Subgroups — C₁₂₇×#13810 #44577483

Order 32512.

1634 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄⋊C₄×C₁₂₇	C₂	1	3	yes	maximal
2	C₂×#376×C₁₂₇	C₂	1	2	yes	maximal
3	C₁₂₇×C₄⋊#95	C₂	1	3	yes	maximal
4	C₁₂₇×#1434	C₂	1	6	yes	maximal
5	C₄×#95×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊C₄×C₁₂₇	C₂²	1	3	yes	—
7	C₂²×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₁₂₇	C₂²	1	3	yes	—
9	C₂²×C₄²×C₁₂₇	C₂²	1	6	yes	—
10	C₄×C₄⋊C₄×C₁₂₇	C₄	1	24	yes	complemented
11	#376×C₁₂₇	—	2	8	no	—
12	C₂×#95×C₁₂₇	C₂²	1	3	yes	—
13	C₂×#95×C₁₂₇	C₂²	1	6	yes	—
14	C₂×#95×C₁₂₇	C₂²	1	6	yes	—
15	C₂×#95×C₁₂₇	C₂²	1	6	yes	—
16	C₂×#95×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₁₂₇	C₂³	1	6	yes	—
18	C₂³×C₄×C₁₂₇	C₂³	1	3	yes	—
19	C₂³×C₄×C₁₂₇	C₂³	1	2	yes	—
20	C₂³×C₄×C₁₂₇	C₂³	1	3	yes	—
21	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₁₂₇	—	2	6	no	—
23	C₂×C₄⋊C₄×C₁₂₇	—	2	6	no	—
24	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	12	yes	—
25	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	12	yes	—
26	C₂×C₄²×C₁₂₇	—	2	12	no	—
27	C₂×C₄²×C₁₂₇	—	2	12	no	—
28	C₂×C₄²×C₁₂₇	—	2	12	no	—
29	C₂×C₄²×C₁₂₇	C₂×C₄	1	12	yes	—
30	C₂×C₄²×C₁₂₇	C₂×C₄	1	24	yes	—
31	#95×C₁₂₇	—	2	12	no	—
32	#95×C₁₂₇	—	2	6	no	—
33	#95×C₁₂₇	—	4	6	no	—
34	#95×C₁₂₇	C₂×C₄	1	24	yes	—
35	#95×C₁₂₇	D₄	1	4	yes	—
36	C₂⁴×C₁₂₇	C₂⁴	1	1	yes	characteristic
37	C₂²×C₄×C₁₂₇	—	2	2	no	—
38	C₂²×C₄×C₁₂₇	—	2	6	no	—
39	C₂²×C₄×C₁₂₇	—	2	6	no	—
40	C₂²×C₄×C₁₂₇	—	2	6	no	—
41	C₂²×C₄×C₁₂₇	—	2	6	no	—
42	C₂²×C₄×C₁₂₇	—	2	6	no	—
43	C₂²×C₄×C₁₂₇	—	2	6	no	—
44	C₂²×C₄×C₁₂₇	—	2	6	no	—
45	C₂²×C₄×C₁₂₇	—	2	6	no	—
46	C₂²×C₄×C₁₂₇	—	2	6	no	—
47	C₂²×C₄×C₁₂₇	—	2	6	no	—
48	C₂²×C₄×C₁₂₇	—	2	3	no	—
49	C₂²×C₄×C₁₂₇	—	2	6	no	—
50	C₂²×C₄×C₁₂₇	—	2	3	no	—
51	C₂²×C₄×C₁₂₇	C₂²×C₄	1	6	yes	—
52	C₂²×C₄×C₁₂₇	C₂²×C₄	1	12	yes	—
53	C₂²×C₄×C₁₂₇	C₂²×C₄	1	12	yes	—
54	C₂²×C₄×C₁₂₇	C₂²×C₄	1	6	yes	—
55	C₂²×C₄×C₁₂₇	C₂²×C₄	1	6	yes	—
56	C₂²×C₄×C₁₂₇	C₂×D₄	1	6	yes	—
57	C₂²×C₄×C₁₂₇	#41	1	6	yes	—
58	C₂²×C₄×C₁₂₇	#41	1	6	yes	—
