Subgroups — C₂×#22550537 #30652198

Order 16256.

1620 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523249	C₂	1	1	yes	maximal, characteristic, complemented
2	#22550537	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16861198	C₂	1	3	yes	maximal, complemented
4	C₂×#16861185	C₂	1	3	yes	maximal, complemented
5	C₂×#16861183	C₂	1	3	yes	maximal, complemented
6	C₂×Q₈×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₂²⋊D₁₂₇	C₂	1	3	yes	maximal, complemented
8	C₂³×Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14523249	—	2	8	no	—
10	C₂×#14523249	C₂²	1	8	yes	complemented
11	C₂²×D₅₀₈	C₂²	1	3	yes	—
12	C₂²×C₂²⋊D₁₂₇	C₂²	1	3	yes	—
13	C₂²×C₄×D₁₂₇	C₂²	1	3	yes	—
14	#16861198	C₂²	1	24	yes	complemented
15	#16861185	C₂²	1	24	yes	complemented
16	#16861183	C₂²	1	24	yes	complemented
17	Q₈×Dic₁₂₇	C₂²	1	8	yes	complemented
18	C₄×C₂²⋊D₁₂₇	C₂²	1	24	yes	complemented
19	C₂×C₄⋊Dic₁₂₇	C₂²	1	3	yes	—
20	C₂×Dic₁₂₇⋊C₄	C₂²	1	3	yes	—
21	C₂×C₂²⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
22	C₂×(C₂×D₁₂₇)⋊C₄	C₂²	1	3	yes	—
23	C₂×(C₂×D₁₂₇)⋊C₄	C₂²	1	6	yes	—
24	C₂×C₄×Dic₁₂₇	C₂²	1	3	yes	—
25	C₂²×Q₈×C₁₂₇	—	2	8	no	—
26	C₂²×Q₈×C₁₂₇	C₂²	1	3	yes	—
27	C₂²×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
28	C₂²×Q₈×C₁₂₇	C₂²	1	8	yes	complemented
29	C₂³×C₄×C₁₂₇	C₂²	1	3	yes	—
30	C₂³×D₁₂₇	C₂³	1	3	yes	—
31	#14523249	—	2	32	no	—
32	C₂×D₅₀₈	—	2	12	no	—
33	C₂×D₅₀₈	C₂³	1	12	yes	—
34	C₂×C₂²⋊D₁₂₇	—	2	12	no	—
35	C₂×C₂²⋊D₁₂₇	C₂³	1	12	yes	—
36	C₂²×Dic₁₂₇	C₂³	1	1	yes	characteristic
37	C₂²×Dic₁₂₇	C₂³	1	3	yes	—
38	C₂²×Dic₁₂₇	C₂³	1	1	yes	characteristic
39	C₂×C₄×D₁₂₇	—	2	12	no	—
40	C₂×C₄×D₁₂₇	C₂³	1	12	yes	—
41	C₄⋊Dic₁₂₇	C₂³	1	12	yes	—
42	Dic₁₂₇⋊C₄	C₂³	1	12	yes	—
43	C₂²⋊Dic₁₂₇	C₂³	1	4	yes	—
44	(C₂×D₁₂₇)⋊C₄	C₂³	1	24	yes	—
45	(C₂×D₁₂₇)⋊C₄	C₂³	1	12	yes	—
46	C₄×Dic₁₂₇	C₂³	1	12	yes	—
47	C₂⁴×C₁₂₇	C₂³	1	1	yes	characteristic
48	C₂×Q₈×C₁₂₇	—	2	32	no	—
49	C₂×Q₈×C₁₂₇	—	2	12	no	—
50	C₂×Q₈×C₁₂₇	D₄	1	8	yes	complemented
51	C₂×Q₈×C₁₂₇	C₂³	1	12	yes	—
