Subgroups — C₁₂₇×#11611 #44575284

Order 32512.

338 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#446×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#2150	C₂	1	1	yes	maximal, characteristic
3	C₄×M₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1338	C₂	1	4	yes	maximal, complemented
5	C₂×C₄×Q₈×C₁₂₇	C₄	1	1	yes	characteristic
6	C₂×M₃₂×C₁₂₇	—	2	1	no	—
7	C₂×M₃₂×C₁₂₇	—	2	1	no	—
8	#446×C₁₂₇	—	2	1	no	—
9	#446×C₁₂₇	—	2	1	no	—
10	#446×C₁₂₇	C₄	1	2	yes	—
11	#446×C₁₂₇	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₁₂₇	—	2	1	no	—
13	C₂×C₄⋊C₈×C₁₂₇	C₄	1	1	yes	characteristic
14	C₂×#97×C₁₂₇	—	2	1	no	—
15	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆×C₁₂₇	—	2	1	no	—
17	C₄⋊C₁₆×C₁₂₇	C₂²	1	2	yes	—
18	#346×C₁₂₇	C₂²	1	2	yes	—
19	C₄×C₁₆×C₁₂₇	—	2	1	no	—
20	C₂²×Q₈×C₁₂₇	C₈	1	1	yes	characteristic
21	C₂²×C₈×C₁₂₇	—	2	1	no	—
22	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
24	C₄×Q₈×C₁₂₇	—	2	1	no	—
25	C₄×Q₈×C₁₂₇	—	2	1	no	—
26	C₄×Q₈×C₁₂₇	C₈	1	2	yes	—
27	C₄×Q₈×C₁₂₇	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₁₂₇	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₂₇	C₈	1	1	yes	characteristic
30	C₂×C₄²×C₁₂₇	—	2	1	no	—
31	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
32	M₃₂×C₁₂₇	—	2	4	no	—
33	M₃₂×C₁₂₇	—	4	2	no	—
34	C₂×C₁₆×C₁₂₇	—	2	2	no	—
35	C₂×C₁₆×C₁₂₇	—	2	2	no	—
36	C₄⋊C₈×C₁₂₇	—	2	2	no	—
37	C₄⋊C₈×C₁₂₇	—	2	2	no	—
38	C₄⋊C₈×C₁₂₇	—	2	1	no	—
39	C₄⋊C₈×C₁₂₇	C₂×C₄	1	2	yes	—
40	#97×C₁₂₇	—	2	2	no	—
41	#97×C₁₂₇	—	2	2	no	—
42	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
45	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
46	C₂×Q₈×C₁₂₇	—	2	1	no	—
47	C₂×Q₈×C₁₂₇	—	4	2	no	—
48	C₂×Q₈×C₁₂₇	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₁₂₇	—	2	1	no	—
50	C₂²×C₄×C₁₂₇	—	2	1	no	—
51	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₁₂₇	M₁₆	1	1	yes	characteristic
54	C₂×C₈×C₁₂₇	—	2	2	no	—
55	C₂×C₈×C₁₂₇	—	2	2	no	—
56	C₂×C₈×C₁₂₇	—	4	2	no	—
57	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₂₇	SD₁₆	1	2	yes	—
60	C₂×C₈×C₁₂₇	Q₁₆	1	2	yes	—
61	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
63	C₄⋊C₄×C₁₂₇	—	2	2	no	—
64	C₄⋊C₄×C₁₂₇	—	2	1	no	—
65	C₄⋊C₄×C₁₂₇	—	2	2	no	—
66	C₄⋊C₄×C₁₂₇	C₂×C₈	1	2	yes	—
67	C₄²×C₁₂₇	—	2	2	no	—
68	C₄²×C₁₂₇	—	2	2	no	—
69	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₁₂₇	C₂×C₈	1	1	yes	characteristic
71	C₄²×C₁₂₇	M₁₆	1	1	yes	characteristic
72	C₄²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
73	C₁₆×C₁₂₇	—	2	4	no	—
74	C₁₆×C₁₂₇	—	4	2	no	—
75	C₂³×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
76	Q₈×C₁₂₇	—	4	2	no	—
77	Q₈×C₁₂₇	—	4	2	no	—
78	C₂×C₄×C₁₂₇	—	2	2	no	—
79	C₂×C₄×C₁₂₇	—	2	2	no	—
80	C₂×C₄×C₁₂₇	—	2	2	no	—
81	C₂×C₄×C₁₂₇	—	4	1	no	—
82	C₂×C₄×C₁₂₇	—	4	2	no	—
83	C₂×C₄×C₁₂₇	—	4	2	no	—
84	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
85	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
87	C₂×C₄×C₁₂₇	#103	1	2	yes	—
88	C₂×C₄×C₁₂₇	#104	1	2	yes	—
89	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
92	C₈×C₁₂₇	—	2	1	no	—
93	C₈×C₁₂₇	—	2	1	no	—
94	C₈×C₁₂₇	—	4	2	no	—
95	C₈×C₁₂₇	#103	1	2	yes	—
96	C₈×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
97	C₈×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
98	C₂²×C₁₂₇	—	2	1	no	—
99	C₂²×C₁₂₇	—	2	1	no	—
100	C₂²×C₁₂₇	Q₈⋊C₈	1	2	yes	—
101	C₂²×C₁₂₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
102	C₄×C₁₂₇	—	2	1	no	—
103	C₄×C₁₂₇	—	2	1	no	—
104	C₄×C₁₂₇	—	2	1	no	—
105	C₄×C₁₂₇	—	4	2	no	—
106	C₄×C₁₂₇	—	4	2	no	—
107	C₄×C₁₂₇	Q₈⋊C₈	1	2	yes	—
108	C₄×C₁₂₇	#349	1	2	yes	—
109	C₄×C₁₂₇	C₂×#103	1	1	yes	characteristic
110	C₄×C₁₂₇	C₂×#104	1	1	yes	characteristic
111	#11611	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂×C₁₂₇	—	2	2	no	—
113	C₂×C₁₂₇	C₂×Q₈⋊C₈	1	1	yes	characteristic
114	C₂×C₁₂₇	C₂×#349	1	1	yes	characteristic
115	C₂×C₁₂₇	#2117	1	1	yes	characteristic
116	C₂×#446	C₂×C₁₂₇	1	1	yes	characteristic
117	#2150	C₂×C₁₂₇	1	1	yes	characteristic
