Subgroups — C₁₂₇×#21668 #44585341

Order 32512.

702 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#2401	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#1924	C₂	1	3	yes	maximal
3	C₁₂₇×#1917	C₂	1	3	yes	maximal, complemented
4	C₁₂₇×#1840	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₄⋊C₄×C₁₂₇	C₂	1	3	yes	maximal, complemented
6	C₈×C₂²⋊C₄×C₁₂₇	C₂	1	3	yes	maximal
7	C₁₂₇×#1726	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	3	yes	—
10	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	3	yes	—
11	C₂×#97×C₁₂₇	C₂²	1	3	yes	—
12	C₂×C₄×C₈×C₁₂₇	C₂²	1	3	yes	—
13	C₂×C₄×C₈×C₁₂₇	C₂²	1	3	yes	—
14	#401×C₁₂₇	C₄	1	1	yes	characteristic
15	#400×C₁₂₇	C₄	1	3	yes	—
16	#397×C₁₂₇	C₄	1	3	yes	—
17	#396×C₁₂₇	C₄	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₁₂₇	C₂²	1	3	yes	—
19	C₄×C₂²⋊C₄×C₁₂₇	C₂²	1	3	yes	—
20	#376×C₁₂₇	C₂²	1	1	yes	characteristic
21	#336×C₁₂₇	C₂²	1	3	yes	—
22	#336×C₁₂₇	C₂²	1	3	yes	—
23	#336×C₁₂₇	C₂²	1	3	yes	—
24	#336×C₁₂₇	C₂²	1	3	yes	—
25	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₁₂₇	—	2	6	no	—
27	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₁₂₇	C₂³	1	3	yes	—
29	C₂²×C₈×C₁₂₇	C₂³	1	3	yes	—
30	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	3	yes	—
32	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	3	yes	—
33	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	3	yes	—
34	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	3	yes	—
35	C₂×C₄²×C₁₂₇	C₂³	1	3	yes	—
36	C₂×C₄²×C₁₂₇	C₂³	1	3	yes	—
37	C₄⋊C₈×C₁₂₇	—	2	6	no	—
38	C₂²⋊C₈×C₁₂₇	—	2	6	no	—
39	#97×C₁₂₇	—	2	6	no	—
40	C₄×C₈×C₁₂₇	—	2	6	no	—
41	C₄×C₈×C₁₂₇	—	2	6	no	—
42	#95×C₁₂₇	C₂×C₄	1	1	yes	characteristic
43	#95×C₁₂₇	C₂×C₄	1	3	yes	—
44	#95×C₁₂₇	C₂×C₄	1	3	yes	—
45	#95×C₁₂₇	C₂×C₄	1	3	yes	—
46	#95×C₁₂₇	C₂×C₄	1	3	yes	—
47	#95×C₁₂₇	C₂×C₄	1	3	yes	—
48	C₂⁴×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₂₇	—	2	3	no	—
50	C₂²×C₄×C₁₂₇	—	2	3	no	—
51	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₂₇	C₂²×C₄	1	3	yes	—
53	C₂²×C₄×C₁₂₇	C₂²×C₄	1	3	yes	—
54	C₂²×C₄×C₁₂₇	C₂²×C₄	1	3	yes	—
55	C₂²×C₄×C₁₂₇	C₂²×C₄	1	3	yes	—
56	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
57	C₂×C₈×C₁₂₇	—	2	3	no	—
58	C₂×C₈×C₁₂₇	—	2	3	no	—
59	C₂×C₈×C₁₂₇	—	2	3	no	—
60	C₂×C₈×C₁₂₇	—	2	3	no	—
61	C₂×C₈×C₁₂₇	—	2	3	no	—
62	C₂×C₈×C₁₂₇	—	2	3	no	—
63	C₂×C₈×C₁₂₇	—	4	4	no	—
64	C₂×C₈×C₁₂₇	#41	1	6	yes	—
65	C₂×C₈×C₁₂₇	#41	1	6	yes	—
66	C₄⋊C₄×C₁₂₇	—	2	6	no	—
67	C₄⋊C₄×C₁₂₇	—	2	6	no	—
68	C₂²⋊C₄×C₁₂₇	—	2	6	no	—
69	C₂²⋊C₄×C₁₂₇	—	2	6	no	—
70	C₄²×C₁₂₇	—	2	3	no	—
71	C₄²×C₁₂₇	—	2	3	no	—
72	C₄²×C₁₂₇	—	2	3	no	—
73	C₄²×C₁₂₇	—	2	3	no	—
74	C₂³×C₁₂₇	—	2	3	no	—
75	C₂³×C₁₂₇	—	2	3	no	—
76	C₂³×C₁₂₇	#131	1	2	yes	—
77	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₂₇	—	2	3	no	—
79	C₂×C₄×C₁₂₇	—	2	3	no	—
