Subgroups — #30647293

Order 16256.

484 conjugacy classes of subgroups.

The subgroup lattice has 180 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	#22549904	C₂	1	4	yes	maximal, complemented
4	#22549902	C₂	1	4	yes	maximal, complemented
5	#22549899	C₂	1	4	yes	maximal, complemented
6	C₂²×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#14523232	—	2	4	no	—
8	C₂×#14523232	C₂²	1	2	yes	—
9	C₂×#14523232	C₂²	1	2	yes	—
10	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
11	#16861168	C₄	1	2	yes	—
12	#16861168	C₄	1	2	yes	—
13	#16861168	C₂²	1	2	yes	—
14	#16861168	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₁₂₇	C₄	1	2	yes	—
16	C₂×C₂²⋊Dic₁₂₇	C₄	1	2	yes	—
17	C₂×C₄×Dic₁₂₇	C₂²	1	2	yes	—
18	Dic₁₂₇⋊C₈	C₂²	1	16	yes	complemented
19	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₁₂₇	—	2	4	no	—
21	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
22	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
23	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
24	C₂²×Dic₁₂₇	C₂×C₄	1	4	yes	—
25	#14523232	—	2	4	no	—
26	#14523232	—	2	4	no	—
27	C₂×C₁₂₇⋊C₈	—	2	4	no	—
28	C₂×C₁₂₇⋊C₈	C₂³	1	4	yes	—
29	C₄⋊Dic₁₂₇	C₂×C₄	1	4	yes	—
30	C₄⋊Dic₁₂₇	C₂×C₄	1	4	yes	—
31	C₂²⋊Dic₁₂₇	C₂×C₄	1	4	yes	—
32	C₂²⋊Dic₁₂₇	C₂×C₄	1	4	yes	—
33	C₄×Dic₁₂₇	C₂×C₄	1	4	yes	—
34	C₄×Dic₁₂₇	C₂³	1	4	yes	—
35	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₂₇	D₄	1	4	yes	—
40	C₂²×C₄×C₁₂₇	Q₈	1	4	yes	—
41	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
44	M₁₆×C₁₂₇	—	2	4	no	—
45	M₁₆×C₁₂₇	—	2	4	no	—
46	C₂×C₈×C₁₂₇	—	2	4	no	—
47	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
48	C₂×Dic₁₂₇	—	4	4	no	—
49	C₂×Dic₁₂₇	C₂²×C₄	1	8	yes	—
50	C₁₂₇⋊C₈	—	2	4	no	—
51	C₂³×C₁₂₇	—	4	2	no	—
52	C₂³×C₁₂₇	C₄⋊C₄	1	4	yes	—
53	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
54	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
55	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
57	C₂×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
58	C₂×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
59	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
66	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
67	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
68	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
69	C₈×C₁₂₇	—	2	4	no	—
70	C₂²×C₁₂₇	—	2	2	no	—
71	C₂²×C₁₂₇	—	2	2	no	—
72	C₂²×C₁₂₇	—	2	2	no	—
73	C₂²×C₁₂₇	—	4	4	no	—
74	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
75	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
76	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
77	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
78	C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
79	C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
80	C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
81	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
82	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
83	Dic₁₂₇	—	2	8	no	—
84	C₂×C₁₂₇	—	2	2	no	—
85	C₂×C₁₂₇	—	2	2	no	—
86	C₂×C₁₂₇	—	2	2	no	—
87	C₂×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₁₂₇	#568	1	2	yes	—
89	#2905	—	127	1	no	maximal, Sylow
90	C₁₂₇	#2905	1	1	yes	characteristic, complemented, Sylow
91	C₂²×M₁₆	—	127	1	no	—
92	C₂²×M₁₆	D₁₂₇	1	1	yes	characteristic
93	C₂×#117	—	127	1	no	—
