Subgroups — C₂×#22550688 #30653491

Order 16256.

1460 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861256	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16861255	C₂	1	1	yes	maximal, characteristic, complemented
3	#22550688	C₂	1	16	yes	maximal, direct product factor
4	C₂×#16861242	C₂	1	3	yes	maximal, complemented
5	C₂×#16861237	C₂	1	3	yes	maximal, complemented
6	C₂×#16861231	C₂	1	3	yes	maximal, complemented
7	C₂²×#14523237	C₂	1	3	yes	maximal, complemented
8	C₂×#131×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#16861256	—	2	4	no	—
10	#16861256	C₂²	1	8	yes	complemented
11	#16861255	—	2	4	no	—
12	#16861255	C₂²	1	8	yes	complemented
13	C₂×#14523249	—	2	1	no	—
14	C₂×#14523248	C₂²	1	3	yes	—
15	C₂×#14523248	C₂²	1	3	yes	complemented
16	C₂×#14523247	—	2	3	no	—
17	C₂²×Dic₂₅₄	C₂²	1	3	yes	complemented
18	C₂×Q₈×D₁₂₇	—	2	1	no	—
19	C₂²×D₅₀₈	C₂²	1	3	yes	—
20	C₂×D₄×D₁₂₇	—	2	3	no	—
21	#16861242	C₂²	1	24	yes	complemented
22	#16861237	C₂²	1	24	yes	complemented
23	#16861231	C₂²	1	24	yes	complemented
24	C₂×Dic₅₀₈	C₂²	1	3	yes	complemented
25	C₂×#14523237	C₂²	1	3	yes	complemented
26	C₂×#14523237	C₂²	1	6	yes	complemented
27	C₂×#14523237	C₂²	1	1	yes	characteristic, complemented
28	C₂×#14523237	C₂²	1	24	yes	complemented
29	C₂×D₁₀₁₆	C₂²	1	3	yes	complemented
30	C₂×#41×C₁₂₇	C₂²	1	1	yes	characteristic
31	#131×C₁₂₇	C₂²	1	8	yes	complemented
32	C₂×M₁₆×C₁₂₇	C₂²	1	3	yes	complemented
33	C₂²×C₈×C₁₂₇	C₂²	1	3	yes	complemented
34	C₂³×D₁₂₇	—	2	3	no	—
35	#14523249	—	2	8	no	—
36	#14523248	—	2	6	no	—
37	#14523248	—	2	6	no	—
38	#14523248	C₂³	1	12	yes	—
39	#14523248	C₂³	1	12	yes	complemented
40	#14523247	—	2	24	no	—
41	C₂×Dic₂₅₄	—	2	3	no	—
42	C₂×Dic₂₅₄	—	2	6	no	—
43	C₂×Dic₂₅₄	C₂³	1	3	yes	—
44	C₂×Dic₂₅₄	C₂³	1	1	yes	characteristic
45	C₂×Dic₂₅₄	C₂³	1	12	yes	complemented
46	Q₈×D₁₂₇	—	2	8	no	—
47	C₂×D₅₀₈	—	2	3	no	—
48	C₂×D₅₀₈	—	2	6	no	—
49	C₂×D₅₀₈	C₂³	1	1	yes	characteristic
50	C₂×D₅₀₈	C₂³	1	3	yes	—
51	C₂×D₅₀₈	C₂³	1	12	yes	—
52	C₂×C₂²⋊D₁₂₇	—	2	3	no	—
53	C₂×C₂²⋊D₁₂₇	—	2	3	no	—
54	D₄×D₁₂₇	—	2	24	no	—
55	C₂²×Dic₁₂₇	—	2	3	no	—
56	C₂×C₄×D₁₂₇	—	2	3	no	—
57	C₂×C₄×D₁₂₇	—	2	3	no	—
58	Dic₅₀₈	C₂³	1	12	yes	complemented
59	#14523237	C₂³	1	12	yes	complemented
60	#14523237	C₂³	1	24	yes	complemented
