Subgroups — #30638446

Order 16256.

424 conjugacy classes of subgroups.

The subgroup lattice has 174 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	C₂×#16861168	C₂	1	1	yes	maximal, characteristic
3	#22549638	C₂	1	4	yes	maximal, complemented
4	C₂×#117×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₂₇⋊C₈	—	2	2	no	—
6	C₂²×C₁₂₇⋊C₈	—	2	2	no	—
7	C₂²×C₁₂₇⋊C₈	C₄	1	4	yes	complemented
8	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
9	#16861168	—	2	2	no	—
10	#16861168	C₄	1	4	yes	complemented
11	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
12	C₂×C₂²⋊Dic₁₂₇	—	2	1	no	—
13	C₂×C₄×Dic₁₂₇	—	2	1	no	—
14	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
15	#117×C₁₂₇	—	2	2	no	—
16	#117×C₁₂₇	C₄	1	4	yes	complemented
17	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₁₂₇	—	2	1	no	—
19	C₂×C₄²×C₁₂₇	—	2	1	no	—
20	C₂²×Dic₁₂₇	—	2	2	no	—
21	C₂×C₁₂₇⋊C₈	—	2	8	no	—
22	C₂×C₁₂₇⋊C₈	—	2	2	no	—
23	C₂×C₁₂₇⋊C₈	—	2	2	no	—
24	C₂×C₁₂₇⋊C₈	C₂×C₄	1	4	yes	—
25	C₄⋊Dic₁₂₇	—	2	2	no	—
26	C₄⋊Dic₁₂₇	C₂×C₄	1	4	yes	—
27	C₂²⋊Dic₁₂₇	—	2	4	no	—
28	C₄×Dic₁₂₇	—	2	4	no	—
29	C₂⁴×C₁₂₇	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₂₇	—	2	2	no	—
31	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
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	1	yes	characteristic
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	C₄⋊C₄×C₁₂₇	—	2	2	no	—
40	C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
41	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
42	C₄²×C₁₂₇	—	2	4	no	—
43	C₂×Dic₁₂₇	—	2	4	no	—
44	C₂×Dic₁₂₇	—	4	4	no	—
45	C₁₂₇⋊C₈	—	2	4	no	—
46	C₂³×C₁₂₇	—	2	4	no	—
47	C₂³×C₁₂₇	C₂²⋊C₄	1	2	yes	—
48	C₂³×C₁₂₇	C₂²⋊C₄	1	2	yes	—
49	C₂³×C₁₂₇	C₄⋊C₄	1	2	yes	—
50	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₂₇	—	2	4	no	—
52	C₂×C₄×C₁₂₇	—	4	4	no	—
53	C₂×C₄×C₁₂₇	C₄²	1	4	yes	—
54	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
55	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
56	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
57	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
58	C₂×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
59	C₂×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
60	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
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×Q₈	1	1	yes	characteristic
66	C₂²×C₁₂₇	—	2	4	no	—
67	C₂²×C₁₂₇	—	2	4	no	—
68	C₂²×C₁₂₇	—	2	4	no	—
69	C₂²×C₁₂₇	#95	1	4	yes	—
70	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₁₂₇	#135	1	4	yes	—
74	C₄×C₁₂₇	—	4	4	no	—
75	C₄×C₁₂₇	#95	1	4	yes	—
76	C₄×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
77	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
80	Dic₁₂₇	—	4	4	no	—
81	C₂×C₁₂₇	—	2	4	no	—
82	C₂×C₁₂₇	C₂×#95	1	1	yes	characteristic
83	C₂×C₁₂₇	#416	1	2	yes	—
84	C₂×C₁₂₇	#416	1	2	yes	—
85	C₂×C₁₂₇	#427	1	2	yes	—
86	#1735	—	127	1	no	maximal, Sylow
87	C₁₂₇	#1735	1	1	yes	characteristic, complemented, Sylow
88	C₂³×C₈	—	127	1	no	—
89	C₂×#117	—	127	1	no	—
90	C₂×#117	D₁₂₇	1	1	yes	characteristic
91	#340	—	127	4	no	—
92	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
93	C₂²×C₈	—	127	4	no	—
94	C₂²×C₈	—	127	2	no	—
95	C₂²×C₈	—	254	2	no	—
96	C₂²×C₈	—	254	2	no	—
97	#117	—	2	2	no	—
98	#117	—	127	4	no	—
99	#117	—	254	2	no	—