118	C₄×M₃₂	C₂×C₁₂₇	1	1	yes	characteristic
119	#1338	C₂×C₁₂₇	1	4	yes	complemented
120	C₁₂₇	#11611	1	1	yes	characteristic, direct product factor, Sylow
121	C₂×C₄×Q₈	C₄×C₁₂₇	1	1	yes	characteristic
122	C₂×M₃₂	—	2	1	no	—
123	C₂×M₃₂	—	2	1	no	—
124	#446	—	2	1	no	—
125	#446	—	2	1	no	—
126	#446	C₄×C₁₂₇	1	2	yes	—
127	#446	C₂²×C₁₂₇	1	2	yes	—
128	C₂×C₄⋊C₈	—	2	1	no	—
129	C₂×C₄⋊C₈	C₄×C₁₂₇	1	1	yes	characteristic
130	C₂×#97	—	2	1	no	—
131	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
132	C₄⋊C₁₆	—	2	1	no	—
133	C₄⋊C₁₆	C₂²×C₁₂₇	1	2	yes	—
134	#346	C₂²×C₁₂₇	1	2	yes	—
135	C₄×C₁₆	—	2	1	no	—
136	C₂²×Q₈	C₈×C₁₂₇	1	1	yes	characteristic
137	C₂²×C₈	—	2	1	no	—
138	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
139	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
140	C₄×Q₈	—	2	1	no	—
141	C₄×Q₈	—	2	1	no	—
142	C₄×Q₈	C₈×C₁₂₇	1	2	yes	—
143	C₄×Q₈	C₂×C₄×C₁₂₇	1	2	yes	—
144	C₂×C₄⋊C₄	—	2	1	no	—
145	C₂×C₄⋊C₄	C₈×C₁₂₇	1	1	yes	characteristic
146	C₂×C₄²	—	2	1	no	—
147	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
148	M₃₂	—	2	4	no	—
149	M₃₂	—	4	2	no	—
150	C₂×C₁₆	—	2	2	no	—
151	C₂×C₁₆	—	2	2	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	2	1	no	—
155	C₄⋊C₈	C₂×C₄×C₁₂₇	1	2	yes	—
156	#97	—	2	2	no	—
157	#97	—	2	2	no	—
158	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
159	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
160	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
161	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
162	C₂×Q₈	—	2	1	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂×Q₈	C₂×C₈×C₁₂₇	1	2	yes	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₈×C₁₂₇	1	1	yes	characteristic
169	C₂²×C₄	M₁₆×C₁₂₇	1	1	yes	characteristic
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
174	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
175	C₂×C₈	SD₁₆×C₁₂₇	1	2	yes	—
176	C₂×C₈	Q₁₆×C₁₂₇	1	2	yes	—
177	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
178	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	C₂×C₈×C₁₂₇	1	2	yes	—
183	C₄²	—	2	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
186	C₄²	C₂×C₈×C₁₂₇	1	1	yes	characteristic
187	C₄²	M₁₆×C₁₂₇	1	1	yes	characteristic
188	C₄²	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
189	C₁₆	—	2	4	no	—
190	C₁₆	—	4	2	no	—
191	C₂³	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
192	Q₈	—	4	2	no	—
193	Q₈	—	4	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
201	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
202	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
203	C₂×C₄	#103×C₁₂₇	1	2	yes	—
204	C₂×C₄	#104×C₁₂₇	1	2	yes	—
205	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
206	C₂×C₄	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
207	C₂×C₄	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
208	C₈	—	2	1	no	—
209	C₈	—	2	1	no	—
210	C₈	—	4	2	no	—
211	C₈	#103×C₁₂₇	1	2	yes	—
212	C₈	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
213	C₈	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	Q₈⋊C₈×C₁₂₇	1	2	yes	—
217	C₂²	C₂×C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	4	2	no	—
222	C₄	—	4	2	no	—
223	C₄	Q₈⋊C₈×C₁₂₇	1	2	yes	—
224	C₄	#349×C₁₂₇	1	2	yes	—
225	C₄	C₂×#103×C₁₂₇	1	1	yes	characteristic
226	C₄	C₂×#104×C₁₂₇	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	C₂×Q₈⋊C₈×C₁₂₇	1	1	yes	characteristic
229	C₂	C₂×#349×C₁₂₇	1	1	yes	characteristic
230	C₂	C₁₂₇×#2117	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂×C₂²⋊C₈
prime radical	C₂³×C₁₂₇	C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₂₇×#1338	C₂	semidirect
111. #11611	C₁₂₇	direct (N × H)
119. #1338	C₂×C₁₂₇	semidirect
120. C₁₂₇	#11611	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#446×C₁₂₇	2
C₁₂₇×#2150	2
C₄×M₃₂×C₁₂₇	2
C₁₂₇×#1338	2
#11611	127

Subgroups — C127×#11611 #44575284

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#11611 #44575284