Subgroups — #30647319

Order 16256.

492 conjugacy classes of subgroups.

The subgroup lattice has 198 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	C₂×#16861168	C₂	1	1	yes	maximal, characteristic
3	#22549919	C₂	1	4	yes	maximal, complemented
4	#22549916	C₂	1	4	yes	maximal, complemented
5	#22549912	C₂	1	4	yes	maximal, complemented
6	C₂×#117×C₁₂₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#14523232	C₄	1	8	yes	complemented
8	C₂×#14523232	C₂²	1	2	yes	—
9	C₂×#14523232	C₂²	1	2	yes	—
10	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
11	#16861168	—	2	1	no	—
12	#16861168	—	2	1	no	—
13	#16861168	C₂²	1	2	yes	—
14	#16861168	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
16	C₂×C₂²⋊Dic₁₂₇	—	2	1	no	—
17	C₂×C₄×Dic₁₂₇	—	2	1	no	—
18	#16861142	C₂²	1	8	yes	complemented
19	#16861140	C₂²	1	8	yes	complemented
20	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
21	#117×C₁₂₇	—	2	1	no	—
22	#117×C₁₂₇	—	2	1	no	—
23	#117×C₁₂₇	C₂²	1	2	yes	—
24	#117×C₁₂₇	C₂²	1	2	yes	—
25	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₁₂₇	—	2	1	no	—
27	C₂×C₄²×C₁₂₇	—	2	1	no	—
28	C₂²×Dic₁₂₇	—	2	2	no	—
29	#14523232	C₂×C₄	1	8	yes	complemented
30	#14523232	C₂×C₄	1	8	yes	complemented
31	C₂×C₁₂₇⋊C₈	C₂×C₄	1	8	yes	complemented
32	C₂×C₁₂₇⋊C₈	C₂³	1	4	yes	—
33	C₄⋊Dic₁₂₇	—	2	2	no	—
34	C₄⋊Dic₁₂₇	C₂³	1	4	yes	—
35	C₂²⋊Dic₁₂₇	—	2	2	no	—
36	C₂²⋊Dic₁₂₇	—	2	2	no	—
37	C₄×Dic₁₂₇	—	2	2	no	—
38	C₄×Dic₁₂₇	—	2	2	no	—
39	C₂⁴×C₁₂₇	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₂₇	—	2	2	no	—
41	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
42	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₂₇	Q₈	1	4	yes	—
46	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
49	C₄⋊C₄×C₁₂₇	—	2	2	no	—
50	C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
51	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
52	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
53	C₄²×C₁₂₇	—	2	2	no	—
54	C₄²×C₁₂₇	—	2	2	no	—
55	C₂×Dic₁₂₇	—	2	4	no	—
56	C₂×Dic₁₂₇	—	4	4	no	—
57	C₁₂₇⋊C₈	C₂²×C₄	1	8	yes	—
58	C₂³×C₁₂₇	—	4	2	no	—
59	C₂³×C₁₂₇	C₄⋊C₄	1	4	yes	—
60	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
61	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
62	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₁₂₇	—	2	4	no	—
64	C₂×C₄×C₁₂₇	—	4	4	no	—
65	C₂×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
66	C₂×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
67	C₂×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
68	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
70	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
75	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
76	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
77	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
78	C₂²×C₁₂₇	—	2	2	no	—
79	C₂²×C₁₂₇	—	2	2	no	—
80	C₂²×C₁₂₇	—	2	2	no	—
81	C₂²×C₁₂₇	—	4	4	no	—
82	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
83	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
84	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
85	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
86	C₄×C₁₂₇	—	4	4	no	—
87	C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
88	C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
89	C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
90	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
91	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
92	Dic₁₂₇	—	4	4	no	—
93	C₂×C₁₂₇	—	2	2	no	—
94	C₂×C₁₂₇	—	2	2	no	—
95	C₂×C₁₂₇	—	2	2	no	—
96	C₂×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₁₂₇	#575	1	2	yes	—
98	#2911	—	127	1	no	maximal, Sylow
99	C₁₂₇	#2911	1	1	yes	characteristic, complemented, Sylow
100	C₂²×M₁₆	—	127	1	no	—
101	C₂×#117	—	127	1	no	—
102	C₂×#117	D₁₂₇	1	1	yes	characteristic
