Subgroups — #30638456

Order 16256.

372 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523232	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14523232	C₂	1	1	yes	maximal, characteristic
3	#22549640	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523232	—	2	1	no	—
6	C₂×#14523232	—	2	2	no	—
7	C₂×#14523232	—	2	2	no	—
8	C₂×#14523232	—	2	2	no	—
9	C₂×#14523232	C₄	1	4	yes	—
10	C₂×#14523232	C₄	1	2	yes	—
11	C₂×#14523232	C₄	1	2	yes	—
12	C₂×#14523232	C₂²	1	2	yes	—
13	C₂²×C₁₂₇⋊C₈	—	2	1	no	—
14	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
15	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₁₂₇	—	2	2	no	—
17	C₂×M₁₆×C₁₂₇	—	2	1	no	—
18	C₂×M₁₆×C₁₂₇	C₄	1	4	yes	—
19	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
20	C₂²×C₈×C₁₂₇	—	2	1	no	—
21	#14523232	—	2	4	no	—
22	#14523232	—	2	4	no	—
23	#14523232	—	2	4	no	—
24	#14523232	—	2	2	no	—
25	#14523232	C₂×C₄	1	4	yes	—
26	C₂×C₁₂₇⋊C₈	—	2	4	no	—
27	C₂×C₁₂₇⋊C₈	—	2	2	no	—
28	C₂×C₁₂₇⋊C₈	—	2	2	no	—
29	C₂×C₁₂₇⋊C₈	—	2	2	no	—
30	C₂×C₁₂₇⋊C₈	C₂×C₄	1	4	yes	—
31	C₂⁴×C₁₂₇	Q₈	1	1	yes	characteristic
32	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
39	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
40	C₂²×C₄×C₁₂₇	Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
42	M₁₆×C₁₂₇	—	2	4	no	—
43	M₁₆×C₁₂₇	—	2	2	no	—
44	M₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
45	C₂×C₈×C₁₂₇	—	2	4	no	—
46	C₂×C₈×C₁₂₇	—	2	2	no	—
47	C₁₂₇⋊C₈	—	2	4	no	—
48	C₁₂₇⋊C₈	—	2	4	no	—
49	C₂³×C₁₂₇	—	4	2	no	—
50	C₂³×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂³×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
52	C₂³×C₁₂₇	C₄⋊C₄	1	2	yes	—
53	C₂³×C₁₂₇	C₄⋊C₄	1	2	yes	—
54	C₂³×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₄×C₁₂₇	C₄²	1	2	yes	—
56	C₂×C₄×C₁₂₇	C₄²	1	2	yes	—
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	2	yes	—
62	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
63	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
64	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
67	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
73	C₈×C₁₂₇	—	2	4	no	—
74	C₂²×C₁₂₇	—	2	2	no	—
75	C₂²×C₁₂₇	—	2	2	no	—
76	C₂²×C₁₂₇	—	2	2	no	—
77	C₂²×C₁₂₇	—	4	2	no	—
78	C₂²×C₁₂₇	—	4	2	no	—
79	C₂²×C₁₂₇	#95	1	2	yes	—
80	C₂²×C₁₂₇	#95	1	2	yes	—
81	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₄×C₁₂₇	#95	1	2	yes	—
85	C₄×C₁₂₇	#95	1	2	yes	—
86	C₄×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
87	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₁₂₇	—	2	2	no	—
91	C₂×C₁₂₇	—	2	2	no	—
92	C₂×C₁₂₇	—	2	1	no	—
93	C₂×C₁₂₇	—	2	1	no	—
94	C₂×C₁₂₇	C₂×#95	1	1	yes	characteristic
95	C₂×C₁₂₇	#430	1	2	yes	—
96	#1738	—	127	1	no	maximal, Sylow
97	C₁₂₇	#1738	1	1	yes	characteristic, complemented, Sylow
98	C₂²×M₁₆	—	127	1	no	—
99	C₂²×M₁₆	—	127	1	no	—
100	C₂²×M₁₆	D₁₂₇	1	1	yes	characteristic
101	#341	—	127	4	no	—
102	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
103	C₂×M₁₆	—	2	2	no	—
104	C₂×M₁₆	—	2	1	no	—
105	C₂×M₁₆	—	127	2	no	—
106	C₂×M₁₆	—	127	2	no	—
107	C₂×M₁₆	—	127	4	no	—
108	C₂×M₁₆	—	127	2	no	—
109	C₂×M₁₆	—	254	2	no	—
110	C₂×M₁₆	—	254	1	no	—
111	C₂×M₁₆	—	254	2	no	—
112	C₂×M₁₆	—	254	2	no	—
113	C₂×M₁₆	Dic₁₂₇	1	4	yes	—
114	C₂×M₁₆	C₂×D₁₂₇	1	2	yes	—
115	C₂²×C₈	—	2	1	no	—
116	C₂²×C₈	—	127	2	no	—