61	#14523237	C₂³	1	4	yes	complemented
62	D₁₀₁₆	C₂³	1	12	yes	complemented
63	#41×C₁₂₇	D₄	1	4	yes	—
64	#41×C₁₂₇	C₂³	1	4	yes	—
65	C₂×Q₈×C₁₂₇	D₄	1	1	yes	characteristic
66	C₂×D₄×C₁₂₇	D₄	1	3	yes	—
67	C₂²×C₄×C₁₂₇	C₂³	1	3	yes	—
68	M₁₆×C₁₂₇	C₂³	1	12	yes	complemented
69	C₂×C₈×C₁₂₇	C₂³	1	3	yes	—
70	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₂₇	C₂³	1	12	yes	complemented
72	C₂²×D₁₂₇	—	2	1	no	—
73	C₂²×D₁₂₇	—	2	3	no	—
74	C₂²×D₁₂₇	—	2	12	no	—
75	C₂²×D₁₂₇	—	8	6	no	—
76	Dic₂₅₄	—	2	12	no	—
77	Dic₂₅₄	—	2	3	no	—
78	Dic₂₅₄	—	2	1	no	—
79	Dic₂₅₄	C₂⁴	1	6	yes	—
80	Dic₂₅₄	C₂⁴	1	2	yes	—
81	D₅₀₈	—	2	12	no	—
82	D₅₀₈	—	2	3	no	—
83	D₅₀₈	—	2	1	no	—
84	D₅₀₈	C₂⁴	1	2	yes	—
85	D₅₀₈	C₂⁴	1	6	yes	—
86	C₂²⋊D₁₂₇	—	2	12	no	—
87	C₂²⋊D₁₂₇	—	2	12	no	—
88	C₂×Dic₁₂₇	—	2	1	no	—
89	C₂×Dic₁₂₇	—	2	3	no	—
90	C₂×Dic₁₂₇	—	2	12	no	—
91	C₄×D₁₂₇	—	2	12	no	—
92	C₄×D₁₂₇	—	2	12	no	—
93	C₂³×C₁₂₇	C₂×D₄	1	3	yes	—
94	Q₈×C₁₂₇	C₂×D₄	1	4	yes	—
95	D₄×C₁₂₇	C₂×D₄	1	12	yes	—
96	C₂×C₄×C₁₂₇	C₂×D₄	1	3	yes	—
97	C₂×C₄×C₁₂₇	C₂×D₄	1	6	yes	—
98	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
99	C₂×C₄×C₁₂₇	C₂⁴	1	6	yes	—
100	C₈×C₁₂₇	C₂⁴	1	2	yes	—
101	C₈×C₁₂₇	C₂⁴	1	6	yes	—
102	C₂²×C₁₂₇	—	2	6	no	—
103	C₂²×C₁₂₇	C₂²×D₄	1	6	yes	—
104	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
105	C₂×D₁₂₇	—	2	2	no	—
106	C₂×D₁₂₇	—	2	6	no	—
107	C₂×D₁₂₇	—	4	2	no	—
108	C₂×D₁₂₇	—	4	6	no	—
109	C₂×D₁₂₇	—	8	12	no	—
110	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
111	C₄×C₁₂₇	C₂²×D₄	1	6	yes	—
112	C₄×C₁₂₇	C₂⁵	1	1	yes	characteristic
113	Dic₁₂₇	—	2	2	no	—
114	Dic₁₂₇	—	2	6	no	—
115	C₂×C₁₂₇	—	2	6	no	—
116	C₂×C₁₂₇	#577	1	2	yes	complemented
117	C₂×C₁₂₇	C₂³×D₄	1	1	yes	characteristic
118	D₁₂₇	—	4	2	no	—
119	D₁₂₇	—	4	6	no	—
120	C₂×#577	—	127	1	no	maximal, Sylow
121	C₁₂₇	C₂×#577	1	1	yes	characteristic, complemented, Sylow
122	C₂×2₋¹⁺⁴	—	127	1	no	—
123	C₂×H(5,𝔽₂)	—	127	1	no	—
124	#577	—	127	16	no	—
125	C₂×#137	—	127	3	no	—
126	C₂×AGL(1,ℤ/8)	—	127	3	no	—
127	C₂×#135	—	127	3	no	—
128	C₂²×SD₁₆	—	127	3	no	—
129	C₂×#131	D₁₂₇	1	1	yes	characteristic, complemented
130	2₋¹⁺⁴	—	127	8	no	—
131	2₋¹⁺⁴	—	254	4	no	—
132	H(5,𝔽₂)	—	127	8	no	—
133	H(5,𝔽₂)	—	254	4	no	—
134	C₂×#41	—	127	3	no	—
135	C₂×#41	—	127	3	no	—
136	C₂×#41	—	254	1	no	—
137	C₂×#41	—	254	3	no	—
