Subgroups — C₁₂₇×#9610 #44573283

Order 32512.

294 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#108×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#1565	C₂	1	2	yes	maximal
3	C₂×#335×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1452	C₂	1	1	yes	maximal, characteristic
5	C₁₂₇×#1283	C₂	1	2	yes	maximal
6	#431×C₁₂₇	C₂²	1	1	yes	characteristic
7	#431×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂×#108×C₁₂₇	C₄	1	2	yes	—
9	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
10	C₄×M₁₆×C₁₂₇	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	#341×C₁₂₇	C₄	1	1	yes	characteristic
13	#341×C₁₂₇	C₄	1	1	yes	characteristic
14	#335×C₁₂₇	—	2	2	no	—
15	#335×C₁₂₇	—	2	2	no	—
16	#334×C₁₂₇	—	2	2	no	—
17	#322×C₁₂₇	—	2	2	no	—
18	C₂×M₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
19	C₂×M₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₁₂₇	—	2	2	no	—
21	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
24	#108×C₁₂₇	—	2	4	no	—
25	C₄⋊C₈×C₁₂₇	—	2	2	no	—
26	C₄⋊C₈×C₁₂₇	—	2	2	no	—
27	C₄⋊C₈×C₁₂₇	—	2	1	no	—
28	C₄⋊C₈×C₁₂₇	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₁₂₇	D₄	1	1	yes	characteristic
30	C₂²⋊C₈×C₁₂₇	—	2	1	no	—
31	C₂²⋊C₈×C₁₂₇	Q₈	1	2	yes	—
32	#97×C₁₂₇	—	2	2	no	—
33	C₄×C₈×C₁₂₇	—	2	1	no	—
34	C₄×C₈×C₁₂₇	—	2	1	no	—
35	C₄×C₈×C₁₂₇	—	2	2	no	—
36	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
37	C₄×C₈×C₁₂₇	Q₈	1	2	yes	—
38	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₂₇	#41	1	1	yes	characteristic
41	M₁₆×C₁₂₇	—	2	2	no	—
42	M₁₆×C₁₂₇	—	2	2	no	—
43	M₁₆×C₁₂₇	—	2	2	no	—
44	M₁₆×C₁₂₇	M₁₆	1	4	yes	—
45	C₂×C₈×C₁₂₇	—	2	2	no	—
46	C₂×C₈×C₁₂₇	—	2	2	no	—
47	C₂×C₈×C₁₂₇	—	2	2	no	—
48	C₂×C₈×C₁₂₇	—	2	2	no	—
49	C₂×C₈×C₁₂₇	—	2	1	no	—
50	C₂×C₈×C₁₂₇	—	2	1	no	—
51	C₂×C₈×C₁₂₇	—	2	2	no	—
52	C₂×C₈×C₁₂₇	—	4	2	no	—
53	C₂×C₈×C₁₂₇	—	8	2	no	—
54	C₂×C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
55	C₂×C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
56	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₁₂₇	#41	1	1	yes	characteristic
60	C₂³×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₁₂₇	—	2	1	no	—
62	C₂×C₄×C₁₂₇	—	2	1	no	—
63	C₂×C₄×C₁₂₇	—	4	1	no	—
64	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₁₂₇	#121	1	1	yes	characteristic
67	C₂×C₄×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₁₂₇	#125	1	1	yes	characteristic
69	C₂×C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₁₂₇	C₂×M₁₆	1	2	yes	—
71	C₂×C₄×C₁₂₇	#131	1	2	yes	—
72	C₈×C₁₂₇	—	2	2	no	—
73	C₈×C₁₂₇	—	2	2	no	—
74	C₈×C₁₂₇	—	2	4	no	—
75	C₈×C₁₂₇	—	4	2	no	—
76	C₈×C₁₂₇	—	4	2	no	—
77	C₈×C₁₂₇	—	4	1	no	—
78	C₈×C₁₂₇	—	4	1	no	—
79	C₈×C₁₂₇	—	8	2	no	—
80	C₂²×C₁₂₇	—	2	1	no	—
81	C₂²×C₁₂₇	—	2	1	no	—
82	C₂²×C₁₂₇	#389	1	1	yes	characteristic
83	C₂²×C₁₂₇	#446	1	2	yes	—
84	C₄×C₁₂₇	—	2	1	no	—
85	C₄×C₁₂₇	—	2	1	no	—
86	C₄×C₁₂₇	—	2	1	no	—
87	C₄×C₁₂₇	#423	1	1	yes	characteristic
88	C₄×C₁₂₇	#424	1	1	yes	characteristic
89	C₄×C₁₂₇	#435	1	2	yes	—
90	C₄×C₁₂₇	#472	1	1	yes	characteristic
91	C₄×C₁₂₇	#473	1	1	yes	characteristic
92	#9610	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂×C₁₂₇	—	2	2	no	—
94	C₂×C₁₂₇	#1940	1	1	yes	characteristic
95	C₂×C₁₂₇	#1952	1	1	yes	characteristic
96	C₂×C₁₂₇	#1953	1	1	yes	characteristic
97	C₄×#108	C₂×C₁₂₇	1	1	yes	characteristic
98	#1565	C₂×C₁₂₇	1	2	yes	—
99	C₂×#335	C₂×C₁₂₇	1	1	yes	characteristic
100	#1452	C₂×C₁₂₇	1	1	yes	characteristic
101	#1283	C₂×C₁₂₇	1	2	yes	—
