Subgroups — C₁₂₇×#14686 #44578359

Order 32512.

2718 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄²×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#384×C₁₂₇	C₂	1	2	yes	maximal, complemented
3	C₂×#380×C₁₂₇	C₂	1	6	yes	maximal
4	C₁₂₇×C₄⋊#95	C₂	1	6	yes	maximal, complemented
5	C₂⁴×C₄×C₁₂₇	C₂²	1	3	yes	—
6	C₂²×C₄⋊C₄×C₁₂₇	C₂²	1	12	yes	—
7	C₂²×C₂²⋊C₄×C₁₂₇	C₂²	1	4	yes	—
8	C₂²×C₄²×C₁₂₇	—	2	8	no	—
9	C₂²×C₄²×C₁₂₇	C₄	1	8	yes	complemented
10	C₂²×C₄²×C₁₂₇	C₂²	1	3	yes	—
11	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
12	#384×C₁₂₇	—	2	8	no	—
13	#380×C₁₂₇	—	2	24	no	—
14	C₂×#95×C₁₂₇	C₂²	1	12	yes	—
15	C₂⁵×C₁₂₇	C₂³	1	1	yes	characteristic
16	C₂³×C₄×C₁₂₇	—	2	12	no	—
17	C₂³×C₄×C₁₂₇	—	2	12	no	—
18	C₂³×C₄×C₁₂₇	C₂×C₄	1	12	yes	—
19	C₂³×C₄×C₁₂₇	D₄	1	12	yes	—
20	C₂³×C₄×C₁₂₇	Q₈	1	12	yes	—
21	C₂³×C₄×C₁₂₇	C₂³	1	8	yes	—
22	C₂³×C₄×C₁₂₇	C₂³	1	3	yes	—
23	C₂³×C₄×C₁₂₇	C₂³	1	3	yes	—
24	C₂×C₄⋊C₄×C₁₂₇	—	2	24	no	—
25	C₂×C₄⋊C₄×C₁₂₇	—	2	24	no	—
26	C₂×C₄⋊C₄×C₁₂₇	—	2	24	no	—
27	C₂×C₂²⋊C₄×C₁₂₇	—	2	24	no	—
28	C₂×C₄²×C₁₂₇	—	2	12	no	—
29	C₂×C₄²×C₁₂₇	—	2	32	no	—
30	C₂×C₄²×C₁₂₇	C₂×C₄	1	12	yes	—
31	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₁₂₇	D₄	1	8	yes	—
33	#95×C₁₂₇	—	2	24	no	—
34	C₂⁴×C₁₂₇	—	2	2	no	—
35	C₂⁴×C₁₂₇	—	2	6	no	—
36	C₂⁴×C₁₂₇	C₂²×C₄	1	2	yes	—
37	C₂⁴×C₁₂₇	C₂×D₄	1	6	yes	—
38	C₂⁴×C₁₂₇	C₂×Q₈	1	6	yes	—
39	C₂⁴×C₁₂₇	C₂⁴	1	1	yes	characteristic
40	C₂²×C₄×C₁₂₇	—	2	6	no	—
41	C₂²×C₄×C₁₂₇	—	2	6	no	—
42	C₂²×C₄×C₁₂₇	—	2	24	no	—
43	C₂²×C₄×C₁₂₇	—	2	24	no	—
44	C₂²×C₄×C₁₂₇	—	2	48	no	—
45	C₂²×C₄×C₁₂₇	—	2	48	no	—
46	C₂²×C₄×C₁₂₇	—	2	24	no	—
47	C₂²×C₄×C₁₂₇	—	2	8	no	—
48	C₂²×C₄×C₁₂₇	—	2	24	no	—
49	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	48	yes	complemented
50	C₂²×C₄×C₁₂₇	C₂²×C₄	1	6	yes	—
51	C₂²×C₄×C₁₂₇	C₂²×C₄	1	6	yes	—
52	C₂²×C₄×C₁₂₇	C₂×D₄	1	6	yes	—
53	C₂²×C₄×C₁₂₇	C₂×D₄	1	12	yes	—
54	C₂²×C₄×C₁₂₇	C₂×D₄	1	6	yes	—
55	C₂²×C₄×C₁₂₇	C₂×Q₈	1	6	yes	—
56	C₂²×C₄×C₁₂₇	C₂×Q₈	1	6	yes	—
57	C₂²×C₄×C₁₂₇	#41	1	12	yes	—
58	C₄⋊C₄×C₁₂₇	—	4	48	no	—
59	C₂²⋊C₄×C₁₂₇	—	4	16	no	—
60	C₄²×C₁₂₇	—	2	24	no	—