138	C₂×#41	C₂×D₁₂₇	1	1	yes	characteristic
139	C₂²×Q₈	—	127	3	no	—
140	C₂²×Q₈	—	254	1	no	—
141	C₂²×D₄	—	127	3	no	—
142	C₂²×D₄	—	254	3	no	—
143	#137	—	127	24	no	—
144	AGL(1,ℤ/8)	—	127	24	no	—
145	#135	—	127	24	no	—
146	C₂×Q₁₆	—	127	3	no	—
147	C₂×SD₁₆	—	127	3	no	—
148	C₂×SD₁₆	—	127	6	no	—
149	C₂×SD₁₆	—	127	1	no	—
150	C₂×SD₁₆	—	127	24	no	—
151	C₂×D₈	—	127	3	no	—
152	#131	C₂×D₁₂₇	1	8	yes	complemented
153	C₂×M₁₆	C₂×D₁₂₇	1	3	yes	complemented
154	C₂²×C₈	C₂×D₁₂₇	1	3	yes	complemented
155	C₂⁴	—	254	3	no	—
156	#41	—	127	12	no	—
157	#41	—	127	12	no	—
158	#41	—	254	8	no	—
159	#41	—	254	24	no	—
160	#41	—	254	6	no	—
161	#41	—	254	6	no	—
162	#41	D₅₀₈	1	4	yes	—
163	#41	C₂²×D₁₂₇	1	4	yes	—
164	C₂×Q₈	—	127	3	no	—
165	C₂×Q₈	—	127	1	no	—
166	C₂×Q₈	—	127	12	no	—
167	C₂×Q₈	—	254	8	no	—
168	C₂×Q₈	—	254	3	no	—
169	C₂×Q₈	—	254	6	no	—
170	C₂×Q₈	D₅₀₈	1	1	yes	characteristic
171	C₂×D₄	—	127	1	no	—
172	C₂×D₄	—	127	3	no	—
173	C₂×D₄	—	127	12	no	—
174	C₂×D₄	—	254	3	no	—
175	C₂×D₄	—	254	3	no	—
176	C₂×D₄	—	254	24	no	—
177	C₂×D₄	—	254	3	no	—
178	C₂×D₄	—	254	6	no	—
179	C₂×D₄	D₅₀₈	1	3	yes	—
180	C₂²×C₄	—	254	3	no	—
181	C₂²×C₄	—	254	3	no	—
182	C₂²×C₄	—	254	3	no	—
183	C₂²×C₄	C₂²×D₁₂₇	1	3	yes	—
184	Q₁₆	—	127	12	no	—
185	SD₁₆	—	127	12	no	—
186	SD₁₆	—	127	24	no	—
187	SD₁₆	—	127	4	no	—
188	D₈	—	127	12	no	—
189	M₁₆	C₂²×D₁₂₇	1	12	yes	complemented
190	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
191	C₂×C₈	C₂²×D₁₂₇	1	3	yes	—
192	C₂×C₈	C₂²×D₁₂₇	1	12	yes	complemented
193	C₂³	—	1016	6	no	—
194	C₂³	—	254	12	no	—
195	C₂³	—	254	1	no	—
196	C₂³	—	254	3	no	—
197	C₂³	C₂×D₅₀₈	1	3	yes	—
198	Q₈	—	127	6	no	—
199	Q₈	—	127	2	no	—
200	Q₈	—	254	12	no	—
201	Q₈	—	254	3	no	—
202	Q₈	—	254	1	no	—
203	Q₈	C₂×D₅₀₈	1	4	yes	—
204	D₄	—	127	2	no	—
205	D₄	—	127	6	no	—
206	D₄	—	254	12	no	—
207	D₄	—	254	12	no	—
208	D₄	—	254	12	no	—
209	D₄	—	254	3	no	—
210	D₄	—	254	1	no	—
211	D₄	C₂×D₅₀₈	1	12	yes	—
212	C₂×C₄	—	254	12	no	—
213	C₂×C₄	—	254	1	no	—
214	C₂×C₄	—	254	3	no	—
215	C₂×C₄	—	254	12	no	—
216	C₂×C₄	—	254	12	no	—
217	C₂×C₄	C₂×D₅₀₈	1	6	yes	—
218	C₂×C₄	C₂×D₅₀₈	1	3	yes	—
219	C₂×C₄	C₂³×D₁₂₇	1	6	yes	—
220	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
221	C₈	C₂³×D₁₂₇	1	2	yes	—
222	C₈	C₂³×D₁₂₇	1	6	yes	—
223	C₂²	—	2	6	no	—
224	C₂²	—	1016	12	no	—
225	C₂²	—	508	6	no	—
