Subgroups — C₁₂₇×#19663 #44583336

Order 32512.

690 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#3230	C₂	1	2	yes	maximal
2	C₁₂₇×#3209	C₂	1	2	yes	maximal
3	C₁₂₇×#3160	C₂	1	1	yes	maximal, characteristic
4	C₂×#498×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₁₂₇×#1723	C₂	1	4	yes	maximal, complemented
6	C₁₂₇×#1707	C₂	1	4	yes	maximal, complemented
7	C₁₂₇×#1450	C₂	1	1	yes	maximal, characteristic
8	#533×C₁₂₇	C₂²	1	1	yes	characteristic
9	#533×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₁₂₇	C₂²	1	2	yes	—
11	#500×C₁₂₇	C₂²	1	2	yes	—
12	#500×C₁₂₇	C₂²	1	4	yes	—
13	#499×C₁₂₇	—	2	2	no	—
14	#498×C₁₂₇	—	2	2	no	—
15	#498×C₁₂₇	C₂²	1	2	yes	—
16	#498×C₁₂₇	C₂²	1	2	yes	—
17	#498×C₁₂₇	C₂²	1	2	yes	—
18	#494×C₁₂₇	C₂²	1	4	yes	—
19	#487×C₁₂₇	C₂²	1	4	yes	—
20	#487×C₁₂₇	C₂²	1	4	yes	—
21	#484×C₁₂₇	—	2	4	no	—
22	#477×C₁₂₇	—	2	4	no	—
23	#462×C₁₂₇	—	2	4	no	—
24	#451×C₁₂₇	—	2	4	no	—
25	#431×C₁₂₇	C₂²	1	2	yes	—
26	#427×C₁₂₇	—	2	2	no	—
27	C₂×#106×C₁₂₇	—	2	2	no	—
28	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
29	#322×C₁₂₇	C₂²	1	2	yes	—
30	C₈²×C₁₂₇	C₂²	1	2	yes	—
31	C₂²×Q₈×C₁₂₇	—	2	2	no	—
32	C₂×Q₁₆×C₁₂₇	—	2	8	no	—
33	C₂²×C₈×C₁₂₇	D₄	1	2	yes	—
34	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
35	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
36	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
37	C₄⋊Q₈×C₁₂₇	C₂³	1	4	yes	—
38	C₄⋊Q₈×C₁₂₇	C₂³	1	2	yes	—
39	C₄⋊Q₈×C₁₂₇	C₂³	1	2	yes	—
40	#125×C₁₂₇	—	2	1	no	—
41	#125×C₁₂₇	—	2	1	no	—
42	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
43	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
44	C₄×Q₈×C₁₂₇	—	2	2	no	—
45	C₄×Q₈×C₁₂₇	—	2	2	no	—
46	#117×C₁₂₇	—	2	1	no	—
47	#117×C₁₂₇	—	2	1	no	—
48	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
50	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
51	#107×C₁₂₇	—	2	4	no	—
52	#107×C₁₂₇	—	2	8	no	—
53	#106×C₁₂₇	—	2	4	no	—
54	#106×C₁₂₇	—	2	4	no	—
55	#106×C₁₂₇	—	2	4	no	—
56	C₄⋊C₈×C₁₂₇	D₄	1	4	yes	—
57	#103×C₁₂₇	—	2	8	no	—
58	#103×C₁₂₇	—	2	8	no	—
59	C₂²⋊C₈×C₁₂₇	D₄	1	4	yes	—
60	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
61	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₁₂₇	C₂³	1	4	yes	—
64	C₂×Q₈×C₁₂₇	—	2	2	no	—
65	C₂×Q₈×C₁₂₇	—	2	4	no	—
66	C₂×Q₈×C₁₂₇	—	2	2	no	—
67	C₂×Q₈×C₁₂₇	—	4	2	no	—
68	C₂×Q₈×C₁₂₇	—	8	4	no	—
69	C₂²×C₄×C₁₂₇	—	4	2	no	—
70	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
72	Q₁₆×C₁₂₇	—	8	8	no	—
73	C₂×C₈×C₁₂₇	Q₁₆	1	8	yes	complemented
74	C₂×C₈×C₁₂₇	C₂×D₄	1	8	yes	—
75	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
77	C₄⋊C₄×C₁₂₇	—	2	2	no	—
78	C₄⋊C₄×C₁₂₇	—	2	4	no	—
79	C₄⋊C₄×C₁₂₇	—	2	4	no	—
80	C₄⋊C₄×C₁₂₇	—	2	2	no	—
