Subgroups — C₂²×#14583010 #25960641

Order 11008.

3654 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₃₄₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13063287	C₂	1	2	yes	maximal, complemented
3	C₂×#14583010	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#13063245	C₂	1	2	yes	maximal, complemented
5	C₂²×#13063235	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#106×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₇₂	C₂²	1	2	yes	—
8	C₂³×C₄×D₄₃	—	2	2	no	—
9	C₂²×D₃₄₄	C₄	1	16	yes	complemented
10	C₂²×D₃₄₄	C₂²	1	12	yes	—
11	C₂³×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×#13063287	—	2	12	no	—
13	C₂×#13063287	C₂²	1	24	yes	complemented
14	C₂²×(C₂×D₄₃)⋊C₄	—	2	2	no	—
15	C₂²×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
16	#14583010	C₂²	1	64	yes	direct product factor
17	C₂×#13063245	C₂²	1	24	yes	complemented
18	C₂×#13063235	C₂²	1	12	yes	complemented
19	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
21	C₂×#106×C₄₃	C₂²	1	12	yes	complemented
22	C₂⁴×D₄₃	—	2	2	no	—
23	C₂²×D₁₇₂	—	2	6	no	—
24	C₂²×D₁₇₂	—	2	8	no	—
25	C₂²×D₁₇₂	C₂×C₄	1	16	yes	complemented
26	C₂²×D₁₇₂	C₂³	1	12	yes	—
27	C₂³×Dic₄₃	D₄	1	2	yes	—
28	C₂²×C₄×D₄₃	—	2	24	no	—
29	C₂²×C₄×D₄₃	—	4	16	no	—
30	C₂×D₃₄₄	C₂×C₄	1	96	yes	complemented
31	C₂×D₃₄₄	C₂³	1	16	yes	—
32	C₂²×C₄₃⋊C₈	—	2	4	no	—
33	C₂²×C₄₃⋊C₈	C₂³	1	6	yes	—
34	#13063287	—	2	48	no	—
35	#13063287	C₂³	1	32	yes	complemented
36	C₂×(C₂×D₄₃)⋊C₄	—	2	24	no	—
37	C₂×C₄×Dic₄₃	D₄	1	6	yes	—
38	C₂×C₄×Dic₄₃	C₂³	1	6	yes	—
39	#13063245	C₂³	1	32	yes	complemented
40	#13063235	C₂³	1	16	yes	complemented
41	C₂³×C₄×C₄₃	—	2	2	no	—
42	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₄₃	C₂×C₄	1	8	yes	complemented
44	C₂²×C₈×C₄₃	C₂³	1	6	yes	—
45	C₂×C₄⋊C₄×C₄₃	—	2	6	no	—
46	C₂×C₄⋊C₄×C₄₃	C₂³	1	12	yes	—
47	#106×C₄₃	C₂³	1	16	yes	complemented
48	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
49	C₂³×D₄₃	—	2	16	no	—
50	C₂³×D₄₃	—	2	12	no	—
51	C₂³×D₄₃	—	4	8	no	—
52	C₂³×D₄₃	—	4	8	no	—
53	C₂²×C₄×C₄₃	—	2	12	no	—
54	C₂²×C₄×C₄₃	—	4	8	no	—
55	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
57	C₂²×C₄×C₄₃	#41	1	4	yes	—
58	C₂²×C₄×C₄₃	C₂⁴	1	3	yes	—
59	C₂²×Dic₄₃	—	2	8	no	—
60	C₂²×Dic₄₃	C₂×D₄	1	12	yes	—
61	C₂×D₁₇₂	—	2	48	no	—
62	C₂×D₁₇₂	—	2	12	no	—
63	C₂×D₁₇₂	—	2	24	no	—
64	C₂×D₁₇₂	C₂²×C₄	1	48	yes	complemented
65	C₂×D₁₇₂	C₂⁴	1	8	yes	—
66	C₂×C₄×D₄₃	—	2	32	no	—
67	C₂×C₄×D₄₃	—	4	96	no	—
68	C₂×C₈×C₄₃	C₂²×C₄	1	24	yes	complemented
