Subgroups — C₁₂₇×#20830 #44584503

Order 32512.

770 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#2920	C₂	1	2	yes	maximal
2	C₂²×#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1849	C₂	1	2	yes	maximal, complemented
4	C₁₂₇×#1838	C₂	1	2	yes	maximal, complemented
5	C₁₂₇×#1810	C₂	1	4	yes	maximal
6	C₁₂₇×#1778	C₂	1	4	yes	maximal, complemented
7	C₂³×C₈×C₁₂₇	C₂²	1	2	yes	—
8	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
9	#432×C₁₂₇	—	2	4	no	—
10	#431×C₁₂₇	—	2	4	no	—
11	C₂×#108×C₁₂₇	C₂²	1	8	yes	—
12	#423×C₁₂₇	C₂²	1	4	yes	—
13	#407×C₁₂₇	C₂²	1	4	yes	—
14	C₄×M₁₆×C₁₂₇	C₂²	1	4	yes	—
15	C₂×#97×C₁₂₇	C₄	1	8	yes	—
16	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
19	#341×C₁₂₇	C₂²	1	8	yes	—
20	C₂³×C₄×C₁₂₇	Q₈	1	2	yes	—
21	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₁₂₇	C₂³	1	8	yes	—
23	C₂²×C₈×C₁₂₇	—	2	8	no	—
24	C₂²×C₈×C₁₂₇	C₂×C₄	1	8	yes	—
25	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
26	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
27	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₁₂₇	D₄	1	4	yes	—
29	C₂×C₄²×C₁₂₇	Q₈	1	2	yes	—
30	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
32	#108×C₁₂₇	—	2	16	no	—
33	C₄⋊C₈×C₁₂₇	—	2	8	no	—
34	C₂²⋊C₈×C₁₂₇	—	2	8	no	—
35	#97×C₁₂₇	—	2	4	no	—
36	#97×C₁₂₇	C₂×C₄	1	4	yes	—
37	#97×C₁₂₇	C₂×C₄	1	8	yes	—
38	#97×C₁₂₇	C₂×C₄	1	4	yes	—
39	C₄×C₈×C₁₂₇	—	2	4	no	—
40	C₂⁴×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₂₇	—	2	4	no	—
42	C₂²×C₄×C₁₂₇	—	4	4	no	—
43	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
49	C₂²×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
50	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
51	M₁₆×C₁₂₇	—	2	16	no	—
52	C₂×C₈×C₁₂₇	—	2	4	no	—
53	C₂×C₈×C₁₂₇	—	2	16	no	—
54	C₂×C₈×C₁₂₇	—	2	8	no	—
55	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
56	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
57	C₂×C₈×C₁₂₇	#41	1	8	yes	—
58	C₄²×C₁₂₇	C₄⋊C₄	1	8	yes	—
59	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
60	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
61	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
62	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
63	C₂³×C₁₂₇	—	2	2	no	—
64	C₂³×C₁₂₇	—	2	4	no	—
65	C₂³×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
66	C₂³×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
67	C₂×C₄×C₁₂₇	—	2	2	no	—
68	C₂×C₄×C₁₂₇	—	2	8	no	—
69	C₂×C₄×C₁₂₇	—	2	4	no	—
70	C₂×C₄×C₁₂₇	—	2	4	no	—
71	C₂×C₄×C₁₂₇	—	4	4	no	—
72	C₂×C₄×C₁₂₇	—	4	2	no	—
73	C₂×C₄×C₁₂₇	—	4	2	no	—
74	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
75	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
76	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
77	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
79	C₂×C₄×C₁₂₇	C₂×#41	1	4	yes	—
80	C₈×C₁₂₇	—	2	4	no	—
81	C₈×C₁₂₇	—	4	8	no	—
82	C₈×C₁₂₇	#117	1	8	yes	—
83	C₂²×C₁₂₇	—	2	8	no	—
84	C₂²×C₁₂₇	—	2	2	no	—
85	C₂²×C₁₂₇	—	2	2	no	—
86	C₂²×C₁₂₇	—	2	2	no	—
87	C₂²×C₁₂₇	#430	1	4	yes	—
88	C₂²×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₁₂₇	#533	1	2	yes	—
90	C₄×C₁₂₇	—	2	4	no	—
91	C₄×C₁₂₇	—	2	4	no	—
92	C₄×C₁₂₇	—	2	2	no	—
93	C₄×C₁₂₇	C₂×#117	1	2	yes	—
94	C₄×C₁₂₇	#532	1	2	yes	—
95	#20830	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂×C₁₂₇	—	2	2	no	—
97	C₂×C₁₂₇	—	2	4	no	—
98	C₂×C₁₂₇	#2285	1	1	yes	characteristic
99	C₂×C₁₂₇	C₂×#430	1	2	yes	—
100	#2920	C₂×C₁₂₇	1	2	yes	—
101	C₂²×#97	C₂×C₁₂₇	1	1	yes	characteristic
102	#1849	C₂×C₁₂₇	1	2	yes	complemented
103	#1838	C₂×C₁₂₇	1	2	yes	complemented
104	#1810	C₂×C₁₂₇	1	4	yes	—
105	#1778	C₂×C₁₂₇	1	4	yes	complemented
106	C₁₂₇	#20830	1	1	yes	characteristic, direct product factor, Sylow
107	C₂³×C₈	C₂²×C₁₂₇	1	2	yes	—
108	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
109	#432	—	2	4	no	—
110	#431	—	2	4	no	—
