Subgroups — C₁₂₇×#9611 #44573284

Order 32512.

382 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#1947	C₂	1	2	yes	maximal
2	C₁₂₇×#1844	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#334×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₁₂₇×#1296	C₂	1	2	yes	maximal
6	C₂×#107×C₁₂₇	—	2	2	no	—
7	C₂×#106×C₁₂₇	—	2	1	no	—
8	C₂×#106×C₁₂₇	—	2	1	no	—
9	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₂₇	—	2	1	no	—
12	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
13	#384×C₁₂₇	C₂²	1	2	yes	—
14	C₄×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
15	#340×C₁₂₇	—	2	2	no	—
16	#336×C₁₂₇	—	2	1	no	—
17	#334×C₁₂₇	C₄	1	4	yes	complemented
18	C₂²×C₈×C₁₂₇	—	2	1	no	—
19	C₂²×C₈×C₁₂₇	—	2	2	no	—
20	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
23	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
24	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
25	C₂×C₄⋊C₄×C₁₂₇	D₄	1	2	yes	—
26	C₂×C₄²×C₁₂₇	—	2	1	no	—
27	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
28	#107×C₁₂₇	—	4	4	no	—
29	#106×C₁₂₇	—	4	2	no	—
30	#106×C₁₂₇	—	4	2	no	—
31	C₄⋊C₈×C₁₂₇	—	2	2	no	—
32	C₄⋊C₈×C₁₂₇	—	2	1	no	—
33	C₄⋊C₈×C₁₂₇	C₂×C₄	1	4	yes	—
34	C₄⋊C₈×C₁₂₇	Q₈	1	2	yes	—
35	C₄×C₈×C₁₂₇	—	4	2	no	—
36	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₁₂₇	Q₈	1	2	yes	—
38	#95×C₁₂₇	—	2	1	no	—
39	C₂²×C₄×C₁₂₇	—	2	2	no	—
40	C₂²×C₄×C₁₂₇	—	4	2	no	—
41	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₂₇	#41	1	1	yes	characteristic
44	C₂×C₈×C₁₂₇	—	2	1	no	—
45	C₂×C₈×C₁₂₇	—	2	1	no	—
46	C₂×C₈×C₁₂₇	—	2	4	no	—
47	C₂×C₈×C₁₂₇	—	2	2	no	—
48	C₂×C₈×C₁₂₇	—	4	1	no	—
49	C₂×C₈×C₁₂₇	—	4	2	no	—
50	C₂×C₈×C₁₂₇	—	4	2	no	—
51	C₂×C₈×C₁₂₇	—	4	1	no	—
52	C₂×C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
54	C₂×C₈×C₁₂₇	D₈	1	2	yes	—
55	C₂×C₈×C₁₂₇	Q₁₆	1	2	yes	—
56	C₄⋊C₄×C₁₂₇	—	2	2	no	—
57	C₄⋊C₄×C₁₂₇	—	4	2	no	—
58	C₄⋊C₄×C₁₂₇	—	4	2	no	—
59	C₄⋊C₄×C₁₂₇	—	4	2	no	—
60	C₄⋊C₄×C₁₂₇	—	4	2	no	—
61	C₄⋊C₄×C₁₂₇	—	8	2	no	—
62	C₄⋊C₄×C₁₂₇	SD₁₆	1	4	yes	—
63	C₄²×C₁₂₇	—	4	1	no	—
64	C₄²×C₁₂₇	—	4	1	no	—
65	C₄²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
68	C₄²×C₁₂₇	#41	1	1	yes	characteristic
69	C₂³×C₁₂₇	#121	1	1	yes	characteristic
70	C₂×C₄×C₁₂₇	—	2	2	no	—
71	C₂×C₄×C₁₂₇	—	2	4	no	—
72	C₂×C₄×C₁₂₇	—	4	2	no	—
73	C₂×C₄×C₁₂₇	—	4	2	no	—
74	C₂×C₄×C₁₂₇	—	8	2	no	—
75	C₂×C₄×C₁₂₇	—	8	2	no	—
76	C₂×C₄×C₁₂₇	—	8	2	no	—
77	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₁₂₇	#125	1	1	yes	characteristic
82	C₂×C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	2	yes	—
85	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
86	C₂×C₄×C₁₂₇	#135	1	2	yes	—
87	C₂×C₄×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
88	C₂×C₄×C₁₂₇	#137	1	1	yes	characteristic
89	C₈×C₁₂₇	—	2	2	no	—
90	C₈×C₁₂₇	—	4	4	no	—
91	C₈×C₁₂₇	—	4	2	no	—
92	C₈×C₁₂₇	#107	1	4	yes	complemented
93	C₂²×C₁₂₇	#389	1	1	yes	characteristic
94	C₂²×C₁₂₇	#459	1	1	yes	characteristic
95	C₂²×C₁₂₇	#462	1	1	yes	characteristic
96	C₂²×C₁₂₇	#465	1	2	yes	—
97	C₂²×C₁₂₇	#471	1	2	yes	—
98	C₄×C₁₂₇	—	2	1	no	—
99	C₄×C₁₂₇	—	2	1	no	—
100	C₄×C₁₂₇	—	4	4	no	—
101	C₄×C₁₂₇	—	16	2	no	—
102	C₄×C₁₂₇	C₂×#107	1	1	yes	characteristic
103	C₄×C₁₂₇	#428	1	1	yes	characteristic
104	C₄×C₁₂₇	C₄×SD₁₆	1	2	yes	—
105	C₄×C₁₂₇	#474	1	1	yes	characteristic
106	C₄×C₁₂₇	#477	1	1	yes	characteristic
107	C₄×C₁₂₇	#498	1	1	yes	characteristic
108	C₄×C₁₂₇	#499	1	1	yes	characteristic
109	#9611	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂×C₁₂₇	#1665	1	2	yes	—
111	C₂×C₁₂₇	#1672	1	2	yes	—
112	C₂×C₁₂₇	#1941	1	1	yes	characteristic
113	C₂×C₁₂₇	#1947	1	1	yes	characteristic
114	C₂×C₁₂₇	#1951	1	1	yes	characteristic
115	#1947	C₂×C₁₂₇	1	2	yes	—
116	#1844	C₂×C₁₂₇	1	1	yes	characteristic
117	C₈×C₄⋊C₄	C₂×C₁₂₇	1	1	yes	characteristic
118	C₂×#334	C₂×C₁₂₇	1	1	yes	characteristic
119	#1296	C₂×C₁₂₇	1	2	yes	—
120	C₁₂₇	#9611	1	1	yes	characteristic, direct product factor, Sylow
