Subgroups — C₂²×#90443 #14267608

Order 1792.

11462 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#6553	C₂	1	2	yes	maximal, complemented
2	C₂²×C₇×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#90443	C₂	1	48	yes	maximal, direct product factor
4	C₂²×#6471	C₂	1	6	yes	maximal, complemented
5	C₂²×#6467	C₂	1	3	yes	maximal, complemented
6	C₂³×D₅₆	C₂	1	3	yes	maximal, complemented
7	C₂²×C₇×#41	C₂²	1	1	yes	characteristic
8	C₂×#6553	—	2	24	no	—
9	C₂×#6553	C₂²	1	48	yes	complemented
10	C₂²×#1161	—	2	2	no	—
11	C₂²×D₄×D₇	—	2	6	no	—
12	C₂²×#1157	C₂²	1	6	yes	complemented
13	C₂³×D₂₈	C₂²	1	6	yes	complemented
14	C₂×C₇×#131	C₂²	1	24	yes	complemented
15	C₂²×C₇×M₁₆	C₂²	1	3	yes	complemented
16	C₂³×C₇×C₈	C₂²	1	3	yes	complemented
17	#90443	C₂²	1	256	yes	direct product factor
18	C₂×#6471	C₂²	1	144	yes	complemented
19	C₂²×Dic₂₈	C₂²	1	1	yes	characteristic, complemented
20	C₂×#6467	C₂²	1	72	yes	complemented
21	C₂²×D₅₆	C₂²	1	3	yes	complemented
22	C₂²×D₅₆	C₂²	1	6	yes	complemented
23	C₂²×D₅₆	C₂²	1	72	yes	complemented
24	C₂²×#1132	C₂²	1	6	yes	complemented
25	C₂²×#576	—	7	1	no	maximal, Sylow
26	C₂⁴×D₇	—	2	6	no	—
27	C₂×C₇×#41	D₄	1	12	yes	—
28	C₂×C₇×#41	C₂³	1	12	yes	—
29	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
30	C₂²×C₇×D₄	D₄	1	3	yes	—
31	C₂³×C₄×C₇	C₂³	1	3	yes	—
32	C₂²×C₂²⋊D₇	—	2	6	no	—
33	#6553	—	2	192	no	—
34	#6553	C₂³	1	128	yes	complemented
35	C₂×#1161	—	2	48	no	—
36	C₂×D₄×D₇	—	2	144	no	—
37	C₂×#1157	—	2	36	no	—
38	C₂×#1157	C₂³	1	72	yes	complemented
39	C₂²×D₂₈	—	2	6	no	—
40	C₂²×D₂₈	—	2	36	no	—
41	C₂²×D₂₈	C₂³	1	6	yes	—
42	C₂²×D₂₈	C₂³	1	72	yes	complemented
43	C₂²×C₄×D₇	—	2	6	no	—
44	C₂²×Dic₁₄	C₂³	1	2	yes	—
45	C₇×#131	C₂³	1	64	yes	complemented
46	C₂×C₇×M₁₆	C₂³	1	36	yes	complemented
47	C₂²×C₇×C₈	C₂³	1	3	yes	—
48	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
49	C₂²×C₇×C₈	C₂³	1	36	yes	complemented
50	#6471	C₂³	1	384	yes	complemented
51	C₂×Dic₂₈	C₂³	1	12	yes	complemented
52	#6467	C₂³	1	192	yes	complemented
53	C₂×D₅₆	C₂³	1	36	yes	complemented
54	C₂×D₅₆	C₂³	1	72	yes	complemented
55	C₂×D₅₆	C₂³	1	192	yes	complemented
56	C₂×#1132	C₂³	1	72	yes	complemented
57	C₂²×H(5,𝔽₂)	—	7	2	no	—
58	C₂×#576	—	7	48	no	—
59	C₂²×AGL(1,ℤ/8)	—	7	6	no	—
60	C₂²×#135	—	7	3	no	—
61	C₂³×D₈	—	7	3	no	—
62	C₂²×#131	D₇	1	1	yes	characteristic, complemented
63	C₂⁴×C₇	C₂×D₄	1	3	yes	—
64	C₂³×D₇	—	2	6	no	—
65	C₂³×D₇	—	2	72	no	—
66	C₂³×D₇	—	8	24	no	—
67	C₇×#41	C₂×D₄	1	48	yes	—
68	C₇×#41	C₂⁴	1	16	yes	—