102	C₁₂₇	#9610	1	1	yes	characteristic, direct product factor, Sylow
103	#431	C₂²×C₁₂₇	1	1	yes	characteristic
104	#431	C₂²×C₁₂₇	1	1	yes	characteristic
105	C₂×#108	C₄×C₁₂₇	1	2	yes	—
106	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
107	C₄×M₁₆	C₂²×C₁₂₇	1	2	yes	—
108	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
109	#341	C₄×C₁₂₇	1	1	yes	characteristic
110	#341	C₄×C₁₂₇	1	1	yes	characteristic
111	#335	—	2	2	no	—
112	#335	—	2	2	no	—
113	#334	—	2	2	no	—
114	#322	—	2	2	no	—
115	C₂×M₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
116	C₂×M₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
117	C₂²×C₈	—	2	2	no	—
118	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
119	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
120	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
121	#108	—	2	4	no	—
122	C₄⋊C₈	—	2	2	no	—
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	2	1	no	—
125	C₄⋊C₈	D₄×C₁₂₇	1	1	yes	characteristic
126	C₄⋊C₈	D₄×C₁₂₇	1	1	yes	characteristic
127	C₂²⋊C₈	—	2	1	no	—
128	C₂²⋊C₈	Q₈×C₁₂₇	1	2	yes	—
129	#97	—	2	2	no	—
130	C₄×C₈	—	2	1	no	—
131	C₄×C₈	—	2	1	no	—
132	C₄×C₈	—	2	2	no	—
133	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
134	C₄×C₈	Q₈×C₁₂₇	1	2	yes	—
135	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
136	C₂²×C₄	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
137	C₂²×C₄	#41×C₁₂₇	1	1	yes	characteristic
138	M₁₆	—	2	2	no	—
139	M₁₆	—	2	2	no	—
140	M₁₆	—	2	2	no	—
141	M₁₆	M₁₆×C₁₂₇	1	4	yes	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	8	2	no	—
151	C₂×C₈	C₄⋊C₄×C₁₂₇	1	2	yes	—
152	C₂×C₈	C₄⋊C₄×C₁₂₇	1	2	yes	—
153	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
154	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
155	C₄²	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
156	C₄²	#41×C₁₂₇	1	1	yes	characteristic
157	C₂³	C₄×Q₈×C₁₂₇	1	1	yes	characteristic
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
162	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
163	C₂×C₄	#121×C₁₂₇	1	1	yes	characteristic
164	C₂×C₄	C₂²⋊Q₈×C₁₂₇	1	1	yes	characteristic
165	C₂×C₄	#125×C₁₂₇	1	1	yes	characteristic
166	C₂×C₄	C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
167	C₂×C₄	C₂×M₁₆×C₁₂₇	1	2	yes	—
168	C₂×C₄	#131×C₁₂₇	1	2	yes	—
169	C₈	—	2	2	no	—
170	C₈	—	2	2	no	—
171	C₈	—	2	4	no	—
172	C₈	—	4	2	no	—
173	C₈	—	4	2	no	—
174	C₈	—	4	1	no	—
175	C₈	—	4	1	no	—
176	C₈	—	8	2	no	—
177	C₂²	—	2	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	#389×C₁₂₇	1	1	yes	characteristic
180	C₂²	#446×C₁₂₇	1	2	yes	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	#423×C₁₂₇	1	1	yes	characteristic
185	C₄	#424×C₁₂₇	1	1	yes	characteristic
186	C₄	#435×C₁₂₇	1	2	yes	—
187	C₄	#472×C₁₂₇	1	1	yes	characteristic
188	C₄	#473×C₁₂₇	1	1	yes	characteristic
189	C₂	—	2	2	no	—
190	C₂	C₁₂₇×#1940	1	1	yes	characteristic
191	C₂	C₁₂₇×#1952	1	1	yes	characteristic
192	C₂	C₁₂₇×#1953	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₂₇	#121
commutator	C₂²×C₄	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	#389
prime radical	C₂³×C₁₂₇	C₄×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
92. #9610	C₁₂₇	direct (N × H)
102. C₁₂₇	#9610	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#108×C₁₂₇	2
C₁₂₇×#1565	2
C₂×#335×C₁₂₇	2
C₁₂₇×#1452	2
C₁₂₇×#1283	2
#9610	127

Subgroups — C127×#9610 #44573283

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#9610 #44573283