Subgroups — C₂²×#14583340 #26003095

Order 11008.

2078 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13063347	C₂	1	1	yes	maximal, characteristic
2	C₂×#14583340	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#13063271	C₂	1	2	yes	maximal, complemented
4	C₂²×#13063268	C₂	1	2	yes	maximal, complemented
5	C₂²×#13063265	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×Q₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#10621898	—	2	1	no	—
8	C₂×#13063347	—	2	6	no	—
9	C₂×#13063347	C₂²	1	12	yes	—
10	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
11	C₂²×#10621873	—	2	1	no	—
12	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
13	#14583340	C₂²	1	64	yes	direct product factor
14	C₂×#13063271	C₂²	1	24	yes	complemented
15	C₂×#13063268	C₂²	1	24	yes	complemented
16	C₂×#13063265	C₂²	1	12	yes	complemented
17	C₂²×C₄₃⋊C₁₆	C₂²	1	2	yes	—
18	C₂³×Q₈×C₄₃	—	2	1	no	—
19	C₂²×Q₁₆×C₄₃	—	2	1	no	—
20	C₂²×Q₁₆×C₄₃	—	2	6	no	—
21	C₂²×Q₁₆×C₄₃	C₂²	1	2	yes	—
22	C₂²×Q₁₆×C₄₃	C₂²	1	12	yes	complemented
23	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×#10621898	—	2	12	no	—
25	C₂×#10621898	—	4	6	no	—
26	C₂²×Dic₈₆	—	2	1	no	—
27	C₂²×D₁₇₂	—	2	1	no	—
28	C₂²×C₂²⋊D₄₃	—	4	1	no	—
29	C₂²×C₄×D₄₃	—	4	1	no	—
30	#13063347	—	2	24	no	—
31	#13063347	C₂³	1	16	yes	—
32	C₂×Dic₁₇₂	—	2	3	no	—
33	C₂×Dic₁₇₂	C₂³	1	6	yes	—
34	C₂×#10621873	—	2	12	no	—
35	C₂×D₃₄₄	—	2	3	no	—
36	C₂×D₃₄₄	C₂³	1	6	yes	—
37	#13063271	C₂³	1	32	yes	complemented
38	#13063268	C₂³	1	32	yes	complemented
39	#13063265	C₂³	1	16	yes	complemented
40	C₂×C₄₃⋊C₁₆	C₂³	1	12	yes	—
41	C₂²×Q₈×C₄₃	—	2	2	no	—
42	C₂²×Q₈×C₄₃	—	2	12	no	—
43	C₂²×Q₈×C₄₃	—	4	1	no	—
44	C₂²×Q₈×C₄₃	—	4	6	no	—
45	C₂³×C₄×C₄₃	—	4	1	no	—
46	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
47	C₂×Q₁₆×C₄₃	—	2	6	no	—
48	C₂×Q₁₆×C₄₃	—	2	12	no	—
49	C₂×Q₁₆×C₄₃	—	2	24	no	—
50	C₂×Q₁₆×C₄₃	C₂³	1	12	yes	—
51	C₂×Q₁₆×C₄₃	C₂³	1	16	yes	complemented
52	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₄₃	D₄	1	6	yes	—
54	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₄₃	C₂³	1	6	yes	—
56	C₂⁴×C₄₃	D₈	1	1	yes	characteristic
57	C₂³×D₄₃	—	4	1	no	—
58	C₂×Q₈×C₄₃	—	2	12	no	—
59	C₂×Q₈×C₄₃	—	2	16	no	—
60	C₂×Q₈×C₄₃	—	4	6	no	—
61	C₂×Q₈×C₄₃	—	4	12	no	—
62	C₂×Q₈×C₄₃	—	4	24	no	—
63	C₂²×C₄×C₄₃	—	4	12	no	—
