Subgroups — C₂×#22549927 #30647344

Order 16256.

528 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×D₁₂₇	C₂	1	1	yes	maximal, characteristic
2	#22549927	C₂	1	8	yes	maximal, direct product factor
3	C₂×Dic₁₂₇⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₂×(C₂×D₁₂₇)⋊C₈	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₁₂₇	C₄	1	1	yes	characteristic
8	C₂²×C₄×D₁₂₇	C₄	1	1	yes	characteristic
9	C₂²×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
10	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄×Dic₁₂₇	C₄	1	1	yes	characteristic
12	C₂×C₄×Dic₁₂₇	C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
14	C₄²×D₁₂₇	C₄	1	4	yes	—
15	C₄²×D₁₂₇	C₂²	1	4	yes	—
16	Dic₁₂₇⋊C₈	C₂²	1	8	yes	complemented
17	(C₂×D₁₂₇)⋊C₈	C₂²	1	8	yes	—
18	C₄×C₁₂₇⋊C₈	C₂²	1	4	yes	complemented
19	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
20	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₁₂₇	C₂²	1	4	yes	complemented
22	C₂³×D₁₂₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×D₁₂₇	—	2	4	no	—
27	C₂×C₄×D₁₂₇	—	2	4	no	—
28	C₂×C₄×D₁₂₇	C₈	1	8	yes	complemented
29	C₂×C₄×D₁₂₇	C₂×C₄	1	4	yes	—
30	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
31	C₂×C₄×D₁₂₇	C₂×C₄	1	4	yes	—
32	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
33	C₂×C₁₂₇⋊C₈	—	2	4	no	—
34	C₂×C₁₂₇⋊C₈	C₂³	1	4	yes	—
35	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
36	C₄×Dic₁₂₇	C₂×C₄	1	4	yes	—
37	C₄×Dic₁₂₇	C₂×C₄	1	4	yes	—
38	C₄×Dic₁₂₇	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
42	C₂×C₈×C₁₂₇	—	2	4	no	—
43	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
44	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
45	C₄²×C₁₂₇	C₂³	1	2	yes	—
46	C₂²×D₁₂₇	—	2	2	no	—
47	C₂²×D₁₂₇	—	2	2	no	—
48	C₂²×D₁₂₇	C₂×C₈	1	4	yes	—
49	C₂²×D₁₂₇	C₂²×C₄	1	2	yes	—
50	C₂×Dic₁₂₇	—	2	2	no	—
51	C₂×Dic₁₂₇	—	2	2	no	—
52	C₂×Dic₁₂₇	C₂×C₈	1	4	yes	—
53	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
54	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
55	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
56	C₄×D₁₂₇	—	2	4	no	—
57	C₄×D₁₂₇	—	2	8	no	—
58	C₄×D₁₂₇	—	2	4	no	—
59	C₄×D₁₂₇	C₂×C₈	1	16	yes	complemented
60	C₁₂₇⋊C₈	—	2	4	no	—
61	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₂₇	C₂×C₈	1	4	yes	—
63	C₂×C₄×C₁₂₇	M₁₆	1	4	yes	—
64	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₁₂₇	#41	1	4	yes	—
68	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
69	C₈×C₁₂₇	—	2	4	no	—
70	C₂²×C₁₂₇	#117	1	2	yes	—
71	C₂²×C₁₂₇	C₂²×C₈	1	2	yes	—
72	C₂²×C₁₂₇	C₂×M₁₆	1	2	yes	—
73	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
74	C₂×D₁₂₇	—	2	2	no	—
75	C₂×D₁₂₇	—	2	2	no	—
76	C₂×D₁₂₇	—	2	8	no	—
77	C₂×D₁₂₇	C₂²×C₈	1	4	yes	—
78	C₄×C₁₂₇	#117	1	2	yes	—
79	C₄×C₁₂₇	C₂²×C₈	1	4	yes	—
80	C₄×C₁₂₇	C₂×M₁₆	1	4	yes	—
81	C₄×C₁₂₇	C₂×#41	1	2	yes	—
82	Dic₁₂₇	—	2	2	no	—
83	Dic₁₂₇	—	2	2	no	—
84	Dic₁₂₇	C₂²×C₈	1	4	yes	—
85	C₂×C₁₂₇	#431	1	4	yes	complemented
86	C₂×C₁₂₇	C₂×#117	1	1	yes	characteristic
87	C₂×C₁₂₇	C₂³×C₈	1	1	yes	characteristic
88	C₂×C₁₂₇	C₂²×M₁₆	1	1	yes	characteristic
89	D₁₂₇	—	2	4	no	—
90	C₂×#431	—	127	1	no	maximal, Sylow
91	C₁₂₇	C₂×#431	1	1	yes	characteristic, complemented, Sylow
92	C₂²×C₄²	—	127	1	no	—
93	#431	—	127	8	no	—
94	C₂×C₄⋊C₈	—	127	2	no	—
95	C₂×C₂²⋊C₈	—	127	2	no	—
96	C₂×C₄×C₈	—	127	1	no	—
97	C₂×C₄×C₈	D₁₂₇	1	1	yes	characteristic, complemented
98	C₂³×C₄	—	127	1	no	—
99	C₂³×C₄	—	127	1	no	—
100	C₂³×C₄	—	127	1	no	—
101	C₂²×C₈	—	127	2	no	—