80	C₂×C₄×C₁₂₇	—	2	3	no	—
81	C₂×C₄×C₁₂₇	—	2	3	no	—
82	C₂×C₄×C₁₂₇	—	2	3	no	—
83	C₂×C₄×C₁₂₇	—	2	3	no	—
84	C₂×C₄×C₁₂₇	—	2	3	no	—
85	C₂×C₄×C₁₂₇	—	2	3	no	—
86	C₂×C₄×C₁₂₇	—	2	3	no	—
87	C₂×C₄×C₁₂₇	—	2	3	no	—
88	C₂×C₄×C₁₂₇	—	2	3	no	—
89	C₂×C₄×C₁₂₇	—	2	3	no	—
90	C₂×C₄×C₁₂₇	—	4	1	no	—
91	C₂×C₄×C₁₂₇	—	4	3	no	—
92	C₂×C₄×C₁₂₇	#131	1	6	yes	—
93	C₂×C₄×C₁₂₇	C₂×#41	1	3	yes	—
94	C₂×C₄×C₁₂₇	C₂×#41	1	3	yes	—
95	C₈×C₁₂₇	—	2	6	no	—
96	C₈×C₁₂₇	—	4	1	no	—
97	C₈×C₁₂₇	—	4	3	no	—
98	C₈×C₁₂₇	#126	1	4	yes	—
99	C₂²×C₁₂₇	—	2	6	no	—
100	C₂²×C₁₂₇	—	4	1	no	—
101	C₂²×C₁₂₇	—	4	3	no	—
102	C₂²×C₁₂₇	C₄×#41	1	3	yes	—
103	C₂²×C₁₂₇	C₂×#131	1	1	yes	characteristic
104	C₂²×C₁₂₇	C₂×#131	1	3	yes	—
105	C₄×C₁₂₇	—	2	6	no	—
106	C₄×C₁₂₇	—	4	1	no	—
107	C₄×C₁₂₇	—	4	3	no	—
108	C₄×C₁₂₇	—	4	3	no	—
109	C₄×C₁₂₇	—	4	3	no	—
110	C₄×C₁₂₇	C₂×#126	1	1	yes	characteristic
111	C₄×C₁₂₇	#529	1	3	yes	—
112	#21668	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂×C₁₂₇	—	4	2	no	—
114	C₂×C₁₂₇	C₄×#126	1	1	yes	characteristic
115	C₂×C₁₂₇	#2966	1	3	yes	—
116	C₂×C₁₂₇	#2966	1	3	yes	—
117	#2401	C₂×C₁₂₇	1	1	yes	characteristic
118	#1924	C₂×C₁₂₇	1	3	yes	—
119	#1917	C₂×C₁₂₇	1	3	yes	complemented
120	#1840	C₂×C₁₂₇	1	1	yes	characteristic, complemented
121	C₈×C₄⋊C₄	C₂×C₁₂₇	1	3	yes	complemented
122	C₈×C₂²⋊C₄	C₂×C₁₂₇	1	3	yes	—
123	#1726	C₂×C₁₂₇	1	1	yes	characteristic, complemented
124	C₁₂₇	#21668	1	1	yes	characteristic, direct product factor, Sylow
125	C₂³×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
126	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	3	yes	—
127	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	3	yes	—
128	C₂×#97	C₂²×C₁₂₇	1	3	yes	—
129	C₂×C₄×C₈	C₂²×C₁₂₇	1	3	yes	—
130	C₂×C₄×C₈	C₂²×C₁₂₇	1	3	yes	—
131	#401	C₄×C₁₂₇	1	1	yes	characteristic
132	#400	C₄×C₁₂₇	1	3	yes	—
133	#397	C₄×C₁₂₇	1	3	yes	—
134	#396	C₄×C₁₂₇	1	1	yes	characteristic
135	C₄×C₄⋊C₄	C₂²×C₁₂₇	1	3	yes	—
136	C₄×C₂²⋊C₄	C₂²×C₁₂₇	1	3	yes	—
137	#376	C₂²×C₁₂₇	1	1	yes	characteristic
138	#336	C₂²×C₁₂₇	1	3	yes	—
139	#336	C₂²×C₁₂₇	1	3	yes	—
140	#336	C₂²×C₁₂₇	1	3	yes	—
141	#336	C₂²×C₁₂₇	1	3	yes	—
142	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
143	C₂²×C₈	—	2	6	no	—
144	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
145	C₂²×C₈	C₂³×C₁₂₇	1	3	yes	—
146	C₂²×C₈	C₂³×C₁₂₇	1	3	yes	—
147	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
148	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	3	yes	—
149	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	3	yes	—
150	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	3	yes	—
151	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	3	yes	—
152	C₂×C₄²	C₂³×C₁₂₇	1	3	yes	—
153	C₂×C₄²	C₂³×C₁₂₇	1	3	yes	—
154	C₄⋊C₈	—	2	6	no	—
155	C₂²⋊C₈	—	2	6	no	—
156	#97	—	2	6	no	—
157	C₄×C₈	—	2	6	no	—
158	C₄×C₈	—	2	6	no	—