59	C₂²×C₄×C₁₂₇	#41	1	2	yes	—
60	C₄⋊C₄×C₁₂₇	—	2	24	no	—
61	C₄⋊C₄×C₁₂₇	C₄²	1	16	yes	—
62	C₄²×C₁₂₇	—	2	24	no	—
63	C₄²×C₁₂₇	—	2	48	no	—
64	C₂³×C₁₂₇	C₂³×C₄	1	3	yes	—
65	C₂³×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
66	C₂³×C₁₂₇	C₂×#41	1	3	yes	—
67	C₂³×C₁₂₇	C₂×#41	1	1	yes	characteristic
68	C₂³×C₁₂₇	C₂×#41	1	3	yes	—
69	C₂³×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
70	C₂³×C₁₂₇	2₋¹⁺⁴	1	3	yes	—
71	C₂×C₄×C₁₂₇	—	2	6	no	—
72	C₂×C₄×C₁₂₇	—	2	12	no	—
73	C₂×C₄×C₁₂₇	—	2	12	no	—
74	C₂×C₄×C₁₂₇	—	2	6	no	—
75	C₂×C₄×C₁₂₇	—	2	12	no	—
76	C₂×C₄×C₁₂₇	—	2	12	no	—
77	C₂×C₄×C₁₂₇	—	2	6	no	—
78	C₂×C₄×C₁₂₇	—	2	12	no	—
79	C₂×C₄×C₁₂₇	—	2	12	no	—
80	C₂×C₄×C₁₂₇	—	2	6	no	—
81	C₂×C₄×C₁₂₇	—	2	6	no	—
82	C₂×C₄×C₁₂₇	—	2	12	no	—
83	C₂×C₄×C₁₂₇	—	4	6	no	—
84	C₂×C₄×C₁₂₇	—	4	6	no	—
85	C₂×C₄×C₁₂₇	—	4	6	no	—
86	C₂×C₄×C₁₂₇	—	4	6	no	—
87	C₂×C₄×C₁₂₇	—	4	6	no	—
88	C₂×C₄×C₁₂₇	—	4	6	no	—
89	C₂×C₄×C₁₂₇	C₂×C₄²	1	8	yes	—
90	C₂×C₄×C₁₂₇	C₂×C₄²	1	4	yes	—
91	C₂×C₄×C₁₂₇	#117	1	12	yes	—
92	C₂×C₄×C₁₂₇	#117	1	12	yes	—
93	C₂×C₄×C₁₂₇	C₄×D₄	1	12	yes	—
94	C₂²×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
95	C₂²×C₁₂₇	C₂×#117	1	3	yes	—
96	C₂²×C₁₂₇	C₂×#117	1	3	yes	—
97	C₂²×C₁₂₇	C₂×C₄×D₄	1	3	yes	—
98	C₂²×C₁₂₇	C₄×#41	1	3	yes	—
99	C₂²×C₁₂₇	#519	1	3	yes	—
100	C₂²×C₁₂₇	#520	1	3	yes	—
101	C₂²×C₁₂₇	#547	1	3	yes	—
102	C₂²×C₁₂₇	#550	1	1	yes	characteristic
103	C₂²×C₁₂₇	#552	1	6	yes	—
104	C₂²×C₁₂₇	#555	1	3	yes	—
105	C₂²×C₁₂₇	#556	1	3	yes	—
106	C₄×C₁₂₇	—	2	12	no	—
107	C₄×C₁₂₇	—	2	8	no	—
108	C₄×C₁₂₇	—	4	6	no	—
109	C₄×C₁₂₇	—	4	12	no	—
110	C₄×C₁₂₇	#376	1	8	yes	complemented
111	#13810	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂×C₁₂₇	C₂×#376	1	1	yes	characteristic
113	C₂×C₁₂₇	C₄²×D₄	1	1	yes	characteristic
114	C₂×C₁₂₇	#2276	1	1	yes	characteristic
115	C₂×C₁₂₇	#2344	1	3	yes	—
116	C₂×C₁₂₇	#2346	1	3	yes	—
117	C₂×C₁₂₇	#2354	1	3	yes	—
118	C₂×C₁₂₇	#2356	1	3	yes	—
119	C₂×C₄×C₄⋊C₄	C₂×C₁₂₇	1	3	yes	—
120	C₂×#376	C₂×C₁₂₇	1	2	yes	—
121	C₄⋊#95	C₂×C₁₂₇	1	3	yes	—
