Subgroups — C₁₂₇×#11681 #44575354

Order 32512.

324 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#3170	C₂	1	1	yes	maximal, characteristic
2	C₄×M₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1566	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1334	C₂	1	4	yes	maximal, complemented
5	#533×C₁₂₇	—	2	1	no	—
6	C₂×M₃₂×C₁₂₇	—	2	1	no	—
7	#500×C₁₂₇	C₄	1	1	yes	characteristic
8	#494×C₁₂₇	C₄	1	1	yes	characteristic
9	#487×C₁₂₇	—	2	1	no	—
10	#467×C₁₂₇	—	2	1	no	—
11	#467×C₁₂₇	C₄	1	2	yes	—
12	C₄×Q₁₆×C₁₂₇	—	2	1	no	—
13	C₄×Q₁₆×C₁₂₇	C₂²	1	2	yes	—
14	#423×C₁₂₇	—	2	1	no	—
15	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
16	#346×C₁₂₇	C₂²	1	2	yes	—
17	C₄×C₁₆×C₁₂₇	—	2	1	no	—
18	#334×C₁₂₇	—	2	1	no	—
19	#334×C₁₂₇	C₂²	1	2	yes	—
20	C₂×Q₁₆×C₁₂₇	—	2	1	no	—
21	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
22	C₂×M₁₆×C₁₂₇	—	4	1	no	—
23	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
25	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
26	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
27	#125×C₁₂₇	—	4	1	no	—
28	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
29	C₂²⋊Q₈×C₁₂₇	—	4	1	no	—
30	C₄×Q₈×C₁₂₇	—	2	2	no	—
31	C₄×Q₈×C₁₂₇	—	4	1	no	—
32	#117×C₁₂₇	—	4	1	no	—
33	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
34	M₃₂×C₁₂₇	—	2	4	no	—
35	C₂×C₁₆×C₁₂₇	—	2	2	no	—
36	#107×C₁₂₇	—	2	1	no	—
37	#107×C₁₂₇	C₂×C₄	1	2	yes	—
38	C₄⋊C₈×C₁₂₇	—	2	2	no	—
39	C₄⋊C₈×C₁₂₇	—	4	1	no	—
40	#103×C₁₂₇	—	2	2	no	—
41	#103×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	2	no	—
47	C₂×Q₈×C₁₂₇	—	4	1	no	—
48	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
51	Q₁₆×C₁₂₇	—	2	4	no	—
52	Q₁₆×C₁₂₇	—	4	2	no	—
53	M₁₆×C₁₂₇	—	8	2	no	—
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	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₁₂₇	—	2	2	no	—
62	C₄⋊C₄×C₁₂₇	—	4	1	no	—
63	C₄⋊C₄×C₁₂₇	—	4	1	no	—
64	C₄⋊C₄×C₁₂₇	—	4	2	no	—
65	C₄⋊C₄×C₁₂₇	—	4	1	no	—
66	C₄⋊C₄×C₁₂₇	—	4	2	no	—
67	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
68	C₄²×C₁₂₇	—	4	2	no	—
69	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₁₂₇	D₈	1	1	yes	characteristic
71	C₄²×C₁₂₇	SD₁₆	1	1	yes	characteristic
72	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
73	C₁₆×C₁₂₇	—	2	4	no	—
74	C₂³×C₁₂₇	#102	1	1	yes	characteristic
75	Q₈×C₁₂₇	—	4	2	no	—
76	Q₈×C₁₂₇	—	4	2	no	—
77	Q₈×C₁₂₇	—	8	2	no	—
78	C₂×C₄×C₁₂₇	—	2	2	no	—
79	C₂×C₄×C₁₂₇	—	2	2	no	—
80	C₂×C₄×C₁₂₇	—	2	1	no	—
81	C₂×C₄×C₁₂₇	—	2	1	no	—
82	C₂×C₄×C₁₂₇	—	4	2	no	—
83	C₂×C₄×C₁₂₇	—	4	2	no	—
84	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
85	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
86	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
87	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
90	C₈×C₁₂₇	—	2	1	no	—
91	C₈×C₁₂₇	—	2	1	no	—
92	C₈×C₁₂₇	—	8	2	no	—
93	C₈×C₁₂₇	#104	1	2	yes	—
94	C₈×C₁₂₇	#131	1	2	yes	—
95	C₂²×C₁₂₇	—	2	1	no	—
96	C₂²×C₁₂₇	—	2	1	no	—
97	C₂²×C₁₂₇	—	2	1	no	—
98	C₂²×C₁₂₇	C₂×#102	1	1	yes	characteristic
99	C₄×C₁₂₇	—	2	1	no	—
100	C₄×C₁₂₇	—	2	1	no	—
101	C₄×C₁₂₇	—	2	1	no	—
102	C₄×C₁₂₇	—	8	2	no	—
103	C₄×C₁₂₇	—	8	2	no	—
104	C₄×C₁₂₇	#360	1	2	yes	—
105	C₄×C₁₂₇	#362	1	2	yes	—
106	C₄×C₁₂₇	#408	1	1	yes	characteristic
107	C₄×C₁₂₇	C₂×#104	1	1	yes	characteristic
108	#11681	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂×C₁₂₇	—	4	1	no	—
110	C₂×C₁₂₇	#1492	1	1	yes	characteristic
111	C₂×C₁₂₇	C₂×#360	1	1	yes	characteristic
112	C₂×C₁₂₇	C₂×#362	1	1	yes	characteristic
113	#3170	C₂×C₁₂₇	1	1	yes	characteristic
114	C₄×M₃₂	C₂×C₁₂₇	1	1	yes	characteristic
115	#1566	C₂×C₁₂₇	1	1	yes	characteristic
116	#1334	C₂×C₁₂₇	1	4	yes	complemented