69	C₂×C₇×Q₈	C₂×D₄	1	12	yes	—
70	C₂×C₇×D₄	C₂×D₄	1	36	yes	—
71	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
72	C₂²×C₄×C₇	C₂×D₄	1	18	yes	—
73	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
74	C₂²×C₄×C₇	C₂⁴	1	18	yes	—
75	C₂×C₂²⋊D₇	—	2	72	no	—
76	C₂²×Dic₇	—	2	2	no	—
77	#1161	—	2	128	no	—
78	D₄×D₇	—	2	384	no	—
79	#1157	—	2	144	no	—
80	#1157	C₂⁴	1	96	yes	complemented
81	C₂×D₂₈	—	2	18	no	—
82	C₂×D₂₈	—	2	72	no	—
83	C₂×D₂₈	—	2	144	no	—
84	C₂×D₂₈	C₂⁴	1	36	yes	—
85	C₂×D₂₈	C₂⁴	1	96	yes	complemented
86	C₂×C₄×D₇	—	2	72	no	—
87	C₂×Dic₁₄	—	2	6	no	—
88	C₂×Dic₁₄	C₂⁴	1	12	yes	—
89	C₇×M₁₆	C₂⁴	1	48	yes	complemented
90	C₂×C₇×C₈	C₂⁴	1	18	yes	—
91	C₂×C₇×C₈	C₂⁴	1	6	yes	—
92	C₂×C₇×C₈	C₂⁴	1	48	yes	complemented
93	Dic₂₈	C₂⁴	1	16	yes	complemented
94	D₅₆	C₂⁴	1	48	yes	complemented
95	D₅₆	C₂⁴	1	96	yes	complemented
96	#1132	C₂⁴	1	96	yes	complemented
97	C₂×H(5,𝔽₂)	—	7	48	no	—
98	C₂×H(5,𝔽₂)	—	14	24	no	—
99	C₂²×#41	—	7	6	no	—
100	C₂²×#41	—	14	2	no	—
101	C₂²×#41	C₂×D₇	1	1	yes	characteristic
102	C₂³×D₄	—	7	6	no	—
103	C₂³×D₄	—	14	6	no	—
104	#576	—	7	256	no	—
105	C₂×AGL(1,ℤ/8)	—	7	144	no	—
106	C₂×#135	—	7	72	no	—
107	C₂²×Q₁₆	—	7	1	no	—
108	C₂²×SD₁₆	—	7	6	no	—
109	C₂²×D₈	—	7	3	no	—
110	C₂²×D₈	—	7	6	no	—
111	C₂²×D₈	—	7	72	no	—
112	C₂×#131	C₂×D₇	1	24	yes	complemented
113	C₂²×M₁₆	C₂×D₇	1	3	yes	complemented
114	C₂³×C₈	C₂×D₇	1	3	yes	complemented
115	C₂³×C₇	—	2	12	no	—
116	C₂³×C₇	C₂²×D₄	1	18	yes	—
117	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
118	C₂²×D₇	—	2	36	no	—
119	C₂²×D₇	—	2	96	no	—
120	C₂²×D₇	—	4	24	no	—
121	C₂²×D₇	—	8	144	no	—
122	C₇×Q₈	C₂²×D₄	1	16	yes	—
123	C₇×D₄	C₂²×D₄	1	48	yes	—
124	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
125	C₂×C₄×C₇	C₂²×D₄	1	18	yes	—
126	C₂×C₄×C₇	C₂²×D₄	1	36	yes	—
127	C₂×C₄×C₇	C₂⁵	1	3	yes	—
128	C₂×C₄×C₇	C₂⁵	1	12	yes	—
129	C₂²⋊D₇	—	2	96	no	—
130	C₂×Dic₇	—	2	12	no	—
131	D₂₈	—	2	36	no	—
132	D₂₈	—	2	96	no	—
133	D₂₈	C₂⁵	1	24	yes	—
134	C₄×D₇	—	2	96	no	—
135	Dic₁₄	—	2	12	no	—
136	Dic₁₄	C₂⁵	1	8	yes	—
137	C₇×C₈	C₂⁵	1	12	yes	—
138	C₇×C₈	C₂⁵	1	4	yes	—
139	C₂⁵	—	14	6	no	—
140	H(5,𝔽₂)	—	7	128	no	—
141	H(5,𝔽₂)	—	14	192	no	—
142	C₂×#41	—	7	72	no	—
143	C₂×#41	—	14	48	no	—
144	C₂×#41	—	14	36	no	—
145	C₂×#41	D₂₈	1	12	yes	—
146	C₂×#41	C₂²×D₇	1	12	yes	—
147	C₂²×Q₈	—	7	2	no	—
148	C₂²×Q₈	D₂₈	1	1	yes	characteristic
149	C₂²×D₄	—	7	6	no	—
150	C₂²×D₄	—	7	72	no	—
151	C₂²×D₄	—	14	6	no	—
152	C₂²×D₄	—	14	144	no	—
153	C₂²×D₄	—	14	6	no	—
154	C₂²×D₄	—	14	36	no	—