122	#1434	C₂×C₁₂₇	1	6	yes	—
123	C₄×#95	C₂×C₁₂₇	1	1	yes	characteristic
124	C₁₂₇	#13810	1	1	yes	characteristic, direct product factor, Sylow
125	C₂²×C₄⋊C₄	C₂²×C₁₂₇	1	3	yes	—
126	C₂²×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
127	C₂²×C₄²	C₂²×C₁₂₇	1	3	yes	—
128	C₂²×C₄²	C₂²×C₁₂₇	1	6	yes	—
129	C₄×C₄⋊C₄	C₄×C₁₂₇	1	24	yes	complemented
130	#376	—	2	8	no	—
131	C₂×#95	C₂²×C₁₂₇	1	3	yes	—
132	C₂×#95	C₂²×C₁₂₇	1	6	yes	—
133	C₂×#95	C₂²×C₁₂₇	1	6	yes	—
134	C₂×#95	C₂²×C₁₂₇	1	6	yes	—
135	C₂×#95	C₂²×C₁₂₇	1	1	yes	characteristic
136	C₂³×C₄	C₂³×C₁₂₇	1	6	yes	—
137	C₂³×C₄	C₂³×C₁₂₇	1	3	yes	—
138	C₂³×C₄	C₂³×C₁₂₇	1	2	yes	—
139	C₂³×C₄	C₂³×C₁₂₇	1	3	yes	—
140	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
141	C₂×C₄⋊C₄	—	2	6	no	—
142	C₂×C₄⋊C₄	—	2	6	no	—
143	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	12	yes	—
144	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	12	yes	—
145	C₂×C₄²	—	2	12	no	—
146	C₂×C₄²	—	2	12	no	—
147	C₂×C₄²	—	2	12	no	—
148	C₂×C₄²	C₂×C₄×C₁₂₇	1	12	yes	—
149	C₂×C₄²	C₂×C₄×C₁₂₇	1	24	yes	—
150	#95	—	2	12	no	—
151	#95	—	2	6	no	—
152	#95	—	4	6	no	—
153	#95	C₂×C₄×C₁₂₇	1	24	yes	—
154	#95	D₄×C₁₂₇	1	4	yes	—
155	C₂⁴	C₂⁴×C₁₂₇	1	1	yes	characteristic
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	6	no	—
158	C₂²×C₄	—	2	6	no	—
159	C₂²×C₄	—	2	6	no	—
160	C₂²×C₄	—	2	6	no	—
161	C₂²×C₄	—	2	6	no	—
162	C₂²×C₄	—	2	6	no	—
163	C₂²×C₄	—	2	6	no	—
164	C₂²×C₄	—	2	6	no	—
165	C₂²×C₄	—	2	6	no	—
166	C₂²×C₄	—	2	6	no	—
167	C₂²×C₄	—	2	3	no	—
168	C₂²×C₄	—	2	6	no	—
169	C₂²×C₄	—	2	3	no	—
170	C₂²×C₄	C₂²×C₄×C₁₂₇	1	6	yes	—
171	C₂²×C₄	C₂²×C₄×C₁₂₇	1	12	yes	—
172	C₂²×C₄	C₂²×C₄×C₁₂₇	1	12	yes	—
173	C₂²×C₄	C₂²×C₄×C₁₂₇	1	6	yes	—
174	C₂²×C₄	C₂²×C₄×C₁₂₇	1	6	yes	—
175	C₂²×C₄	C₂×D₄×C₁₂₇	1	6	yes	—
176	C₂²×C₄	#41×C₁₂₇	1	6	yes	—
177	C₂²×C₄	#41×C₁₂₇	1	6	yes	—
178	C₂²×C₄	#41×C₁₂₇	1	2	yes	—
179	C₄⋊C₄	—	2	24	no	—
180	C₄⋊C₄	C₄²×C₁₂₇	1	16	yes	—
181	C₄²	—	2	24	no	—
182	C₄²	—	2	48	no	—
183	C₂³	C₂³×C₄×C₁₂₇	1	3	yes	—
