	2	yes	—
124	C₂²×C₄₃	C₄×C₄⋊C₄	1	2	yes	—
125	C₂²×C₄₃	#421	1	4	yes	—
126	C₄×C₄₃	—	2	4	no	—
127	C₄×C₄₃	—	2	4	no	—
128	C₄×C₄₃	—	4	8	no	—
129	C₄×C₄₃	C₄³	1	1	yes	characteristic
130	C₄×C₄₃	#376	1	1	yes	characteristic
131	C₄×C₄₃	C₄×C₂²⋊C₄	1	2	yes	—
132	C₄×C₄₃	C₄×C₄⋊C₄	1	2	yes	—
133	C₄×C₄₃	C₄×C₄⋊C₄	1	2	yes	—
134	Dic₄₃	—	4	8	no	—
135	C₂×C₄×M₁₆	—	43	1	no	—
136	C₄×#117	D₄₃	1	1	yes	characteristic
137	C₂×#376	—	43	1	no	—
138	C₂×#339	—	43	4	no	—
139	C₂×C₄₃	—	2	2	no	—
140	C₂×C₄₃	—	2	2	no	—
141	C₂×C₄₃	C₄×#95	1	1	yes	characteristic
142	C₂×C₄₃	#1733	1	2	yes	—
143	C₂×C₄₃	#1759	1	4	yes	—
144	C₂²×M₁₆	—	43	2	no	—
145	C₂×#117	C₂×D₄₃	1	2	yes	—
146	C₂²×C₄²	—	43	2	no	—
147	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
148	C₄×M₁₆	—	43	4	no	—
149	C₄×M₁₆	—	86	2	no	—
150	C₂×#97	—	86	1	no	—
151	C₂×C₄×C₈	—	86	1	no	—
152	C₄×C₄⋊C₄	Dic₄₃	1	4	yes	complemented
153	C₄×C₂²⋊C₄	—	2	2	no	—
154	#376	—	2	1	no	—
155	#376	—	43	4	no	—
156	#376	—	86	2	no	—
157	C₂×#95	—	86	2	no	—
158	C₄³	—	2	1	no	—
159	#339	—	43	16	no	—
160	C₄₃	#10701	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₄	—	86	2	no	—
162	C₂³×C₄	C₂²⋊D₄₃	1	2	yes	—
163	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
164	C₂×M₁₆	—	43	8	no	—
165	C₂×M₁₆	—	43	4	no	—
166	C₂×M₁₆	—	86	2	no	—
167	C₂×M₁₆	—	86	4	no	—
168	C₂²×C₈	—	86	2	no	—
169	#117	—	2	4	no	—
170	#117	C₄×D₄₃	1	8	yes	—
171	C₂×C₄⋊C₄	C₂×Dic₄₃	1	4	yes	—
172	C₂×C₂²⋊C₄	—	2	2	no	—
173	C₂×C₄²	—	2	2	no	—
174	C₂×C₄²	—	2	4	no	—
175	C₂×C₄²	—	43	8	no	—
176	C₂×C₄²	—	43	4	no	—
177	C₂×C₄²	—	86	2	no	—
178	C₂×C₄²	—	86	4	no	—
179	C₂×C₄²	Dic₈₆	1	2	yes	—
180	C₂×C₄²	C₄×D₄₃	1	2	yes	—
181	C₂×C₄²	C₄×D₄₃	1	2	yes	—
182	C₂×C₄²	D₁₇₂	1	2	yes	—
183	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
184	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
185	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
186	#97	—	86	4	no	—
187	C₄×C₈	—	86	4	no	—
188	#95	—	2	4	no	—
189	#95	—	86	8	no	—
190	C₂⁴	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	4	4	no	—
193	C₂²×C₄	—	86	4	no	—
194	C₂²×C₄	—	86	8	no	—
195	C₂²×C₄	—	172	8	no	—
196	C₂²×C₄	Dic₄₃⋊C₄	1	4	yes	—
197	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	4	yes	—
198	C₂²×C₄	C₂²⋊Dic₄₃	1	2	yes	—
199	C₂²×C₄	C₂²⋊Dic₄₃	1	2	yes	—
200	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₄×D₄₃	1	4	yes	—
204	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
205	C₂²×C₄	#10621898	1	4	yes	—
206	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
208	M₁₆	—	43	8	no	—
209	M₁₆	—	86	4	no	—
210	M₁₆	—	86	8	no	—
211	C₂×C₈	—	86	4	no	—
212	C₂×C₈	—	86	8	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	C₄×Dic₄₃	1	8	yes	—
215	C₂²⋊C₄	—	2	8	no	—
