Subgroups — C₁₂₇×#12025 #44575698

Order 32512.

224 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	C₄×M₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#1568	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1567	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1372	C₂	1	2	yes	maximal, complemented
5	C₁₂₇×#1371	C₂	1	2	yes	maximal, complemented
6	C₂×M₃₂×C₁₂₇	C₄	1	2	yes	—
7	#432×C₁₂₇	—	2	1	no	—
8	#423×C₁₂₇	—	2	1	no	—
9	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
10	#346×C₁₂₇	C₄	1	1	yes	characteristic
11	#346×C₁₂₇	C₄	1	1	yes	characteristic
12	C₄×C₁₆×C₁₂₇	C₂²	1	2	yes	—
13	#335×C₁₂₇	—	2	1	no	—
14	#335×C₁₂₇	C₂²	1	2	yes	—
15	#335×C₁₂₇	C₂²	1	1	yes	characteristic
16	#335×C₁₂₇	C₂²	1	1	yes	characteristic
17	#334×C₁₂₇	—	2	1	no	—
18	C₂×M₁₆×C₁₂₇	—	4	1	no	—
19	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
22	M₃₂×C₁₂₇	—	2	4	no	—
23	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
25	C₄⋊C₈×C₁₂₇	—	2	1	no	—
26	C₄⋊C₈×C₁₂₇	—	2	1	no	—
27	C₄⋊C₈×C₁₂₇	—	2	2	no	—
28	C₄⋊C₈×C₁₂₇	—	4	1	no	—
29	C₂²⋊C₈×C₁₂₇	—	4	1	no	—
30	#97×C₁₂₇	—	4	1	no	—
31	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
33	C₄×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
34	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
37	C₂²×C₄×C₁₂₇	Q₁₆	1	1	yes	characteristic
38	M₁₆×C₁₂₇	—	8	2	no	—
39	C₂×C₈×C₁₂₇	—	2	2	no	—
40	C₂×C₈×C₁₂₇	—	2	2	no	—
41	C₂×C₈×C₁₂₇	—	4	1	no	—
42	C₂×C₈×C₁₂₇	—	4	1	no	—
43	C₂×C₈×C₁₂₇	—	4	2	no	—
44	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₁₂₇	D₈	1	1	yes	characteristic
50	C₄²×C₁₂₇	Q₁₆	1	1	yes	characteristic
51	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
52	C₁₆×C₁₂₇	—	2	2	no	—
53	C₁₆×C₁₂₇	M₁₆	1	4	yes	complemented
54	C₂³×C₁₂₇	#107	1	1	yes	characteristic
55	C₂×C₄×C₁₂₇	—	2	1	no	—
56	C₂×C₄×C₁₂₇	—	2	1	no	—
57	C₂×C₄×C₁₂₇	—	2	1	no	—
58	C₂×C₄×C₁₂₇	—	2	1	no	—
59	C₂×C₄×C₁₂₇	—	4	1	no	—
60	C₂×C₄×C₁₂₇	#107	1	1	yes	characteristic
61	C₂×C₄×C₁₂₇	#107	1	1	yes	characteristic
62	C₂×C₄×C₁₂₇	#107	1	1	yes	characteristic
63	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
66	C₈×C₁₂₇	—	2	1	no	—
67	C₈×C₁₂₇	—	2	1	no	—
68	C₈×C₁₂₇	—	8	1	no	—
69	C₈×C₁₂₇	—	8	2	no	—
70	C₈×C₁₂₇	—	8	1	no	—
71	C₈×C₁₂₇	#108	1	2	yes	—
72	C₈×C₁₂₇	C₂×M₁₆	1	2	yes	—
73	C₂²×C₁₂₇	—	2	1	no	—
74	C₂²×C₁₂₇	—	2	1	no	—
75	C₂²×C₁₂₇	—	2	1	no	—
76	C₂²×C₁₂₇	C₂×#107	1	1	yes	characteristic
77	C₄×C₁₂₇	—	2	1	no	—
78	C₄×C₁₂₇	—	2	1	no	—
79	C₄×C₁₂₇	—	2	1	no	—
80	C₄×C₁₂₇	—	2	1	no	—
81	C₄×C₁₂₇	#423	1	1	yes	characteristic
82	C₄×C₁₂₇	C₂×#108	1	1	yes	characteristic
83	C₄×C₁₂₇	#509	1	1	yes	characteristic
84	C₄×C₁₂₇	#510	1	1	yes	characteristic
85	#12025	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂×C₁₂₇	—	4	1	no	—
87	C₂×C₁₂₇	#1567	1	1	yes	characteristic
88	C₂×C₁₂₇	#2159	1	1	yes	characteristic
89	C₂×C₁₂₇	#2161	1	1	yes	characteristic
90	C₄×M₃₂	C₂×C₁₂₇	1	1	yes	characteristic
91	#1568	C₂×C₁₂₇	1	1	yes	characteristic
92	#1567	C₂×C₁₂₇	1	1	yes	characteristic
93	#1372	C₂×C₁₂₇	1	2	yes	complemented
94	#1371	C₂×C₁₂₇	1	2	yes	complemented
95	C₁₂₇	#12025	1	1	yes	characteristic, direct product factor, Sylow
96	C₂×M₃₂	C₄×C₁₂₇	1	2	yes	—
97	#432	—	2	1	no	—
98	#423	—	2	1	no	—
99	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
100	#346	C₄×C₁₂₇	1	1	yes	characteristic
