Subgroups — C₁₂₇×#2569 #30635521

Order 16256.

456 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#400×C₁₂₇	C₂	1	2	yes	maximal, complemented
2	#398×C₁₂₇	C₂	1	2	yes	maximal, complemented
3	#387×C₁₂₇	C₂	1	2	yes	maximal, complemented
4	#387×C₁₂₇	C₂	1	2	yes	maximal, complemented
5	#385×C₁₂₇	C₂	1	2	yes	maximal
6	#380×C₁₂₇	C₂	1	1	yes	maximal, characteristic
7	#380×C₁₂₇	C₂	1	2	yes	maximal
8	C₄×C₂²⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#95×C₁₂₇	C₂	1	1	yes	maximal, characteristic
10	C₂³×C₄×C₁₂₇	C₂²	1	2	yes	—
11	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
20	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
21	#95×C₁₂₇	C₂²	1	2	yes	—
22	#95×C₁₂₇	C₂²	1	1	yes	characteristic
23	#95×C₁₂₇	C₂²	1	1	yes	characteristic
24	#95×C₁₂₇	C₂²	1	2	yes	—
25	#95×C₁₂₇	C₂²	1	1	yes	characteristic
26	#95×C₁₂₇	C₂²	1	1	yes	characteristic
27	#95×C₁₂₇	C₂²	1	2	yes	—
28	#95×C₁₂₇	C₂²	1	2	yes	—
29	#95×C₁₂₇	C₂²	1	2	yes	—
30	#95×C₁₂₇	C₂²	1	2	yes	—
31	C₂⁴×C₁₂₇	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₁₂₇	—	2	4	no	—
33	C₂²×C₄×C₁₂₇	—	2	4	no	—
34	C₂²×C₄×C₁₂₇	—	4	2	no	—
35	C₂²×C₄×C₁₂₇	—	4	2	no	—
36	C₂²×C₄×C₁₂₇	Q₈	1	4	yes	—
37	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
45	C₄⋊C₄×C₁₂₇	—	2	4	no	—
46	C₄⋊C₄×C₁₂₇	—	2	4	no	—
47	C₄⋊C₄×C₁₂₇	—	4	2	no	—
48	C₄⋊C₄×C₁₂₇	—	4	2	no	—
49	C₄⋊C₄×C₁₂₇	—	4	2	no	—
50	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
51	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
52	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
53	C₄²×C₁₂₇	—	4	2	no	—
54	C₂³×C₁₂₇	—	2	2	no	—
55	C₂³×C₁₂₇	—	2	2	no	—
56	C₂³×C₁₂₇	C₂×Q₈	1	2	yes	—
57	C₂³×C₁₂₇	#41	1	2	yes	—
58	C₂³×C₁₂₇	#41	1	2	yes	—
59	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
60	C₂×C₄×C₁₂₇	—	2	2	no	—
61	C₂×C₄×C₁₂₇	—	2	2	no	—
62	C₂×C₄×C₁₂₇	—	2	2	no	—
63	C₂×C₄×C₁₂₇	—	2	2	no	—
64	C₂×C₄×C₁₂₇	—	2	2	no	—
65	C₂×C₄×C₁₂₇	—	2	2	no	—
66	C₂×C₄×C₁₂₇	—	2	2	no	—
67	C₂×C₄×C₁₂₇	—	2	2	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	—	2	2	no	—
73	C₂×C₄×C₁₂₇	—	2	2	no	—
74	C₂×C₄×C₁₂₇	—	2	2	no	—
75	C₂×C₄×C₁₂₇	—	2	2	no	—
76	C₂×C₄×C₁₂₇	—	2	2	no	—
77	C₂×C₄×C₁₂₇	—	2	2	no	—
78	C₂×C₄×C₁₂₇	—	2	2	no	—
79	C₂×C₄×C₁₂₇	—	4	4	no	—
80	C₂×C₄×C₁₂₇	—	4	8	no	—
81	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
82	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
83	C₂×C₄×C₁₂₇	#41	1	4	yes	—
84	C₂²×C₁₂₇	—	2	2	no	—
85	C₂²×C₁₂₇	—	2	2	no	—
86	C₂²×C₁₂₇	—	2	4	no	—
87	C₂²×C₁₂₇	—	2	4	no	—
88	C₂²×C₁₂₇	#125	1	4	yes	—
89	C₂²×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
90	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
91	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
92	C₂²×C₁₂₇	C₂×#41	1	2	yes	—
93	C₂²×C₁₂₇	H(5,𝔽₂)	1	2	yes	—
94	C₄×C₁₂₇	—	2	2	no	—
95	C₄×C₁₂₇	—	2	2	no	—
96	C₄×C₁₂₇	—	2	4	no	—
97	C₄×C₁₂₇	—	4	2	no	—
98	C₄×C₁₂₇	—	4	2	no	—
99	C₄×C₁₂₇	—	4	2	no	—
100	C₄×C₁₂₇	—	4	2	no	—
101	C₄×C₁₂₇	—	4	2	no	—
102	C₂×C₁₂₇	—	2	4	no	—
103	C₂×C₁₂₇	C₂×#125	1	1	yes	characteristic
104	C₂×C₁₂₇	#533	1	1	yes	characteristic
105	C₂×C₁₂₇	#543	1	1	yes	characteristic
106	C₂×C₁₂₇	#551	1	2	yes	—
107	C₂×C₁₂₇	#553	1	2	yes	—
108	#2569	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₁₂₇	#2569	1	1	yes	characteristic, direct product factor, Sylow
110	#400	C₂×C₁₂₇	1	2	yes	complemented
111	#398	C₂×C₁₂₇	1	2	yes	complemented
112	#387	C₂×C₁₂₇	1	2	yes	complemented
113	#387	C₂×C₁₂₇	1	2	yes	complemented
114	#385	C₂×C₁₂₇	1	2	yes	—
115	#380	C₂×C₁₂₇	1	1	yes	characteristic
116	#380	C₂×C₁₂₇	1	2	yes	—