121	C₂×#107	—	2	2	no	—
122	C₂×#106	—	2	1	no	—
123	C₂×#106	—	2	1	no	—
124	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
125	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	1	yes	characteristic
126	C₂×C₄×C₈	—	2	1	no	—
127	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
128	#384	C₂²×C₁₂₇	1	2	yes	—
129	C₄×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
130	#340	—	2	2	no	—
131	#336	—	2	1	no	—
132	#334	C₄×C₁₂₇	1	4	yes	complemented
133	C₂²×C₈	—	2	1	no	—
134	C₂²×C₈	—	2	2	no	—
135	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
136	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	D₄×C₁₂₇	1	2	yes	—
141	C₂×C₄²	—	2	1	no	—
142	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
143	#107	—	4	4	no	—
144	#106	—	4	2	no	—
145	#106	—	4	2	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	2	1	no	—
148	C₄⋊C₈	C₂×C₄×C₁₂₇	1	4	yes	—
149	C₄⋊C₈	Q₈×C₁₂₇	1	2	yes	—
150	C₄×C₈	—	4	2	no	—
151	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
152	C₄×C₈	Q₈×C₁₂₇	1	2	yes	—
153	#95	—	2	1	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
157	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
158	C₂²×C₄	#41×C₁₂₇	1	1	yes	characteristic
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	4	1	no	—
164	C₂×C₈	—	4	2	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	C₄⋊C₄×C₁₂₇	1	2	yes	—
168	C₂×C₈	C₄⋊C₄×C₁₂₇	1	2	yes	—
169	C₂×C₈	D₈×C₁₂₇	1	2	yes	—
170	C₂×C₈	Q₁₆×C₁₂₇	1	2	yes	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	8	2	no	—
177	C₄⋊C₄	SD₁₆×C₁₂₇	1	4	yes	—
178	C₄²	—	4	1	no	—
179	C₄²	—	4	1	no	—
180	C₄²	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
181	C₄²	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
182	C₄²	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
183	C₄²	#41×C₁₂₇	1	1	yes	characteristic
184	C₂³	#121×C₁₂₇	1	1	yes	characteristic
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	8	2	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
193	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
194	C₂×C₄	C₄×Q₈×C₁₂₇	1	1	yes	characteristic
195	C₂×C₄	C₂²⋊Q₈×C₁₂₇	1	1	yes	characteristic
196	C₂×C₄	#125×C₁₂₇	1	1	yes	characteristic
197	C₂×C₄	C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
198	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
199	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	2	yes	—
200	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
201	C₂×C₄	#135×C₁₂₇	1	2	yes	—
202	C₂×C₄	AGL(1,ℤ/8)×C₁₂₇	1	1	yes	characteristic
203	C₂×C₄	#137×C₁₂₇	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	4	4	no	—
206	C₈	—	4	2	no	—
207	C₈	#107×C₁₂₇	1	4	yes	complemented
208	C₂²	#389×C₁₂₇	1	1	yes	characteristic
209	C₂²	#459×C₁₂₇	1	1	yes	characteristic
210	C₂²	#462×C₁₂₇	1	1	yes	characteristic
211	C₂²	#465×C₁₂₇	1	2	yes	—
212	C₂²	#471×C₁₂₇	1	2	yes	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	4	4	no	—
216	C₄	—	16	2	no	—
217	C₄	C₂×#107×C₁₂₇	1	1	yes	characteristic
218	C₄	#428×C₁₂₇	1	1	yes	characteristic
219	C₄	C₄×SD₁₆×C₁₂₇	1	2	yes	—
220	C₄	#474×C₁₂₇	1	1	yes	characteristic
221	C₄	#477×C₁₂₇	1	1	yes	characteristic
222	C₄	#498×C₁₂₇	1	1	yes	characteristic
223	C₄	#499×C₁₂₇	1	1	yes	characteristic
224	C₂	C₁₂₇×#1665	1	2	yes	—
225	C₂	C₁₂₇×#1672	1	2	yes	—
226	C₂	C₁₂₇×#1941	1	1	yes	characteristic
227	C₂	C₁₂₇×#1947	1	1	yes	characteristic
228	C₂	C₁₂₇×#1951	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
17. #334×C₁₂₇	C₄	semidirect
92. C₈×C₁₂₇	#107	semidirect
109. #9611	C₁₂₇	direct (N × H)
120. C₁₂₇	#9611	direct (N × H)
132. #334	C₄×C₁₂₇	semidirect
207. C₈	#107×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#1947	2
C₁₂₇×#1844	2
C₈×C₄⋊C₄×C₁₂₇	2
C₂×#334×C₁₂₇	2
C₁₂₇×#1296	2
#9611	127

Subgroups — C127×#9611 #44573284

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#9611 #44573284