69	C₂×C₈×C₄₃	C₂⁴	1	4	yes	—
70	C₄⋊C₄×C₄₃	—	2	12	no	—
71	C₄⋊C₄×C₄₃	C₂⁴	1	8	yes	—
72	(C₂×D₄₃)⋊C₄	—	2	32	no	—
73	C₄×Dic₄₃	C₂×D₄	1	12	yes	—
74	C₄×Dic₄₃	C₂⁴	1	4	yes	—
75	C₂×C₄₃⋊C₈	—	2	12	no	—
76	C₂×C₄₃⋊C₈	C₂⁴	1	4	yes	—
77	D₃₄₄	C₂²×C₄	1	64	yes	complemented
78	C₂³×C₄₃	C₄×D₄	1	4	yes	—
79	C₂³×C₄₃	AGL(1,ℤ/8)	1	8	yes	—
80	C₂³×C₄₃	C₂²×D₄	1	3	yes	—
81	C₂²×D₄₃	—	2	48	no	—
82	C₂²×D₄₃	—	2	8	no	—
83	C₂²×D₄₃	—	4	24	no	—
84	C₂²×D₄₃	—	4	24	no	—
85	C₂²×D₄₃	—	4	64	no	—
86	C₂×C₄×C₄₃	—	2	8	no	—
87	C₂×C₄×C₄₃	—	4	24	no	—
88	C₂×C₄×C₄₃	C₄×D₄	1	12	yes	—
89	C₂×C₄×C₄₃	C₂³×C₄	1	6	yes	—
90	C₂×C₄×C₄₃	C₂²×D₄	1	3	yes	—
91	C₂×C₄×C₄₃	C₂×#41	1	6	yes	—
92	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
93	C₂×Dic₄₃	—	2	24	no	—
94	C₂×Dic₄₃	C₂²×D₄	1	8	yes	—
95	D₁₇₂	—	2	48	no	—
96	D₁₇₂	—	2	8	no	—
97	D₁₇₂	C₂³×C₄	1	16	yes	complemented
98	C₄×D₄₃	—	4	64	no	—
99	C₈×C₄₃	C₂³×C₄	1	8	yes	complemented
100	C₄₃⋊C₈	—	2	4	no	—
101	C₂²×#442	—	43	1	no	maximal, Sylow
102	C₂²×C₄₃	#442	1	16	yes	complemented
103	C₂²×C₄₃	C₂×C₄×D₄	1	6	yes	—
104	C₂²×C₄₃	C₂×AGL(1,ℤ/8)	1	12	yes	—
105	C₂²×C₄₃	C₂³×D₄	1	1	yes	characteristic
106	C₂×D₄₃	—	2	16	no	—
107	C₂×D₄₃	—	4	96	no	—
108	C₂×D₄₃	—	4	8	no	—
109	C₂×D₄₃	—	4	8	no	—
110	C₄×C₄₃	—	4	8	no	—
111	C₄×C₄₃	C₂×C₄×D₄	1	6	yes	—
112	C₄×C₄₃	C₂⁴×C₄	1	1	yes	characteristic
113	C₄×C₄₃	C₂²×#41	1	1	yes	characteristic
114	Dic₄₃	—	2	8	no	—
115	C₂³×D₈	—	43	1	no	—
116	C₂²×C₄×D₄	—	43	2	no	—
117	C₂×#442	—	43	24	no	—
118	C₂²×#106	D₄₃	1	1	yes	characteristic, complemented
119	C₂²×#102	—	43	2	no	—
120	C₂²×#97	—	43	1	no	—
121	C₂×C₄₃	C₂×#442	1	12	yes	complemented
122	C₂×C₄₃	C₂²×C₄×D₄	1	1	yes	characteristic
123	C₂×C₄₃	C₂²×AGL(1,ℤ/8)	1	2	yes	—
124	D₄₃	—	4	16	no	—
125	C₂³×D₄	—	43	2	no	—
126	C₂⁴×C₄	—	86	2	no	—
127	C₂²×D₈	—	43	12	no	—
128	C₂²×D₈	—	43	16	no	—
129	C₂³×C₈	—	43	1	no	—
130	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
131	C₂×C₄×D₄	—	43	24	no	—
132	C₂×C₄×D₄	—	86	12	no	—
133	C₂²×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
134	C₂²×C₂²⋊C₄	—	86	2	no	—
135	C₂²×C₄²	—	43	1	no	—
136	#442	—	43	64	no	—
137	C₂×#106	C₂×D₄₃	1	12	yes	complemented
138	C₂×#102	—	43	24	no	—
139	C₂×#97	—	43	12	no	—
140	C₄₃	C₂²×#442	1	1	yes	characteristic, complemented, Sylow
141	C₂⁵	—	86	2	no	—
142	C₂²×D₄	—	43	16	no	—
143	C₂²×D₄	—	43	12	no	—
144	C₂²×D₄	—	86	6	no	—
145	C₂²×D₄	—	86	8	no	—
146	C₂³×C₄	—	2	2	no	—
147	C₂³×C₄	—	43	2	no	—
148	C₂³×C₄	—	86	24	no	—
149	C₂³×C₄	—	172	16	no	—