155	C₂²×D₄	D₂₈	1	3	yes	—
156	C₂³×C₄	—	14	6	no	—
157	C₂³×C₄	C₂²×D₇	1	3	yes	—
158	AGL(1,ℤ/8)	—	7	384	no	—
159	#135	—	7	192	no	—
160	C₂×Q₁₆	—	7	12	no	—
161	C₂×SD₁₆	—	7	72	no	—
162	C₂×D₈	—	7	36	no	—
163	C₂×D₈	—	7	72	no	—
164	C₂×D₈	—	7	192	no	—
165	#131	C₂²×D₇	1	64	yes	complemented
166	C₂×M₁₆	C₂²×D₇	1	36	yes	complemented
167	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₇	1	3	yes	—
169	C₂²×C₈	C₂²×D₇	1	36	yes	complemented
170	C₂²×C₇	—	2	36	no	—
171	C₂²×C₇	#576	1	4	yes	complemented
172	C₂²×C₇	C₂³×D₄	1	12	yes	—
173	C₂²×C₇	C₂³×D₄	1	3	yes	—
174	C₂×D₇	—	2	24	no	—
175	C₂×D₇	—	4	72	no	—
176	C₂×D₇	—	8	96	no	—
177	C₄×C₇	C₂³×D₄	1	3	yes	—
178	C₄×C₇	C₂³×D₄	1	12	yes	—
179	C₄×C₇	C₂⁶	1	1	yes	characteristic
180	Dic₇	—	2	8	no	—
181	C₂⁴	—	14	6	no	—
182	C₂⁴	—	14	72	no	—
183	C₂⁴	—	56	24	no	—
184	C₂⁴	C₂×D₂₈	1	3	yes	—
185	#41	—	7	96	no	—
186	#41	—	14	128	no	—
187	#41	—	14	144	no	—
188	#41	C₂×D₂₈	1	48	yes	—
189	#41	C₂³×D₇	1	16	yes	—
190	C₂×Q₈	—	7	12	no	—
191	C₂×Q₈	—	14	6	no	—
192	C₂×Q₈	C₂×D₂₈	1	12	yes	—
193	C₂×D₄	—	7	36	no	—
194	C₂×D₄	—	7	96	no	—
195	C₂×D₄	—	14	72	no	—
196	C₂×D₄	—	14	384	no	—
197	C₂×D₄	—	14	72	no	—
198	C₂×D₄	—	14	18	no	—
199	C₂×D₄	—	14	144	no	—
200	C₂×D₄	C₂×D₂₈	1	36	yes	—
201	C₂²×C₄	—	14	2	no	—
202	C₂²×C₄	—	14	72	no	—
203	C₂²×C₄	C₂×D₂₈	1	18	yes	—
204	C₂²×C₄	C₂×D₂₈	1	3	yes	—
205	C₂²×C₄	C₂³×D₇	1	18	yes	—
206	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
207	Q₁₆	—	7	16	no	—
208	SD₁₆	—	7	96	no	—
209	D₈	—	7	48	no	—
210	D₈	—	7	96	no	—
211	M₁₆	C₂³×D₇	1	48	yes	complemented
212	C₂×C₈	C₂³×D₇	1	6	yes	—
213	C₂×C₈	C₂³×D₇	1	18	yes	—
214	C₂×C₈	C₂³×D₇	1	48	yes	complemented
215	C₂×C₇	—	2	12	no	—
216	C₂×C₇	C₂×#576	1	6	yes	complemented
217	C₂×C₇	C₂⁴×D₄	1	1	yes	characteristic
218	D₇	—	4	24	no	—
219	C₂³	—	2	12	no	—
220	C₂³	—	14	36	no	—
221	C₂³	—	14	96	no	—
222	C₂³	—	28	24	no	—
223	C₂³	—	56	144	no	—
224	C₂³	C₂²×D₂₈	1	18	yes	—
225	C₂³	C₂²×D₂₈	1	1	yes	characteristic
226	Q₈	—	7	8	no	—
227	Q₈	—	14	12	no	—
228	Q₈	C₂²×D₂₈	1	16	yes	—
229	D₄	—	7	24	no	—
230	D₄	—	14	96	no	—
231	D₄	—	14	96	no	—
232	D₄	—	14	36	no	—
233	D₄	C₂²×D₂₈	1	48	yes	—
234	C₂×C₄	—	14	12	no	—
235	C₂×C₄	—	14	96	no	—
236	C₂×C₄	C₂²×D₂₈	1	18	yes	—
237	C₂×C₄	C₂²×D₂₈	1	3	yes	—
238	C₂×C₄	C₂²×D₂₈	1	36	yes	—
239	C₂×C₄	C₂⁴×D₇	1	3	yes	—
240	C₂×C₄	C₂⁴×D₇	1	12	yes	—
241	C₈	C₂⁴×D₇	1	4	yes	—
242	C₈	C₂⁴×D₇	1	12	yes	—
243	C₇	C₂²×#576	1	1	yes	characteristic, complemented, Sylow
244	C₂²	—	2	36	no	—
245	C₂²	—	14	24	no	—
246	C₂²	—	28	72	no	—