61	C₄²×C₁₂₇	C₂²⋊C₄	1	16	yes	complemented
62	C₂³×C₁₂₇	—	2	2	no	—
63	C₂³×C₁₂₇	—	2	6	no	—
64	C₂³×C₁₂₇	—	2	12	no	—
65	C₂³×C₁₂₇	—	2	12	no	—
66	C₂³×C₁₂₇	—	2	12	no	—
67	C₂³×C₁₂₇	—	2	12	no	—
68	C₂³×C₁₂₇	C₂×C₄⋊C₄	1	12	yes	—
69	C₂³×C₁₂₇	#127	1	4	yes	—
70	C₂³×C₁₂₇	C₄⋊Q₈	1	12	yes	—
71	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
72	C₂³×C₁₂₇	C₂²×D₄	1	2	yes	—
73	C₂³×C₁₂₇	C₂²×D₄	1	3	yes	—
74	C₂³×C₁₂₇	C₂²×Q₈	1	3	yes	—
75	C₂³×C₁₂₇	C₂×#41	1	6	yes	—
76	C₂×C₄×C₁₂₇	—	2	6	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	96	no	—
81	C₂×C₄×C₁₂₇	—	4	24	no	—
82	C₂×C₄×C₁₂₇	—	4	8	no	—
83	C₂×C₄×C₁₂₇	—	4	24	no	—
84	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	12	yes	—
85	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	12	yes	—
86	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	12	yes	—
87	C₂×C₄×C₁₂₇	#117	1	12	yes	—
88	C₂×C₄×C₁₂₇	#121	1	24	yes	—
89	C₂×C₄×C₁₂₇	C₂²⋊Q₈	1	24	yes	—
90	C₂²×C₁₂₇	—	2	8	no	—
91	C₂²×C₁₂₇	—	2	24	no	—
92	C₂²×C₁₂₇	—	2	12	no	—
93	C₂²×C₁₂₇	—	2	12	no	—
94	C₂²×C₁₂₇	#384	1	8	yes	complemented
95	C₂²×C₁₂₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
96	C₂²×C₁₂₇	C₂²×C₄⋊C₄	1	3	yes	—
97	C₂²×C₁₂₇	C₂×#117	1	3	yes	—
98	C₂²×C₁₂₇	C₂×#121	1	6	yes	—
99	C₂²×C₁₂₇	C₂×C₂²⋊Q₈	1	6	yes	—
100	C₂²×C₁₂₇	C₂×#127	1	1	yes	characteristic
101	C₂²×C₁₂₇	C₂×C₄⋊Q₈	1	3	yes	—
102	C₂²×C₁₂₇	#532	1	6	yes	—
103	C₂²×C₁₂₇	#533	1	6	yes	—
104	C₄×C₁₂₇	—	2	12	no	—
105	C₄×C₁₂₇	—	8	8	no	—
106	C₄×C₁₂₇	#380	1	24	yes	complemented
107	#14686	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂×C₁₂₇	—	2	8	no	—
109	C₂×C₁₂₇	C₂×#380	1	3	yes	—
110	C₂×C₁₂₇	C₂×#384	1	1	yes	characteristic
111	C₂×C₁₂₇	#2285	1	3	yes	—
112	C₂×C₁₂₇	#2856	1	2	yes	—
113	C₂×C₁₂₇	#2857	1	6	yes	—
114	C₂³×C₄²	C₂×C₁₂₇	1	1	yes	characteristic
115	C₂×#384	C₂×C₁₂₇	1	2	yes	complemented
116	C₂×#380	C₂×C₁₂₇	1	6	yes	—
117	C₄⋊#95	C₂×C₁₂₇	1	6	yes	complemented
118	C₁₂₇	#14686	1	1	yes	characteristic, direct product factor, Sylow
119	C₂⁴×C₄	C₂²×C₁₂₇	1	3	yes	—
120	C₂²×C₄⋊C₄	C₂²×C₁₂₇	1	12	yes	—
121	C₂²×C₂²⋊C₄	C₂²×C₁₂₇	1	4	yes	—
122	C₂²×C₄²	—	2	8	no	—
