Subgroups — C₁₂₇×#7172 #44570845

Order 32512.

246 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₄×#108×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1564	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₆×C₁₂₇	—	2	1	no	—
5	#431×C₁₂₇	—	2	1	no	—
6	C₂×#108×C₁₂₇	C₄	1	2	yes	—
7	C₄×M₁₆×C₁₂₇	—	2	1	no	—
8	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₄×C₁₆×C₁₂₇	—	2	1	no	—
10	C₄×C₁₆×C₁₂₇	C₄	1	2	yes	—
11	#341×C₁₂₇	—	2	1	no	—
12	#335×C₁₂₇	C₄	1	2	yes	—
13	#334×C₁₂₇	—	2	1	no	—
14	C₂×M₁₆×C₁₂₇	—	2	2	no	—
15	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₁₂₇	—	2	4	no	—
19	C₂×C₁₆×C₁₂₇	—	2	2	no	—
20	#108×C₁₂₇	—	2	2	no	—
21	#108×C₁₂₇	C₈	1	4	yes	complemented
22	C₄⋊C₈×C₁₂₇	—	2	2	no	—
23	C₂²⋊C₈×C₁₂₇	—	4	1	no	—
24	#97×C₁₂₇	—	4	1	no	—
25	C₄×C₈×C₁₂₇	—	4	1	no	—
26	C₄×C₈×C₁₂₇	—	4	1	no	—
27	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
30	C₄×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
31	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
33	C₂²×C₄×C₁₂₇	Q₁₆	1	1	yes	characteristic
34	M₁₆×C₁₂₇	—	2	4	no	—
35	M₁₆×C₁₂₇	—	4	2	no	—
36	C₂×C₈×C₁₂₇	—	2	2	no	—
37	C₂×C₈×C₁₂₇	—	4	2	no	—
38	C₂×C₈×C₁₂₇	—	4	2	no	—
39	C₂×C₈×C₁₂₇	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₂₇	C₂×C₈	1	2	yes	—
44	C₂×C₈×C₁₂₇	M₁₆	1	2	yes	—
45	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₁₂₇	D₈	1	1	yes	characteristic
48	C₄²×C₁₂₇	SD₁₆	1	1	yes	characteristic
49	C₁₆×C₁₂₇	—	2	4	no	—
50	C₂³×C₁₂₇	#103	1	1	yes	characteristic
51	C₂×C₄×C₁₂₇	—	2	2	no	—
52	C₂×C₄×C₁₂₇	—	2	2	no	—
53	C₂×C₄×C₁₂₇	#95	1	1	yes	characteristic
54	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	2	yes	—
55	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
56	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
57	C₂×C₄×C₁₂₇	#103	1	1	yes	characteristic
58	C₂×C₄×C₁₂₇	#104	1	2	yes	—
59	C₂×C₄×C₁₂₇	#106	1	1	yes	characteristic
60	C₂×C₄×C₁₂₇	#107	1	1	yes	characteristic
61	C₈×C₁₂₇	—	2	1	no	—
62	C₈×C₁₂₇	—	2	1	no	—
63	C₈×C₁₂₇	—	4	4	no	—
64	C₈×C₁₂₇	—	8	2	no	—
65	C₈×C₁₂₇	C₄×C₈	1	1	yes	characteristic
66	C₈×C₁₂₇	#97	1	1	yes	characteristic
67	C₈×C₁₂₇	C₄⋊C₈	1	2	yes	—
68	C₂²×C₁₂₇	—	2	1	no	—
69	C₂²×C₁₂₇	—	2	1	no	—
70	C₂²×C₁₂₇	Q₈⋊C₈	1	2	yes	—
71	C₂²×C₁₂₇	#340	1	1	yes	characteristic
72	C₄×C₁₂₇	—	2	1	no	—
73	C₄×C₁₂₇	—	2	1	no	—
74	C₄×C₁₂₇	D₄⋊C₈	1	2	yes	—
75	C₄×C₁₂₇	#336	1	1	yes	characteristic
76	C₄×C₁₂₇	#339	1	1	yes	characteristic
77	C₄×C₁₂₇	#359	1	2	yes	—
78	C₄×C₁₂₇	#368	1	2	yes	—
79	#7172	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂×C₁₂₇	—	2	2	no	—
81	C₂×C₁₂₇	#1279	1	1	yes	characteristic
82	C₂×C₁₂₇	#1381	1	1	yes	characteristic
83	C₂×C₁₂₇	#1382	1	1	yes	characteristic
84	C₂×C₄×C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
85	C₄×#108	C₂×C₁₂₇	1	1	yes	characteristic
86	#1564	C₂×C₁₂₇	1	1	yes	characteristic
87	C₁₂₇	#7172	1	1	yes	characteristic, direct product factor, Sylow
88	C₂²×C₁₆	—	2	1	no	—
89	#431	—	2	1	no	—
90	C₂×#108	C₄×C₁₂₇	1	2	yes	—
91	C₄×M₁₆	—	2	1	no	—
92	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
93	C₄×C₁₆	—	2	1	no	—
94	C₄×C₁₆	C₄×C₁₂₇	1	2	yes	—
95	#341	—	2	1	no	—
