Subgroups — #30641086

Order 16256.

696 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861168	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×Dic₁₂₇	C₂	1	4	yes	maximal, complemented
4	C₄×C₄⋊Dic₁₂₇	C₂	1	4	yes	maximal, complemented
5	C₂²⋊C₄×Dic₁₂₇	C₂	1	4	yes	maximal
6	C₂×#117×C₁₂₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₂₇	C₂²	1	2	yes	—
8	#16861168	C₄	1	8	yes	complemented
9	C₂×C₄⋊Dic₁₂₇	C₄	1	8	yes	complemented
10	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
13	C₂×C₂²⋊Dic₁₂₇	C₂²	1	2	yes	—
14	C₂×C₄×Dic₁₂₇	C₂²	1	8	yes	—
15	C₂×C₄×Dic₁₂₇	C₂²	1	2	yes	—
16	C₁₂₇⋊#95	C₂²	1	8	yes	—
17	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
18	#117×C₁₂₇	C₄	1	8	yes	complemented
19	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
21	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
22	C₂²×Dic₁₂₇	—	4	4	no	—
23	C₂²×Dic₁₂₇	C₂×C₄	1	8	yes	—
24	C₂²×Dic₁₂₇	C₂³	1	4	yes	—
25	C₂²×Dic₁₂₇	C₂³	1	4	yes	—
26	C₄⋊Dic₁₂₇	C₂×C₄	1	8	yes	—
27	C₄⋊Dic₁₂₇	C₂×C₄	1	8	yes	—
28	C₄⋊Dic₁₂₇	C₂×C₄	1	8	yes	complemented
29	C₂²⋊Dic₁₂₇	C₂×C₄	1	8	yes	—
30	C₄×Dic₁₂₇	—	4	8	no	—
31	C₄×Dic₁₂₇	C₂×C₄	1	8	yes	complemented
32	C₂⁴×C₁₂₇	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
34	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
35	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
36	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
37	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₁₂₇	C₂×C₄	1	8	yes	complemented
40	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	8	yes	—
41	C₄²×C₁₂₇	C₂×C₄	1	8	yes	complemented
42	C₂×Dic₁₂₇	—	2	4	no	—
43	C₂×Dic₁₂₇	—	2	8	no	—
44	C₂×Dic₁₂₇	—	2	4	no	—
45	C₂×Dic₁₂₇	—	4	8	no	—
46	C₂×Dic₁₂₇	C₂²×C₄	1	8	yes	—
47	C₂×Dic₁₂₇	C₂²×C₄	1	8	yes	—
48	C₂³×C₁₂₇	—	2	2	no	—
49	C₂³×C₁₂₇	—	2	2	no	—
50	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
51	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
52	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
53	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
54	C₂×C₄×C₁₂₇	—	2	4	no	—
55	C₂×C₄×C₁₂₇	—	2	4	no	—
56	C₂×C₄×C₁₂₇	C₄²	1	16	yes	complemented
57	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₁₂₇	C₂²×C₄	1	8	yes	—
62	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₁₂₇	—	2	2	no	—
65	C₂²×C₁₂₇	—	2	2	no	—
66	C₂²×C₁₂₇	—	2	4	no	—
67	C₂²×C₁₂₇	—	2	4	no	—
68	C₂²×C₁₂₇	C₂×C₄²	1	4	yes	—
69	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
70	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
71	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
72	C₂²×C₁₂₇	H(5,𝔽₂)	1	2	yes	—
73	C₂²×C₁₂₇	2₋¹⁺⁴	1	2	yes	—
74	C₄×C₁₂₇	—	2	8	no	—
75	C₄×C₁₂₇	C₂×C₄²	1	4	yes	—
76	C₄×C₁₂₇	C₂×C₄²	1	4	yes	—
77	Dic₁₂₇	—	2	8	no	—
78	Dic₁₂₇	—	2	8	no	—
79	C₂×C₁₂₇	—	2	4	no	—
80	C₂×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
81	C₂×C₁₂₇	#518	1	1	yes	characteristic
82	C₂×C₁₂₇	#519	1	1	yes	characteristic
83	C₂×C₁₂₇	#520	1	2	yes	—
84	C₂×C₁₂₇	#520	1	2	yes	—
85	#2274	—	127	1	no	maximal, Sylow
86	C₁₂₇	#2274	1	1	yes	characteristic, complemented, Sylow
87	C₂×#117	—	127	1	no	—
88	C₂×#117	D₁₂₇	1	1	yes	characteristic
89	C₂²×C₄⋊C₄	—	127	1	no	—
90	C₄×C₄⋊C₄	—	127	4	no	—
91	C₄×C₄⋊C₄	—	127	4	no	—
92	C₄×C₂²⋊C₄	—	127	4	no	—
93	C₂³×C₄	—	127	2	no	—
94	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