100	#117	Dic₁₂₇	1	4	yes	complemented
101	C₂×C₄⋊C₄	—	127	2	no	—
102	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
103	C₂×C₂²⋊C₄	—	2	1	no	—
104	C₂×C₂²⋊C₄	—	254	1	no	—
105	C₂×C₄²	—	2	1	no	—
106	C₂×C₄²	—	254	1	no	—
107	C₂⁴	C₂²⋊D₁₂₇	1	1	yes	characteristic
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	254	2	no	—
110	C₂²×C₄	C₄×D₁₂₇	1	2	yes	—
111	C₂²×C₄	C₄×D₁₂₇	1	2	yes	—
112	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
113	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
114	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
115	C₂²×C₄	D₅₀₈	1	2	yes	—
116	C₂²×C₄	Dic₂₅₄	1	2	yes	—
117	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
118	C₂×C₈	—	127	4	no	—
119	C₂×C₈	—	254	8	no	—
120	C₂×C₈	—	254	2	no	—
121	C₂×C₈	—	254	2	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	127	4	no	—
124	C₄⋊C₄	—	254	2	no	—
125	C₄⋊C₄	C₂×Dic₁₂₇	1	4	yes	—
126	C₂²⋊C₄	—	2	4	no	—
127	C₂²⋊C₄	—	254	4	no	—
128	C₄²	—	2	4	no	—
129	C₄²	—	254	4	no	—
130	C₂³	—	2	4	no	—
131	C₂³	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
132	C₂³	C₂²⋊Dic₁₂₇	1	2	yes	—
133	C₂³	Dic₁₂₇⋊C₄	1	2	yes	—
134	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	—	508	4	no	—
138	C₂×C₄	—	254	4	no	—
139	C₂×C₄	C₄×Dic₁₂₇	1	4	yes	—
140	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
141	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	4	yes	—
142	C₂×C₄	C₂²⋊Dic₁₂₇	1	2	yes	—
143	C₂×C₄	Dic₁₂₇⋊C₄	1	2	yes	—
144	C₂×C₄	Dic₁₂₇⋊C₄	1	4	yes	—
145	C₂×C₄	C₄⋊Dic₁₂₇	1	4	yes	—
146	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
147	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
148	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
149	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
150	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
151	C₂×C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
152	C₈	—	254	4	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	2	4	no	—
155	C₂²	—	2	4	no	—
156	C₂²	C₁₂₇⋊#95	1	4	yes	—
157	C₂²	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
158	C₂²	C₂×C₂²⋊Dic₁₂₇	1	1	yes	characteristic
159	C₂²	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
160	C₂²	#16861229	1	4	yes	—
161	C₄	—	4	4	no	—
162	C₄	—	508	4	no	—
163	C₄	C₁₂₇⋊#95	1	4	yes	—
164	C₄	C₂×C₄×Dic₁₂₇	1	1	yes	characteristic
165	C₄	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
166	C₄	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
167	C₄	C₂×C₄⋊Dic₁₂₇	1	1	yes	characteristic
168	C₂	—	2	4	no	—
169	C₂	C₂×C₁₂₇⋊#95	1	1	yes	characteristic
170	C₂	#22549870	1	2	yes	—
171	C₂	#22549872	1	2	yes	—
172	C₂	#22549912	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊D₁₂₇
commutator	C₄×C₁₂₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂×#117×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂×D₄
prime radical	C₂⁴×C₁₂₇	D₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂×#117	D₁₂₇
hypocenter	C₁₂₇	#1735
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #22549638	C₂	semidirect
7. C₂²×C₁₂₇⋊C₈	C₄	semidirect
10. #16861168	C₄	semidirect
16. #117×C₁₂₇	C₄	semidirect
87. C₁₂₇	#1735	semidirect
100. #117	Dic₁₂₇	semidirect

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
C₂×#16861168	2
#22549638	2
C₂×#117×C₁₂₇	2
#1735	127