81	C₄⋊C₄×C₁₂₇	—	2	2	no	—
82	C₄⋊C₄×C₁₂₇	—	2	2	no	—
83	C₄⋊C₄×C₁₂₇	—	4	2	no	—
84	C₄⋊C₄×C₁₂₇	—	4	2	no	—
85	C₄⋊C₄×C₁₂₇	—	4	2	no	—
86	C₄⋊C₄×C₁₂₇	—	4	2	no	—
87	C₂²⋊C₄×C₁₂₇	—	4	1	no	—
88	C₂²⋊C₄×C₁₂₇	—	4	1	no	—
89	C₄²×C₁₂₇	—	4	1	no	—
90	C₄²×C₁₂₇	—	4	1	no	—
91	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
93	C₄²×C₁₂₇	C₂⁴	1	1	yes	characteristic
94	C₂³×C₁₂₇	#127	1	1	yes	characteristic
95	Q₈×C₁₂₇	—	4	4	no	—
96	Q₈×C₁₂₇	—	4	8	no	—
97	Q₈×C₁₂₇	—	4	4	no	—
98	Q₈×C₁₂₇	—	8	4	no	—
99	C₂×C₄×C₁₂₇	—	2	4	no	—
100	C₂×C₄×C₁₂₇	—	4	2	no	—
101	C₂×C₄×C₁₂₇	—	4	2	no	—
102	C₂×C₄×C₁₂₇	—	4	4	no	—
103	C₂×C₄×C₁₂₇	—	8	4	no	—
104	C₂×C₄×C₁₂₇	#127	1	2	yes	—
105	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
106	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	4	yes	—
107	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
109	C₈×C₁₂₇	—	2	8	no	—
110	C₈×C₁₂₇	C₂×Q₁₆	1	4	yes	—
111	C₈×C₁₂₇	C₂×Q₁₆	1	4	yes	—
112	C₂²×C₁₂₇	—	2	2	no	—
113	C₂²×C₁₂₇	#494	1	2	yes	—
114	C₂²×C₁₂₇	C₂×#127	1	1	yes	characteristic
115	C₄×C₁₂₇	—	2	2	no	—
116	C₄×C₁₂₇	—	8	4	no	—
117	C₄×C₁₂₇	—	8	2	no	—
118	C₄×C₁₂₇	—	8	2	no	—
119	C₄×C₁₂₇	#494	1	2	yes	—
120	C₄×C₁₂₇	C₂²×Q₁₆	1	2	yes	—
121	C₄×C₁₂₇	#577	1	4	yes	—
122	#19663	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂×C₁₂₇	—	2	2	no	—
124	C₂×C₁₂₇	C₂×#494	1	1	yes	characteristic
125	C₂×C₁₂₇	#3154	1	2	yes	—
126	#3230	C₂×C₁₂₇	1	2	yes	—
127	#3209	C₂×C₁₂₇	1	2	yes	—
128	#3160	C₂×C₁₂₇	1	1	yes	characteristic
129	C₂×#498	C₂×C₁₂₇	1	1	yes	characteristic
130	#1723	C₂×C₁₂₇	1	4	yes	complemented
131	#1707	C₂×C₁₂₇	1	4	yes	complemented
132	#1450	C₂×C₁₂₇	1	1	yes	characteristic
133	C₁₂₇	#19663	1	1	yes	characteristic, direct product factor, Sylow
134	#533	C₂²×C₁₂₇	1	1	yes	characteristic
135	#533	C₂²×C₁₂₇	1	1	yes	characteristic
136	C₂×C₄⋊Q₈	C₂²×C₁₂₇	1	2	yes	—
137	#500	C₂²×C₁₂₇	1	2	yes	—
138	#500	C₂²×C₁₂₇	1	4	yes	—
139	#499	—	2	2	no	—
140	#498	—	2	2	no	—
141	#498	C₂²×C₁₂₇	1	2	yes	—
142	#498	C₂²×C₁₂₇	1	2	yes	—
143	#498	C₂²×C₁₂₇	1	2	yes	—
144	#494	C₂²×C₁₂₇	1	4	yes	—
145	#487	C₂²×C₁₂₇	1	4	yes	—
146	#487	C₂²×C₁₂₇	1	4	yes	—
147	#484	—	2	4	no	—
148	#477	—	2	4	no	—
149	#462	—	2	4	no	—
150	#451	—	2	4	no	—
151	#431	C₂²×C₁₂₇	1	2	yes	—
152	#427	—	2	2	no	—
153	C₂×#106	—	2	2	no	—
154	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
155	#322	C₂²×C₁₂₇	1	2	yes	—
156	C₈²	C₂²×C₁₂₇	1	2	yes	—
157	C₂²×Q₈	—	2	2	no	—
158	C₂×Q₁₆	—	2	8	no	—
159	C₂²×C₈	D₄×C₁₂₇	1	2	yes	—
160	C₄⋊Q₈	—	2	2	no	—
161	C₄⋊Q₈	—	2	2	no	—
162	C₄⋊Q₈	—	2	2	no	—
163	C₄⋊Q₈	C₂³×C₁₂₇	1	4	yes	—
164	C₄⋊Q₈	C₂³×C₁₂₇	1	2	yes	—
165	C₄⋊Q₈	C₂³×C₁₂₇	1	2	yes	—
166	#125	—	2	1	no	—
167	#125	—	2	1	no	—
168	C₂²⋊Q₈	—	2	2	no	—
169	C₂²⋊Q₈	—	2	2	no	—
