Subgroups — C₁₂₇×#20429 #44584102

Order 32512.

1150 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#423×C₁₂₇	C₂	1	2	yes	maximal
2	C₂×C₄×M₁₆×C₁₂₇	C₂	1	2	yes	maximal
3	C₂×#376×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1768	C₂	1	8	yes	maximal, complemented
5	C₁₂₇×#1728	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₁₂₇	C₂²	1	4	yes	—
7	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
10	#423×C₁₂₇	C₄	1	16	yes	complemented
11	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	8	yes	—
12	C₄×M₁₆×C₁₂₇	—	2	8	no	—
13	C₂×#97×C₁₂₇	C₂²	1	4	yes	—
14	C₂×C₄×C₈×C₁₂₇	C₂²	1	4	yes	—
15	#376×C₁₂₇	C₄	1	4	yes	—
16	#376×C₁₂₇	C₂²	1	4	yes	—
17	C₂×#95×C₁₂₇	C₄	1	4	yes	—
18	#336×C₁₂₇	C₂²	1	8	yes	—
19	C₂³×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
23	C₂×M₁₆×C₁₂₇	—	2	8	no	—
24	C₂×M₁₆×C₁₂₇	—	2	8	no	—
25	C₂×M₁₆×C₁₂₇	C₂×C₄	1	16	yes	—
26	C₂²×C₈×C₁₂₇	C₂³	1	8	yes	—
27	C₂×C₄²×C₁₂₇	—	2	4	no	—
28	C₂×C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₂₇	C₂×C₄	1	8	yes	—
31	C₂×C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₂₇	D₄	1	4	yes	—
33	C₂×C₄²×C₁₂₇	Q₈	1	4	yes	—
34	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
35	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
36	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
37	C₄⋊C₈×C₁₂₇	C₂×C₄	1	32	yes	complemented
38	#97×C₁₂₇	—	2	8	no	—
39	C₄×C₈×C₁₂₇	—	2	8	no	—
40	#95×C₁₂₇	C₂×C₄	1	16	yes	—
41	C₂⁴×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₂₇	—	2	4	no	—
43	C₂²×C₄×C₁₂₇	—	4	4	no	—
44	C₂²×C₄×C₁₂₇	—	4	4	no	—
45	C₂²×C₄×C₁₂₇	—	4	4	no	—
46	C₂²×C₄×C₁₂₇	C₄²	1	8	yes	—
47	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	8	yes	—
48	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
49	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
50	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
51	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
53	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
55	C₂²×C₄×C₁₂₇	#41	1	4	yes	—
56	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
57	M₁₆×C₁₂₇	—	2	32	no	—
58	C₂×C₈×C₁₂₇	—	2	8	no	—
59	C₂×C₈×C₁₂₇	—	2	8	no	—
60	C₂×C₈×C₁₂₇	C₂²×C₄	1	16	yes	—
61	C₄²×C₁₂₇	—	2	4	no	—
62	C₄²×C₁₂₇	—	2	4	no	—
63	C₄²×C₁₂₇	C₄²	1	8	yes	—
64	C₄²×C₁₂₇	C₄⋊C₄	1	8	yes	—
65	C₄²×C₁₂₇	C₂²×C₄	1	8	yes	—
66	C₄²×C₁₂₇	C₂×D₄	1	4	yes	—
67	C₄²×C₁₂₇	C₂×Q₈	1	4	yes	—
68	C₂³×C₁₂₇	—	2	4	no	—
69	C₂³×C₁₂₇	C₂×C₄²	1	2	yes	—
70	C₂³×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
71	C₂³×C₁₂₇	#117	1	2	yes	—
72	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₂₇	—	2	8	no	—
74	C₂×C₄×C₁₂₇	—	2	8	no	—
75	C₂×C₄×C₁₂₇	—	2	4	no	—
76	C₂×C₄×C₁₂₇	—	2	8	no	—
77	C₂×C₄×C₁₂₇	—	4	8	no	—
78	C₂×C₄×C₁₂₇	—	4	8	no	—
79	C₂×C₄×C₁₂₇	—	4	8	no	—
80	C₂×C₄×C₁₂₇	C₂×C₄²	1	8	yes	—
81	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
82	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	8	yes	—
83	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