94	#424	—	127	4	no	—
95	#424	—	127	4	no	—
96	#423	—	127	4	no	—
97	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
98	C₂×M₁₆	—	2	4	no	—
99	C₂×M₁₆	—	127	2	no	—
100	C₂×M₁₆	—	127	2	no	—
101	C₂×M₁₆	—	254	4	no	—
102	C₂×M₁₆	C₂×D₁₂₇	1	2	yes	—
103	C₂×M₁₆	C₂×D₁₂₇	1	2	yes	—
104	C₂²×C₈	—	127	2	no	—
105	C₂²×C₈	C₂×D₁₂₇	1	2	yes	—
106	#117	—	127	2	no	—
107	#117	—	127	2	no	—
108	#117	—	127	2	no	—
109	#117	—	127	2	no	—
110	C₂×C₄⋊C₄	—	127	2	no	—
111	C₂×C₂²⋊C₄	—	127	2	no	—
112	C₂×C₄²	—	127	2	no	—
113	C₄⋊C₈	—	127	16	no	—
114	C₂⁴	C₄×D₁₂₇	1	1	yes	characteristic
115	C₂²×C₄	—	127	4	no	—
116	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
117	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
118	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
119	C₂²×C₄	C₂²⋊D₁₂₇	1	4	yes	—
120	C₂²×C₄	Dic₂₅₄	1	4	yes	—
121	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
122	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
123	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
124	M₁₆	—	2	4	no	—
125	M₁₆	—	2	4	no	—
126	M₁₆	—	254	4	no	—
127	M₁₆	—	254	4	no	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	127	4	no	—
130	C₂×C₈	—	254	4	no	—
131	C₂×C₈	C₂²×D₁₂₇	1	4	yes	—
132	C₄⋊C₄	—	127	4	no	—
133	C₄⋊C₄	—	127	4	no	—
134	C₂²⋊C₄	—	127	4	no	—
135	C₂²⋊C₄	—	127	4	no	—
136	C₄²	—	127	4	no	—
137	C₄²	—	127	4	no	—
138	C₂³	—	4	2	no	—
139	C₂³	Dic₁₂₇⋊C₄	1	4	yes	—
140	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
141	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
142	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
143	C₂×C₄	—	127	8	no	—
144	C₂×C₄	—	508	4	no	—
145	C₂×C₄	Dic₁₂₇⋊C₄	1	4	yes	—
146	C₂×C₄	Dic₁₂₇⋊C₄	1	4	yes	—
147	C₂×C₄	Dic₁₂₇⋊C₄	1	4	yes	—
148	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
149	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
150	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
151	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
152	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
153	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
154	C₂×C₄	C₂×Dic₂₅₄	1	2	yes	—
155	C₂×C₄	C₂×Dic₂₅₄	1	2	yes	—
156	C₂×C₄	C₂×Dic₂₅₄	1	2	yes	—
157	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
158	C₈	—	2	4	no	—
159	C₈	—	254	4	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	4	4	no	—
164	C₂²	C₂×Dic₁₂₇⋊C₄	1	2	yes	—
165	C₂²	C₂×Dic₁₂₇⋊C₄	1	2	yes	—
166	C₂²	C₂×Dic₁₂₇⋊C₄	1	2	yes	—
167	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
168	C₄	—	254	8	no	—
169	C₄	C₂×Dic₁₂₇⋊C₄	1	2	yes	—
170	C₄	C₂×Dic₁₂₇⋊C₄	1	2	yes	—
171	C₄	C₂×Dic₁₂₇⋊C₄	1	2	yes	—
172	C₄	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
173	C₄	C₂²×Dic₂₅₄	1	1	yes	characteristic
174	C₂	—	2	2	no	—
175	C₂	—	2	2	no	—
176	C₂	—	2	2	no	—
177	C₂	C₂²×Dic₁₂₇⋊C₄	1	1	yes	characteristic
178	C₂	#22550642	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Fitting	C₂²×M₁₆×C₁₂₇	C₂
prime radical	C₂⁴×C₁₂₇	C₂×C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₂²×M₁₆	D₁₂₇
hypocenter	C₁₂₇	#2905
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #22549904	C₂	semidirect
4. #22549902	C₂	semidirect
5. #22549899	C₂	semidirect
18. Dic₁₂₇⋊C₈	C₂²	semidirect
90. C₁₂₇	#2905	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
#22549904	2
#22549902	2
#22549899	2
C₂²×M₁₆×C₁₂₇	2
#2905	127