95	#117	—	127	8	no	—
96	#117	Dic₁₂₇	1	8	yes	complemented
97	C₂×C₄⋊C₄	—	127	8	no	—
98	C₂×C₄⋊C₄	—	127	2	no	—
99	C₂×C₄⋊C₄	—	127	2	no	—
100	C₂×C₄⋊C₄	—	127	2	no	—
101	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
102	C₂×C₂²⋊C₄	—	127	2	no	—
103	C₂×C₂²⋊C₄	C₂×D₁₂₇	1	2	yes	—
104	C₂×C₄²	—	127	2	no	—
105	C₂×C₄²	—	127	8	no	—
106	C₂×C₄²	C₂×D₁₂₇	1	2	yes	—
107	#95	—	127	8	no	—
108	C₂⁴	C₂²×D₁₂₇	1	1	yes	characteristic
109	C₂²×C₄	—	127	8	no	—
110	C₂²×C₄	—	127	4	no	—
111	C₂²×C₄	—	127	4	no	—
112	C₂²×C₄	—	508	4	no	—
113	C₂²×C₄	C₄×D₁₂₇	1	4	yes	—
114	C₂²×C₄	C₄×D₁₂₇	1	4	yes	—
115	C₂²×C₄	C₂×Dic₁₂₇	1	4	yes	—
116	C₂²×C₄	C₂²×D₁₂₇	1	4	yes	—
117	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
118	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
119	C₄⋊C₄	—	127	8	no	—
120	C₄⋊C₄	—	127	8	no	—
121	C₄⋊C₄	—	127	8	no	—
122	C₄⋊C₄	C₂×Dic₁₂₇	1	8	yes	complemented
123	C₂²⋊C₄	—	127	8	no	—
124	C₂²⋊C₄	C₂×Dic₁₂₇	1	8	yes	—
125	C₄²	—	127	8	no	—
126	C₄²	—	508	8	no	—
127	C₄²	C₂×Dic₁₂₇	1	8	yes	complemented
128	C₂³	—	2	2	no	—
129	C₂³	—	2	2	no	—
130	C₂³	C₂×C₄×D₁₂₇	1	2	yes	—
131	C₂³	C₂×C₄×D₁₂₇	1	2	yes	—
132	C₂³	C₂²×Dic₁₂₇	1	2	yes	—
133	C₂³	C₂³×D₁₂₇	1	1	yes	characteristic
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	127	8	no	—
137	C₂×C₄	—	127	8	no	—
138	C₂×C₄	—	508	8	no	—
139	C₂×C₄	—	254	4	no	—
140	C₂×C₄	—	254	8	no	—
141	C₂×C₄	—	254	4	no	—
142	C₂×C₄	C₄×Dic₁₂₇	1	16	yes	complemented
143	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
144	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
145	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
146	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
147	C₂×C₄	C₂²×Dic₁₂₇	1	8	yes	—
148	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
149	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
150	C₂²	—	2	2	no	—
151	C₂²	—	2	2	no	—
152	C₂²	—	2	4	no	—
153	C₂²	—	2	4	no	—
154	C₂²	C₂×C₄×Dic₁₂₇	1	4	yes	—
155	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
156	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
157	C₂²	C₂³×Dic₁₂₇	1	1	yes	characteristic
158	C₂²	#16861255	1	2	yes	—
159	C₂²	#16861256	1	2	yes	—
160	C₄	—	2	8	no	—
161	C₄	—	254	8	no	—
162	C₄	—	254	8	no	—
163	C₄	C₂×C₄×Dic₁₂₇	1	4	yes	—
164	C₄	C₂×C₄×Dic₁₂₇	1	4	yes	—
165	C₂	—	2	4	no	—
166	C₂	C₂²×C₄×Dic₁₂₇	1	1	yes	characteristic
167	C₂	#22550350	1	1	yes	characteristic
168	C₂	#22550353	1	1	yes	characteristic
169	C₂	#22550356	1	2	yes	—
170	C₂	#22550359	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₂₇
commutator	C₂×C₁₂₇	C₂²×C₄²
Fitting	C₂×#117×C₁₂₇	C₂
socle, cosocle	C₂³×C₁₂₇	C₂⁴
prime radical	C₂⁴×C₁₂₇	C₂³
hypercenter	C₂×#117	D₁₂₇
hypocenter	C₁₂₇	#2274
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. C₄⋊C₄×Dic₁₂₇	C₂	semidirect
4. C₄×C₄⋊Dic₁₂₇	C₂	semidirect
8. #16861168	C₄	semidirect
9. C₂×C₄⋊Dic₁₂₇	C₄	semidirect
18. #117×C₁₂₇	C₄	semidirect
28. C₄⋊Dic₁₂₇	C₂×C₄	semidirect
31. C₄×Dic₁₂₇	C₂×C₄	semidirect
39. C₄⋊C₄×C₁₂₇	C₂×C₄	semidirect
41. C₄²×C₁₂₇	C₂×C₄	semidirect
56. C₂×C₄×C₁₂₇	C₄²	semidirect
86. C₁₂₇	#2274	semidirect
96. #117	Dic₁₂₇	semidirect
122. C₄⋊C₄	C₂×Dic₁₂₇	semidirect
127. C₄²	C₂×Dic₁₂₇	semidirect
142. C₂×C₄	C₄×Dic₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861168	2
C₂²×C₄⋊Dic₁₂₇	2
C₄⋊C₄×Dic₁₂₇	2
C₄×C₄⋊Dic₁₂₇	2
C₂²⋊C₄×Dic₁₂₇	2
C₂×#117×C₁₂₇	2
#2274	127