Subgroups — #30647363

Order 16256.

484 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	#22549933	C₂	1	4	yes	maximal, complemented
2	#22549929	C₂	1	4	yes	maximal, complemented
3	#22549900	C₂	1	2	yes	maximal, complemented
4	#22549840	C₂	1	2	yes	maximal
5	C₄×#14523232	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄²×C₁₂₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#14523232	C₂²	1	4	yes	—
8	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	8	yes	complemented
9	C₂²⋊(C₁₂₇⋊C₈)	C₂²	1	8	yes	—
10	#16861120	C₂²	1	4	yes	complemented
11	C₄×C₁₂₇⋊C₈	C₂²	1	4	yes	complemented
12	C₂³×C₄×C₁₂₇	C₄	1	2	yes	—
13	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₂₇	C₄	1	2	yes	—
15	C₂×C₄²×C₁₂₇	C₄	1	2	yes	—
16	C₂×C₄²×C₁₂₇	C₄	1	2	yes	—
17	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
20	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
21	#14523232	—	2	8	no	—
22	C₂×C₁₂₇⋊C₈	C₂³	1	8	yes	—
23	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₂₇	—	2	8	no	—
25	C₂²×C₄×C₁₂₇	—	2	4	no	—
26	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
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	C₂³	1	2	yes	—
35	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
36	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
37	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
38	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
39	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
40	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
41	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
42	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
44	C₄²×C₁₂₇	C₂³	1	2	yes	—
45	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
46	C₁₂₇⋊C₈	—	2	8	no	—
47	C₂³×C₁₂₇	—	2	2	no	—
48	C₂³×C₁₂₇	—	2	4	no	—
49	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
50	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₁₂₇	—	2	2	no	—
52	C₂×C₄×C₁₂₇	—	2	4	no	—
53	C₂×C₄×C₁₂₇	—	2	4	no	—
54	C₂×C₄×C₁₂₇	—	2	4	no	—
55	C₂×C₄×C₁₂₇	—	2	4	no	—
56	C₂×C₄×C₁₂₇	—	2	4	no	—
57	C₂×C₄×C₁₂₇	—	2	4	no	—
58	C₂×C₄×C₁₂₇	—	2	4	no	—
59	C₂×C₄×C₁₂₇	M₁₆	1	8	yes	complemented
60	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₁₂₇	#41	1	4	yes	—
67	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
68	C₂²×C₁₂₇	—	2	8	no	—
69	C₂²×C₁₂₇	—	2	2	no	—
70	C₂²×C₁₂₇	—	2	2	no	—
71	C₂²×C₁₂₇	—	2	2	no	—
72	C₂²×C₁₂₇	#117	1	2	yes	—
73	C₂²×C₁₂₇	C₂×M₁₆	1	4	yes	—
74	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
75	C₄×C₁₂₇	—	2	2	no	—
76	C₄×C₁₂₇	—	2	2	no	—
77	C₄×C₁₂₇	—	2	2	no	—
78	C₄×C₁₂₇	#117	1	2	yes	—
79	C₄×C₁₂₇	C₂×M₁₆	1	4	yes	—
80	C₄×C₁₂₇	C₂×M₁₆	1	4	yes	—
81	C₄×C₁₂₇	C₂×#41	1	2	yes	—
82	C₂×C₁₂₇	—	2	2	no	—
83	C₂×C₁₂₇	—	2	4	no	—
84	C₂×C₁₂₇	C₂×#117	1	1	yes	characteristic
85	C₂×C₁₂₇	C₂²×M₁₆	1	2	yes	—
86	#2920	—	127	1	no	maximal, Sylow
87	C₁₂₇	#2920	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₄²	D₁₂₇	1	1	yes	characteristic
89	#432	—	127	4	no	—
90	#431	—	127	4	no	—
91	#423	—	127	2	no	—
92	#407	—	127	2	no	—
93	C₄×M₁₆	—	127	2	no	—
94	C₂³×C₄	Dic₁₂₇	1	2	yes	—
95	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
96	C₂×M₁₆	—	127	4	no	—
97	C₂×C₄²	Dic₁₂₇	1	2	yes	—
98	C₂×C₄²	Dic₁₂₇	1	2	yes	—
99	C₂×C₄²	Dic₁₂₇	1	2	yes	—
100	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
