Subgroups — C₁₂₇×#2924 #30635876

Order 16256.

402 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#117×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	#433×C₁₂₇	C₂	1	4	yes	maximal, complemented
3	#432×C₁₂₇	C₂	1	2	yes	maximal, complemented
4	#431×C₁₂₇	C₂	1	2	yes	maximal, complemented
5	#424×C₁₂₇	C₂	1	2	yes	maximal, complemented
6	#407×C₁₂₇	C₂	1	1	yes	maximal, characteristic
7	C₂×C₂²⋊C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	#405×C₁₂₇	C₂	1	2	yes	maximal, complemented
9	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
11	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
12	#117×C₁₂₇	C₄	1	1	yes	characteristic
13	#117×C₁₂₇	C₄	1	1	yes	characteristic
14	#117×C₁₂₇	C₄	1	2	yes	—
15	#117×C₁₂₇	C₂²	1	1	yes	characteristic
16	#117×C₁₂₇	C₂²	1	1	yes	characteristic
17	#117×C₁₂₇	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₁₂₇	C₄	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₂₇	C₄	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₁₂₇	C₄	1	2	yes	—
21	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
22	C₄⋊C₈×C₁₂₇	C₂²	1	4	yes	complemented
23	C₄⋊C₈×C₁₂₇	C₂²	1	4	yes	complemented
24	C₂²⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
25	C₂²⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
26	C₂²⋊C₈×C₁₂₇	C₂²	1	2	yes	—
27	C₂²⋊C₈×C₁₂₇	C₂²	1	2	yes	—
28	C₂²⋊C₈×C₁₂₇	C₂²	1	2	yes	—
29	#97×C₁₂₇	C₂²	1	4	yes	complemented
30	C₄×C₈×C₁₂₇	C₂²	1	4	yes	complemented
31	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₂₇	—	2	4	no	—
33	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
41	M₁₆×C₁₂₇	—	2	4	no	—
42	C₂×C₈×C₁₂₇	—	2	4	no	—
43	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
44	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
45	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
46	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
47	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
48	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
49	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
50	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
51	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
52	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
53	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
54	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
55	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
56	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
57	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
58	C₄²×C₁₂₇	C₂³	1	2	yes	—
59	C₄²×C₁₂₇	C₂³	1	2	yes	—
60	C₂³×C₁₂₇	—	2	2	no	—
61	C₂³×C₁₂₇	—	2	2	no	—
62	C₂³×C₁₂₇	—	4	1	no	—
63	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
64	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
65	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₂₇	—	2	2	no	—
67	C₂×C₄×C₁₂₇	—	2	4	no	—
68	C₂×C₄×C₁₂₇	—	2	2	no	—
69	C₂×C₄×C₁₂₇	—	2	2	no	—
70	C₂×C₄×C₁₂₇	—	2	2	no	—
71	C₂×C₄×C₁₂₇	—	2	2	no	—
72	C₂×C₄×C₁₂₇	—	4	2	no	—
73	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₂₇	#41	1	4	yes	—
81	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
82	C₈×C₁₂₇	—	2	2	no	—
83	C₈×C₁₂₇	—	2	2	no	—
84	C₈×C₁₂₇	—	2	4	no	—
85	C₂²×C₁₂₇	—	2	1	no	—
86	C₂²×C₁₂₇	—	2	1	no	—
87	C₂²×C₁₂₇	—	2	1	no	—
88	C₂²×C₁₂₇	—	2	1	no	—
89	C₂²×C₁₂₇	—	2	1	no	—
90	C₂²×C₁₂₇	—	2	1	no	—
91	C₂²×C₁₂₇	—	2	1	no	—
92	C₂²×C₁₂₇	—	4	4	no	—
93	C₂²×C₁₂₇	#117	1	2	yes	—
94	C₂²×C₁₂₇	#131	1	2	yes	—
95	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
96	C₄×C₁₂₇	—	2	1	no	—
97	C₄×C₁₂₇	—	2	1	no	—
98	C₄×C₁₂₇	—	2	2	no	—
99	C₄×C₁₂₇	—	2	2	no	—
100	C₄×C₁₂₇	—	2	2	no	—
101	C₄×C₁₂₇	—	2	2	no	—
102	C₄×C₁₂₇	#117	1	2	yes	—
103	C₄×C₁₂₇	C₂×#41	1	2	yes	—
104	C₂×C₁₂₇	—	2	2	no	—
105	C₂×C₁₂₇	—	2	2	no	—
106	C₂×C₁₂₇	—	4	1	no	—
107	C₂×C₁₂₇	C₂×#117	1	1	yes	characteristic
108	C₂×C₁₂₇	C₂×#131	1	1	yes	characteristic
109	C₂×C₁₂₇	#568	1	1	yes	characteristic
110	#2924	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₁₂₇	#2924	1	1	yes	characteristic, direct product factor, Sylow
112	C₂×#117	C₂×C₁₂₇	1	1	yes	characteristic
113	#433	C₂×C₁₂₇	1	4	yes	complemented
114	#432	C₂×C₁₂₇	1	2	yes	complemented
115	#431	C₂×C₁₂₇	1	2	yes	complemented
116	#424	C₂×C₁₂₇	1	2	yes	complemented
117	#407	C₂×C₁₂₇	1	1	yes	characteristic
118	C₂×C₂²⋊C₈	C₂×C₁₂₇	1	1	yes	characteristic
119	#405	C₂×C₁₂₇	1	2	yes	complemented
120	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
121	C₂×M₁₆	C₂²×C₁₂₇	1	2	yes	—
