Subgroups — #30647710

Order 16256.

362 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550410	C₂	1	1	yes	maximal, characteristic
2	#22550005	C₂	1	2	yes	maximal, complemented
3	#22549951	C₂	1	2	yes	maximal, complemented
4	#22549950	C₂	1	2	yes	maximal, complemented
5	#22549946	C₂	1	2	yes	maximal, complemented
6	#22549938	C₂	1	2	yes	maximal, complemented
7	#22549833	C₂	1	2	yes	maximal, complemented
8	C₄×#14523232	C₂	1	1	yes	maximal, characteristic, complemented
9	#432×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14523232	C₂²	1	2	yes	—
11	C₂×#14523231	C₂²	1	2	yes	—
12	C₂×C₈×D₁₂₇	C₂²	1	2	yes	—
13	#16861203	C₄	1	2	yes	—
14	#16861201	C₄	1	1	yes	characteristic
15	#16861197	C₄	1	1	yes	characteristic
16	#16861192	C₄	1	2	yes	—
17	#16861188	C₄	1	1	yes	characteristic
18	#16861183	C₄	1	1	yes	characteristic
19	C₄×Dic₂₅₄	C₂²	1	1	yes	characteristic
20	C₄×D₅₀₈	C₂²	1	1	yes	characteristic
21	C₄×C₂²⋊D₁₂₇	C₂²	1	2	yes	—
22	#16861166	C₂²	1	2	yes	—
23	Dic₁₂₇⋊C₈	C₂²	1	2	yes	—
24	Dic₁₂₇⋊C₈	C₂²	1	2	yes	—
25	(C₂×D₁₂₇)⋊C₈	C₂²	1	2	yes	—
26	(C₂×D₁₂₇)⋊C₈	C₂²	1	2	yes	—
27	#16861120	C₂²	1	2	yes	—
28	#16861119	C₂²	1	2	yes	—
29	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
30	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
31	C₈×Dic₁₂₇	C₂²	1	2	yes	—
32	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
33	C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
34	C₂²⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
35	C₂²⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
36	#97×C₁₂₇	C₂²	1	2	yes	—
37	C₂×Dic₂₅₄	C₂×C₄	1	1	yes	characteristic
38	C₂×D₅₀₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₂²⋊D₁₂₇	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
41	#14523232	—	2	4	no	—
42	#14523231	—	2	4	no	—
43	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
44	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
45	C₈×D₁₂₇	—	2	4	no	—
46	C₄⋊Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
47	C₄⋊Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
48	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
49	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
50	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
51	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
52	C₂²⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
53	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	2	yes	—
54	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	2	yes	—
55	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	2	yes	—
56	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	2	yes	—
57	C₄×Dic₁₂₇	C₂³	1	2	yes	—
58	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
62	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
63	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
64	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
65	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
66	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
67	C₂²×D₁₂₇	C₂²×C₄	1	2	yes	—
68	Dic₂₅₄	—	4	1	no	—
69	D₅₀₈	—	4	1	no	—
70	C₂²⋊D₁₂₇	—	4	2	no	—
71	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
72	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
73	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
74	C₄×D₁₂₇	—	2	2	no	—
75	C₄×D₁₂₇	—	2	2	no	—
76	C₁₂₇⋊C₈	—	2	2	no	—
77	C₁₂₇⋊C₈	#41	1	4	yes	—
78	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₂₇	—	2	2	no	—
80	C₂×C₄×C₁₂₇	—	4	1	no	—
81	C₂×C₄×C₁₂₇	—	4	1	no	—
82	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
88	C₈×C₁₂₇	—	2	2	no	—
89	C₈×C₁₂₇	—	2	2	no	—
90	C₂²×C₁₂₇	—	2	1	no	—
91	C₂²×C₁₂₇	—	2	1	no	—
92	C₂²×C₁₂₇	—	2	1	no	—
93	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
94	C₂×D₁₂₇	—	2	2	no	—
95	C₂×D₁₂₇	—	2	2	no	—
96	C₂×D₁₂₇	—	2	2	no	—
97	C₄×C₁₂₇	—	2	1	no	—
98	C₄×C₁₂₇	—	2	1	no	—
99	C₄×C₁₂₇	—	2	1	no	—
100	C₄×C₁₂₇	—	2	1	no	—
101	C₄×C₁₂₇	—	2	1	no	—
102	C₄×C₁₂₇	C₂×#41	1	2	yes	—
103	Dic₁₂₇	—	2	2	no	—
104	Dic₁₂₇	—	2	2	no	—