123	C₂²×C₄²	C₄×C₁₂₇	1	8	yes	complemented
124	C₂²×C₄²	C₂²×C₁₂₇	1	3	yes	—
125	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
126	#384	—	2	8	no	—
127	#380	—	2	24	no	—
128	C₂×#95	C₂²×C₁₂₇	1	12	yes	—
129	C₂⁵	C₂³×C₁₂₇	1	1	yes	characteristic
130	C₂³×C₄	—	2	12	no	—
131	C₂³×C₄	—	2	12	no	—
132	C₂³×C₄	C₂×C₄×C₁₂₇	1	12	yes	—
133	C₂³×C₄	D₄×C₁₂₇	1	12	yes	—
134	C₂³×C₄	Q₈×C₁₂₇	1	12	yes	—
135	C₂³×C₄	C₂³×C₁₂₇	1	8	yes	—
136	C₂³×C₄	C₂³×C₁₂₇	1	3	yes	—
137	C₂³×C₄	C₂³×C₁₂₇	1	3	yes	—
138	C₂×C₄⋊C₄	—	2	24	no	—
139	C₂×C₄⋊C₄	—	2	24	no	—
140	C₂×C₄⋊C₄	—	2	24	no	—
141	C₂×C₂²⋊C₄	—	2	24	no	—
142	C₂×C₄²	—	2	12	no	—
143	C₂×C₄²	—	2	32	no	—
144	C₂×C₄²	C₂×C₄×C₁₂₇	1	12	yes	—
145	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
146	C₂×C₄²	D₄×C₁₂₇	1	8	yes	—
147	#95	—	2	24	no	—
148	C₂⁴	—	2	2	no	—
149	C₂⁴	—	2	6	no	—
150	C₂⁴	C₂²×C₄×C₁₂₇	1	2	yes	—
151	C₂⁴	C₂×D₄×C₁₂₇	1	6	yes	—
152	C₂⁴	C₂×Q₈×C₁₂₇	1	6	yes	—
153	C₂⁴	C₂⁴×C₁₂₇	1	1	yes	characteristic
154	C₂²×C₄	—	2	6	no	—
155	C₂²×C₄	—	2	6	no	—
156	C₂²×C₄	—	2	24	no	—
157	C₂²×C₄	—	2	24	no	—
158	C₂²×C₄	—	2	48	no	—
159	C₂²×C₄	—	2	48	no	—
160	C₂²×C₄	—	2	24	no	—
161	C₂²×C₄	—	2	8	no	—
162	C₂²×C₄	—	2	24	no	—
163	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	48	yes	complemented
164	C₂²×C₄	C₂²×C₄×C₁₂₇	1	6	yes	—
165	C₂²×C₄	C₂²×C₄×C₁₂₇	1	6	yes	—
166	C₂²×C₄	C₂×D₄×C₁₂₇	1	6	yes	—
167	C₂²×C₄	C₂×D₄×C₁₂₇	1	12	yes	—
168	C₂²×C₄	C₂×D₄×C₁₂₇	1	6	yes	—
169	C₂²×C₄	C₂×Q₈×C₁₂₇	1	6	yes	—
170	C₂²×C₄	C₂×Q₈×C₁₂₇	1	6	yes	—
171	C₂²×C₄	#41×C₁₂₇	1	12	yes	—
172	C₄⋊C₄	—	4	48	no	—
173	C₂²⋊C₄	—	4	16	no	—
174	C₄²	—	2	24	no	—
175	C₄²	C₂²⋊C₄×C₁₂₇	1	16	yes	complemented
176	C₂³	—	2	2	no	—
177	C₂³	—	2	6	no	—
178	C₂³	—	2	12	no	—
179	C₂³	—	2	12	no	—
180	C₂³	—	2	12	no	—
181	C₂³	—	2	12	no	—
182	C₂³	C₂×C₄⋊C₄×C₁₂₇	1	12	yes	—
183	C₂³	#127×C₁₂₇	1	4	yes	—
184	C₂³	C₄⋊Q₈×C₁₂₇	1	12	yes	—
185	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
186	C₂³	C₂²×D₄×C₁₂₇	1	2	yes	—
187	C₂³	C₂²×D₄×C₁₂₇	1	3	yes	—
188	C₂³	C₂²×Q₈×C₁₂₇	1	3	yes	—
189	C₂³	C₂×#41×C₁₂₇	1	6	yes	—
190	C₂×C₄	—	2	6	no	—
191	C₂×C₄	—	2	6	no	—
