Subgroups — C₁₂₇×#19667 #44583340

Order 32512.

780 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#3207	C₂	1	4	yes	maximal, complemented
2	C₁₂₇×#3169	C₂	1	2	yes	maximal, complemented
3	C₁₂₇×#1717	C₂	1	4	yes	maximal, complemented
4	C₁₂₇×#1712	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₂₇×#1710	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₂₇×#1706	C₂	1	2	yes	maximal, complemented
7	C₁₂₇×#1450	C₂	1	1	yes	maximal, characteristic, complemented
8	#532×C₁₂₇	C₂²	1	4	yes	—
9	#500×C₁₂₇	C₂²	1	2	yes	—
10	#493×C₁₂₇	C₂²	1	2	yes	—
11	#493×C₁₂₇	C₂²	1	4	yes	—
12	#492×C₁₂₇	C₂²	1	4	yes	—
13	#487×C₁₂₇	C₂²	1	4	yes	—
14	#486×C₁₂₇	C₂²	1	4	yes	—
15	#486×C₁₂₇	C₂²	1	4	yes	—
16	#466×C₁₂₇	—	2	4	no	—
17	#463×C₁₂₇	—	2	8	no	—
18	#449×C₁₂₇	—	2	8	no	—
19	C₄×D₈×C₁₂₇	—	2	4	no	—
20	#431×C₁₂₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
22	#322×C₁₂₇	C₂²	1	2	yes	—
23	C₈²×C₁₂₇	C₂²	1	1	yes	characteristic
24	C₈²×C₁₂₇	C₂²	1	1	yes	characteristic
25	C₂×#41×C₁₂₇	—	2	4	no	—
26	C₂×SD₁₆×C₁₂₇	—	2	8	no	—
27	C₂×D₈×C₁₂₇	—	2	4	no	—
28	C₂×D₈×C₁₂₇	—	2	8	no	—
29	C₂²×C₈×C₁₂₇	D₄	1	2	yes	—
30	C₄⋊Q₈×C₁₂₇	C₂³	1	4	yes	—
31	#127×C₁₂₇	C₂³	1	4	yes	—
32	#124×C₁₂₇	—	2	4	no	—
33	#121×C₁₂₇	—	2	4	no	—
34	#121×C₁₂₇	—	2	4	no	—
35	C₄×D₄×C₁₂₇	—	2	4	no	—
36	C₄×D₄×C₁₂₇	—	2	4	no	—
37	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
38	#107×C₁₂₇	—	2	4	no	—
39	C₄⋊C₈×C₁₂₇	D₄	1	4	yes	—
40	#103×C₁₂₇	—	2	8	no	—
41	#102×C₁₂₇	—	2	8	no	—
42	#102×C₁₂₇	—	2	8	no	—
43	C₂²⋊C₈×C₁₂₇	D₄	1	4	yes	—
44	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
45	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
47	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
48	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
49	#41×C₁₂₇	—	8	8	no	—
50	C₂×Q₈×C₁₂₇	—	2	4	no	—
51	C₂×D₄×C₁₂₇	—	2	4	no	—
52	C₂×D₄×C₁₂₇	—	2	4	no	—
53	C₂×D₄×C₁₂₇	—	2	4	no	—
54	C₂×D₄×C₁₂₇	—	4	4	no	—
55	C₂²×C₄×C₁₂₇	—	4	4	no	—
56	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
58	SD₁₆×C₁₂₇	—	8	8	no	—
59	D₈×C₁₂₇	—	4	8	no	—
60	D₈×C₁₂₇	—	8	8	no	—
61	C₂×C₈×C₁₂₇	—	2	4	no	—
62	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
64	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
65	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
66	C₄⋊C₄×C₁₂₇	—	2	4	no	—
67	C₄⋊C₄×C₁₂₇	—	2	4	no	—
68	C₂²⋊C₄×C₁₂₇	—	4	4	no	—
69	C₂²⋊C₄×C₁₂₇	—	4	4	no	—
70	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
72	C₄²×C₁₂₇	C₂⁴	1	1	yes	characteristic
73	C₂³×C₁₂₇	—	4	4	no	—
74	C₂³×C₁₂₇	#127	1	1	yes	characteristic
75	Q₈×C₁₂₇	—	4	8	no	—