170	C₄×Q₈	—	2	2	no	—
171	C₄×Q₈	—	2	2	no	—
172	#117	—	2	1	no	—
173	#117	—	2	1	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
177	#107	—	2	4	no	—
178	#107	—	2	8	no	—
179	#106	—	2	4	no	—
180	#106	—	2	4	no	—
181	#106	—	2	4	no	—
182	C₄⋊C₈	D₄×C₁₂₇	1	4	yes	—
183	#103	—	2	8	no	—
184	#103	—	2	8	no	—
185	C₂²⋊C₈	D₄×C₁₂₇	1	4	yes	—
186	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
187	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
188	C₄×C₈	C₂³×C₁₂₇	1	4	yes	—
189	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	—	2	4	no	—
192	C₂×Q₈	—	2	2	no	—
193	C₂×Q₈	—	4	2	no	—
194	C₂×Q₈	—	8	4	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
197	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
198	Q₁₆	—	8	8	no	—
199	C₂×C₈	Q₁₆×C₁₂₇	1	8	yes	complemented
200	C₂×C₈	C₂×D₄×C₁₂₇	1	8	yes	—
201	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
202	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₂²⋊C₄	—	4	1	no	—
214	C₂²⋊C₄	—	4	1	no	—
215	C₄²	—	4	1	no	—
216	C₄²	—	4	1	no	—
217	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
218	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
219	C₄²	C₂⁴×C₁₂₇	1	1	yes	characteristic
220	C₂³	#127×C₁₂₇	1	1	yes	characteristic
221	Q₈	—	4	4	no	—
222	Q₈	—	4	8	no	—
223	Q₈	—	4	4	no	—
224	Q₈	—	8	4	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	8	4	no	—
230	C₂×C₄	#127×C₁₂₇	1	2	yes	—
231	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
232	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	4	yes	—
233	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
234	C₂×C₄	C₂²×D₄×C₁₂₇	1	2	yes	—
235	C₈	—	2	8	no	—
236	C₈	C₂×Q₁₆×C₁₂₇	1	4	yes	—
237	C₈	C₂×Q₁₆×C₁₂₇	1	4	yes	—
238	C₂²	—	2	2	no	—
239	C₂²	#494×C₁₂₇	1	2	yes	—
240	C₂²	C₂×#127×C₁₂₇	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	8	4	no	—
243	C₄	—	8	2	no	—
244	C₄	—	8	2	no	—
245	C₄	#494×C₁₂₇	1	2	yes	—
246	C₄	C₂²×Q₁₆×C₁₂₇	1	2	yes	—
247	C₄	#577×C₁₂₇	1	4	yes	—
248	C₂	—	2	2	no	—
249	C₂	C₂×#494×C₁₂₇	1	1	yes	characteristic
250	C₂	C₁₂₇×#3154	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₂₇
prime radical	C₂³×C₁₂₇	#127
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
5. C₁₂₇×#1723	C₂	semidirect
6. C₁₂₇×#1707	C₂	semidirect
73. C₂×C₈×C₁₂₇	Q₁₆	semidirect
122. #19663	C₁₂₇	direct (N × H)
130. #1723	C₂×C₁₂₇	semidirect
131. #1707	C₂×C₁₂₇	semidirect
133. C₁₂₇	#19663	direct (N × H)
199. C₂×C₈	Q₁₆×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#3230	2
C₁₂₇×#3209	2
C₁₂₇×#3160	2
C₂×#498×C₁₂₇	2
C₁₂₇×#1723	2
C₁₂₇×#1707	2
C₁₂₇×#1450	2
#19663	127

Subgroups — C127×#19663 #44583336

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#19663 #44583336