52	C₂×Q₈×C₁₂₇	C₂³	1	4	yes	—
53	C₂²×C₄×C₁₂₇	—	2	12	no	—
54	C₂²×C₄×C₁₂₇	C₂³	1	3	yes	—
55	C₂²×C₄×C₁₂₇	C₂³	1	12	yes	—
56	C₂²×C₄×C₁₂₇	C₂³	1	3	yes	—
57	C₂²×D₁₂₇	—	2	6	no	—
58	C₂²×D₁₂₇	—	2	6	no	—
59	C₂²×D₁₂₇	—	2	6	no	—
60	C₂²×D₁₂₇	C₂⁴	1	6	yes	—
61	D₅₀₈	—	2	24	no	—
62	C₂²⋊D₁₂₇	—	2	24	no	—
63	C₂×Dic₁₂₇	—	2	2	no	—
64	C₂×Dic₁₂₇	—	2	6	no	—
65	C₂×Dic₁₂₇	—	2	2	no	—
66	C₂×Dic₁₂₇	C₂⁴	1	2	yes	—
67	C₂×Dic₁₂₇	C₂⁴	1	6	yes	—
68	C₂×Dic₁₂₇	C₂⁴	1	2	yes	—
69	C₄×D₁₂₇	—	2	24	no	—
70	C₂³×C₁₂₇	—	2	2	no	—
71	C₂³×C₁₂₇	—	2	2	no	—
72	C₂³×C₁₂₇	#41	1	4	yes	complemented
73	C₂³×C₁₂₇	C₂⁴	1	2	yes	—
74	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
75	Q₈×C₁₂₇	—	2	24	no	—
76	Q₈×C₁₂₇	C₂×D₄	1	16	yes	complemented
77	C₂×C₄×C₁₂₇	—	2	6	no	—
78	C₂×C₄×C₁₂₇	—	2	24	no	—
79	C₂×C₄×C₁₂₇	C₂×D₄	1	12	yes	—
80	C₂×C₄×C₁₂₇	C₂⁴	1	6	yes	—
81	C₂×C₄×C₁₂₇	C₂⁴	1	6	yes	—
82	C₂²×C₁₂₇	—	2	8	no	—
83	C₂²×C₁₂₇	—	2	2	no	—
84	C₂²×C₁₂₇	—	2	2	no	—
85	C₂²×C₁₂₇	C₂²×D₄	1	2	yes	—
86	C₂²×C₁₂₇	C₂×#41	1	2	yes	—
87	C₂²×C₁₂₇	C₂×#41	1	4	yes	complemented
88	C₂²×C₁₂₇	2₋¹⁺⁴	1	2	yes	—
89	C₂²×C₁₂₇	C₂⁵	1	1	yes	characteristic
90	C₂×D₁₂₇	—	2	6	no	—
91	C₂×D₁₂₇	—	2	6	no	—
92	C₂×D₁₂₇	—	2	6	no	—
93	C₂×D₁₂₇	—	4	12	no	—
94	C₄×C₁₂₇	—	2	6	no	—
95	C₄×C₁₂₇	C₂²×D₄	1	12	yes	—
96	Dic₁₂₇	—	2	2	no	—
97	Dic₁₂₇	—	2	6	no	—
98	Dic₁₂₇	—	2	2	no	—
99	C₂×C₁₂₇	—	2	4	no	—
100	C₂×C₁₂₇	#548	1	4	yes	complemented
101	C₂×C₁₂₇	C₂³×D₄	1	1	yes	characteristic
102	C₂×C₁₂₇	C₂²×#41	1	1	yes	characteristic
103	C₂×C₁₂₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
104	D₁₂₇	—	4	6	no	—
105	C₂×#548	—	127	1	no	maximal, Sylow
106	C₁₂₇	C₂×#548	1	1	yes	characteristic, complemented, Sylow
107	C₂²×#41	—	127	1	no	—
108	C₂³×Q₈	D₁₂₇	1	1	yes	characteristic, complemented
109	#548	—	127	16	no	—
110	C₂×#124	—	127	3	no	—
111	C₂×C₂²⋊Q₈	—	127	3	no	—
112	C₂×#121	—	127	3	no	—
113	C₂×C₄×Q₈	—	127	1	no	—
114	C₂×C₄×D₄	—	127	3	no	—
115	C₂×#41	—	127	8	no	—
116	C₂×#41	—	254	8	no	—
117	C₂²×Q₈	—	2	8	no	—
118	C₂²×Q₈	C₂×D₁₂₇	1	3	yes	—