150	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
151	C₂×D₈	—	43	96	no	—
152	C₂×D₈	—	43	16	no	—
153	C₂²×C₈	—	43	6	no	—
154	C₂²×C₈	—	86	4	no	—
155	C₂²×C₈	C₄×D₄₃	1	8	yes	complemented
156	C₂²×C₈	C₂²×D₄₃	1	6	yes	—
157	C₄×D₄	—	43	32	no	—
158	C₄×D₄	—	86	48	no	—
159	C₂×C₄⋊C₄	—	2	6	no	—
160	C₂×C₄⋊C₄	C₂²×D₄₃	1	12	yes	—
161	C₂×C₂²⋊C₄	—	86	24	no	—
162	C₂×C₄²	—	43	6	no	—
163	C₂×C₄²	—	43	6	no	—
164	#106	C₂²×D₄₃	1	16	yes	complemented
165	#102	—	43	32	no	—
166	#97	—	43	16	no	—
167	C₂⁴	—	86	16	no	—
168	C₂⁴	—	86	12	no	—
169	C₂⁴	—	172	8	no	—
170	C₂⁴	—	172	8	no	—
171	C₂⁴	D₄×D₄₃	1	1	yes	characteristic
172	C₂×D₄	—	43	48	no	—
173	C₂×D₄	—	43	8	no	—
174	C₂×D₄	—	86	48	no	—
175	C₂×D₄	—	86	12	no	—
176	C₂×D₄	—	86	24	no	—
177	C₂²×C₄	—	2	12	no	—
178	C₂²×C₄	—	4	8	no	—
179	C₂²×C₄	—	43	12	no	—
180	C₂²×C₄	—	86	32	no	—
181	C₂²×C₄	—	86	8	no	—
182	C₂²×C₄	—	172	96	no	—
183	C₂²×C₄	C₂×C₄×D₄₃	1	4	yes	—
184	C₂²×C₄	D₄×D₄₃	1	3	yes	—
185	C₂²×C₄	#10621902	1	4	yes	—
186	C₂²×C₄	C₂³×D₄₃	1	3	yes	—
187	D₈	—	43	64	no	—
188	C₂×C₈	—	43	4	no	—
189	C₂×C₈	—	86	12	no	—
190	C₂×C₈	C₂×C₄×D₄₃	1	24	yes	complemented
191	C₂×C₈	C₂³×D₄₃	1	4	yes	—
192	C₄⋊C₄	—	2	12	no	—
193	C₄⋊C₄	C₂³×D₄₃	1	8	yes	—
194	C₂²⋊C₄	—	86	32	no	—
195	C₄²	—	43	12	no	—
196	C₄²	—	43	4	no	—
197	C₂³	—	86	48	no	—
198	C₂³	—	86	8	no	—
199	C₂³	—	172	24	no	—
200	C₂³	—	172	24	no	—
201	C₂³	—	172	64	no	—
202	C₂³	#13063287	1	4	yes	—
203	C₂³	#13063352	1	8	yes	—
204	C₂³	C₂×D₄×D₄₃	1	3	yes	—
205	D₄	—	43	16	no	—
206	D₄	—	86	48	no	—
207	D₄	—	86	8	no	—
208	C₂×C₄	—	2	8	no	—
209	C₂×C₄	—	4	24	no	—
210	C₂×C₄	—	43	8	no	—
211	C₂×C₄	—	86	24	no	—
212	C₂×C₄	—	172	64	no	—
213	C₂×C₄	#13063287	1	12	yes	—
214	C₂×C₄	C₂²×C₄×D₄₃	1	6	yes	—
215	C₂×C₄	C₂×D₄×D₄₃	1	3	yes	—
216	C₂×C₄	C₂×#10621902	1	6	yes	—
217	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
218	C₈	—	86	4	no	—
219	C₈	C₂²×C₄×D₄₃	1	8	yes	complemented
220	C₂²	—	86	16	no	—
221	C₂²	—	172	8	no	—
222	C₂²	—	172	8	no	—
223	C₂²	—	172	96	no	—
224	C₂²	#14583010	1	16	yes	direct product factor
225	C₂²	C₂×#13063287	1	6	yes	—
226	C₂²	C₂×#13063352	1	12	yes	—
227	C₂²	C₂²×D₄×D₄₃	1	1	yes	characteristic
228	C₄	—	4	8	no	—
229	C₄	—	86	8	no	—
230	C₄	C₂×#13063287	1	6	yes	—
231	C₄	C₂³×C₄×D₄₃	1	1	yes	characteristic
232	C₄	C₂²×#10621902	1	1	yes	characteristic
233	C₂	—	172	16	no	—
234	C₂	C₂×#14583010	1	12	yes	direct product factor
235	C₂	C₂²×#13063287	1	1	yes	characteristic
236	C₂	C₂²×#13063352	1	2	yes	—