226	C₂²	—	508	2	no	—
227	C₂²	—	254	2	no	—
228	C₂²	—	254	6	no	—
229	C₂²	C₂²×D₅₀₈	1	6	yes	—
230	C₂²	C₂²×D₅₀₈	1	1	yes	characteristic
231	C₄	—	254	2	no	—
232	C₄	—	254	6	no	—
233	C₄	C₂²×D₅₀₈	1	6	yes	—
234	C₄	C₂²×D₅₀₈	1	1	yes	characteristic
235	C₄	C₂⁴×D₁₂₇	1	1	yes	characteristic
236	C₂	—	2	6	no	—
237	C₂	—	508	6	no	—
238	C₂	—	508	2	no	—
239	C₂	#22550688	1	2	yes	direct product factor
240	C₂	C₂³×D₅₀₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₀₈
commutator, cosocle	C₄×C₁₂₇	C₂⁵
Frattini	C₄	C₂⁴×D₁₂₇
Fitting	C₂×#131×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
hypercenter	C₂×#131	D₁₂₇
hypocenter	C₁₂₇	C₂×#577
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. C₂×#16861256	C₂	semidirect
2. C₂×#16861255	C₂	semidirect
3. #22550688	C₂	direct (N × H)
4. C₂×#16861242	C₂	semidirect
5. C₂×#16861237	C₂	semidirect
6. C₂×#16861231	C₂	semidirect
7. C₂²×#14523237	C₂	semidirect
8. C₂×#131×C₁₂₇	C₂	semidirect
10. #16861256	C₂²	semidirect
12. #16861255	C₂²	semidirect
15. C₂×#14523248	C₂²	semidirect
17. C₂²×Dic₂₅₄	C₂²	semidirect
21. #16861242	C₂²	semidirect
22. #16861237	C₂²	semidirect
23. #16861231	C₂²	semidirect
24. C₂×Dic₅₀₈	C₂²	semidirect
25. C₂×#14523237	C₂²	semidirect
26. C₂×#14523237	C₂²	semidirect
27. C₂×#14523237	C₂²	semidirect
28. C₂×#14523237	C₂²	semidirect
29. C₂×D₁₀₁₆	C₂²	semidirect
31. #131×C₁₂₇	C₂²	semidirect
32. C₂×M₁₆×C₁₂₇	C₂²	semidirect
33. C₂²×C₈×C₁₂₇	C₂²	semidirect
39. #14523248	C₂³	semidirect
45. C₂×Dic₂₅₄	C₂³	semidirect
58. Dic₅₀₈	C₂³	semidirect
59. #14523237	C₂³	semidirect
60. #14523237	C₂³	semidirect
61. #14523237	C₂³	semidirect
62. D₁₀₁₆	C₂³	semidirect
68. M₁₆×C₁₂₇	C₂³	semidirect
71. C₂×C₈×C₁₂₇	C₂³	semidirect
116. C₂×C₁₂₇	#577	semidirect
121. C₁₂₇	C₂×#577	semidirect
129. C₂×#131	D₁₂₇	semidirect
152. #131	C₂×D₁₂₇	semidirect
153. C₂×M₁₆	C₂×D₁₂₇	semidirect
154. C₂²×C₈	C₂×D₁₂₇	semidirect
189. M₁₆	C₂²×D₁₂₇	semidirect
192. C₂×C₈	C₂²×D₁₂₇	semidirect
239. C₂	#22550688	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16861256	2
C₂×#16861255	2
#22550688	2
C₂×#16861242	2
C₂×#16861237	2
C₂×#16861231	2
C₂²×#14523237	2
C₂×#131×C₁₂₇	2
C₂×#577	127

Subgroups — C2×#22550688 #30653491

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22550688 #30653491