76	D₄×C₁₂₇	—	4	8	no	—
77	D₄×C₁₂₇	—	4	8	no	—
78	D₄×C₁₂₇	—	4	8	no	—
79	D₄×C₁₂₇	—	8	8	no	—
80	C₂×C₄×C₁₂₇	—	2	2	no	—
81	C₂×C₄×C₁₂₇	—	2	4	no	—
82	C₂×C₄×C₁₂₇	—	4	4	no	—
83	C₂×C₄×C₁₂₇	—	8	8	no	—
84	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
85	C₂×C₄×C₁₂₇	#127	1	2	yes	—
86	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
88	C₈×C₁₂₇	—	2	2	no	—
89	C₈×C₁₂₇	—	2	4	no	—
90	C₈×C₁₂₇	—	2	4	no	—
91	C₈×C₁₂₇	#135	1	4	yes	—
92	C₂²×C₁₂₇	—	2	1	no	—
93	C₂²×C₁₂₇	—	2	2	no	—
94	C₂²×C₁₂₇	—	8	4	no	—
95	C₂²×C₁₂₇	—	8	4	no	—
96	C₂²×C₁₂₇	—	8	4	no	—
97	C₂²×C₁₂₇	C₂×#127	1	1	yes	characteristic
98	C₄×C₁₂₇	—	2	1	no	—
99	C₄×C₁₂₇	—	2	2	no	—
100	C₄×C₁₂₇	—	8	4	no	—
101	C₄×C₁₂₇	C₂×#135	1	2	yes	—
102	C₄×C₁₂₇	#576	1	4	yes	—
103	#19667	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂×C₁₂₇	—	4	1	no	—
105	C₂×C₁₂₇	—	16	4	no	—
106	C₂×C₁₂₇	#3147	1	1	yes	characteristic
107	C₂×C₁₂₇	#3155	1	2	yes	—
108	#3207	C₂×C₁₂₇	1	4	yes	complemented
109	#3169	C₂×C₁₂₇	1	2	yes	complemented
110	#1717	C₂×C₁₂₇	1	4	yes	complemented
111	#1712	C₂×C₁₂₇	1	1	yes	characteristic, complemented
112	#1710	C₂×C₁₂₇	1	1	yes	characteristic, complemented
113	#1706	C₂×C₁₂₇	1	2	yes	complemented
114	#1450	C₂×C₁₂₇	1	1	yes	characteristic, complemented
115	C₁₂₇	#19667	1	1	yes	characteristic, direct product factor, Sylow
116	#532	C₂²×C₁₂₇	1	4	yes	—
117	#500	C₂²×C₁₂₇	1	2	yes	—
118	#493	C₂²×C₁₂₇	1	2	yes	—
119	#493	C₂²×C₁₂₇	1	4	yes	—
120	#492	C₂²×C₁₂₇	1	4	yes	—
121	#487	C₂²×C₁₂₇	1	4	yes	—
122	#486	C₂²×C₁₂₇	1	4	yes	—
123	#486	C₂²×C₁₂₇	1	4	yes	—
124	#466	—	2	4	no	—
125	#463	—	2	8	no	—
126	#449	—	2	8	no	—
127	C₄×D₈	—	2	4	no	—
128	#431	C₂²×C₁₂₇	1	2	yes	—
129	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
130	#322	C₂²×C₁₂₇	1	2	yes	—
131	C₈²	C₂²×C₁₂₇	1	1	yes	characteristic
132	C₈²	C₂²×C₁₂₇	1	1	yes	characteristic
133	C₂×#41	—	2	4	no	—
134	C₂×SD₁₆	—	2	8	no	—
135	C₂×D₈	—	2	4	no	—
136	C₂×D₈	—	2	8	no	—
137	C₂²×C₈	D₄×C₁₂₇	1	2	yes	—
138	C₄⋊Q₈	C₂³×C₁₂₇	1	4	yes	—
139	#127	C₂³×C₁₂₇	1	4	yes	—
140	#124	—	2	4	no	—
141	#121	—	2	4	no	—
142	#121	—	2	4	no	—
143	C₄×D₄	—	2	4	no	—
144	C₄×D₄	—	2	4	no	—
145	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
146	#107	—	2	4	no	—
147	C₄⋊C₈	D₄×C₁₂₇	1	4	yes	—
148	#103	—	2	8	no	—
149	#102	—	2	8	no	—
150	#102	—	2	8	no	—
151	C₂²⋊C₈	D₄×C₁₂₇	1	4	yes	—
152	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
153	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
154	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
155	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
