Subgroups — C₂×#22549630 #30638420

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	C₂³×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#22549630	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₂₇	C₄	1	2	yes	—
6	C₂²×C₁₂₇⋊C₈	—	2	2	no	—
7	C₂²×C₁₂₇⋊C₈	—	2	2	no	—
8	C₂²×C₁₂₇⋊C₈	C₄	1	4	yes	complemented
9	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×Dic₁₂₇	—	2	2	no	—
11	C₂×C₄×Dic₁₂₇	—	2	2	no	—
12	C₂×C₄×Dic₁₂₇	C₄	1	2	yes	—
13	C₂×C₄×Dic₁₂₇	C₂²	1	2	yes	—
14	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₂₇	—	2	2	no	—
16	C₂²×C₈×C₁₂₇	—	2	2	no	—
17	C₂²×C₈×C₁₂₇	C₄	1	4	yes	complemented
18	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
19	C₂²×Dic₁₂₇	—	2	4	no	—
20	C₂²×Dic₁₂₇	—	2	4	no	—
21	C₂²×Dic₁₂₇	C₈	1	8	yes	complemented
22	C₂²×Dic₁₂₇	C₂×C₄	1	4	yes	—
23	C₂×C₁₂₇⋊C₈	—	2	8	no	—
24	C₂×C₁₂₇⋊C₈	—	2	2	no	—
25	C₂×C₁₂₇⋊C₈	—	2	2	no	—
26	C₂×C₁₂₇⋊C₈	C₂×C₄	1	4	yes	complemented
27	C₄×Dic₁₂₇	—	2	4	no	—
28	C₄×Dic₁₂₇	—	2	2	no	—
29	C₄×Dic₁₂₇	—	2	2	no	—
30	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
35	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
36	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
37	C₂²×C₄×C₁₂₇	Q₈	1	2	yes	—
38	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
39	C₂×C₈×C₁₂₇	—	2	8	no	—
40	C₂×C₈×C₁₂₇	—	2	2	no	—
41	C₂×C₈×C₁₂₇	—	2	2	no	—
42	C₂×C₈×C₁₂₇	C₂×C₄	1	4	yes	complemented
43	C₂×Dic₁₂₇	—	2	16	no	—
44	C₂×Dic₁₂₇	—	2	4	no	—
45	C₂×Dic₁₂₇	—	2	4	no	—
46	C₂×Dic₁₂₇	C₂×C₈	1	8	yes	complemented
47	C₁₂₇⋊C₈	—	2	4	no	—
48	C₂³×C₁₂₇	C₄²	1	2	yes	—
49	C₂³×C₁₂₇	C₂²⋊C₄	1	2	yes	—
50	C₂³×C₁₂₇	C₄⋊C₄	1	2	yes	—
51	C₂³×C₁₂₇	C₂×C₈	1	4	yes	—
52	C₂³×C₁₂₇	M₁₆	1	4	yes	—
53	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₂₇	C₄²	1	2	yes	—
55	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
56	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
57	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
58	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
59	C₂×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
60	C₂×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
61	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
67	C₈×C₁₂₇	—	2	4	no	—
68	C₂²×C₁₂₇	#95	1	4	yes	—
69	C₂²×C₁₂₇	C₄×C₈	1	4	yes	—
70	C₂²×C₁₂₇	#97	1	4	yes	—
71	C₂²×C₁₂₇	C₂²⋊C₈	1	8	yes	—
72	C₂²×C₁₂₇	C₄⋊C₈	1	8	yes	—
73	C₂²×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
74	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₁₂₇	C₂²×C₈	1	2	yes	—
77	C₂²×C₁₂₇	C₂×M₁₆	1	2	yes	—
78	C₄×C₁₂₇	#95	1	4	yes	—
79	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
83	Dic₁₂₇	—	2	8	no	—
84	C₂×C₁₂₇	#336	1	8	yes	complemented
85	C₂×C₁₂₇	C₂×#95	1	1	yes	characteristic
86	C₂×C₁₂₇	C₂×C₄×C₈	1	1	yes	characteristic
87	C₂×C₁₂₇	C₂×#97	1	1	yes	characteristic
88	C₂×C₁₂₇	C₂×C₂²⋊C₈	1	2	yes	—
89	C₂×C₁₂₇	C₂×C₄⋊C₈	1	2	yes	—
90	C₂×#336	—	127	1	no	maximal, Sylow
91	C₁₂₇	C₂×#336	1	1	yes	characteristic, complemented, Sylow
92	C₂³×C₈	—	127	1	no	—
93	C₂³×C₈	D₁₂₇	1	1	yes	characteristic
94	C₂²×C₄²	—	127	1	no	—
95	#336	—	127	4	no	—
96	C₂³×C₄	—	127	2	no	—
97	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
98	C₂²×C₈	—	2	2	no	—
99	C₂²×C₈	—	2	2	no	—
100	C₂²×C₈	—	127	4	no	—
101	C₂²×C₈	—	127	2	no	—
102	C₂²×C₈	—	254	2	no	—
103	C₂²×C₈	—	254	2	no	—
104	C₂²×C₈	Dic₁₂₇	1	4	yes	complemented
105	C₂²×C₈	C₂×D₁₂₇	1	2	yes	—
106	C₂×C₄²	—	127	2	no	—
107	C₂×C₄²	—	127	2	no	—
108	C₂×C₄²	—	254	2	no	—
109	C₂×C₄²	—	254	2	no	—
110	C₂⁴	C₄×D₁₂₇	1	1	yes	characteristic
111	C₂²×C₄	—	127	8	no	—
112	C₂²×C₄	—	127	4	no	—