117	C₂²×C₈	—	254	1	no	—
118	C₂⁴	Dic₂₅₄	1	1	yes	characteristic
119	C₂²×C₄	C₄×D₁₂₇	1	2	yes	—
120	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
121	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
122	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
123	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
124	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
125	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
126	C₂²×C₄	D₅₀₈	1	2	yes	—
127	C₂²×C₄	Dic₂₅₄	1	1	yes	characteristic
128	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
129	M₁₆	—	2	4	no	—
130	M₁₆	—	2	2	no	—
131	M₁₆	—	127	4	no	—
132	M₁₆	—	254	4	no	—
133	M₁₆	—	254	4	no	—
134	M₁₆	—	254	4	no	—
135	M₁₆	—	254	2	no	—
136	M₁₆	C₂×Dic₁₂₇	1	4	yes	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	127	4	no	—
140	C₂×C₈	—	254	4	no	—
141	C₂×C₈	—	254	2	no	—
142	C₂×C₈	—	254	2	no	—
143	C₂×C₈	—	254	2	no	—
144	C₂³	—	4	2	no	—
145	C₂³	Dic₁₂₇⋊C₄	1	1	yes	characteristic
146	C₂³	Dic₁₂₇⋊C₄	1	1	yes	characteristic
147	C₂³	Dic₁₂₇⋊C₄	1	2	yes	—
148	C₂³	C₄⋊Dic₁₂₇	1	2	yes	—
149	C₂³	C₂×Dic₂₅₄	1	1	yes	characteristic
150	C₂×C₄	C₄×Dic₁₂₇	1	2	yes	—
151	C₂×C₄	C₄×Dic₁₂₇	1	2	yes	—
152	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
153	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
154	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
155	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
156	C₂×C₄	C₂²⋊Dic₁₂₇	1	2	yes	—
157	C₂×C₄	C₂²⋊Dic₁₂₇	1	2	yes	—
158	C₂×C₄	Dic₁₂₇⋊C₄	1	2	yes	—
159	C₂×C₄	Dic₁₂₇⋊C₄	1	1	yes	characteristic
160	C₂×C₄	Dic₁₂₇⋊C₄	1	1	yes	characteristic
161	C₂×C₄	C₄⋊Dic₁₂₇	1	2	yes	—
162	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
163	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
164	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
165	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
166	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
167	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
168	C₈	—	2	4	no	—
169	C₈	—	254	4	no	—
170	C₈	—	254	4	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	2	2	no	—
173	C₂²	—	2	2	no	—
174	C₂²	—	4	2	no	—
175	C₂²	—	4	2	no	—
176	C₂²	C₁₂₇⋊#95	1	2	yes	—
177	C₂²	C₁₂₇⋊#95	1	2	yes	—
178	C₂²	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
179	C₂²	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
180	C₂²	C₂×C₄⋊Dic₁₂₇	1	1	yes	characteristic
181	C₄	C₁₂₇⋊#95	1	2	yes	—
182	C₄	C₁₂₇⋊#95	1	2	yes	—
183	C₄	C₂×C₄×Dic₁₂₇	1	1	yes	characteristic
184	C₄	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
185	C₄	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
186	C₄	C₂×C₂²⋊Dic₁₂₇	1	1	yes	characteristic
187	C₂	—	2	2	no	—
188	C₂	—	2	2	no	—
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	C₂×C₁₂₇⋊#95	1	1	yes	characteristic
192	C₂	#22549924	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₂₇
commutator	C₄×C₁₂₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂²×M₁₆×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₂₇	Q₈
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂²×M₁₆	D₁₂₇
hypocenter	C₁₂₇	#1738
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #22549640	C₂	semidirect
97. C₁₂₇	#1738	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14523232	2
#22549640	2
C₂²×M₁₆×C₁₂₇	2
#1738	127