192	C₂×C₄	—	2	12	no	—
193	C₂×C₄	—	2	12	no	—
194	C₂×C₄	—	2	96	no	—
195	C₂×C₄	—	4	24	no	—
196	C₂×C₄	—	4	8	no	—
197	C₂×C₄	—	4	24	no	—
198	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	12	yes	—
199	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	12	yes	—
200	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	12	yes	—
201	C₂×C₄	#117×C₁₂₇	1	12	yes	—
202	C₂×C₄	#121×C₁₂₇	1	24	yes	—
203	C₂×C₄	C₂²⋊Q₈×C₁₂₇	1	24	yes	—
204	C₂²	—	2	8	no	—
205	C₂²	—	2	24	no	—
206	C₂²	—	2	12	no	—
207	C₂²	—	2	12	no	—
208	C₂²	#384×C₁₂₇	1	8	yes	complemented
209	C₂²	C₂²×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
210	C₂²	C₂²×C₄⋊C₄×C₁₂₇	1	3	yes	—
211	C₂²	C₂×#117×C₁₂₇	1	3	yes	—
212	C₂²	C₂×#121×C₁₂₇	1	6	yes	—
213	C₂²	C₂×C₂²⋊Q₈×C₁₂₇	1	6	yes	—
214	C₂²	C₂×#127×C₁₂₇	1	1	yes	characteristic
215	C₂²	C₂×C₄⋊Q₈×C₁₂₇	1	3	yes	—
216	C₂²	#532×C₁₂₇	1	6	yes	—
217	C₂²	#533×C₁₂₇	1	6	yes	—
218	C₄	—	2	12	no	—
219	C₄	—	8	8	no	—
220	C₄	#380×C₁₂₇	1	24	yes	complemented
221	C₂	—	2	8	no	—
222	C₂	C₂×#380×C₁₂₇	1	3	yes	—
223	C₂	C₂×#384×C₁₂₇	1	1	yes	characteristic
224	C₂	C₁₂₇×#2285	1	3	yes	—
225	C₂	C₁₂₇×#2856	1	2	yes	—
226	C₂	C₁₂₇×#2857	1	6	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₂₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂⁴	C₂⁴×C₁₂₇
prime radical	C₂⁵×C₁₂₇	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#384×C₁₂₇	C₂	semidirect
4. C₁₂₇×C₄⋊#95	C₂	semidirect
9. C₂²×C₄²×C₁₂₇	C₄	semidirect
49. C₂²×C₄×C₁₂₇	C₄⋊C₄	semidirect
61. C₄²×C₁₂₇	C₂²⋊C₄	semidirect
94. C₂²×C₁₂₇	#384	semidirect
106. C₄×C₁₂₇	#380	semidirect
107. #14686	C₁₂₇	direct (N × H)
115. C₂×#384	C₂×C₁₂₇	semidirect
117. C₄⋊#95	C₂×C₁₂₇	semidirect
118. C₁₂₇	#14686	direct (N × H)
123. C₂²×C₄²	C₄×C₁₂₇	semidirect
163. C₂²×C₄	C₄⋊C₄×C₁₂₇	semidirect
175. C₄²	C₂²⋊C₄×C₁₂₇	semidirect
208. C₂²	#384×C₁₂₇	semidirect
220. C₄	#380×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₄²×C₁₂₇	2
C₂×#384×C₁₂₇	2
C₂×#380×C₁₂₇	2
C₁₂₇×C₄⋊#95	2
#14686	127

Subgroups — C127×#14686 #44578359

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#14686 #44578359