117	C₄×C₂²⋊C₄	C₂×C₁₂₇	1	1	yes	characteristic
118	C₂×#95	C₂×C₁₂₇	1	1	yes	characteristic
119	C₂³×C₄	C₂²×C₁₂₇	1	2	yes	—
120	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
121	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
122	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
123	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
124	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
125	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
126	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
127	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
128	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	2	yes	—
129	C₂×C₄²	C₂²×C₁₂₇	1	2	yes	—
130	#95	C₂²×C₁₂₇	1	2	yes	—
131	#95	C₂²×C₁₂₇	1	1	yes	characteristic
132	#95	C₂²×C₁₂₇	1	1	yes	characteristic
133	#95	C₂²×C₁₂₇	1	2	yes	—
134	#95	C₂²×C₁₂₇	1	1	yes	characteristic
135	#95	C₂²×C₁₂₇	1	1	yes	characteristic
136	#95	C₂²×C₁₂₇	1	2	yes	—
137	#95	C₂²×C₁₂₇	1	2	yes	—
138	#95	C₂²×C₁₂₇	1	2	yes	—
139	#95	C₂²×C₁₂₇	1	2	yes	—
140	C₂⁴	C₂³×C₁₂₇	1	1	yes	characteristic
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	Q₈×C₁₂₇	1	4	yes	—
146	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
147	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
148	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
149	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
150	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
151	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
152	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
153	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₂²⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	4	no	—
161	C₂²⋊C₄	—	2	2	no	—
162	C₄²	—	4	2	no	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	2	no	—
165	C₂³	C₂×Q₈×C₁₂₇	1	2	yes	—
166	C₂³	#41×C₁₂₇	1	2	yes	—
167	C₂³	#41×C₁₂₇	1	2	yes	—
168	C₂³	C₂⁴×C₁₂₇	1	1	yes	characteristic
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	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	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	4	no	—
189	C₂×C₄	—	4	8	no	—
190	C₂×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
191	C₂×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
192	C₂×C₄	#41×C₁₂₇	1	4	yes	—
193	C₂²	—	2	2	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	2	4	no	—
196	C₂²	—	2	4	no	—
197	C₂²	#125×C₁₂₇	1	4	yes	—
198	C₂²	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
199	C₂²	C₂×#41×C₁₂₇	1	1	yes	characteristic
200	C₂²	C₂×#41×C₁₂₇	1	1	yes	characteristic
201	C₂²	C₂×#41×C₁₂₇	1	2	yes	—
202	C₂²	H(5,𝔽₂)×C₁₂₇	1	2	yes	—
203	C₄	—	2	2	no	—
204	C₄	—	2	2	no	—
205	C₄	—	2	4	no	—
206	C₄	—	4	2	no	—
207	C₄	—	4	2	no	—
208	C₄	—	4	2	no	—
209	C₄	—	4	2	no	—
210	C₄	—	4	2	no	—
211	C₂	—	2	4	no	—
212	C₂	C₂×#125×C₁₂₇	1	1	yes	characteristic
213	C₂	#533×C₁₂₇	1	1	yes	characteristic
214	C₂	#543×C₁₂₇	1	1	yes	characteristic
215	C₂	#551×C₁₂₇	1	2	yes	—
216	C₂	#553×C₁₂₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. #400×C₁₂₇	C₂	semidirect
2. #398×C₁₂₇	C₂	semidirect
3. #387×C₁₂₇	C₂	semidirect
4. #387×C₁₂₇	C₂	semidirect
108. #2569	C₁₂₇	direct (N × H)
109. C₁₂₇	#2569	direct (N × H)
110. #400	C₂×C₁₂₇	semidirect
111. #398	C₂×C₁₂₇	semidirect
112. #387	C₂×C₁₂₇	semidirect
113. #387	C₂×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#400×C₁₂₇	2
#398×C₁₂₇	2
#387×C₁₂₇	2
#385×C₁₂₇	2
#380×C₁₂₇	2
C₄×C₂²⋊C₄×C₁₂₇	2
C₂×#95×C₁₂₇	2
#2569	127

Subgroups — C127×#2569 #30635521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#2569 #30635521