Subgroups — #30647646

Order 16256.

452 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861231	C₂	1	1	yes	maximal, characteristic
2	#22550642	C₂	1	2	yes	maximal, complemented
3	#22549997	C₂	1	4	yes	maximal, complemented
4	#22549992	C₂	1	4	yes	maximal, complemented
5	#22549892	C₂	1	2	yes	maximal, complemented
6	#22549831	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#108×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14523248	C₂²	1	2	yes	—
9	#16861231	C₄	1	2	yes	complemented
10	#16861231	C₄	1	2	yes	complemented
11	#16861231	C₂²	1	2	yes	—
12	#16861231	C₂²	1	2	yes	—
13	C₂×Dic₅₀₈	C₄	1	1	yes	characteristic, complemented
14	C₂×#14523237	C₄	1	2	yes	complemented
15	C₂×D₁₀₁₆	C₄	1	1	yes	characteristic, complemented
16	#16861217	C₂²	1	8	yes	complemented
17	#16861215	—	2	4	no	—
18	C₂×#14523232	C₂²	1	1	yes	characteristic
19	M₁₆×D₁₂₇	—	2	4	no	—
20	#16861168	C₂²	1	1	yes	characteristic
21	#16861135	C₂²	1	8	yes	complemented
22	#16861119	C₂²	1	2	yes	complemented
23	C₈×Dic₁₂₇	C₂²	1	2	yes	complemented
24	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
25	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
26	#108×C₁₂₇	C₂²	1	2	yes	complemented
27	#108×C₁₂₇	C₂²	1	2	yes	complemented
28	#14523248	—	2	2	no	—
29	#14523248	—	2	2	no	—
30	#14523248	C₂×C₄	1	4	yes	—
31	#14523248	C₂³	1	4	yes	—
32	C₂×Dic₂₅₄	C₂×C₄	1	2	yes	—
33	C₂×D₅₀₈	C₂×C₄	1	2	yes	—
34	C₂×C₂²⋊D₁₂₇	—	2	2	no	—
35	C₂×C₄×D₁₂₇	—	2	2	no	—
36	Dic₅₀₈	C₂×C₄	1	2	yes	—
37	Dic₅₀₈	C₂×C₄	1	2	yes	—
38	#14523237	C₂×C₄	1	4	yes	—
39	#14523237	C₂×C₄	1	4	yes	—
40	D₁₀₁₆	C₂×C₄	1	2	yes	—
41	D₁₀₁₆	C₂×C₄	1	2	yes	—
42	#14523232	D₄	1	2	yes	—
43	#14523232	D₄	1	2	yes	—
44	#14523231	—	2	8	no	—
45	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
46	C₈×D₁₂₇	—	2	8	no	—
47	C₄⋊Dic₁₂₇	—	2	1	no	—
48	C₂²⋊Dic₁₂₇	—	2	1	no	—
49	C₄×Dic₁₂₇	C₂³	1	2	yes	—
50	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
51	M₁₆×C₁₂₇	—	2	2	no	—
52	M₁₆×C₁₂₇	C₂³	1	4	yes	—
53	C₂×C₈×C₁₂₇	—	2	2	no	—
54	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
60	C₂²×D₁₂₇	—	2	2	no	—
61	Dic₂₅₄	—	2	2	no	—
62	Dic₂₅₄	C₂²×C₄	1	4	yes	—
63	D₅₀₈	—	2	2	no	—
64	D₅₀₈	C₂²×C₄	1	4	yes	—
65	C₂²⋊D₁₂₇	—	2	4	no	—
66	C₂²⋊D₁₂₇	—	2	4	no	—
67	C₂×Dic₁₂₇	—	2	2	no	—
68	C₂×Dic₁₂₇	—	2	2	no	—
69	C₄×D₁₂₇	—	2	4	no	—
70	C₄×D₁₂₇	—	2	4	no	—
71	C₁₂₇⋊C₈	C₂×D₄	1	4	yes	—
72	C₂³×C₁₂₇	#41	1	1	yes	characteristic
73	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
78	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
79	C₈×C₁₂₇	—	2	4	no	—
80	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
81	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
82	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
83	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₁₂₇	—	4	1	no	—
85	C₂²×C₁₂₇	C₄×D₄	1	1	yes	characteristic
86	C₂²×C₁₂₇	C₄×D₄	1	1	yes	characteristic
87	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
88	C₂×D₁₂₇	—	2	4	no	—
89	C₂×D₁₂₇	—	8	2	no	—
90	C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
91	C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
92	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
93	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
94	Dic₁₂₇	—	2	4	no	—
95	Dic₁₂₇	—	4	2	no	—
96	C₂×C₁₂₇	—	2	1	no	—
97	C₂×C₁₂₇	—	2	1	no	—
98	C₂×C₁₂₇	—	2	1	no	—
99	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
100	D₁₂₇	—	4	4	no	—
101	#2957	—	127	1	no	maximal, Sylow
102	C₁₂₇	#2957	1	1	yes	characteristic, complemented, Sylow
103	C₂×#135	—	127	1	no	—
104	#568	—	127	2	no	—
105	#444	—	127	4	no	—
106	#443	—	127	4	no	—
107	C₂×#108	D₁₂₇	1	1	yes	characteristic, complemented
108	#421	—	127	2	no	—
109	#405	—	127	1	no	—
110	C₂×#41	—	127	2	no	—
111	#135	—	127	2	no	—
112	#135	—	127	2	no	—
113	#135	—	127	2	no	—
114	#135	—	127	2	no	—
115	C₂×Q₁₆	—	127	1	no	—
116	C₂×SD₁₆	—	127	2	no	—
