Subgroups — C₂×#22550687 #30653490

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	#22550687	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16861239	C₂	1	3	yes	maximal, complemented
4	C₂×#16861235	C₂	1	3	yes	maximal, complemented
5	C₂×#16861229	C₂	1	3	yes	maximal, complemented
6	C₂²×#14523236	C₂	1	3	yes	maximal, complemented
7	C₂×#16861216	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×H(5,𝔽₂)×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#16861256	—	2	4	no	—
10	#16861256	C₂²	1	8	yes	complemented
11	C₂×#14523248	C₂²	1	3	yes	complemented
12	C₂×#14523247	—	2	3	no	—
13	C₂²×Dic₂₅₄	C₂²	1	3	yes	complemented
14	C₂×Q₈×D₁₂₇	—	2	1	no	—
15	#16861239	C₂²	1	24	yes	complemented
16	#16861235	C₂²	1	24	yes	complemented
17	#16861229	C₂²	1	24	yes	complemented
18	C₂×#14523238	C₂²	1	3	yes	complemented
19	C₂×#14523236	C₂²	1	3	yes	complemented
20	C₂×#14523236	C₂²	1	6	yes	complemented
21	C₂×#14523236	C₂²	1	24	yes	complemented
22	C₂×#14523235	C₂²	1	1	yes	characteristic, complemented
23	C₂×#14523233	C₂²	1	3	yes	complemented
24	#16861216	C₂²	1	8	yes	complemented
25	C₂×#14523232	C₂²	1	3	yes	complemented
26	C₂²×C₁₂₇⋊C₈	C₂²	1	3	yes	complemented
27	H(5,𝔽₂)×C₁₂₇	—	2	4	no	—
28	H(5,𝔽₂)×C₁₂₇	C₂²	1	8	yes	complemented
29	C₂×#41×C₁₂₇	—	2	1	no	—
30	C₂×#41×C₁₂₇	C₂²	1	3	yes	—
31	C₂×#41×C₁₂₇	C₂²	1	1	yes	characteristic
32	C₂²×D₄×C₁₂₇	—	2	3	no	—
33	C₂²×D₄×C₁₂₇	C₂²	1	3	yes	—
34	#14523248	—	2	6	no	—
35	#14523248	C₂³	1	12	yes	complemented
36	#14523247	—	2	24	no	—
37	C₂×Dic₂₅₄	—	2	3	no	—
38	C₂×Dic₂₅₄	—	2	6	no	—
39	C₂×Dic₂₅₄	C₂³	1	3	yes	—
40	C₂×Dic₂₅₄	C₂³	1	12	yes	complemented
41	Q₈×D₁₂₇	—	2	8	no	—
42	C₂×D₅₀₈	C₂³	1	1	yes	characteristic
43	C₂×C₂²⋊D₁₂₇	—	2	3	no	—
44	C₂²×Dic₁₂₇	—	2	3	no	—
45	C₂×C₄×D₁₂₇	—	2	3	no	—
46	#14523238	C₂³	1	12	yes	complemented
47	#14523236	C₂³	1	12	yes	complemented
48	#14523236	C₂³	1	24	yes	complemented
49	#14523235	C₂³	1	4	yes	complemented
50	#14523233	C₂³	1	12	yes	complemented
51	#14523232	C₂³	1	12	yes	complemented
52	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
53	C₂×C₁₂₇⋊C₈	C₂³	1	3	yes	—
54	C₂×C₁₂₇⋊C₈	C₂³	1	12	yes	complemented
55	C₂⁴×C₁₂₇	—	2	3	no	—
56	#41×C₁₂₇	—	2	8	no	—