156	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
157	#41	—	8	8	no	—
158	C₂×Q₈	—	2	4	no	—
159	C₂×D₄	—	2	4	no	—
160	C₂×D₄	—	2	4	no	—
161	C₂×D₄	—	2	4	no	—
162	C₂×D₄	—	4	4	no	—
163	C₂²×C₄	—	4	4	no	—
164	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
165	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
166	SD₁₆	—	8	8	no	—
167	D₈	—	4	8	no	—
168	D₈	—	8	8	no	—
169	C₂×C₈	—	2	4	no	—
170	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
171	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
172	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
173	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
174	C₄⋊C₄	—	2	4	no	—
175	C₄⋊C₄	—	2	4	no	—
176	C₂²⋊C₄	—	4	4	no	—
177	C₂²⋊C₄	—	4	4	no	—
178	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
179	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
180	C₄²	C₂⁴×C₁₂₇	1	1	yes	characteristic
181	C₂³	—	4	4	no	—
182	C₂³	#127×C₁₂₇	1	1	yes	characteristic
183	Q₈	—	4	8	no	—
184	D₄	—	4	8	no	—
185	D₄	—	4	8	no	—
186	D₄	—	4	8	no	—
187	D₄	—	8	8	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	8	8	no	—
192	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
193	C₂×C₄	#127×C₁₂₇	1	2	yes	—
194	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
195	C₂×C₄	C₂²×D₄×C₁₂₇	1	2	yes	—
196	C₈	—	2	2	no	—
197	C₈	—	2	4	no	—
198	C₈	—	2	4	no	—
199	C₈	#135×C₁₂₇	1	4	yes	—
200	C₂²	—	2	1	no	—
201	C₂²	—	2	2	no	—
202	C₂²	—	8	4	no	—
203	C₂²	—	8	4	no	—
204	C₂²	—	8	4	no	—
205	C₂²	C₂×#127×C₁₂₇	1	1	yes	characteristic
206	C₄	—	2	1	no	—
207	C₄	—	2	2	no	—
208	C₄	—	8	4	no	—
209	C₄	C₂×#135×C₁₂₇	1	2	yes	—
210	C₄	#576×C₁₂₇	1	4	yes	—
211	C₂	—	4	1	no	—
212	C₂	—	16	4	no	—
213	C₂	C₁₂₇×#3147	1	1	yes	characteristic
214	C₂	C₁₂₇×#3155	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₂₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₂₇×#3207	C₂	semidirect
2. C₁₂₇×#3169	C₂	semidirect
3. C₁₂₇×#1717	C₂	semidirect
4. C₁₂₇×#1712	C₂	semidirect
5. C₁₂₇×#1710	C₂	semidirect
6. C₁₂₇×#1706	C₂	semidirect
7. C₁₂₇×#1450	C₂	semidirect
103. #19667	C₁₂₇	direct (N × H)
108. #3207	C₂×C₁₂₇	semidirect
109. #3169	C₂×C₁₂₇	semidirect
110. #1717	C₂×C₁₂₇	semidirect
111. #1712	C₂×C₁₂₇	semidirect
112. #1710	C₂×C₁₂₇	semidirect
113. #1706	C₂×C₁₂₇	semidirect
114. #1450	C₂×C₁₂₇	semidirect
115. C₁₂₇	#19667	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×#3207	2
C₁₂₇×#3169	2
C₁₂₇×#1717	2
C₁₂₇×#1712	2
C₁₂₇×#1710	2
C₁₂₇×#1706	2
C₁₂₇×#1450	2
#19667	127

Subgroups — C127×#19667 #44583340

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#19667 #44583340