247	C₂²	—	56	96	no	—
248	C₂²	#90443	1	4	yes	direct product factor
249	C₂²	C₂³×D₂₈	1	12	yes	—
250	C₂²	C₂³×D₂₈	1	3	yes	—
251	C₄	—	14	8	no	—
252	C₄	C₂³×D₂₈	1	12	yes	—
253	C₄	C₂³×D₂₈	1	3	yes	—
254	C₄	C₂⁵×D₇	1	1	yes	characteristic
255	C₂	—	2	12	no	—
256	C₂	—	28	24	no	—
257	C₂	C₂×#90443	1	6	yes	direct product factor
258	C₂	C₂⁴×D₂₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₂₈
commutator, cosocle	C₄×C₇	C₂⁶
Frattini	C₄	C₂⁵×D₇
Fitting	C₂²×C₇×#131	C₂
socle	C₂³×C₇	C₂²×D₄
hypercenter	C₂²×#131	D₇
hypocenter	C₇	C₂²×#576
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131

Complements

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

N	H ≅ G/N	type
1. C₂²×#6553	C₂	semidirect
2. C₂²×C₇×#131	C₂	semidirect
3. C₂×#90443	C₂	direct (N × H)
4. C₂²×#6471	C₂	semidirect
5. C₂²×#6467	C₂	semidirect
6. C₂³×D₅₆	C₂	semidirect
9. C₂×#6553	C₂²	semidirect
12. C₂²×#1157	C₂²	semidirect
13. C₂³×D₂₈	C₂²	semidirect
14. C₂×C₇×#131	C₂²	semidirect
15. C₂²×C₇×M₁₆	C₂²	semidirect
16. C₂³×C₇×C₈	C₂²	semidirect
17. #90443	C₂²	direct (N × H)
18. C₂×#6471	C₂²	semidirect
19. C₂²×Dic₂₈	C₂²	semidirect
20. C₂×#6467	C₂²	semidirect
21. C₂²×D₅₆	C₂²	semidirect
22. C₂²×D₅₆	C₂²	semidirect
23. C₂²×D₅₆	C₂²	semidirect
24. C₂²×#1132	C₂²	semidirect
34. #6553	C₂³	semidirect
38. C₂×#1157	C₂³	semidirect
42. C₂²×D₂₈	C₂³	semidirect
45. C₇×#131	C₂³	semidirect
46. C₂×C₇×M₁₆	C₂³	semidirect
49. C₂²×C₇×C₈	C₂³	semidirect
50. #6471	C₂³	semidirect
51. C₂×Dic₂₈	C₂³	semidirect
52. #6467	C₂³	semidirect
53. C₂×D₅₆	C₂³	semidirect
54. C₂×D₅₆	C₂³	semidirect
55. C₂×D₅₆	C₂³	semidirect
56. C₂×#1132	C₂³	semidirect
62. C₂²×#131	D₇	semidirect
80. #1157	C₂⁴	semidirect
85. C₂×D₂₈	C₂⁴	semidirect
89. C₇×M₁₆	C₂⁴	semidirect
92. C₂×C₇×C₈	C₂⁴	semidirect
93. Dic₂₈	C₂⁴	semidirect
94. D₅₆	C₂⁴	semidirect
95. D₅₆	C₂⁴	semidirect
96. #1132	C₂⁴	semidirect
112. C₂×#131	C₂×D₇	semidirect
113. C₂²×M₁₆	C₂×D₇	semidirect
114. C₂³×C₈	C₂×D₇	semidirect
165. #131	C₂²×D₇	semidirect
166. C₂×M₁₆	C₂²×D₇	semidirect
169. C₂²×C₈	C₂²×D₇	semidirect
171. C₂²×C₇	#576	semidirect
211. M₁₆	C₂³×D₇	semidirect
214. C₂×C₈	C₂³×D₇	semidirect
216. C₂×C₇	C₂×#576	semidirect
243. C₇	C₂²×#576	semidirect
248. C₂²	#90443	direct (N × H)
257. C₂	C₂×#90443	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×#6553	2
C₂²×C₇×#131	2
C₂×#90443	2
C₂²×#6471	2
C₂²×#6467	2
C₂³×D₅₆	2
C₂²×#576	7

Subgroups — C22×#90443 #14267608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#90443 #14267608