103	#428	—	127	4	no	—
104	#428	—	127	4	no	—
105	#427	—	127	4	no	—
106	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
107	C₂×M₁₆	—	127	8	no	—
108	C₂×M₁₆	—	127	2	no	—
109	C₂×M₁₆	—	127	2	no	—
110	C₂²×C₈	—	127	2	no	—
111	#117	—	2	1	no	—
112	#117	—	2	1	no	—
113	#117	—	127	2	no	—
114	#117	—	127	2	no	—
115	#117	—	254	1	no	—
116	#117	—	254	1	no	—
117	#117	C₂×D₁₂₇	1	2	yes	—
118	#117	C₂×D₁₂₇	1	2	yes	—
119	C₂×C₄⋊C₄	—	127	2	no	—
120	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
121	C₂×C₂²⋊C₄	—	2	1	no	—
122	C₂×C₂²⋊C₄	—	254	1	no	—
123	C₂×C₄²	—	2	1	no	—
124	C₂×C₄²	—	254	1	no	—
125	#107	—	127	8	no	—
126	#106	—	127	8	no	—
127	C₂⁴	C₂²⋊D₁₂₇	1	1	yes	characteristic
128	C₂²×C₄	—	2	2	no	—
129	C₂²×C₄	—	254	2	no	—
130	C₂²×C₄	C₄×D₁₂₇	1	4	yes	—
131	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
132	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
133	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
134	C₂²×C₄	Dic₂₅₄	1	4	yes	—
135	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
136	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
137	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
138	M₁₆	—	127	8	no	—
139	M₁₆	—	127	8	no	—
140	C₂×C₈	—	127	8	no	—
141	C₂×C₈	—	127	4	no	—
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	—	127	4	no	—
144	C₄⋊C₄	—	254	2	no	—
145	C₄⋊C₄	C₂²×D₁₂₇	1	4	yes	—
146	C₂²⋊C₄	—	2	2	no	—
147	C₂²⋊C₄	—	2	2	no	—
148	C₂²⋊C₄	—	254	2	no	—
149	C₂²⋊C₄	—	254	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	254	2	no	—
153	C₄²	—	254	2	no	—
154	C₂³	—	4	2	no	—
155	C₂³	Dic₁₂₇⋊C₄	1	4	yes	—
156	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
157	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
158	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	508	4	no	—
162	C₂×C₄	—	254	4	no	—
163	C₂×C₄	Dic₁₂₇⋊C₄	1	4	yes	—
164	C₂×C₄	Dic₁₂₇⋊C₄	1	4	yes	—
165	C₂×C₄	Dic₁₂₇⋊C₄	1	4	yes	—
166	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
167	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
168	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
169	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
170	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
171	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
172	C₂×C₄	C₂×Dic₂₅₄	1	2	yes	—
173	C₂×C₄	C₂×Dic₂₅₄	1	2	yes	—
174	C₂×C₄	C₂×Dic₂₅₄	1	2	yes	—
175	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
176	C₈	—	127	8	no	—
177	C₂²	—	2	2	no	—
178	C₂²	—	2	2	no	—
179	C₂²	—	2	2	no	—
180	C₂²	—	4	4	no	—
181	C₂²	C₂×Dic₁₂₇⋊C₄	1	2	yes	—
182	C₂²	C₂×Dic₁₂₇⋊C₄	1	2	yes	—
183	C₂²	C₂×Dic₁₂₇⋊C₄	1	2	yes	—
184	C₂²	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
185	C₄	—	4	4	no	—
186	C₄	—	508	4	no	—
187	C₄	C₂×Dic₁₂₇⋊C₄	1	2	yes	—
188	C₄	C₂×Dic₁₂₇⋊C₄	1	2	yes	—
189	C₄	C₂×Dic₁₂₇⋊C₄	1	2	yes	—
190	C₄	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
191	C₄	C₂²×Dic₂₅₄	1	1	yes	characteristic
192	C₂	—	2	2	no	—
193	C₂	—	2	2	no	—
194	C₂	—	2	2	no	—
195	C₂	C₂²×Dic₁₂₇⋊C₄	1	1	yes	characteristic
196	C₂	#22550674	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₂₇
commutator	C₄×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂×#117×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	C₂⁴×C₁₂₇	D₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₂×#117	D₁₂₇
hypocenter	C₁₂₇	#2911
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #22549919	C₂	semidirect
4. #22549916	C₂	semidirect
5. #22549912	C₂	semidirect
7. C₂×#14523232	C₄	semidirect
18. #16861142	C₂²	semidirect
19. #16861140	C₂²	semidirect
29. #14523232	C₂×C₄	semidirect
30. #14523232	C₂×C₄	semidirect
31. C₂×C₁₂₇⋊C₈	C₂×C₄	semidirect
99. C₁₂₇	#2911	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
C₂×#16861168	2
#22549919	2
#22549916	2
#22549912	2
C₂×#117×C₁₂₇	2
#2911	127