119	C₂²×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
120	C₂²×Q₈	C₂×D₁₂₇	1	8	yes	complemented
121	C₂²×D₄	—	127	3	no	—
122	C₂²×D₄	—	127	3	no	—
123	C₂³×C₄	—	127	3	no	—
124	C₂³×C₄	C₂×D₁₂₇	1	3	yes	—
125	#124	—	127	24	no	—
126	C₂²⋊Q₈	—	127	24	no	—
127	#121	—	127	24	no	—
128	C₄×Q₈	—	127	8	no	—
129	C₄×D₄	—	127	24	no	—
130	C₂×C₄⋊C₄	—	127	3	no	—
131	C₂×C₄⋊C₄	—	127	3	no	—
132	C₂×C₂²⋊C₄	—	127	1	no	—
133	C₂×C₂²⋊C₄	—	127	3	no	—
134	C₂×C₂²⋊C₄	—	127	6	no	—
135	C₂×C₄²	—	127	3	no	—
136	C₂⁴	—	127	3	no	—
137	C₂⁴	C₂²×D₁₂₇	1	1	yes	characteristic
138	#41	—	254	32	no	—
139	C₂×Q₈	—	2	32	no	—
140	C₂×Q₈	—	2	12	no	—
141	C₂×Q₈	C₂²⋊D₁₂₇	1	8	yes	complemented
142	C₂×Q₈	C₂²×D₁₂₇	1	12	yes	—
143	C₂×Q₈	C₂²×D₁₂₇	1	4	yes	—
144	C₂×D₄	—	127	12	no	—
145	C₂×D₄	—	127	12	no	—
146	C₂×D₄	—	254	12	no	—
147	C₂×D₄	—	254	12	no	—
148	C₂²×C₄	—	2	12	no	—
149	C₂²×C₄	—	127	1	no	—
150	C₂²×C₄	—	127	3	no	—
151	C₂²×C₄	—	127	1	no	—
152	C₂²×C₄	—	127	12	no	—
153	C₂²×C₄	—	254	12	no	—
154	C₂²×C₄	C₂²×D₁₂₇	1	12	yes	—
155	C₂²×C₄	C₂²×D₁₂₇	1	3	yes	—
156	C₂²×C₄	C₂²×D₁₂₇	1	3	yes	—
157	C₄⋊C₄	—	127	12	no	—
158	C₄⋊C₄	—	127	12	no	—
159	C₂²⋊C₄	—	127	4	no	—
160	C₂²⋊C₄	—	127	12	no	—
161	C₂²⋊C₄	—	127	24	no	—
162	C₄²	—	127	12	no	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	2	no	—
165	C₂³	—	127	6	no	—
166	C₂³	—	254	6	no	—
167	C₂³	—	254	6	no	—
168	C₂³	—	254	6	no	—
169	C₂³	#14523249	1	4	yes	complemented
170	C₂³	C₂³×D₁₂₇	1	2	yes	—
171	C₂³	C₂³×D₁₂₇	1	1	yes	characteristic
172	Q₈	—	2	24	no	—
173	Q₈	C₂×C₂²⋊D₁₂₇	1	16	yes	complemented
174	D₄	—	254	24	no	—
175	D₄	—	254	24	no	—
176	C₂×C₄	—	2	6	no	—
177	C₂×C₄	—	2	24	no	—
178	C₂×C₄	—	127	2	no	—
179	C₂×C₄	—	127	6	no	—
180	C₂×C₄	—	127	2	no	—
181	C₂×C₄	—	254	2	no	—
182	C₂×C₄	—	254	6	no	—
183	C₂×C₄	—	254	2	no	—
184	C₂×C₄	—	254	24	no	—
185	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	12	yes	—
186	C₂×C₄	C₂³×D₁₂₇	1	6	yes	—
187	C₂×C₄	C₂³×D₁₂₇	1	6	yes	—
188	C₂²	—	2	8	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	508	12	no	—
192	C₂²	—	254	6	no	—
