Subgroups — #30647681

Order 16256.

388 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₅₄	C₂	1	1	yes	maximal, characteristic
2	#22550005	C₂	1	4	yes	maximal, complemented
3	#22550003	C₂	1	2	yes	maximal, complemented
4	#22550000	C₂	1	2	yes	maximal, complemented
5	#22549899	C₂	1	2	yes	maximal
6	C₄×#14523232	C₂	1	1	yes	maximal, characteristic
7	M₁₆×Dic₁₂₇	C₂	1	2	yes	maximal
8	#423×C₁₂₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₅₄	C₄	1	1	yes	characteristic
10	C₂×#14523232	C₂²	1	1	yes	characteristic
11	C₂×#14523232	C₂²	1	1	yes	characteristic
12	C₄×Dic₂₅₄	C₄	1	2	yes	—
13	C₄×Dic₂₅₄	C₄	1	1	yes	characteristic
14	C₄×Dic₂₅₄	C₄	1	1	yes	characteristic
15	C₄×Dic₂₅₄	C₂²	1	1	yes	characteristic
16	C₄×Dic₂₅₄	C₂²	1	1	yes	characteristic
17	C₄×Dic₂₅₄	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₁₂₇	C₄	1	1	yes	characteristic
19	C₂×Dic₁₂₇⋊C₄	C₄	1	2	yes	—
20	C₂×C₄×Dic₁₂₇	C₂²	1	2	yes	—
21	Dic₁₂₇⋊C₈	C₂²	1	4	yes	—
22	Dic₁₂₇⋊C₈	C₂²	1	4	yes	—
23	#16861120	C₂²	1	2	yes	—
24	#16861119	C₂²	1	4	yes	—
25	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
26	C₈×Dic₁₂₇	C₂²	1	4	yes	—
27	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
28	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
29	C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
30	C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
31	C₂×Dic₂₅₄	—	2	4	no	—
32	C₂×Dic₂₅₄	C₂×C₄	1	2	yes	—
33	C₂×Dic₂₅₄	C₂×C₄	1	1	yes	characteristic
34	C₂×Dic₂₅₄	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₁₂₇	C₂×C₄	1	2	yes	—
36	C₂²×Dic₁₂₇	C₂×C₄	1	2	yes	—
37	#14523232	—	2	2	no	—
38	#14523232	Q₈	1	4	yes	complemented
39	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
40	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
41	C₄⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
42	C₄⋊Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
43	C₄⋊Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
44	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
45	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
46	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
47	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
48	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
49	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
50	C₄×Dic₁₂₇	C₂³	1	2	yes	—
51	C₄×Dic₁₂₇	C₂³	1	2	yes	—
52	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
55	M₁₆×C₁₂₇	—	2	4	no	—
56	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
57	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
58	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
59	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
60	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
61	Dic₂₅₄	—	2	2	no	—
62	Dic₂₅₄	—	2	4	no	—
63	Dic₂₅₄	M₁₆	1	4	yes	complemented
64	C₂×Dic₁₂₇	—	2	4	no	—
65	C₂×Dic₁₂₇	—	4	2	no	—
66	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
67	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
68	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
69	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
70	C₁₂₇⋊C₈	—	2	2	no	—
71	C₁₂₇⋊C₈	C₂×Q₈	1	4	yes	complemented
72	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₂₇	—	2	2	no	—
74	C₂×C₄×C₁₂₇	—	4	1	no	—
75	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
81	C₂×C₄×C₁₂₇	#41	1	2	yes	—
82	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
83	C₈×C₁₂₇	—	2	4	no	—
84	C₂²×C₁₂₇	—	2	1	no	—
85	C₂²×C₁₂₇	—	2	1	no	—
86	C₂²×C₁₂₇	C₄×Q₈	1	2	yes	—
87	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
88	C₄×C₁₂₇	—	2	1	no	—
89	C₄×C₁₂₇	—	2	1	no	—
90	C₄×C₁₂₇	—	2	1	no	—
91	C₄×C₁₂₇	C₄×Q₈	1	2	yes	—
92	C₄×C₁₂₇	C₂×M₁₆	1	2	yes	—
93	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
