Subgroups — C₂×#7641846 #72255587

Order 1024.

161717 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#7641846	C₂	1	64	yes	maximal, direct product factor
2	C₂×#37475	C₂	1	24	yes	maximal, complemented
3	C₂×#34982	C₂	1	36	yes	maximal, complemented
4	C₂³×#379	C₂	1	3	yes	maximal
5	C₂⁸	C₂²	1	1	yes	characteristic
6	C₂²×#543	C₂²	1	12	yes	complemented
7	C₂⁴×C₂²⋊C₄	C₂²	1	18	yes	—
8	#37475	C₂²	1	768	yes	complemented
9	C₂×#2842	C₂²	1	24	yes	complemented
10	C₂×#2840	C₂²	1	8	yes	complemented
11	C₂×#2848	C₂²	1	48	yes	complemented
12	#34982	C₂²	1	1152	yes	complemented
13	C₂×#2437	C₂²	1	288	yes	complemented
14	C₂×#2626	C₂²	1	144	yes	complemented
15	C₂×#2315	C₂²	1	36	yes	complemented
16	C₂²×#379	C₂²	1	36	yes	—
17	C₂²×#379	C₂²	1	36	yes	—
18	C₂²×#379	C₂²	1	96	yes	—
19	C₂⁷	—	4	48	no	—
20	C₂⁷	Q₈	1	32	yes	complemented
21	C₂⁷	C₂³	1	9	yes	—
22	C₂⁷	C₂³	1	6	yes	—
23	C₂⁷	C₂³	1	16	yes	—
24	C₂⁵×C₄	C₂³	1	12	yes	—
25	C₂×#543	—	4	96	no	—
26	C₂×#543	C₂³	1	192	yes	complemented
27	C₂²×C₂²⋊Q₈	C₂³	1	144	yes	complemented
28	C₂³×C₂²⋊C₄	—	4	144	no	—
29	C₂³×C₂²⋊C₄	C₂³	1	36	yes	—
30	C₂³×C₂²⋊C₄	C₂³	1	144	yes	—
31	C₂³×C₂²⋊C₄	C₂³	1	288	yes	—
32	C₂³×C₂²⋊C₄	C₂³	1	36	yes	—
33	#2848	C₂³	1	768	yes	complemented
34	#2842	C₂³	1	384	yes	complemented
35	#2840	C₂³	1	128	yes	complemented
36	#2626	C₂³	1	2304	yes	complemented
37	#2437	C₂³	1	4608	yes	complemented
38	C₂×#399	C₂³	1	96	yes	complemented
39	C₂×#396	C₂³	1	192	yes	complemented
40	C₂×#393	C₂³	1	96	yes	complemented
41	#2315	C₂³	1	576	yes	complemented
42	C₂×#385	C₂³	1	288	yes	complemented
43	C₂×#380	C₂³	1	144	yes	complemented
44	C₂×#379	C₂³	1	24	yes	—
45	C₂×#379	C₂³	1	24	yes	—
46	C₂×#379	C₂³	1	72	yes	—
47	C₂×#379	C₂³	1	576	yes	—
48	C₂×#379	C₂³	1	576	yes	—
49	C₂×#379	C₂³	1	72	yes	—
50	C₂⁶	—	2	144	no	—
51	C₂⁶	—	4	768	no	—
52	C₂⁶	—	4	144	no	—
53	C₂⁶	—	4	144	no	—
54	C₂⁶	—	4	384	no	—
55	C₂⁶	—	4	1024	no	—
56	C₂⁶	C₂×Q₈	1	144	yes	—
57	C₂⁶	C₂×Q₈	1	96	yes	—
58	C₂⁶	C₂×Q₈	1	256	yes	complemented
59	C₂⁶	C₂⁴	1	3	yes	—
60	C₂⁶	C₂⁴	1	18	yes	—
61	C₂⁶	C₂⁴	1	9	yes	—
62	C₂⁶	C₂⁴	1	72	yes	—
63	C₂⁶	C₂⁴	1	2	yes	—
64	C₂⁶	C₂⁴	1	48	yes	—