113	C₂²×C₄	—	254	4	no	—
114	C₂²×C₄	—	254	4	no	—
115	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
116	C₂²×C₄	C₄×D₁₂₇	1	2	yes	—
117	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
118	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
119	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
120	C₂²×C₄	D₅₀₈	1	2	yes	—
121	C₂²×C₄	Dic₂₅₄	1	2	yes	—
122	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
123	C₂×C₈	—	2	8	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	127	4	no	—
127	C₂×C₈	—	254	8	no	—
128	C₂×C₈	—	254	2	no	—
129	C₂×C₈	—	254	2	no	—
130	C₂×C₈	C₂×Dic₁₂₇	1	4	yes	complemented
131	C₄²	—	254	4	no	—
132	C₄²	—	254	2	no	—
133	C₄²	—	254	2	no	—
134	C₂³	C₄×Dic₁₂₇	1	2	yes	—
135	C₂³	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
136	C₂³	Dic₁₂₇⋊C₄	1	2	yes	—
137	C₂³	C₈×D₁₂₇	1	4	yes	—
138	C₂³	#14523231	1	4	yes	—
139	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
140	C₂×C₄	—	127	8	no	—
141	C₂×C₄	—	254	4	no	—
142	C₂×C₄	—	254	16	no	—
143	C₂×C₄	—	254	4	no	—
144	C₂×C₄	C₄×Dic₁₂₇	1	2	yes	—
145	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
146	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	4	yes	—
147	C₂×C₄	C₂²⋊Dic₁₂₇	1	4	yes	—
148	C₂×C₄	Dic₁₂₇⋊C₄	1	2	yes	—
149	C₂×C₄	Dic₁₂₇⋊C₄	1	4	yes	—
150	C₂×C₄	C₄⋊Dic₁₂₇	1	4	yes	—
151	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
152	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
153	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
154	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
155	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
156	C₂×C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
157	C₈	—	2	4	no	—
158	C₈	—	254	4	no	—
159	C₂²	C₁₂₇⋊#95	1	4	yes	—
160	C₂²	C₈×Dic₁₂₇	1	4	yes	—
161	C₂²	#16861119	1	4	yes	—
162	C₂²	(C₂×D₁₂₇)⋊C₈	1	8	yes	—
163	C₂²	Dic₁₂₇⋊C₈	1	8	yes	—
164	C₂²	C₂×C₄×Dic₁₂₇	1	1	yes	characteristic
165	C₂²	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
166	C₂²	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
167	C₂²	C₂×C₈×D₁₂₇	1	2	yes	—
168	C₂²	C₂×#14523231	1	2	yes	—
169	C₄	—	254	8	no	—
170	C₄	C₁₂₇⋊#95	1	4	yes	—
171	C₄	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
172	C₄	C₂×C₂²⋊Dic₁₂₇	1	1	yes	characteristic
173	C₄	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
174	C₄	C₂×C₄⋊Dic₁₂₇	1	1	yes	characteristic
175	C₂	#22549630	1	8	yes	direct product factor
176	C₂	C₂×C₁₂₇⋊#95	1	1	yes	characteristic
177	C₂	C₂×C₈×Dic₁₂₇	1	1	yes	characteristic
178	C₂	C₂×#16861119	1	1	yes	characteristic
179	C₂	C₂×(C₂×D₁₂₇)⋊C₈	1	2	yes	—
180	C₂	C₂×Dic₁₂₇⋊C₈	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂×D₁₂₇
commutator	C₂×C₁₂₇	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂³×C₈×C₁₂₇	C₂
socle, prime radical	C₂⁴×C₁₂₇	C₂×C₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂³×C₈	D₁₂₇
hypocenter	C₁₂₇	C₂×#336
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
3. #22549630	C₂	direct (N × H)
8. C₂²×C₁₂₇⋊C₈	C₄	semidirect
17. C₂²×C₈×C₁₂₇	C₄	semidirect
21. C₂²×Dic₁₂₇	C₈	semidirect
26. C₂×C₁₂₇⋊C₈	C₂×C₄	semidirect
42. C₂×C₈×C₁₂₇	C₂×C₄	semidirect
46. C₂×Dic₁₂₇	C₂×C₈	semidirect
84. C₂×C₁₂₇	#336	semidirect
91. C₁₂₇	C₂×#336	semidirect
104. C₂²×C₈	Dic₁₂₇	semidirect
130. C₂×C₈	C₂×Dic₁₂₇	semidirect
175. C₂	#22549630	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₁₂₇⋊C₈	2
C₂²×C₄×Dic₁₂₇	2
#22549630	2
C₂³×C₈×C₁₂₇	2
C₂×#336	127

Subgroups — C2×#22549630 #30638420

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22549630 #30638420