57	#41×C₁₂₇	—	2	6	no	—
58	#41×C₁₂₇	D₄	1	4	yes	—
59	#41×C₁₂₇	C₂³	1	12	yes	—
60	#41×C₁₂₇	C₂³	1	4	yes	—
61	C₂×Q₈×C₁₂₇	D₄	1	1	yes	characteristic
62	C₂×Q₈×C₁₂₇	C₂³	1	1	yes	characteristic
63	C₂×D₄×C₁₂₇	—	2	3	no	—
64	C₂×D₄×C₁₂₇	—	2	24	no	—
65	C₂×D₄×C₁₂₇	—	2	3	no	—
66	C₂×D₄×C₁₂₇	—	2	6	no	—
67	C₂×D₄×C₁₂₇	D₄	1	3	yes	—
68	C₂×D₄×C₁₂₇	C₂³	1	3	yes	—
69	C₂×D₄×C₁₂₇	C₂³	1	12	yes	—
70	C₂²×C₄×C₁₂₇	—	2	3	no	—
71	C₂²×C₄×C₁₂₇	C₂³	1	3	yes	—
72	C₂²×D₁₂₇	—	2	1	no	—
73	Dic₂₅₄	—	2	12	no	—
74	Dic₂₅₄	—	2	3	no	—
75	Dic₂₅₄	C₂⁴	1	6	yes	—
76	D₅₀₈	—	2	1	no	—
77	D₅₀₈	C₂⁴	1	2	yes	—
78	C₂²⋊D₁₂₇	—	2	12	no	—
79	C₂×Dic₁₂₇	—	2	3	no	—
80	C₂×Dic₁₂₇	—	2	12	no	—
81	C₄×D₁₂₇	—	2	12	no	—
82	C₁₂₇⋊C₈	C₂⁴	1	2	yes	—
83	C₁₂₇⋊C₈	C₂⁴	1	6	yes	—
84	C₂³×C₁₂₇	—	2	3	no	—
85	C₂³×C₁₂₇	—	2	12	no	—
86	C₂³×C₁₂₇	—	8	6	no	—
87	C₂³×C₁₂₇	C₂×D₄	1	3	yes	—
88	Q₈×C₁₂₇	—	2	1	no	—
89	Q₈×C₁₂₇	C₂×D₄	1	4	yes	—
90	Q₈×C₁₂₇	C₂⁴	1	2	yes	—
91	D₄×C₁₂₇	—	2	12	no	—
92	D₄×C₁₂₇	—	2	12	no	—
93	D₄×C₁₂₇	—	2	3	no	—
94	D₄×C₁₂₇	C₂×D₄	1	12	yes	—
95	D₄×C₁₂₇	C₂⁴	1	6	yes	—
96	C₂×C₄×C₁₂₇	—	2	1	no	—
97	C₂×C₄×C₁₂₇	—	2	12	no	—
98	C₂×C₄×C₁₂₇	C₂×D₄	1	3	yes	—
99	C₂×C₄×C₁₂₇	C₂×D₄	1	6	yes	—
100	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
101	C₂×C₄×C₁₂₇	C₂⁴	1	6	yes	—
102	C₂²×C₁₂₇	—	2	6	no	—
103	C₂²×C₁₂₇	—	2	6	no	—
104	C₂²×C₁₂₇	—	4	6	no	—
105	C₂²×C₁₂₇	—	8	12	no	—
106	C₂²×C₁₂₇	C₂²×D₄	1	6	yes	—
107	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×D₁₂₇	—	2	2	no	—
109	C₂×D₁₂₇	—	4	2	no	—
110	C₄×C₁₂₇	—	2	2	no	—
111	C₄×C₁₂₇	C₂²×D₄	1	6	yes	—
112	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
113	C₄×C₁₂₇	C₂⁵	1	1	yes	characteristic
114	Dic₁₂₇	—	2	6	no	—
115	C₂×C₁₂₇	—	2	6	no	—
116	C₂×C₁₂₇	—	4	6	no	—
117	C₂×C₁₂₇	#577	1	2	yes	complemented
118	C₂×C₁₂₇	C₂³×D₄	1	1	yes	characteristic
119	D₁₂₇	—	4	2	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)	D₁₂₇	1	1	yes	characteristic, complemented
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	—	127	1	no	—
130	2₋¹⁺⁴	—	127	8	no	—
131	2₋¹⁺⁴	—	254	4	no	—
132	H(5,𝔽₂)	—	2	4	no	—
133	H(5,𝔽₂)	C₂×D₁₂₇	1	8	yes	complemented
134	C₂×#41	—	2	1	no	—
135	C₂×#41	—	127	3	no	—
136	C₂×#41	—	254	3	no	—
137	C₂×#41	C₂×D₁₂₇	1	3	yes	—