94	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
95	Dic₁₂₇	—	2	2	no	—
96	Dic₁₂₇	—	2	2	no	—
97	Dic₁₂₇	—	2	2	no	—
98	Dic₁₂₇	C₂×M₁₆	1	4	yes	—
99	C₂×C₁₂₇	—	2	2	no	—
100	C₂×C₁₂₇	C₂×C₄×Q₈	1	1	yes	characteristic
101	C₂×C₁₂₇	C₂²×M₁₆	1	1	yes	characteristic
102	C₂×C₁₂₇	#568	1	1	yes	characteristic
103	Q₈×M₁₆	—	127	1	no	maximal, Sylow
104	C₁₂₇	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
105	C₂×C₄×Q₈	—	127	1	no	—
106	#446	—	127	2	no	—
107	#446	—	127	4	no	—
108	C₈×Q₈	—	127	2	no	—
109	#423	—	127	2	no	—
110	#423	D₁₂₇	1	1	yes	characteristic
111	C₄×M₁₆	—	127	2	no	—
112	C₄×M₁₆	—	127	1	no	—
113	C₂²×Q₈	—	127	1	no	—
114	C₂×M₁₆	—	127	1	no	—
115	C₂×M₁₆	—	127	1	no	—
116	C₂×M₁₆	C₂×D₁₂₇	1	2	yes	—
117	C₄×Q₈	—	127	1	no	—
118	C₄×Q₈	—	127	2	no	—
119	C₄×Q₈	—	127	1	no	—
120	C₄×Q₈	—	127	1	no	—
121	C₄×Q₈	—	127	2	no	—
122	C₄×Q₈	—	127	1	no	—
123	C₂×C₄⋊C₄	—	127	2	no	—
124	C₂×C₄⋊C₄	—	127	1	no	—
125	C₂×C₄²	—	127	2	no	—
126	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
127	C₄⋊C₈	—	127	4	no	—
128	C₄⋊C₈	—	127	4	no	—
129	C₄⋊C₈	C₂×D₁₂₇	1	2	yes	—
130	C₄⋊C₈	C₂×D₁₂₇	1	2	yes	—
131	#97	—	127	4	no	—
132	#97	—	127	2	no	—
133	C₄×C₈	—	127	4	no	—
134	C₄×C₈	—	127	2	no	—
135	C₂×Q₈	—	127	1	no	—
136	C₂×Q₈	—	127	2	no	—
137	C₂×Q₈	—	127	1	no	—
138	C₂×Q₈	—	254	4	no	—
139	C₂²×C₄	—	127	2	no	—
140	C₂²×C₄	—	127	2	no	—
141	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
142	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
143	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
144	M₁₆	—	2	4	no	—
145	M₁₆	—	127	4	no	—
146	M₁₆	—	254	2	no	—
147	C₂×C₈	—	127	2	no	—
148	C₂×C₈	—	127	2	no	—
149	C₂×C₈	C₂²×D₁₂₇	1	4	yes	—
150	C₄⋊C₄	—	127	2	no	—
151	C₄⋊C₄	—	127	2	no	—
152	C₄⋊C₄	—	127	2	no	—
153	C₄⋊C₄	—	127	2	no	—
154	C₄⋊C₄	—	127	2	no	—
155	C₄⋊C₄	—	127	1	no	—
156	C₄⋊C₄	—	127	1	no	—
157	C₄²	—	127	2	no	—
158	C₄²	—	127	2	no	—
159	C₄²	—	127	2	no	—
160	C₄²	—	127	2	no	—
161	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
162	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
163	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
164	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
165	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
166	Q₈	—	127	4	no	—
167	Q₈	—	254	4	no	—
168	Q₈	—	254	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	127	2	no	—
172	C₂×C₄	—	127	2	no	—
173	C₂×C₄	—	127	2	no	—
174	C₂×C₄	—	127	2	no	—
175	C₂×C₄	—	508	2	no	—
176	C₂×C₄	—	254	4	no	—
177	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
182	C₂×C₄	Q₈×D₁₂₇	1	2	yes	—
183	C₂×C₄	#14523247	1	2	yes	—
184	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
185	C₈	—	2	4	no	—
186	C₈	—	127	4	no	—
187	C₈	—	254	2	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	#16861174	1	2	yes	—
191	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	127	4	no	—
196	C₄	—	254	2	no	—
197	C₄	—	254	2	no	—
198	C₄	—	254	2	no	—
199	C₄	#16861174	1	2	yes	—
200	C₄	M₁₆×D₁₂₇	1	2	yes	—
201	C₄	C₂×Q₈×D₁₂₇	1	1	yes	characteristic
202	C₄	C₂×#14523247	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	C₂×#16861174	1	1	yes	characteristic
205	C₂	C₂×M₁₆×D₁₂₇	1	1	yes	characteristic
206	C₂	#22550642	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Fitting	#423×C₁₂₇	C₂
prime radical	C₂³×C₁₂₇	C₂²×C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#423	D₁₂₇
hypocenter	C₁₂₇	Q₈×M₁₆
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
2. #22550005	C₂	semidirect
3. #22550003	C₂	semidirect
4. #22550000	C₂	semidirect
38. #14523232	Q₈	semidirect
63. Dic₂₅₄	M₁₆	semidirect
71. C₁₂₇⋊C₈	C₂×Q₈	semidirect
104. C₁₂₇	Q₈×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₅₄	2
#22550005	2
#22550003	2
#22550000	2
#22549899	2
C₄×#14523232	2
M₁₆×Dic₁₂₇	2
#423×C₁₂₇	2
Q₈×M₁₆	127