64	C₂²×C₄×C₄₃	—	4	2	no	—
65	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	D₈	1	6	yes	—
67	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	C₂×D₄	1	6	yes	—
69	C₂×C₂²⋊D₄₃	—	4	12	no	—
70	C₂²×Dic₄₃	—	4	1	no	—
71	#10621898	—	2	16	no	—
72	#10621898	—	4	24	no	—
73	C₂×D₁₇₂	—	2	6	no	—
74	C₂×D₁₇₂	—	4	3	no	—
75	C₂×C₄×D₄₃	—	4	12	no	—
76	C₂×Dic₈₆	—	2	6	no	—
77	C₂×Dic₈₆	—	4	3	no	—
78	Q₁₆×C₄₃	—	2	12	no	—
79	Q₁₆×C₄₃	—	2	16	no	—
80	Q₁₆×C₄₃	C₂⁴	1	8	yes	—
81	C₂×C₈×C₄₃	C₂×D₄	1	3	yes	—
82	C₂×C₈×C₄₃	C₂×D₄	1	6	yes	—
83	C₂×C₈×C₄₃	C₂×D₄	1	12	yes	—
84	C₂×C₈×C₄₃	C₂⁴	1	3	yes	—
85	C₂×C₈×C₄₃	C₂⁴	1	4	yes	—
86	Dic₁₇₂	—	2	6	no	—
87	Dic₁₇₂	C₂⁴	1	4	yes	—
88	D₃₄₄	—	2	6	no	—
89	D₃₄₄	C₂⁴	1	4	yes	—
90	#10621873	—	2	16	no	—
91	C₄₃⋊C₁₆	C₂⁴	1	8	yes	—
92	C₂³×C₄₃	—	2	4	no	—
93	C₂³×C₄₃	C₂×D₈	1	6	yes	—
94	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
95	C₂²×D₄₃	—	4	6	no	—
96	C₂²×D₄₃	—	8	4	no	—
97	Q₈×C₄₃	—	2	8	no	—
98	Q₈×C₄₃	—	4	12	no	—
99	Q₈×C₄₃	—	4	16	no	—
100	C₂×C₄×C₄₃	—	4	16	no	—
101	C₂×C₄×C₄₃	—	4	12	no	—
102	C₂×C₄×C₄₃	C₂×D₈	1	3	yes	—
103	C₂×C₄×C₄₃	C₂×D₈	1	6	yes	—
104	C₂×C₄×C₄₃	C₂×D₈	1	12	yes	—
105	C₂×C₄×C₄₃	C₂²×D₄	1	3	yes	—
106	C₂×C₄×C₄₃	C₂²×D₄	1	4	yes	—
107	C₂²⋊D₄₃	—	4	16	no	—
108	C₂×Dic₄₃	—	4	6	no	—
109	D₁₇₂	—	2	4	no	—
110	D₁₇₂	—	4	6	no	—
111	C₄×D₄₃	—	4	16	no	—
112	Dic₈₆	—	2	4	no	—
113	Dic₈₆	—	4	6	no	—
114	C₈×C₄₃	C₂²×D₄	1	4	yes	—
115	C₈×C₄₃	C₂²×D₄	1	3	yes	—
116	C₈×C₄₃	C₂⁵	1	1	yes	characteristic
117	C₂²×#510	—	43	1	no	maximal, Sylow
118	C₂²×C₄₃	—	2	12	no	—
119	C₂²×C₄₃	#510	1	4	yes	complemented
120	C₂²×C₄₃	C₂²×D₈	1	4	yes	—
121	C₂²×C₄₃	C₂²×D₈	1	3	yes	—
122	C₂×D₄₃	—	4	4	no	—
123	C₂×D₄₃	—	8	12	no	—
124	C₄×C₄₃	—	4	8	no	—
125	C₄×C₄₃	C₂²×D₈	1	3	yes	—
126	C₄×C₄₃	C₂²×D₈	1	4	yes	—
127	C₄×C₄₃	C₂³×D₄	1	1	yes	characteristic
128	Dic₄₃	—	4	4	no	—
129	C₂²×#135	—	43	1	no	—
130	C₂³×Q₁₆	D₄₃	1	1	yes	characteristic, complemented
131	C₂×#510	—	43	24	no	—
132	C₂²×Q₃₂	—	43	2	no	—
133	C₂²×SD₃₂	—	43	2	no	—
134	C₂²×M₃₂	—	43	1	no	—
135	C₂×C₄₃	—	2	4	no	—
136	C₂×C₄₃	C₂×#510	1	6	yes	complemented
137	C₂×C₄₃	C₂³×D₈	1	1	yes	characteristic
138	D₄₃	—	8	4	no	—
139	C₂²×#41	—	86	1	no	—
140	C₂³×Q₈	—	2	1	no	—
141	C₂×#135	—	43	12	no	—
142	C₂×#135	—	86	6	no	—
143	C₂²×Q₁₆	—	2	1	no	—
144	C₂²×Q₁₆	—	2	6	no	—
145	C₂²×Q₁₆	—	43	1	no	—
146	C₂²×Q₁₆	C₂×D₄₃	1	2	yes	—