65	C₂⁶	C₂⁴	1	3	yes	—
66	C₂³×Q₈	C₂⁴	1	16	yes	complemented
67	C₂⁴×C₄	—	4	144	no	—
68	C₂⁴×C₄	C₂⁴	1	72	yes	—
69	C₂⁴×C₄	C₂⁴	1	96	yes	—
70	#543	—	4	768	no	—
71	C₂×C₂²⋊Q₈	—	4	576	no	—
72	C₂×C₂²⋊Q₈	C₂⁴	1	1152	yes	complemented
73	C₂²×C₄⋊C₄	C₂⁴	1	144	yes	complemented
74	C₂²×C₂²⋊C₄	—	2	288	no	—
75	C₂²×C₂²⋊C₄	—	4	576	no	—
76	C₂²×C₂²⋊C₄	—	4	1152	no	—
77	C₂²×C₂²⋊C₄	—	4	144	no	—
78	C₂²×C₂²⋊C₄	C₂⁴	1	288	yes	—
79	C₂²×C₂²⋊C₄	C₂⁴	1	144	yes	—
80	C₂²×C₂²⋊C₄	C₂⁴	1	288	yes	—
81	C₂²×C₂²⋊C₄	C₂⁴	1	144	yes	—
82	C₂²×C₂²⋊C₄	C₂⁴	1	1152	yes	—
83	#399	C₂⁴	1	768	yes	complemented
84	#396	C₂⁴	1	1536	yes	complemented
85	#393	C₂⁴	1	768	yes	complemented
86	#385	C₂⁴	1	2304	yes	complemented
87	#380	C₂⁴	1	1152	yes	complemented
88	#379	C₂⁴	1	192	yes	—
89	#379	C₂⁴	1	192	yes	—
90	#379	C₂⁴	1	576	yes	—
91	#379	C₂⁴	1	576	yes	—
92	C₂×#95	C₂⁴	1	96	yes	complemented
93	C₂⁵	—	2	72	no	—
94	C₂⁵	—	2	1152	no	—
95	C₂⁵	—	2	288	no	—
96	C₂⁵	—	2	576	no	—
97	C₂⁵	—	2	72	no	—
98	C₂⁵	—	4	1152	no	—
99	C₂⁵	—	4	1152	no	—
100	C₂⁵	—	4	3072	no	—
101	C₂⁵	—	4	24	no	—
102	C₂⁵	—	4	72	no	—
103	C₂⁵	—	4	576	no	—
104	C₂⁵	—	4	72	no	—
105	C₂⁵	—	4	576	no	—
106	C₂⁵	—	4	24	no	—
107	C₂⁵	—	4	2304	no	—
108	C₂⁵	—	4	1536	no	—
109	C₂⁵	—	4	8192	no	—
110	C₂⁵	—	4	4096	no	—
111	C₂⁵	C₂²×Q₈	1	24	yes	—
112	C₂⁵	C₂²×Q₈	1	144	yes	—
113	C₂⁵	C₂²×Q₈	1	72	yes	—
114	C₂⁵	C₂²×Q₈	1	576	yes	—
115	C₂⁵	C₂²×Q₈	1	16	yes	—
116	C₂⁵	C₂²×Q₈	1	384	yes	—
117	C₂⁵	C₂²×Q₈	1	24	yes	—
118	C₂⁵	H(5,𝔽₂)	1	48	yes	—
119	C₂⁵	C₂⁵	1	12	yes	—
120	C₂⁵	C₂⁵	1	6	yes	—
121	C₂⁵	C₂⁵	1	72	yes	—
122	C₂⁵	C₂⁵	1	9	yes	—
123	C₂⁵	C₂⁵	1	36	yes	—
124	C₂⁵	C₂⁵	1	8	yes	—
125	C₂⁵	C₂⁵	1	12	yes	—
126	C₂²×Q₈	—	4	96	no	—
127	C₂²×Q₈	C₂⁵	1	64	yes	complemented
128	C₂³×C₄	—	2	288	no	—
129	C₂³×C₄	—	4	288	no	—
130	C₂³×C₄	—	4	576	no	—
131	C₂³×C₄	—	8	384	no	—
132	C₂³×C₄	C₂⁵	1	48	yes	—
133	C₂³×C₄	C₂⁵	1	288	yes	—
134	C₂²⋊Q₈	—	4	2304	no	—
135	C₂×C₄⋊C₄	—	4	288	no	—
136	C₂×C₄⋊C₄	C₂⁵	1	576	yes	complemented
137	C₂×C₂²⋊C₄	—	2	576	no	—
138	C₂×C₂²⋊C₄	—	2	1152	no	—
139	C₂×C₂²⋊C₄	—	4	2304	no	—
140	C₂×C₂²⋊C₄	—	4	288	no	—
141	C₂×C₂²⋊C₄	—	4	576	no	—
142	C₂×C₂²⋊C₄	C₂⁵	1	576	yes	—
143	C₂×C₂²⋊C₄	C₂⁵	1	576	yes	—