96	#335	C₄×C₁₂₇	1	2	yes	—
97	#334	—	2	1	no	—
98	C₂×M₁₆	—	2	2	no	—
99	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
100	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
101	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
102	C₂×C₁₆	—	2	4	no	—
103	C₂×C₁₆	—	2	2	no	—
104	#108	—	2	2	no	—
105	#108	C₈×C₁₂₇	1	4	yes	complemented
106	C₄⋊C₈	—	2	2	no	—
107	C₂²⋊C₈	—	4	1	no	—
108	#97	—	4	1	no	—
109	C₄×C₈	—	4	1	no	—
110	C₄×C₈	—	4	1	no	—
111	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
112	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
113	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
114	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
115	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
116	C₂²×C₄	SD₁₆×C₁₂₇	1	1	yes	characteristic
117	C₂²×C₄	Q₁₆×C₁₂₇	1	1	yes	characteristic
118	M₁₆	—	2	4	no	—
119	M₁₆	—	4	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	—	4	2	no	—
123	C₂×C₈	C₄²×C₁₂₇	1	1	yes	characteristic
124	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
125	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
126	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
127	C₂×C₈	C₂×C₈×C₁₂₇	1	2	yes	—
128	C₂×C₈	M₁₆×C₁₂₇	1	2	yes	—
129	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
130	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
131	C₄²	D₈×C₁₂₇	1	1	yes	characteristic
132	C₄²	SD₁₆×C₁₂₇	1	1	yes	characteristic
133	C₁₆	—	2	4	no	—
134	C₂³	#103×C₁₂₇	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	#95×C₁₂₇	1	1	yes	characteristic
138	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	2	yes	—
139	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
140	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
141	C₂×C₄	#103×C₁₂₇	1	1	yes	characteristic
142	C₂×C₄	#104×C₁₂₇	1	2	yes	—
143	C₂×C₄	#106×C₁₂₇	1	1	yes	characteristic
144	C₂×C₄	#107×C₁₂₇	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	4	4	no	—
148	C₈	—	8	2	no	—
149	C₈	C₄×C₈×C₁₂₇	1	1	yes	characteristic
150	C₈	#97×C₁₂₇	1	1	yes	characteristic
151	C₈	C₄⋊C₈×C₁₂₇	1	2	yes	—
152	C₂²	—	2	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	Q₈⋊C₈×C₁₂₇	1	2	yes	—
155	C₂²	#340×C₁₂₇	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	D₄⋊C₈×C₁₂₇	1	2	yes	—
159	C₄	#336×C₁₂₇	1	1	yes	characteristic
160	C₄	#339×C₁₂₇	1	1	yes	characteristic
161	C₄	#359×C₁₂₇	1	2	yes	—
162	C₄	#368×C₁₂₇	1	2	yes	—
163	C₂	—	2	2	no	—
164	C₂	C₁₂₇×#1279	1	1	yes	characteristic
165	C₂	C₁₂₇×#1381	1	1	yes	characteristic
166	C₂	C₁₂₇×#1382	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₁₂₇	D₈
commutator	C₈	C₄×C₈×C₁₂₇
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₂₇
socle	C₂²×C₁₂₇	#340
prime radical	C₂³×C₁₂₇	#103
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
21. #108×C₁₂₇	C₈	semidirect
79. #7172	C₁₂₇	direct (N × H)
87. C₁₂₇	#7172	direct (N × H)
105. #108	C₈×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₁₆×C₁₂₇	2
C₄×#108×C₁₂₇	2
C₁₂₇×#1564	2
#7172	127

Subgroups — C127×#7172 #44570845

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#7172 #44570845