147	C₂²×Q₁₆	C₂×D₄₃	1	12	yes	complemented
148	C₂²×SD₁₆	—	86	1	no	—
149	C₂²×D₈	—	43	1	no	—
150	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
151	#510	—	43	64	no	—
152	C₂×Q₃₂	—	43	24	no	—
153	C₂×SD₃₂	—	43	24	no	—
154	C₂×M₃₂	—	43	12	no	—
155	C₂²×C₁₆	—	43	2	no	—
156	C₄₃	C₂²×#510	1	1	yes	characteristic, complemented, Sylow
157	C₂×#41	—	86	12	no	—
158	C₂×#41	—	172	6	no	—
159	C₂²×Q₈	—	2	2	no	—
160	C₂²×Q₈	—	2	12	no	—
161	C₂²×Q₈	—	4	6	no	—
162	C₂²×Q₈	—	4	1	no	—
163	C₂²×Q₈	—	86	1	no	—
164	C₂²×D₄	—	86	1	no	—
165	C₂²×D₄	—	172	1	no	—
166	C₂³×C₄	—	4	1	no	—
167	C₂³×C₄	—	172	1	no	—
168	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
169	#135	—	43	16	no	—
170	#135	—	86	24	no	—
171	C₂×Q₁₆	—	2	6	no	—
172	C₂×Q₁₆	—	2	12	no	—
173	C₂×Q₁₆	—	2	24	no	—
174	C₂×Q₁₆	—	43	6	no	—
175	C₂×Q₁₆	—	86	3	no	—
176	C₂×Q₁₆	C₂²×D₄₃	1	12	yes	—
177	C₂×Q₁₆	C₂²×D₄₃	1	16	yes	complemented
178	C₂×SD₁₆	—	86	12	no	—
179	C₂×D₈	—	43	6	no	—
180	C₂×D₈	—	86	3	no	—
181	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
182	C₂²×C₈	C₂²⋊D₄₃	1	6	yes	—
183	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₄₃	1	6	yes	—
185	Q₃₂	—	43	32	no	—
186	SD₃₂	—	43	32	no	—
187	M₃₂	—	43	16	no	—
188	C₂×C₁₆	—	43	12	no	—
189	C₂⁴	—	172	1	no	—
190	C₂⁴	#10621882	1	1	yes	characteristic
191	#41	—	86	16	no	—
192	#41	—	172	24	no	—
193	C₂×Q₈	—	2	12	no	—
194	C₂×Q₈	—	2	16	no	—
195	C₂×Q₈	—	4	12	no	—
196	C₂×Q₈	—	4	6	no	—
197	C₂×Q₈	—	4	24	no	—
198	C₂×Q₈	—	86	6	no	—
199	C₂×Q₈	—	172	3	no	—
200	C₂×D₄	—	86	6	no	—
201	C₂×D₄	—	172	12	no	—
202	C₂×D₄	—	172	3	no	—
203	C₂²×C₄	—	4	12	no	—
204	C₂²×C₄	—	4	2	no	—
205	C₂²×C₄	—	172	1	no	—
206	C₂²×C₄	—	172	12	no	—
207	C₂²×C₄	#10621882	1	6	yes	—
208	C₂²×C₄	#10621882	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₂²⋊D₄₃	1	6	yes	—
210	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
211	Q₁₆	—	2	12	no	—
212	Q₁₆	—	2	16	no	—
213	Q₁₆	—	43	4	no	—
214	Q₁₆	—	86	6	no	—
215	Q₁₆	C₂³×D₄₃	1	8	yes	—
216	SD₁₆	—	86	16	no	—
217	D₈	—	43	4	no	—
218	D₈	—	86	6	no	—
219	C₂×C₈	C₂×C₂²⋊D₄₃	1	3	yes	—
220	C₂×C₈	C₂×C₂²⋊D₄₃	1	6	yes	—
221	C₂×C₈	C₂×C₂²⋊D₄₃	1	12	yes	—
222	C₂×C₈	C₂³×D₄₃	1	3	yes	—
223	C₂×C₈	C₂³×D₄₃	1	4	yes	—
224	C₁₆	—	43	8	no	—
225	C₂³	—	2	4	no	—
226	C₂³	—	344	4	no	—
227	C₂³	—	172	6	no	—
228	C₂³	C₂×#10621882	1	6	yes	—
229	C₂³	C₂×#10621882	1	1	yes	characteristic