144	#95	C₂⁵	1	384	yes	complemented
145	C₂⁴	—	2	384	no	—
146	C₂⁴	—	2	144	no	—
147	C₂⁴	—	2	72	no	—
148	C₂⁴	—	2	288	no	—
149	C₂⁴	—	2	576	no	—
150	C₂⁴	—	2	144	no	—
151	C₂⁴	—	2	72	no	—
152	C₂⁴	—	2	1152	no	—
153	C₂⁴	—	2	288	no	—
154	C₂⁴	—	2	2304	no	—
155	C₂⁴	—	4	96	no	—
156	C₂⁴	—	4	288	no	—
157	C₂⁴	—	4	288	no	—
158	C₂⁴	—	4	2304	no	—
159	C₂⁴	—	4	96	no	—
160	C₂⁴	—	4	2304	no	—
161	C₂⁴	—	4	192	no	—
162	C₂⁴	—	4	48	no	—
163	C₂⁴	—	4	144	no	—
164	C₂⁴	—	4	144	no	—
165	C₂⁴	—	4	48	no	—
166	C₂⁴	—	4	1152	no	—
167	C₂⁴	—	4	4608	no	—
168	C₂⁴	—	4	9216	no	—
169	C₂⁴	—	4	576	no	—
170	C₂⁴	—	4	6144	no	—
171	C₂⁴	—	4	3072	no	—
172	C₂⁴	—	4	16384	no	—
173	C₂⁴	—	4	128	no	—
174	C₂⁴	#543	1	192	yes	complemented
175	C₂⁴	C₂³×Q₈	1	48	yes	—
176	C₂⁴	C₂³×Q₈	1	24	yes	—
177	C₂⁴	C₂³×Q₈	1	288	yes	—
178	C₂⁴	C₂³×Q₈	1	36	yes	—
179	C₂⁴	C₂³×Q₈	1	144	yes	—
180	C₂⁴	C₂³×Q₈	1	32	yes	—
181	C₂⁴	C₂³×Q₈	1	48	yes	—
182	C₂⁴	C₂×H(5,𝔽₂)	1	72	yes	—
183	C₂⁴	C₂×H(5,𝔽₂)	1	96	yes	—
184	C₂⁴	C₂⁶	1	12	yes	—
185	C₂⁴	C₂⁶	1	18	yes	—
186	C₂⁴	C₂⁶	1	1	yes	characteristic
187	C₂×Q₈	—	4	192	no	—
188	C₂×Q₈	—	16	64	no	—
189	C₂²×C₄	—	2	288	no	—
190	C₂²×C₄	—	2	576	no	—
191	C₂²×C₄	—	4	576	no	—
192	C₂²×C₄	—	8	576	no	—
193	C₂²×C₄	—	8	768	no	—
194	C₂²×C₄	C₂⁶	1	96	yes	—
195	C₄⋊C₄	—	4	576	no	—
196	C₂²⋊C₄	—	2	1152	no	—
197	C₂²⋊C₄	—	4	576	no	—
198	C₂³	—	2	384	no	—
199	C₂³	—	2	768	no	—
200	C₂³	—	2	288	no	—
201	C₂³	—	2	72	no	—
202	C₂³	—	2	288	no	—
203	C₂³	—	2	144	no	—
204	C₂³	—	2	72	no	—
205	C₂³	—	2	1152	no	—
206	C₂³	—	2	288	no	—
207	C₂³	—	2	144	no	—
208	C₂³	—	4	384	no	—
209	C₂³	—	4	96	no	—
210	C₂³	—	4	288	no	—
211	C₂³	—	4	288	no	—
212	C₂³	—	4	96	no	—
213	C₂³	—	4	192	no	—
214	C₂³	—	4	1152	no	—
215	C₂³	—	4	2304	no	—
216	C₂³	—	4	9216	no	—
217	C₂³	—	4	576	no	—
218	C₂³	—	4	1152	no	—
219	C₂³	—	4	6144	no	—
220	C₂³	—	4	128	no	—
221	C₂³	—	4	256	no	—
222	C₂³	—	4	96	no	—
223	C₂³	C₂×#543	1	144	yes	—
224	C₂³	C₂×#543	1	192	yes	complemented
225	C₂³	C₂⁴×Q₈	1	24	yes	—
226	C₂³	C₂⁴×Q₈	1	36	yes	—
227	C₂³	C₂⁴×Q₈	1	2	yes	—
228	C₂³	C₂²×H(5,𝔽₂)	1	72	yes	—
229	C₂³	C₂²×H(5,𝔽₂)	1	12	yes	—
230	C₂³	C₂⁷	1	1	yes	characteristic
231	Q₈	—	16	64	no	—