117	C₁₂₇	#11681	1	1	yes	characteristic, direct product factor, Sylow
118	#533	—	2	1	no	—
119	C₂×M₃₂	—	2	1	no	—
120	#500	C₄×C₁₂₇	1	1	yes	characteristic
121	#494	C₄×C₁₂₇	1	1	yes	characteristic
122	#487	—	2	1	no	—
123	#467	—	2	1	no	—
124	#467	C₄×C₁₂₇	1	2	yes	—
125	C₄×Q₁₆	—	2	1	no	—
126	C₄×Q₁₆	C₂²×C₁₂₇	1	2	yes	—
127	#423	—	2	1	no	—
128	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
129	#346	C₂²×C₁₂₇	1	2	yes	—
130	C₄×C₁₆	—	2	1	no	—
131	#334	—	2	1	no	—
132	#334	C₂²×C₁₂₇	1	2	yes	—
133	C₂×Q₁₆	—	2	1	no	—
134	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
135	C₂×M₁₆	—	4	1	no	—
136	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
137	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
138	C₄⋊Q₈	—	2	1	no	—
139	C₄⋊Q₈	—	2	1	no	—
140	#125	—	4	1	no	—
141	C₂²⋊Q₈	—	2	2	no	—
142	C₂²⋊Q₈	—	4	1	no	—
143	C₄×Q₈	—	2	2	no	—
144	C₄×Q₈	—	4	1	no	—
145	#117	—	4	1	no	—
146	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
147	M₃₂	—	2	4	no	—
148	C₂×C₁₆	—	2	2	no	—
149	#107	—	2	1	no	—
150	#107	C₂×C₄×C₁₂₇	1	2	yes	—
151	C₄⋊C₈	—	2	2	no	—
152	C₄⋊C₈	—	4	1	no	—
153	#103	—	2	2	no	—
154	#103	—	2	2	no	—
155	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
156	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
157	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
158	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	—	4	1	no	—
161	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
162	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
163	C₂²×C₄	SD₁₆×C₁₂₇	1	1	yes	characteristic
164	Q₁₆	—	2	4	no	—
165	Q₁₆	—	4	2	no	—
166	M₁₆	—	8	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
171	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
172	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
173	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	4	1	no	—
176	C₄⋊C₄	—	4	1	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	1	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₂²⋊C₄	—	4	2	no	—
181	C₄²	—	4	2	no	—
182	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
183	C₄²	D₈×C₁₂₇	1	1	yes	characteristic
184	C₄²	SD₁₆×C₁₂₇	1	1	yes	characteristic
185	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
186	C₁₆	—	2	4	no	—
187	C₂³	#102×C₁₂₇	1	1	yes	characteristic
188	Q₈	—	4	2	no	—
189	Q₈	—	4	2	no	—
190	Q₈	—	8	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
198	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
199	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
200	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
201	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
202	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
203	C₈	—	2	1	no	—
204	C₈	—	2	1	no	—
205	C₈	—	8	2	no	—
206	C₈	#104×C₁₂₇	1	2	yes	—
207	C₈	#131×C₁₂₇	1	2	yes	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	C₂×#102×C₁₂₇	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	8	2	no	—
216	C₄	—	8	2	no	—
217	C₄	#360×C₁₂₇	1	2	yes	—
218	C₄	#362×C₁₂₇	1	2	yes	—
219	C₄	#408×C₁₂₇	1	1	yes	characteristic
220	C₄	C₂×#104×C₁₂₇	1	1	yes	characteristic
221	C₂	—	4	1	no	—
222	C₂	C₁₂₇×#1492	1	1	yes	characteristic
223	C₂	C₂×#360×C₁₂₇	1	1	yes	characteristic
224	C₂	C₂×#362×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
prime radical	C₂³×C₁₂₇	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₂₇×#1334	C₂	semidirect
108. #11681	C₁₂₇	direct (N × H)
116. #1334	C₂×C₁₂₇	semidirect
117. C₁₂₇	#11681	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×#3170	2
C₄×M₃₂×C₁₂₇	2
C₁₂₇×#1566	2
C₁₂₇×#1334	2
#11681	127

Subgroups — C127×#11681 #44575354

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#11681 #44575354