84	C₂×C₄×C₁₂₇	#117	1	2	yes	—
85	C₂×C₄×C₁₂₇	C₄×D₄	1	8	yes	—
86	C₂×C₄×C₁₂₇	C₄×Q₈	1	8	yes	—
87	C₂×C₄×C₁₂₇	C₂³×C₄	1	2	yes	—
88	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
90	C₂×C₄×C₁₂₇	C₂×#41	1	2	yes	—
91	C₈×C₁₂₇	—	2	16	no	—
92	C₂²×C₁₂₇	—	2	4	no	—
93	C₂²×C₁₂₇	—	2	4	no	—
94	C₂²×C₁₂₇	—	2	4	no	—
95	C₂²×C₁₂₇	C₄×C₄⋊C₄	1	4	yes	—
96	C₂²×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
97	C₂²×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
98	C₂²×C₁₂₇	C₂×#117	1	1	yes	characteristic
99	C₂²×C₁₂₇	#568	1	4	yes	—
100	C₄×C₁₂₇	—	2	8	no	—
101	C₄×C₁₂₇	—	2	8	no	—
102	C₄×C₁₂₇	—	4	4	no	—
103	C₄×C₁₂₇	C₄×C₄⋊C₄	1	4	yes	—
104	C₄×C₁₂₇	C₂×C₄×D₄	1	2	yes	—
105	C₄×C₁₂₇	C₂×C₄×Q₈	1	2	yes	—
106	#20429	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂×C₁₂₇	—	2	4	no	—
108	C₂×C₁₂₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₁₂₇	#2873	1	2	yes	—
110	C₂×C₁₂₇	#2905	1	2	yes	—
111	C₂×C₁₂₇	#2921	1	2	yes	—
112	C₂×#423	C₂×C₁₂₇	1	2	yes	—
113	C₂×C₄×M₁₆	C₂×C₁₂₇	1	2	yes	—
114	C₂×#376	C₂×C₁₂₇	1	1	yes	characteristic
115	#1768	C₂×C₁₂₇	1	8	yes	complemented
116	#1728	C₂×C₁₂₇	1	2	yes	—
117	C₁₂₇	#20429	1	1	yes	characteristic, direct product factor, Sylow
118	C₂²×M₁₆	C₂²×C₁₂₇	1	4	yes	—
119	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
120	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
121	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
122	#423	C₄×C₁₂₇	1	16	yes	complemented
123	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	8	yes	—
124	C₄×M₁₆	—	2	8	no	—
125	C₂×#97	C₂²×C₁₂₇	1	4	yes	—
126	C₂×C₄×C₈	C₂²×C₁₂₇	1	4	yes	—
127	#376	C₄×C₁₂₇	1	4	yes	—
128	#376	C₂²×C₁₂₇	1	4	yes	—
129	C₂×#95	C₄×C₁₂₇	1	4	yes	—
130	#336	C₂²×C₁₂₇	1	8	yes	—
131	C₂³×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
132	C₂³×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
133	C₂³×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
134	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
135	C₂×M₁₆	—	2	8	no	—
136	C₂×M₁₆	—	2	8	no	—
137	C₂×M₁₆	C₂×C₄×C₁₂₇	1	16	yes	—
138	C₂²×C₈	C₂³×C₁₂₇	1	8	yes	—
139	C₂×C₄²	—	2	4	no	—
140	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
141	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
142	C₂×C₄²	C₂×C₄×C₁₂₇	1	8	yes	—
143	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
144	C₂×C₄²	D₄×C₁₂₇	1	4	yes	—
145	C₂×C₄²	Q₈×C₁₂₇	1	4	yes	—
146	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
147	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
148	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
149	C₄⋊C₈	C₂×C₄×C₁₂₇	1	32	yes	complemented
150	#97	—	2	8	no	—
151	C₄×C₈	—	2	8	no	—
152	#95	C₂×C₄×C₁₂₇	1	16	yes	—
153	C₂⁴	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	4	4	no	—
156	C₂²×C₄	—	4	4	no	—
157	C₂²×C₄	—	4	4	no	—
158	C₂²×C₄	C₄²×C₁₂₇	1	8	yes	—
159	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	8	yes	—
160	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