	subgroup	G/H
center	C₂⁴	D₄×D₄₃
commutator	C₄×C₄₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₂²×#106×C₄₃	C₂
socle	C₂⁴×C₄₃	C₂×D₄
prime radical	C₂³×D₃₄₄	C₂
cosocle	C₂×C₄×C₄₃	C₂⁵
hypercenter	C₂²×#106	D₄₃
hypocenter	C₄₃	C₂²×#442
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#106

Complements

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

N	H ≅ G/N	type
2. C₂²×#13063287	C₂	semidirect
3. C₂×#14583010	C₂	direct (N × H)
4. C₂²×#13063245	C₂	semidirect
5. C₂²×#13063235	C₂	semidirect
6. C₂²×#106×C₄₃	C₂	semidirect
9. C₂²×D₃₄₄	C₄	semidirect
13. C₂×#13063287	C₂²	semidirect
16. #14583010	C₂²	direct (N × H)
17. C₂×#13063245	C₂²	semidirect
18. C₂×#13063235	C₂²	semidirect
21. C₂×#106×C₄₃	C₂²	semidirect
25. C₂²×D₁₇₂	C₂×C₄	semidirect
30. C₂×D₃₄₄	C₂×C₄	semidirect
35. #13063287	C₂³	semidirect
39. #13063245	C₂³	semidirect
40. #13063235	C₂³	semidirect
43. C₂²×C₈×C₄₃	C₂×C₄	semidirect
47. #106×C₄₃	C₂³	semidirect
64. C₂×D₁₇₂	C₂²×C₄	semidirect
68. C₂×C₈×C₄₃	C₂²×C₄	semidirect
77. D₃₄₄	C₂²×C₄	semidirect
97. D₁₇₂	C₂³×C₄	semidirect
99. C₈×C₄₃	C₂³×C₄	semidirect
102. C₂²×C₄₃	#442	semidirect
118. C₂²×#106	D₄₃	semidirect
121. C₂×C₄₃	C₂×#442	semidirect
137. C₂×#106	C₂×D₄₃	semidirect
140. C₄₃	C₂²×#442	semidirect
155. C₂²×C₈	C₄×D₄₃	semidirect
164. #106	C₂²×D₄₃	semidirect
190. C₂×C₈	C₂×C₄×D₄₃	semidirect
219. C₈	C₂²×C₄×D₄₃	semidirect
224. C₂²	#14583010	direct (N × H)
234. C₂	C₂×#14583010	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₃₄₄	2
C₂²×#13063287	2
C₂×#14583010	2
C₂²×#13063245	2
C₂²×#13063235	2
C₂²×#106×C₄₃	2
C₂²×#442	43

Subgroups — C22×#14583010 #25960641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#14583010 #25960641