184	C₂³	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
185	C₂³	C₂×#41×C₁₂₇	1	3	yes	—
186	C₂³	C₂×#41×C₁₂₇	1	1	yes	characteristic
187	C₂³	C₂×#41×C₁₂₇	1	3	yes	—
188	C₂³	H(5,𝔽₂)×C₁₂₇	1	1	yes	characteristic
189	C₂³	2₋¹⁺⁴×C₁₂₇	1	3	yes	—
190	C₂×C₄	—	2	6	no	—
191	C₂×C₄	—	2	12	no	—
192	C₂×C₄	—	2	12	no	—
193	C₂×C₄	—	2	6	no	—
194	C₂×C₄	—	2	12	no	—
195	C₂×C₄	—	2	12	no	—
196	C₂×C₄	—	2	6	no	—
197	C₂×C₄	—	2	12	no	—
198	C₂×C₄	—	2	12	no	—
199	C₂×C₄	—	2	6	no	—
200	C₂×C₄	—	2	6	no	—
201	C₂×C₄	—	2	12	no	—
202	C₂×C₄	—	4	6	no	—
203	C₂×C₄	—	4	6	no	—
204	C₂×C₄	—	4	6	no	—
205	C₂×C₄	—	4	6	no	—
206	C₂×C₄	—	4	6	no	—
207	C₂×C₄	—	4	6	no	—
208	C₂×C₄	C₂×C₄²×C₁₂₇	1	8	yes	—
209	C₂×C₄	C₂×C₄²×C₁₂₇	1	4	yes	—
210	C₂×C₄	#117×C₁₂₇	1	12	yes	—
211	C₂×C₄	#117×C₁₂₇	1	12	yes	—
212	C₂×C₄	C₄×D₄×C₁₂₇	1	12	yes	—
213	C₂²	C₂²×C₄²×C₁₂₇	1	1	yes	characteristic
214	C₂²	C₂×#117×C₁₂₇	1	3	yes	—
215	C₂²	C₂×#117×C₁₂₇	1	3	yes	—
216	C₂²	C₂×C₄×D₄×C₁₂₇	1	3	yes	—
217	C₂²	C₄×#41×C₁₂₇	1	3	yes	—
218	C₂²	#519×C₁₂₇	1	3	yes	—
219	C₂²	#520×C₁₂₇	1	3	yes	—
220	C₂²	#547×C₁₂₇	1	3	yes	—
221	C₂²	#550×C₁₂₇	1	1	yes	characteristic
222	C₂²	#552×C₁₂₇	1	6	yes	—
223	C₂²	#555×C₁₂₇	1	3	yes	—
224	C₂²	#556×C₁₂₇	1	3	yes	—
225	C₄	—	2	12	no	—
226	C₄	—	2	8	no	—
227	C₄	—	4	6	no	—
228	C₄	—	4	12	no	—
229	C₄	#376×C₁₂₇	1	8	yes	complemented
230	C₂	C₂×#376×C₁₂₇	1	1	yes	characteristic
231	C₂	C₄²×D₄×C₁₂₇	1	1	yes	characteristic
232	C₂	C₁₂₇×#2276	1	1	yes	characteristic
233	C₂	C₁₂₇×#2344	1	3	yes	—
234	C₂	C₁₂₇×#2346	1	3	yes	—
235	C₂	C₁₂₇×#2354	1	3	yes	—
236	C₂	C₁₂₇×#2356	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
10. C₄×C₄⋊C₄×C₁₂₇	C₄	semidirect
110. C₄×C₁₂₇	#376	semidirect
111. #13810	C₁₂₇	direct (N × H)
124. C₁₂₇	#13810	direct (N × H)
129. C₄×C₄⋊C₄	C₄×C₁₂₇	semidirect
229. C₄	#376×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₄⋊C₄×C₁₂₇	2
C₂×#376×C₁₂₇	2
C₁₂₇×C₄⋊#95	2
C₁₂₇×#1434	2
C₄×#95×C₁₂₇	2
#13810	127

Subgroups — C127×#13810 #44577483

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#13810 #44577483