138	C₂×#41	C₂×D₁₂₇	1	1	yes	characteristic
139	C₂²×Q₈	—	127	3	no	—
140	C₂²×Q₈	—	254	1	no	—
141	C₂²×D₄	—	2	3	no	—
142	C₂²×D₄	C₂×D₁₂₇	1	3	yes	—
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	1	no	—
148	C₂×SD₁₆	—	127	3	no	—
149	C₂×SD₁₆	—	127	6	no	—
150	C₂×SD₁₆	—	127	24	no	—
151	C₂×D₈	—	127	3	no	—
152	#131	—	127	8	no	—
153	C₂×M₁₆	—	127	3	no	—
154	C₂²×C₈	—	127	3	no	—
155	C₂⁴	—	2	3	no	—
156	#41	—	2	8	no	—
157	#41	—	2	6	no	—
158	#41	—	127	12	no	—
159	#41	—	254	24	no	—
160	#41	—	254	6	no	—
161	#41	C₂²⋊D₁₂₇	1	4	yes	—
162	#41	C₂²×D₁₂₇	1	12	yes	—
163	#41	C₂²×D₁₂₇	1	4	yes	—
164	C₂×Q₈	—	127	3	no	—
165	C₂×Q₈	—	127	12	no	—
166	C₂×Q₈	—	254	8	no	—
167	C₂×Q₈	—	254	3	no	—
168	C₂×Q₈	—	254	6	no	—
169	C₂×Q₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
170	C₂×Q₈	C₂²×D₁₂₇	1	1	yes	characteristic
171	C₂×D₄	—	2	3	no	—
172	C₂×D₄	—	2	24	no	—
173	C₂×D₄	—	2	3	no	—
174	C₂×D₄	—	2	6	no	—
175	C₂×D₄	—	127	1	no	—
176	C₂×D₄	—	254	3	no	—
177	C₂×D₄	C₂²⋊D₁₂₇	1	3	yes	—
178	C₂×D₄	C₂²×D₁₂₇	1	3	yes	—
179	C₂×D₄	C₂²×D₁₂₇	1	12	yes	—
180	C₂²×C₄	—	2	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	4	no	—
186	SD₁₆	—	127	12	no	—
187	SD₁₆	—	127	24	no	—
188	D₈	—	127	12	no	—
189	M₁₆	—	127	12	no	—
190	C₂×C₈	—	127	1	no	—
191	C₂×C₈	—	127	3	no	—
192	C₂×C₈	—	127	12	no	—
193	C₂³	—	2	3	no	—
194	C₂³	—	2	12	no	—
195	C₂³	—	8	6	no	—
196	C₂³	—	254	1	no	—
197	C₂³	C₂×C₂²⋊D₁₂₇	1	3	yes	—
198	Q₈	—	2	1	no	—
199	Q₈	—	127	6	no	—
200	Q₈	—	254	12	no	—
201	Q₈	—	254	3	no	—
202	Q₈	C₂×C₂²⋊D₁₂₇	1	4	yes	—
203	Q₈	C₂³×D₁₂₇	1	2	yes	—
204	D₄	—	2	12	no	—
205	D₄	—	2	12	no	—
206	D₄	—	2	3	no	—
207	D₄	—	127	2	no	—
208	D₄	—	254	12	no	—
209	D₄	—	254	1	no	—
210	D₄	C₂×C₂²⋊D₁₂₇	1	12	yes	—
211	D₄	C₂³×D₁₂₇	1	6	yes	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	12	no	—
214	C₂×C₄	—	254	12	no	—
215	C₂×C₄	—	254	3	no	—
216	C₂×C₄	—	254	12	no	—
217	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	6	yes	—
218	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	3	yes	—
219	C₂×C₄	C₂³×D₁₂₇	1	6	yes	—
220	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
221	C₈	—	127	2	no	—
222	C₈	—	127	6	no	—
223	C₂²	—	2	6	no	—
224	C₂²	—	2	6	no	—
225	C₂²	—	4	6	no	—