230	Q₈	—	2	8	no	—
231	Q₈	—	4	16	no	—
232	Q₈	—	4	12	no	—
233	Q₈	—	86	4	no	—
234	Q₈	—	172	6	no	—
235	D₄	—	86	4	no	—
236	D₄	—	172	16	no	—
237	D₄	—	172	6	no	—
238	C₂×C₄	—	4	16	no	—
239	C₂×C₄	—	4	12	no	—
240	C₂×C₄	—	172	6	no	—
241	C₂×C₄	—	172	16	no	—
242	C₂×C₄	C₂×#10621882	1	6	yes	—
243	C₂×C₄	C₂×#10621882	1	3	yes	—
244	C₂×C₄	C₂×#10621882	1	12	yes	—
245	C₂×C₄	C₂²×C₂²⋊D₄₃	1	3	yes	—
246	C₂×C₄	C₂²×C₂²⋊D₄₃	1	4	yes	—
247	C₈	C₂²×C₂²⋊D₄₃	1	3	yes	—
248	C₈	C₂²×C₂²⋊D₄₃	1	4	yes	—
249	C₈	C₂⁴×D₄₃	1	1	yes	characteristic
250	C₂²	—	2	12	no	—
251	C₂²	—	344	12	no	—
252	C₂²	—	172	4	no	—
253	C₂²	#14583340	1	4	yes	direct product factor
254	C₂²	C₂²×#10621882	1	4	yes	—
255	C₂²	C₂²×#10621882	1	3	yes	—
256	C₄	—	4	8	no	—
257	C₄	—	172	4	no	—
258	C₄	C₂²×#10621882	1	4	yes	—
259	C₄	C₂²×#10621882	1	3	yes	—
260	C₄	C₂³×C₂²⋊D₄₃	1	1	yes	characteristic
261	C₂	—	2	4	no	—
262	C₂	—	344	4	no	—
263	C₂	C₂×#14583340	1	6	yes	direct product factor
264	C₂	C₂³×#10621882	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#10621882
commutator, cosocle	C₈×C₄₃	C₂⁵
Frattini	C₈	C₂⁴×D₄₃
Fitting	C₂³×Q₁₆×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂²×#13063347	C₂
hypercenter	C₂³×Q₁₆	D₄₃
hypocenter	C₄₃	C₂²×#510
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×Q₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#14583340	C₂	direct (N × H)
3. C₂²×#13063271	C₂	semidirect
4. C₂²×#13063268	C₂	semidirect
5. C₂²×#13063265	C₂	semidirect
6. C₂³×Q₁₆×C₄₃	C₂	semidirect
13. #14583340	C₂²	direct (N × H)
14. C₂×#13063271	C₂²	semidirect
15. C₂×#13063268	C₂²	semidirect
16. C₂×#13063265	C₂²	semidirect
22. C₂²×Q₁₆×C₄₃	C₂²	semidirect
37. #13063271	C₂³	semidirect
38. #13063268	C₂³	semidirect
39. #13063265	C₂³	semidirect
51. C₂×Q₁₆×C₄₃	C₂³	semidirect
119. C₂²×C₄₃	#510	semidirect
130. C₂³×Q₁₆	D₄₃	semidirect
136. C₂×C₄₃	C₂×#510	semidirect
147. C₂²×Q₁₆	C₂×D₄₃	semidirect
156. C₄₃	C₂²×#510	semidirect
177. C₂×Q₁₆	C₂²×D₄₃	semidirect
253. C₂²	#14583340	direct (N × H)
263. C₂	C₂×#14583340	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13063347	2
C₂×#14583340	2
C₂²×#13063271	2
C₂²×#13063268	2
C₂²×#13063265	2
C₂³×Q₁₆×C₄₃	2
C₂²×#510	43

Subgroups — C22×#14583340 #26003095

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#14583340 #26003095