111	C₂×#108	C₂²×C₁₂₇	1	8	yes	—
112	#423	C₂²×C₁₂₇	1	4	yes	—
113	#407	C₂²×C₁₂₇	1	4	yes	—
114	C₄×M₁₆	C₂²×C₁₂₇	1	4	yes	—
115	C₂×#97	C₄×C₁₂₇	1	8	yes	—
116	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
117	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
118	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
119	#341	C₂²×C₁₂₇	1	8	yes	—
120	C₂³×C₄	Q₈×C₁₂₇	1	2	yes	—
121	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
122	C₂×M₁₆	C₂³×C₁₂₇	1	8	yes	—
123	C₂²×C₈	—	2	8	no	—
124	C₂²×C₈	C₂×C₄×C₁₂₇	1	8	yes	—
125	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
126	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
127	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
128	C₂×C₄²	D₄×C₁₂₇	1	4	yes	—
129	C₂×C₄²	Q₈×C₁₂₇	1	2	yes	—
130	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
131	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
132	#108	—	2	16	no	—
133	C₄⋊C₈	—	2	8	no	—
134	C₂²⋊C₈	—	2	8	no	—
135	#97	—	2	4	no	—
136	#97	C₂×C₄×C₁₂₇	1	4	yes	—
137	#97	C₂×C₄×C₁₂₇	1	8	yes	—
138	#97	C₂×C₄×C₁₂₇	1	4	yes	—
139	C₄×C₈	—	2	4	no	—
140	C₂⁴	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	4	4	no	—
143	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	4	yes	—
144	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	4	yes	—
145	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
146	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
147	C₂²×C₄	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
148	C₂²×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
149	C₂²×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
150	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
151	M₁₆	—	2	16	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	2	16	no	—
154	C₂×C₈	—	2	8	no	—
155	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
156	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
157	C₂×C₈	#41×C₁₂₇	1	8	yes	—
158	C₄²	C₄⋊C₄×C₁₂₇	1	8	yes	—
159	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
160	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
161	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
162	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	4	no	—
165	C₂³	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
166	C₂³	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
175	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	4	yes	—
176	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	4	yes	—
177	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
178	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
179	C₂×C₄	C₂×#41×C₁₂₇	1	4	yes	—
180	C₈	—	2	4	no	—
181	C₈	—	4	8	no	—
182	C₈	#117×C₁₂₇	1	8	yes	—
183	C₂²	—	2	8	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	2	2	no	—
186	C₂²	—	2	2	no	—
187	C₂²	#430×C₁₂₇	1	4	yes	—
188	C₂²	C₂²×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
189	C₂²	#533×C₁₂₇	1	2	yes	—
190	C₄	—	2	4	no	—
191	C₄	—	2	4	no	—
192	C₄	—	2	2	no	—
193	C₄	C₂×#117×C₁₂₇	1	2	yes	—
194	C₄	#532×C₁₂₇	1	2	yes	—
195	C₂	—	2	2	no	—
196	C₂	—	2	4	no	—
197	C₂	C₁₂₇×#2285	1	1	yes	characteristic
198	C₂	C₂×#430×C₁₂₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle	C₂²×C₁₂₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₁₂₇	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂²×C₄²×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₂₇×#1849	C₂	semidirect
4. C₁₂₇×#1838	C₂	semidirect
6. C₁₂₇×#1778	C₂	semidirect
95. #20830	C₁₂₇	direct (N × H)
102. #1849	C₂×C₁₂₇	semidirect
103. #1838	C₂×C₁₂₇	semidirect
105. #1778	C₂×C₁₂₇	semidirect
106. C₁₂₇	#20830	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₂₇×#2920	2
C₂²×#97×C₁₂₇	2
C₁₂₇×#1849	2
C₁₂₇×#1838	2
C₁₂₇×#1810	2
C₁₂₇×#1778	2
#20830	127

Subgroups — C127×#20830 #44584503

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#20830 #44584503