101	C₂×C₄²	C₂×D₁₂₇	1	2	yes	—
102	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
103	C₂×C₄²	C₂×D₁₂₇	1	2	yes	—
104	C₄⋊C₈	—	127	8	no	—
105	C₂²⋊C₈	—	127	8	no	—
106	#97	—	127	4	no	—
107	C₄×C₈	—	127	4	no	—
108	C₂⁴	C₂×Dic₁₂₇	1	1	yes	characteristic
109	C₂²×C₄	—	2	8	no	—
110	C₂²×C₄	—	2	4	no	—
111	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
112	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
113	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
114	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
115	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
116	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
117	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
118	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
119	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
120	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
121	M₁₆	—	254	8	no	—
122	C₂×C₈	—	127	8	no	—
123	C₄²	C₂×Dic₁₂₇	1	2	yes	—
124	C₄²	C₂×Dic₁₂₇	1	2	yes	—
125	C₄²	C₂×Dic₁₂₇	1	2	yes	—
126	C₄²	C₂×Dic₁₂₇	1	2	yes	—
127	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
128	C₄²	C₂×Dic₁₂₇	1	2	yes	—
129	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
130	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
131	C₄²	C₂²×D₁₂₇	1	2	yes	—
132	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
133	C₂³	—	2	4	no	—
134	C₂³	—	2	2	no	—
135	C₂³	C₂²×Dic₁₂₇	1	2	yes	—
136	C₂³	C₂²×Dic₁₂₇	1	1	yes	characteristic
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	#14523232	1	8	yes	complemented
146	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
147	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
148	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
149	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
150	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
151	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
152	C₂×C₄	#14523248	1	4	yes	—
153	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
154	C₈	—	254	8	no	—
155	C₂²	—	2	8	no	—
156	C₂²	—	2	2	no	—
157	C₂²	—	2	2	no	—
158	C₂²	—	2	2	no	—
159	C₂²	#16861168	1	2	yes	—
160	C₂²	C₂×#14523232	1	4	yes	—
161	C₂²	C₂³×Dic₁₂₇	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	2	2	no	—
165	C₄	#16861168	1	2	yes	—
166	C₄	C₂×#14523232	1	4	yes	—
167	C₄	C₂×#14523232	1	4	yes	—
168	C₄	C₂×#14523248	1	2	yes	—
169	C₂	—	2	4	no	—
170	C₂	—	2	2	no	—
171	C₂	C₂×#16861168	1	1	yes	characteristic
172	C₂	C₂²×#14523232	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂²×C₄²×C₁₂₇	C₂
prime radical	C₂⁴×C₁₂₇	C₂×C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₂²×C₄²	D₁₂₇
hypocenter	C₁₂₇	#2920
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄² ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. #22549933	C₂	semidirect
2. #22549929	C₂	semidirect
3. #22549900	C₂	semidirect
5. C₄×#14523232	C₂	semidirect
8. C₄⋊(C₁₂₇⋊C₈)	C₂²	semidirect
10. #16861120	C₂²	semidirect
11. C₄×C₁₂₇⋊C₈	C₂²	semidirect
59. C₂×C₄×C₁₂₇	M₁₆	semidirect
87. C₁₂₇	#2920	semidirect
145. C₂×C₄	#14523232	semidirect

Maximal subgroups

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

subgroup	index
#22549933	2
#22549929	2
#22549900	2
#22549840	2
C₄×#14523232	2
C₂²×C₄²×C₁₂₇	2
#2920	127