122	C₂²×C₈	C₂²×C₁₂₇	1	2	yes	—
123	#117	C₄×C₁₂₇	1	1	yes	characteristic
124	#117	C₄×C₁₂₇	1	1	yes	characteristic
125	#117	C₄×C₁₂₇	1	2	yes	—
126	#117	C₂²×C₁₂₇	1	1	yes	characteristic
127	#117	C₂²×C₁₂₇	1	1	yes	characteristic
128	#117	C₂²×C₁₂₇	1	2	yes	—
129	C₂×C₄⋊C₄	C₄×C₁₂₇	1	1	yes	characteristic
130	C₂×C₄⋊C₄	C₄×C₁₂₇	1	1	yes	characteristic
131	C₂×C₂²⋊C₄	C₄×C₁₂₇	1	2	yes	—
132	C₂×C₄²	C₂²×C₁₂₇	1	2	yes	—
133	C₄⋊C₈	C₂²×C₁₂₇	1	4	yes	complemented
134	C₄⋊C₈	C₂²×C₁₂₇	1	4	yes	complemented
135	C₂²⋊C₈	C₂²×C₁₂₇	1	1	yes	characteristic
136	C₂²⋊C₈	C₂²×C₁₂₇	1	1	yes	characteristic
137	C₂²⋊C₈	C₂²×C₁₂₇	1	2	yes	—
138	C₂²⋊C₈	C₂²×C₁₂₇	1	2	yes	—
139	C₂²⋊C₈	C₂²×C₁₂₇	1	2	yes	—
140	#97	C₂²×C₁₂₇	1	4	yes	complemented
141	C₄×C₈	C₂²×C₁₂₇	1	4	yes	complemented
142	C₂⁴	C₂×C₄×C₁₂₇	1	1	yes	characteristic
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
145	C₂²×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
149	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
150	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
151	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
152	M₁₆	—	2	4	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	C₂³×C₁₂₇	1	2	yes	—
155	C₂×C₈	C₂³×C₁₂₇	1	4	yes	—
156	C₂×C₈	C₂³×C₁₂₇	1	2	yes	—
157	C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
158	C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
159	C₄⋊C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
160	C₄⋊C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
161	C₄⋊C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
162	C₄⋊C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
163	C₂²⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
164	C₂²⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
165	C₂²⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
166	C₂²⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
167	C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
168	C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
169	C₄²	C₂³×C₁₂₇	1	2	yes	—
170	C₄²	C₂³×C₁₂₇	1	2	yes	—
171	C₂³	—	2	2	no	—
172	C₂³	—	2	2	no	—
173	C₂³	—	4	1	no	—
174	C₂³	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
175	C₂³	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
176	C₂³	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
185	C₂×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
186	C₂×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
187	C₂×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
188	C₂×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
189	C₂×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
190	C₂×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
191	C₂×C₄	#41×C₁₂₇	1	4	yes	—
192	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
193	C₈	—	2	2	no	—
194	C₈	—	2	2	no	—
195	C₈	—	2	4	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	—	2	1	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	4	4	no	—
204	C₂²	#117×C₁₂₇	1	2	yes	—
205	C₂²	#131×C₁₂₇	1	2	yes	—
206	C₂²	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	2	2	no	—
210	C₄	—	2	2	no	—
211	C₄	—	2	2	no	—
212	C₄	—	2	2	no	—
213	C₄	#117×C₁₂₇	1	2	yes	—
214	C₄	C₂×#41×C₁₂₇	1	2	yes	—
215	C₂	—	2	2	no	—
216	C₂	—	2	2	no	—
217	C₂	—	4	1	no	—
218	C₂	C₂×#117×C₁₂₇	1	1	yes	characteristic
219	C₂	C₂×#131×C₁₂₇	1	1	yes	characteristic
220	C₂	#568×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂⁴
commutator	C₂²	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₂₇
socle	C₂²×C₁₂₇	C₂³×C₄
prime radical	C₂⁴×C₁₂₇	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #433×C₁₂₇	C₂	semidirect
3. #432×C₁₂₇	C₂	semidirect
4. #431×C₁₂₇	C₂	semidirect
5. #424×C₁₂₇	C₂	semidirect
8. #405×C₁₂₇	C₂	semidirect
22. C₄⋊C₈×C₁₂₇	C₂²	semidirect
23. C₄⋊C₈×C₁₂₇	C₂²	semidirect
29. #97×C₁₂₇	C₂²	semidirect
30. C₄×C₈×C₁₂₇	C₂²	semidirect
110. #2924	C₁₂₇	direct (N × H)
111. C₁₂₇	#2924	direct (N × H)
113. #433	C₂×C₁₂₇	semidirect
114. #432	C₂×C₁₂₇	semidirect
115. #431	C₂×C₁₂₇	semidirect
116. #424	C₂×C₁₂₇	semidirect
119. #405	C₂×C₁₂₇	semidirect
133. C₄⋊C₈	C₂²×C₁₂₇	semidirect
134. C₄⋊C₈	C₂²×C₁₂₇	semidirect
140. #97	C₂²×C₁₂₇	semidirect
141. C₄×C₈	C₂²×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#117×C₁₂₇	2
#433×C₁₂₇	2
#432×C₁₂₇	2
#431×C₁₂₇	2
#424×C₁₂₇	2
#407×C₁₂₇	2
C₂×C₂²⋊C₈×C₁₂₇	2
#405×C₁₂₇	2
#2924	127

Subgroups — C127×#2924 #30635876

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#2924 #30635876