232	C₂×C₄	—	2	288	no	—
233	C₂×C₄	—	4	192	no	—
234	C₂×C₄	—	8	576	no	—
235	C₂²	—	2	384	no	—
236	C₂²	—	2	72	no	—
237	C₂²	—	2	144	no	—
238	C₂²	—	2	288	no	—
239	C₂²	—	2	72	no	—
240	C₂²	—	4	192	no	—
241	C₂²	—	4	1152	no	—
242	C₂²	—	4	576	no	—
243	C₂²	—	4	128	no	—
244	C₂²	—	4	48	no	—
245	C₂²	—	4	96	no	—
246	C₂²	C₂²×#543	1	72	yes	—
247	C₂²	C₂²×#543	1	12	yes	—
248	C₂²	#62817	1	16	yes	—
249	C₂²	C₂⁵×Q₈	1	1	yes	characteristic
250	C₂²	C₂³×H(5,𝔽₂)	1	6	yes	—
251	C₄	—	4	96	no	—
252	C₂	—	2	72	no	—
253	C₂	—	4	24	no	—
254	C₂	#7641846	1	8	yes	direct product factor
255	C₂	C₂³×#543	1	3	yes	—
256	C₂	C₂×#62817	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂⁸	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #7641846	C₂	direct (N × H)
2. C₂×#37475	C₂	semidirect
3. C₂×#34982	C₂	semidirect
6. C₂²×#543	C₂²	semidirect
8. #37475	C₂²	semidirect
9. C₂×#2842	C₂²	semidirect
10. C₂×#2840	C₂²	semidirect
11. C₂×#2848	C₂²	semidirect
12. #34982	C₂²	semidirect
13. C₂×#2437	C₂²	semidirect
14. C₂×#2626	C₂²	semidirect
15. C₂×#2315	C₂²	semidirect
20. C₂⁷	Q₈	semidirect
26. C₂×#543	C₂³	semidirect
27. C₂²×C₂²⋊Q₈	C₂³	semidirect
33. #2848	C₂³	semidirect
34. #2842	C₂³	semidirect
35. #2840	C₂³	semidirect
36. #2626	C₂³	semidirect
37. #2437	C₂³	semidirect
38. C₂×#399	C₂³	semidirect
39. C₂×#396	C₂³	semidirect
40. C₂×#393	C₂³	semidirect
41. #2315	C₂³	semidirect
42. C₂×#385	C₂³	semidirect
43. C₂×#380	C₂³	semidirect
58. C₂⁶	C₂×Q₈	semidirect
66. C₂³×Q₈	C₂⁴	semidirect
72. C₂×C₂²⋊Q₈	C₂⁴	semidirect
73. C₂²×C₄⋊C₄	C₂⁴	semidirect
83. #399	C₂⁴	semidirect
84. #396	C₂⁴	semidirect
85. #393	C₂⁴	semidirect
86. #385	C₂⁴	semidirect
87. #380	C₂⁴	semidirect
92. C₂×#95	C₂⁴	semidirect
127. C₂²×Q₈	C₂⁵	semidirect
136. C₂×C₄⋊C₄	C₂⁵	semidirect
144. #95	C₂⁵	semidirect
174. C₂⁴	#543	semidirect
224. C₂³	C₂×#543	semidirect
254. C₂	#7641846	direct (N × H)

Maximal subgroups

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

subgroup	index
#7641846	2
C₂×#37475	2
C₂×#34982	2
C₂³×#379	2

Subgroups — C2×#7641846 #72255587

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#7641846 #72255587