226	C₂²	—	8	12	no	—
227	C₂²	—	508	2	no	—
228	C₂²	—	254	2	no	—
229	C₂²	C₂²×C₂²⋊D₁₂₇	1	6	yes	—
230	C₂²	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	254	6	no	—
233	C₄	C₂²×C₂²⋊D₁₂₇	1	6	yes	—
234	C₄	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
235	C₄	C₂⁴×D₁₂₇	1	1	yes	characteristic
236	C₂	—	2	6	no	—
237	C₂	—	4	6	no	—
238	C₂	—	508	2	no	—
239	C₂	#22550687	1	2	yes	direct product factor
240	C₂	C₂³×C₂²⋊D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₂₇
commutator, cosocle	C₄×C₁₂₇	C₂⁵
Frattini	C₄	C₂⁴×D₁₂₇
Fitting	C₂×H(5,𝔽₂)×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
hypercenter	C₂×H(5,𝔽₂)	D₁₂₇
hypocenter	C₁₂₇	C₂×#577
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂×#16861256	C₂	semidirect
2. #22550687	C₂	direct (N × H)
3. C₂×#16861239	C₂	semidirect
4. C₂×#16861235	C₂	semidirect
5. C₂×#16861229	C₂	semidirect
6. C₂²×#14523236	C₂	semidirect
7. C₂×#16861216	C₂	semidirect
8. C₂×H(5,𝔽₂)×C₁₂₇	C₂	semidirect
10. #16861256	C₂²	semidirect
11. C₂×#14523248	C₂²	semidirect
13. C₂²×Dic₂₅₄	C₂²	semidirect
15. #16861239	C₂²	semidirect
16. #16861235	C₂²	semidirect
17. #16861229	C₂²	semidirect
18. C₂×#14523238	C₂²	semidirect
19. C₂×#14523236	C₂²	semidirect
20. C₂×#14523236	C₂²	semidirect
21. C₂×#14523236	C₂²	semidirect
22. C₂×#14523235	C₂²	semidirect
23. C₂×#14523233	C₂²	semidirect
24. #16861216	C₂²	semidirect
25. C₂×#14523232	C₂²	semidirect
26. C₂²×C₁₂₇⋊C₈	C₂²	semidirect
28. H(5,𝔽₂)×C₁₂₇	C₂²	semidirect
35. #14523248	C₂³	semidirect
40. C₂×Dic₂₅₄	C₂³	semidirect
46. #14523238	C₂³	semidirect
47. #14523236	C₂³	semidirect
48. #14523236	C₂³	semidirect
49. #14523235	C₂³	semidirect
50. #14523233	C₂³	semidirect
51. #14523232	C₂³	semidirect
54. C₂×C₁₂₇⋊C₈	C₂³	semidirect
117. C₂×C₁₂₇	#577	semidirect
121. C₁₂₇	C₂×#577	semidirect
123. C₂×H(5,𝔽₂)	D₁₂₇	semidirect
133. H(5,𝔽₂)	C₂×D₁₂₇	semidirect
239. C₂	#22550687	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
#22550687	2
C₂×#16861239	2
C₂×#16861235	2
C₂×#16861229	2
C₂²×#14523236	2
C₂×#16861216	2
C₂×H(5,𝔽₂)×C₁₂₇	2
C₂×#577	127

Subgroups — C2×#22550687 #30653490

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22550687 #30653490