193	C₂²	—	254	6	no	—
194	C₂²	—	254	6	no	—
195	C₂²	C₂²×C₂²⋊D₁₂₇	1	2	yes	—
196	C₂²	C₂×#14523249	1	2	yes	—
197	C₂²	C₂×#14523249	1	4	yes	complemented
198	C₂²	#16861257	1	2	yes	—
199	C₂²	C₂⁴×D₁₂₇	1	1	yes	characteristic
200	C₄	—	2	6	no	—
201	C₄	—	254	2	no	—
202	C₄	—	254	6	no	—
203	C₄	—	254	2	no	—
204	C₄	C₂²×C₂²⋊D₁₂₇	1	12	yes	—
205	C₂	—	2	4	no	—
206	C₂	—	508	6	no	—
207	C₂	#22550537	1	4	yes	direct product factor
208	C₂	C₂³×C₂²⋊D₁₂₇	1	1	yes	characteristic
209	C₂	C₂²×#14523249	1	1	yes	characteristic
210	C₂	C₂×#16861257	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₂₇
commutator, cosocle	C₂²×C₁₂₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₂₇
Fitting	C₂³×Q₈×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂⁴
hypercenter	C₂³×Q₈	D₁₂₇
hypocenter	C₁₂₇	C₂×#548
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#14523249	C₂	semidirect
2. #22550537	C₂	direct (N × H)
3. C₂×#16861198	C₂	semidirect
4. C₂×#16861185	C₂	semidirect
5. C₂×#16861183	C₂	semidirect
6. C₂×Q₈×Dic₁₂₇	C₂	semidirect
7. C₂×C₄×C₂²⋊D₁₂₇	C₂	semidirect
8. C₂³×Q₈×C₁₂₇	C₂	semidirect
10. C₂×#14523249	C₂²	semidirect
14. #16861198	C₂²	semidirect
15. #16861185	C₂²	semidirect
16. #16861183	C₂²	semidirect
17. Q₈×Dic₁₂₇	C₂²	semidirect
18. C₄×C₂²⋊D₁₂₇	C₂²	semidirect
28. C₂²×Q₈×C₁₂₇	C₂²	semidirect
50. C₂×Q₈×C₁₂₇	D₄	semidirect
72. C₂³×C₁₂₇	#41	semidirect
76. Q₈×C₁₂₇	C₂×D₄	semidirect
87. C₂²×C₁₂₇	C₂×#41	semidirect
100. C₂×C₁₂₇	#548	semidirect
106. C₁₂₇	C₂×#548	semidirect
108. C₂³×Q₈	D₁₂₇	semidirect
120. C₂²×Q₈	C₂×D₁₂₇	semidirect
141. C₂×Q₈	C₂²⋊D₁₂₇	semidirect
169. C₂³	#14523249	semidirect
173. Q₈	C₂×C₂²⋊D₁₂₇	semidirect
197. C₂²	C₂×#14523249	semidirect
207. C₂	#22550537	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×#14523249	2
#22550537	2
C₂×#16861198	2
C₂×#16861185	2
C₂×#16861183	2
C₂×Q₈×Dic₁₂₇	2
C₂×C₄×C₂²⋊D₁₂₇	2
C₂³×Q₈×C₁₂₇	2
C₂×#548	127

Subgroups — C2×#22550537 #30652198

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22550537 #30652198