216	C₄²	—	2	8	no	—
217	C₄²	—	4	8	no	—
218	C₄²	—	43	8	no	—
219	C₄²	—	86	4	no	—
220	C₄²	—	86	8	no	—
221	C₄²	C₄×Dic₄₃	1	4	yes	—
222	C₄²	Dic₄₃⋊C₄	1	2	yes	—
223	C₄²	Dic₄₃⋊C₄	1	2	yes	—
224	C₄²	C₄⋊Dic₄₃	1	4	yes	—
225	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
226	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
227	C₂³	—	2	4	no	—
228	C₂³	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
229	C₂³	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
230	C₂³	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
231	C₂³	C₄×C₂²⋊D₄₃	1	4	yes	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	8	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	8	no	—
238	C₂×C₄	—	86	8	no	—
239	C₂×C₄	—	172	8	no	—
240	C₂×C₄	—	172	16	no	—
241	C₂×C₄	C₄₃⋊#95	1	4	yes	—
242	C₂×C₄	C₄₃⋊#95	1	4	yes	—
243	C₂×C₄	C₄²×D₄₃	1	2	yes	—
244	C₂×C₄	C₄²×D₄₃	1	2	yes	—
245	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
246	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
247	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
248	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
249	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
250	C₂×C₄	#13063282	1	2	yes	—
251	C₂×C₄	#13063282	1	2	yes	—
252	C₂×C₄	#13063284	1	2	yes	—
253	C₂×C₄	C₄×C₂²⋊D₄₃	1	4	yes	—
254	C₂×C₄	C₄×D₁₇₂	1	4	yes	—
255	C₂×C₄	C₄×Dic₈₆	1	4	yes	—
256	C₈	—	86	8	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	8	no	—
260	C₂²	C₂×C₄₃⋊#95	1	1	yes	characteristic
261	C₂²	C₄×C₂²⋊Dic₄₃	1	2	yes	—
262	C₂²	C₄×(C₂×D₄₃)⋊C₄	1	2	yes	—
263	C₂²	C₄×Dic₄₃⋊C₄	1	2	yes	—
264	C₂²	#14582916	1	4	yes	—
265	C₄	—	2	4	no	—
266	C₄	—	2	4	no	—
267	C₄	—	4	8	no	—
268	C₄	—	172	8	no	—
269	C₄	C₄²×Dic₄₃	1	1	yes	characteristic
270	C₄	#14582745	1	1	yes	characteristic
271	C₄	C₄×(C₂×D₄₃)⋊C₄	1	2	yes	—
272	C₄	C₄×C₄⋊Dic₄₃	1	2	yes	—
273	C₄	C₄×Dic₄₃⋊C₄	1	2	yes	—
274	C₂	—	2	2	no	—
275	C₂	—	2	2	no	—
276	C₂	C₄×C₄₃⋊#95	1	1	yes	characteristic
277	C₂	#19134304	1	2	yes	—
278	C₂	#19134405	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₃
commutator	C₄×C₄₃	C₄³
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₄×#117×C₄₃	C₂
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₃	C₂×D₄
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	C₄×#117	D₄₃
hypocenter	C₄₃	#10701
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
8. C₄×#10621877	C₄	semidirect
12. #14582745	C₄	semidirect
14. #14582658	C₄	semidirect
17. C₄×C₄⋊C₄×C₄₃	C₄	semidirect
81. C₄×Dic₄₃	C₄²	semidirect
84. #10621877	C₄²	semidirect
152. C₄×C₄⋊C₄	Dic₄₃	semidirect
160. C₄₃	#10701	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#10621877	2
C₂×#14582745	2
C₂×#14582658	2
C₄×#117×C₄₃	2
#10701	43