161	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
162	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
163	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
164	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
165	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
166	C₂²×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
167	C₂²×C₄	#41×C₁₂₇	1	4	yes	—
168	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
169	M₁₆	—	2	32	no	—
170	C₂×C₈	—	2	8	no	—
171	C₂×C₈	—	2	8	no	—
172	C₂×C₈	C₂²×C₄×C₁₂₇	1	16	yes	—
173	C₄²	—	2	4	no	—
174	C₄²	—	2	4	no	—
175	C₄²	C₄²×C₁₂₇	1	8	yes	—
176	C₄²	C₄⋊C₄×C₁₂₇	1	8	yes	—
177	C₄²	C₂²×C₄×C₁₂₇	1	8	yes	—
178	C₄²	C₂×D₄×C₁₂₇	1	4	yes	—
179	C₄²	C₂×Q₈×C₁₂₇	1	4	yes	—
180	C₂³	—	2	4	no	—
181	C₂³	C₂×C₄²×C₁₂₇	1	2	yes	—
182	C₂³	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
183	C₂³	#117×C₁₂₇	1	2	yes	—
184	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
185	C₂×C₄	—	2	8	no	—
186	C₂×C₄	—	2	8	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	8	no	—
189	C₂×C₄	—	4	8	no	—
190	C₂×C₄	—	4	8	no	—
191	C₂×C₄	—	4	8	no	—
192	C₂×C₄	C₂×C₄²×C₁₂₇	1	8	yes	—
193	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
194	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	8	yes	—
195	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
196	C₂×C₄	#117×C₁₂₇	1	2	yes	—
197	C₂×C₄	C₄×D₄×C₁₂₇	1	8	yes	—
198	C₂×C₄	C₄×Q₈×C₁₂₇	1	8	yes	—
199	C₂×C₄	C₂³×C₄×C₁₂₇	1	2	yes	—
200	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
201	C₂×C₄	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
202	C₂×C₄	C₂×#41×C₁₂₇	1	2	yes	—
203	C₈	—	2	16	no	—
204	C₂²	—	2	4	no	—
205	C₂²	—	2	4	no	—
206	C₂²	—	2	4	no	—
207	C₂²	C₄×C₄⋊C₄×C₁₂₇	1	4	yes	—
208	C₂²	C₂²×C₄²×C₁₂₇	1	1	yes	characteristic
209	C₂²	C₂²×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
210	C₂²	C₂×#117×C₁₂₇	1	1	yes	characteristic
211	C₂²	#568×C₁₂₇	1	4	yes	—
212	C₄	—	2	8	no	—
213	C₄	—	2	8	no	—
214	C₄	—	4	4	no	—
215	C₄	C₄×C₄⋊C₄×C₁₂₇	1	4	yes	—
216	C₄	C₂×C₄×D₄×C₁₂₇	1	2	yes	—
217	C₄	C₂×C₄×Q₈×C₁₂₇	1	2	yes	—
218	C₂	—	2	4	no	—
219	C₂	C₂×C₄×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
220	C₂	C₁₂₇×#2873	1	2	yes	—
221	C₂	C₁₂₇×#2905	1	2	yes	—
222	C₂	C₁₂₇×#2921	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₂₇	C₂⁴
commutator	C₂²	C₂²×C₄²×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle	C₂³×C₁₂₇	C₂³×C₄
prime radical	C₂⁴×C₁₂₇	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₂₇×#1768	C₂	semidirect
10. #423×C₁₂₇	C₄	semidirect
37. C₄⋊C₈×C₁₂₇	C₂×C₄	semidirect
106. #20429	C₁₂₇	direct (N × H)
115. #1768	C₂×C₁₂₇	semidirect
117. C₁₂₇	#20429	direct (N × H)
122. #423	C₄×C₁₂₇	semidirect
149. C₄⋊C₈	C₂×C₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#423×C₁₂₇	2
C₂×C₄×M₁₆×C₁₂₇	2
C₂×#376×C₁₂₇	2
C₁₂₇×#1768	2
C₁₂₇×#1728	2
#20429	127

Subgroups — C127×#20429 #44584102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#20429 #44584102