101	#346	C₄×C₁₂₇	1	1	yes	characteristic
102	C₄×C₁₆	C₂²×C₁₂₇	1	2	yes	—
103	#335	—	2	1	no	—
104	#335	C₂²×C₁₂₇	1	2	yes	—
105	#335	C₂²×C₁₂₇	1	1	yes	characteristic
106	#335	C₂²×C₁₂₇	1	1	yes	characteristic
107	#334	—	2	1	no	—
108	C₂×M₁₆	—	4	1	no	—
109	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
110	C₂²×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
111	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
112	M₃₂	—	2	4	no	—
113	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
114	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
115	C₄⋊C₈	—	2	1	no	—
116	C₄⋊C₈	—	2	1	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	4	1	no	—
119	C₂²⋊C₈	—	4	1	no	—
120	#97	—	4	1	no	—
121	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
122	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
123	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
124	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
125	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
126	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
127	C₂²×C₄	Q₁₆×C₁₂₇	1	1	yes	characteristic
128	M₁₆	—	8	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	—	4	1	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
135	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
136	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
137	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
138	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
139	C₄²	D₈×C₁₂₇	1	1	yes	characteristic
140	C₄²	Q₁₆×C₁₂₇	1	1	yes	characteristic
141	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
142	C₁₆	—	2	2	no	—
143	C₁₆	M₁₆×C₁₂₇	1	4	yes	complemented
144	C₂³	#107×C₁₂₇	1	1	yes	characteristic
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	#107×C₁₂₇	1	1	yes	characteristic
151	C₂×C₄	#107×C₁₂₇	1	1	yes	characteristic
152	C₂×C₄	#107×C₁₂₇	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
154	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
155	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
156	C₈	—	2	1	no	—
157	C₈	—	2	1	no	—
158	C₈	—	8	2	no	—
159	C₈	—	8	1	no	—
160	C₈	—	8	1	no	—
161	C₈	#108×C₁₂₇	1	2	yes	—
162	C₈	C₂×M₁₆×C₁₂₇	1	2	yes	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	C₂×#107×C₁₂₇	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	#423×C₁₂₇	1	1	yes	characteristic
172	C₄	C₂×#108×C₁₂₇	1	1	yes	characteristic
173	C₄	#509×C₁₂₇	1	1	yes	characteristic
174	C₄	#510×C₁₂₇	1	1	yes	characteristic
175	C₂	—	4	1	no	—
176	C₂	C₁₂₇×#1567	1	1	yes	characteristic
177	C₂	C₁₂₇×#2159	1	1	yes	characteristic
178	C₂	C₁₂₇×#2161	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂×#107
prime radical	C₂³×C₁₂₇	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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
4. C₁₂₇×#1372	C₂	semidirect
5. C₁₂₇×#1371	C₂	semidirect
53. C₁₆×C₁₂₇	M₁₆	semidirect
85. #12025	C₁₂₇	direct (N × H)
93. #1372	C₂×C₁₂₇	semidirect
94. #1371	C₂×C₁₂₇	semidirect
95. C₁₂₇	#12025	direct (N × H)
143. C₁₆	M₁₆×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×M₃₂×C₁₂₇	2
C₁₂₇×#1568	2
C₁₂₇×#1567	2
C₁₂₇×#1372	2
C₁₂₇×#1371	2
#12025	127

Subgroups — C127×#12025 #44575698

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12025 #44575698