105	Dic₁₂₇	—	2	2	no	—
106	C₂×C₁₂₇	—	4	1	no	—
107	C₂×C₁₂₇	C₄×#41	1	1	yes	characteristic
108	C₂×C₁₂₇	#568	1	1	yes	characteristic
109	C₂×C₁₂₇	#568	1	1	yes	characteristic
110	D₁₂₇	—	4	2	no	—
111	#2967	—	127	1	no	maximal, Sylow
112	C₁₂₇	#2967	1	1	yes	characteristic, complemented, Sylow
113	#529	—	127	1	no	—
114	#446	—	127	2	no	—
115	#436	—	127	2	no	—
116	#435	—	127	2	no	—
117	#435	—	127	2	no	—
118	#433	—	127	2	no	—
119	#432	D₁₂₇	1	1	yes	characteristic, complemented
120	#405	—	127	2	no	—
121	C₄×M₁₆	—	127	1	no	—
122	C₂×M₁₆	—	127	2	no	—
123	C₂×M₁₆	—	127	2	no	—
124	C₂²×C₈	—	127	2	no	—
125	#126	—	127	2	no	—
126	#125	—	127	1	no	—
127	#124	—	127	1	no	—
128	#123	—	127	2	no	—
129	C₂²⋊Q₈	—	127	1	no	—
130	#121	—	127	1	no	—
131	C₄×Q₈	—	127	1	no	—
132	C₄×D₄	—	127	2	no	—
133	C₄×D₄	—	127	1	no	—
134	#117	—	127	2	no	—
135	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
136	C₄⋊C₈	—	127	2	no	—
137	C₄⋊C₈	—	127	2	no	—
138	C₄⋊C₈	C₂×D₁₂₇	1	2	yes	—
139	C₂²⋊C₈	—	127	2	no	—
140	C₂²⋊C₈	—	127	2	no	—
141	C₂²⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
142	C₂²⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
143	#97	—	127	2	no	—
144	#97	—	127	2	no	—
145	#97	C₂×D₁₂₇	1	2	yes	—
146	C₄×C₈	—	127	2	no	—
147	C₄×C₈	—	127	1	no	—
148	C₄×C₈	—	127	1	no	—
149	C₂×Q₈	—	127	1	no	—
150	C₂×D₄	—	127	2	no	—
151	C₂×D₄	—	127	1	no	—
152	C₂²×C₄	—	127	2	no	—
153	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
154	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
155	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
156	M₁₆	—	254	4	no	—
157	M₁₆	—	254	4	no	—
158	C₂×C₈	—	127	2	no	—
159	C₂×C₈	—	127	2	no	—
160	C₂×C₈	—	254	4	no	—
161	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
162	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
163	C₄⋊C₄	—	127	2	no	—
164	C₄⋊C₄	—	127	2	no	—
165	C₄⋊C₄	—	127	2	no	—
166	C₄⋊C₄	—	127	2	no	—
167	C₄⋊C₄	—	127	1	no	—
168	C₄⋊C₄	—	127	1	no	—
169	C₂²⋊C₄	—	127	2	no	—
170	C₂²⋊C₄	—	127	2	no	—
171	C₂²⋊C₄	—	127	2	no	—
172	C₂²⋊C₄	—	127	2	no	—
173	C₂²⋊C₄	—	127	2	no	—
174	C₄²	—	127	2	no	—
175	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
176	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
177	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
178	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
179	C₂³	—	127	2	no	—
180	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
181	Q₈	—	508	1	no	—
182	D₄	—	508	1	no	—
183	D₄	—	508	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	—	4	1	no	—
187	C₂×C₄	—	127	2	no	—
188	C₂×C₄	—	127	2	no	—
189	C₂×C₄	—	127	2	no	—
190	C₂×C₄	—	254	2	no	—
191	C₂×C₄	—	254	2	no	—
192	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
195	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
196	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
197	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	—	2	2	no	—
200	C₈	—	127	4	no	—
201	C₈	—	254	2	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	254	2	no	—
206	C₂²	—	254	2	no	—
207	C₂²	—	254	2	no	—
208	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	254	2	no	—
215	C₄	—	254	2	no	—
216	C₄	—	254	2	no	—
217	C₄	#41×D₁₂₇	1	2	yes	—
218	C₂	—	4	1	no	—
219	C₂	—	508	2	no	—
220	C₂	#22550344	1	1	yes	characteristic
221	C₂	#22550642	1	1	yes	characteristic
222	C₂	#22550642	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Fitting	#432×C₁₂₇	C₂
prime radical	#16861183	C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#432	D₁₂₇
hypocenter	C₁₂₇	#2967
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
2. #22550005	C₂	semidirect
3. #22549951	C₂	semidirect
4. #22549950	C₂	semidirect
5. #22549946	C₂	semidirect
6. #22549938	C₂	semidirect
7. #22549833	C₂	semidirect
8. C₄×#14523232	C₂	semidirect
9. #432×C₁₂₇	C₂	semidirect
112. C₁₂₇	#2967	semidirect
119. #432	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550410	2
#22550005	2
#22549951	2
#22549950	2
#22549946	2
#22549938	2
#22549833	2
C₄×#14523232	2
#432×C₁₂₇	2
#2967	127