159	#95	C₂×C₄×C₁₂₇	1	1	yes	characteristic
160	#95	C₂×C₄×C₁₂₇	1	3	yes	—
161	#95	C₂×C₄×C₁₂₇	1	3	yes	—
162	#95	C₂×C₄×C₁₂₇	1	3	yes	—
163	#95	C₂×C₄×C₁₂₇	1	3	yes	—
164	#95	C₂×C₄×C₁₂₇	1	3	yes	—
165	C₂⁴	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
166	C₂²×C₄	—	2	3	no	—
167	C₂²×C₄	—	2	3	no	—
168	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
169	C₂²×C₄	C₂²×C₄×C₁₂₇	1	3	yes	—
170	C₂²×C₄	C₂²×C₄×C₁₂₇	1	3	yes	—
171	C₂²×C₄	C₂²×C₄×C₁₂₇	1	3	yes	—
172	C₂²×C₄	C₂²×C₄×C₁₂₇	1	3	yes	—
173	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
174	C₂×C₈	—	2	3	no	—
175	C₂×C₈	—	2	3	no	—
176	C₂×C₈	—	2	3	no	—
177	C₂×C₈	—	2	3	no	—
178	C₂×C₈	—	2	3	no	—
179	C₂×C₈	—	2	3	no	—
180	C₂×C₈	—	4	4	no	—
181	C₂×C₈	#41×C₁₂₇	1	6	yes	—
182	C₂×C₈	#41×C₁₂₇	1	6	yes	—
183	C₄⋊C₄	—	2	6	no	—
184	C₄⋊C₄	—	2	6	no	—
185	C₂²⋊C₄	—	2	6	no	—
186	C₂²⋊C₄	—	2	6	no	—
187	C₄²	—	2	3	no	—
188	C₄²	—	2	3	no	—
189	C₄²	—	2	3	no	—
190	C₄²	—	2	3	no	—
191	C₂³	—	2	3	no	—
192	C₂³	—	2	3	no	—
193	C₂³	#131×C₁₂₇	1	2	yes	—
194	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
195	C₂×C₄	—	2	3	no	—
196	C₂×C₄	—	2	3	no	—
197	C₂×C₄	—	2	3	no	—
198	C₂×C₄	—	2	3	no	—
199	C₂×C₄	—	2	3	no	—
200	C₂×C₄	—	2	3	no	—
201	C₂×C₄	—	2	3	no	—
202	C₂×C₄	—	2	3	no	—
203	C₂×C₄	—	2	3	no	—
204	C₂×C₄	—	2	3	no	—
205	C₂×C₄	—	2	3	no	—
206	C₂×C₄	—	2	3	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	4	3	no	—
209	C₂×C₄	#131×C₁₂₇	1	6	yes	—
210	C₂×C₄	C₂×#41×C₁₂₇	1	3	yes	—
211	C₂×C₄	C₂×#41×C₁₂₇	1	3	yes	—
212	C₈	—	2	6	no	—
213	C₈	—	4	1	no	—
214	C₈	—	4	3	no	—
215	C₈	#126×C₁₂₇	1	4	yes	—
216	C₂²	—	2	6	no	—
217	C₂²	—	4	3	no	—
218	C₂²	—	4	1	no	—
219	C₂²	C₄×#41×C₁₂₇	1	3	yes	—
220	C₂²	C₂×#131×C₁₂₇	1	1	yes	characteristic
221	C₂²	C₂×#131×C₁₂₇	1	3	yes	—
222	C₄	—	2	6	no	—
223	C₄	—	4	3	no	—
224	C₄	—	4	3	no	—
225	C₄	—	4	3	no	—
226	C₄	—	4	1	no	—
227	C₄	C₂×#126×C₁₂₇	1	1	yes	characteristic
228	C₄	#529×C₁₂₇	1	3	yes	—
229	C₂	—	4	2	no	—
230	C₂	C₄×#126×C₁₂₇	1	1	yes	characteristic
231	C₂	C₁₂₇×#2966	1	3	yes	—
232	C₂	C₁₂₇×#2966	1	3	yes	—

	subgroup	G/H
center	C₂²×C₈×C₁₂₇	C₂³
commutator	C₂³	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle	C₂³×C₁₂₇	C₂³×C₄
prime radical	C₂⁴×C₁₂₇	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₂₇×#1917	C₂	semidirect
4. C₁₂₇×#1840	C₂	semidirect
5. C₈×C₄⋊C₄×C₁₂₇	C₂	semidirect
7. C₁₂₇×#1726	C₂	semidirect
112. #21668	C₁₂₇	direct (N × H)
119. #1917	C₂×C₁₂₇	semidirect
120. #1840	C₂×C₁₂₇	semidirect
121. C₈×C₄⋊C₄	C₂×C₁₂₇	semidirect
123. #1726	C₂×C₁₂₇	semidirect
124. C₁₂₇	#21668	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×#2401	2
C₁₂₇×#1924	2
C₁₂₇×#1917	2
C₁₂₇×#1840	2
C₈×C₄⋊C₄×C₁₂₇	2
C₈×C₂²⋊C₄×C₁₂₇	2
C₁₂₇×#1726	2
#21668	127

Subgroups — C127×#21668 #44585341

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#21668 #44585341