102	C₂²×C₈	C₂×D₁₂₇	1	2	yes	—
103	C₂×C₄²	—	127	1	no	—
104	C₂×C₄²	—	127	1	no	—
105	C₂×C₄²	—	127	1	no	—
106	C₂×C₄²	—	127	4	no	—
107	C₂×C₄²	—	127	4	no	—
108	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
109	C₄⋊C₈	—	127	8	no	—
110	C₂²⋊C₈	—	127	8	no	—
111	C₄×C₈	—	127	4	no	—
112	C₄×C₈	C₂×D₁₂₇	1	4	yes	complemented
113	C₂⁴	—	127	1	no	—
114	C₂²×C₄	—	127	1	no	—
115	C₂²×C₄	—	127	8	no	—
116	C₂²×C₄	—	127	4	no	—
117	C₂²×C₄	—	127	1	no	—
118	C₂²×C₄	—	127	1	no	—
119	C₂²×C₄	—	127	4	no	—
120	C₂²×C₄	—	127	2	no	—
121	C₂²×C₄	—	127	2	no	—
122	C₂²×C₄	—	254	4	no	—
123	C₂²×C₄	—	254	4	no	—
124	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
125	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
126	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
127	C₂×C₈	—	2	4	no	—
128	C₂×C₈	—	127	4	no	—
129	C₂×C₈	—	254	4	no	—
130	C₂×C₈	C₂²×D₁₂₇	1	4	yes	—
131	C₄²	—	127	4	no	—
132	C₄²	—	127	2	no	—
133	C₄²	—	127	4	no	—
134	C₄²	—	127	2	no	—
135	C₄²	C₄×D₁₂₇	1	2	yes	—
136	C₄²	C₂²×D₁₂₇	1	2	yes	—
137	C₂³	—	127	4	no	—
138	C₂³	—	127	2	no	—
139	C₂³	—	254	2	no	—
140	C₂³	—	254	2	no	—
141	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
142	C₂×C₄	—	127	4	no	—
143	C₂×C₄	—	127	2	no	—
144	C₂×C₄	—	127	16	no	—
145	C₂×C₄	—	127	2	no	—
146	C₂×C₄	—	127	2	no	—
147	C₂×C₄	—	254	2	no	—
148	C₂×C₄	—	254	2	no	—
149	C₂×C₄	—	254	8	no	—
150	C₂×C₄	—	254	4	no	—
151	C₂×C₄	—	254	4	no	—
152	C₂×C₄	C₈×D₁₂₇	1	4	yes	—
153	C₂×C₄	#14523231	1	4	yes	—
154	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
155	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
156	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
157	C₂×C₄	#14523248	1	4	yes	—
158	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
159	C₈	—	2	4	no	—
160	C₈	—	254	4	no	—
161	C₂²	—	127	4	no	—
162	C₂²	—	254	2	no	—
163	C₂²	—	254	2	no	—
164	C₂²	—	254	8	no	—
165	C₂²	#16861166	1	2	yes	—
166	C₂²	C₂×C₈×D₁₂₇	1	2	yes	—
167	C₂²	C₂×#14523231	1	2	yes	—
168	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
169	C₄	—	127	4	no	—
170	C₄	—	254	2	no	—
171	C₄	—	254	2	no	—
172	C₄	#16861166	1	2	yes	—
173	C₄	C₂×C₈×D₁₂₇	1	4	yes	—
174	C₄	C₂×#14523231	1	4	yes	—
175	C₄	C₂×#14523248	1	2	yes	—
176	C₂	—	254	4	no	—
177	C₂	#22549927	1	4	yes	direct product factor
178	C₂	C₂×#16861166	1	1	yes	characteristic
179	C₂	C₂²×C₈×D₁₂₇	1	1	yes	characteristic
180	C₂	C₂²×#14523231	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂×D₁₂₇
commutator	C₂×C₁₂₇	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂×C₄×C₈×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂²×C₄
prime radical	C₂³×D₁₂₇	C₂×C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₂×C₄×C₈	D₁₂₇
hypocenter	C₁₂₇	C₂×#431
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #22549927	C₂	direct (N × H)
3. C₂×Dic₁₂₇⋊C₈	C₂	semidirect
5. C₂×C₄×C₁₂₇⋊C₈	C₂	semidirect
6. C₂×C₄×C₈×C₁₂₇	C₂	semidirect
16. Dic₁₂₇⋊C₈	C₂²	semidirect
18. C₄×C₁₂₇⋊C₈	C₂²	semidirect
21. C₄×C₈×C₁₂₇	C₂²	semidirect
28. C₂×C₄×D₁₂₇	C₈	semidirect
59. C₄×D₁₂₇	C₂×C₈	semidirect
85. C₂×C₁₂₇	#431	semidirect
91. C₁₂₇	C₂×#431	semidirect
97. C₂×C₄×C₈	D₁₂₇	semidirect
112. C₄×C₈	C₂×D₁₂₇	semidirect
177. C₂	#22549927	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×C₄²×D₁₂₇	2
#22549927	2
C₂×Dic₁₂₇⋊C₈	2
C₂×(C₂×D₁₂₇)⋊C₈	2
C₂×C₄×C₁₂₇⋊C₈	2
C₂×C₄×C₈×C₁₂₇	2
C₂×#431	127

Subgroups — C2×#22549927 #30647344

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22549927 #30647344