117	C₂×D₈	—	127	1	no	—
118	#131	—	127	8	no	—
119	#131	—	254	4	no	—
120	C₂×M₁₆	—	127	1	no	—
121	C₂×M₁₆	—	254	4	no	—
122	C₂×M₁₆	C₂×D₁₂₇	1	2	yes	—
123	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
124	#117	—	127	1	no	—
125	#108	C₂×D₁₂₇	1	2	yes	complemented
126	#108	C₂×D₁₂₇	1	2	yes	complemented
127	#104	—	127	8	no	—
128	#97	—	127	2	no	—
129	C₄×C₈	—	127	2	no	—
130	#41	—	127	4	no	—
131	#41	—	127	4	no	—
132	#41	—	254	2	no	—
133	#41	—	254	2	no	—
134	C₂×Q₈	—	127	2	no	—
135	C₂×D₄	—	127	2	no	—
136	C₂×D₄	—	254	2	no	—
137	C₂²×C₄	—	254	2	no	—
138	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
139	Q₁₆	—	127	2	no	—
140	Q₁₆	—	127	2	no	—
141	SD₁₆	—	127	4	no	—
142	SD₁₆	—	127	4	no	—
143	D₈	—	127	2	no	—
144	D₈	—	127	2	no	—
145	M₁₆	—	2	2	no	—
146	M₁₆	—	127	2	no	—
147	M₁₆	—	127	2	no	—
148	M₁₆	—	254	8	no	—
149	M₁₆	C₂²×D₁₂₇	1	4	yes	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	127	2	no	—
152	C₂×C₈	—	254	8	no	—
153	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
154	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
155	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
156	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
157	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
158	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
159	C₄⋊C₄	—	254	1	no	—
160	C₂²⋊C₄	—	254	1	no	—
161	C₄²	—	127	2	no	—
162	C₂³	—	254	2	no	—
163	C₂³	#14523249	1	1	yes	characteristic
164	Q₈	—	127	4	no	—
165	Q₈	—	254	2	no	—
166	D₄	—	127	4	no	—
167	D₄	—	254	4	no	—
168	D₄	—	254	4	no	—
169	D₄	—	254	2	no	—
170	C₂×C₄	—	254	2	no	—
171	C₂×C₄	—	254	2	no	—
172	C₂×C₄	—	254	4	no	—
173	C₂×C₄	—	254	4	no	—
174	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
175	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
176	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
177	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
178	C₂×C₄	#14523249	1	1	yes	characteristic
179	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
180	C₈	—	2	4	no	—
181	C₈	—	127	4	no	—
182	C₈	C₂×C₄×D₁₂₇	1	1	yes	characteristic
183	C₈	C₂×C₄×D₁₂₇	1	1	yes	characteristic
184	C₈	C₂×C₄×D₁₂₇	1	1	yes	characteristic
185	C₈	C₂×C₄×D₁₂₇	1	1	yes	characteristic
186	C₂²	—	4	1	no	—
187	C₂²	—	1016	2	no	—
188	C₂²	—	254	4	no	—
189	C₂²	#16861171	1	1	yes	characteristic
190	C₂²	#16861171	1	1	yes	characteristic
191	C₂²	C₂×#14523249	1	1	yes	characteristic
192	C₄	—	508	2	no	—
193	C₄	—	254	4	no	—
194	C₄	#16861171	1	1	yes	characteristic
195	C₄	#16861171	1	1	yes	characteristic
196	C₄	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
197	C₄	C₂×D₄×D₁₂₇	1	1	yes	characteristic
198	C₂	—	2	1	no	—
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	—	508	4	no	—
202	C₂	C₂×#16861171	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₁₂₇
commutator	C₄×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂×#108×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂×C₄×D₄
prime radical	C₂×#16861231	C₂
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₂×#108	D₁₂₇
hypocenter	C₁₂₇	#2957
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
2. #22550642	C₂	semidirect
3. #22549997	C₂	semidirect
4. #22549992	C₂	semidirect
5. #22549892	C₂	semidirect
6. #22549831	C₂	semidirect
7. C₂×#108×C₁₂₇	C₂	semidirect
9. #16861231	C₄	semidirect
10. #16861231	C₄	semidirect
13. C₂×Dic₅₀₈	C₄	semidirect
14. C₂×#14523237	C₄	semidirect
15. C₂×D₁₀₁₆	C₄	semidirect
16. #16861217	C₂²	semidirect
21. #16861135	C₂²	semidirect
22. #16861119	C₂²	semidirect
23. C₈×Dic₁₂₇	C₂²	semidirect
26. #108×C₁₂₇	C₂²	semidirect
27. #108×C₁₂₇	C₂²	semidirect
102. C₁₂₇	#2957	semidirect
107. C₂×#108	D₁₂₇	semidirect
125. #108	C₂×D₁₂₇	semidirect
126. #108	C₂×D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861231	2
#22550642	2
#22549997	2
#22549992	2
#22549892	2
#22549831	2
C₂×#108×C₁₂₇	2
#2957	127