Subgroups — C₁₂₇×#10833 #44574506

Order 32512.

950 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₈×C₁₂₇	C₂	1	2	yes	maximal
2	C₂×#376×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#336×C₁₂₇	C₂	1	4	yes	maximal
4	C₂³×C₈×C₁₂₇	C₂²	1	4	yes	—
5	C₂²×C₄²×C₁₂₇	C₂²	1	2	yes	—
6	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₈×C₁₂₇	—	2	4	no	—
8	C₂×C₄⋊C₈×C₁₂₇	—	2	4	no	—
9	C₂×C₄⋊C₈×C₁₂₇	C₄	1	8	yes	complemented
10	#376×C₁₂₇	—	2	4	no	—
11	C₂×#95×C₁₂₇	C₄	1	4	yes	—
12	#336×C₁₂₇	—	2	8	no	—
13	C₂³×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
14	C₂³×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
15	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
16	C₂²×C₈×C₁₂₇	—	2	8	no	—
17	C₂²×C₈×C₁₂₇	—	2	4	no	—
18	C₂²×C₈×C₁₂₇	—	2	4	no	—
19	C₂²×C₈×C₁₂₇	—	2	8	no	—
20	C₂²×C₈×C₁₂₇	C₂×C₄	1	8	yes	—
21	C₂×C₄²×C₁₂₇	—	2	4	no	—
22	C₂×C₄²×C₁₂₇	—	2	4	no	—
23	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₁₂₇	D₄	1	4	yes	—
25	C₂×C₄²×C₁₂₇	D₄	1	2	yes	—
26	C₂×C₄²×C₁₂₇	D₄	1	4	yes	—
27	C₂×C₄²×C₁₂₇	Q₈	1	4	yes	—
28	C₂×C₄²×C₁₂₇	Q₈	1	2	yes	—
29	C₄⋊C₈×C₁₂₇	—	2	16	no	—
30	#95×C₁₂₇	—	2	8	no	—
31	C₂⁴×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₂₇	—	2	8	no	—
33	C₂²×C₄×C₁₂₇	—	2	4	no	—
34	C₂²×C₄×C₁₂₇	—	2	4	no	—
35	C₂²×C₄×C₁₂₇	—	2	16	no	—
36	C₂²×C₄×C₁₂₇	C₄²	1	4	yes	—
37	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
38	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	8	yes	—
39	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
40	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
41	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
46	C₂²×C₄×C₁₂₇	#41	1	4	yes	—
47	C₂×C₈×C₁₂₇	—	2	8	no	—
48	C₂×C₈×C₁₂₇	—	2	16	no	—
49	C₂×C₈×C₁₂₇	—	4	8	no	—
50	C₂×C₈×C₁₂₇	—	4	4	no	—
51	C₂×C₈×C₁₂₇	—	4	4	no	—
52	C₄²×C₁₂₇	—	2	8	no	—
53	C₄²×C₁₂₇	—	2	8	no	—
54	C₄²×C₁₂₇	C₂²⋊C₄	1	8	yes	—
55	C₄²×C₁₂₇	C₄⋊C₄	1	8	yes	—
56	C₂³×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
57	C₂³×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂³×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂³×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
60	C₂³×C₁₂₇	#117	1	2	yes	—
61	C₂³×C₁₂₇	#131	1	8	yes	—
62	C₂×C₄×C₁₂₇	—	2	8	no	—
63	C₂×C₄×C₁₂₇	—	2	8	no	—
64	C₂×C₄×C₁₂₇	—	2	8	no	—
65	C₂×C₄×C₁₂₇	—	2	8	no	—
66	C₂×C₄×C₁₂₇	—	2	16	no	—
67	C₂×C₄×C₁₂₇	—	4	8	no	—
68	C₂×C₄×C₁₂₇	—	4	8	no	—
69	C₂×C₄×C₁₂₇	#95	1	8	yes	—
70	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
71	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₁₂₇	C₄×D₄	1	4	yes	—
73	C₂×C₄×C₁₂₇	C₄×Q₈	1	4	yes	—
74	C₂×C₄×C₁₂₇	#121	1	4	yes	—
75	C₂×C₄×C₁₂₇	C₂²⋊Q₈	1	4	yes	—
76	C₂×C₄×C₁₂₇	#124	1	2	yes	—
77	C₂×C₄×C₁₂₇	#125	1	2	yes	—
78	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
79	C₂×C₄×C₁₂₇	C₄⋊Q₈	1	2	yes	—
80	C₂×C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
81	C₈×C₁₂₇	—	4	8	no	—
82	C₂²×C₁₂₇	C₂×#95	1	1	yes	characteristic
83	C₂²×C₁₂₇	C₄×C₄⋊C₄	1	2	yes	—
84	C₂²×C₁₂₇	#380	1	2	yes	—
85	C₂²×C₁₂₇	#382	1	1	yes	characteristic
86	C₂²×C₁₂₇	#384	1	1	yes	characteristic
87	C₂²×C₁₂₇	#405	1	4	yes	—
88	C₂²×C₁₂₇	#408	1	4	yes	—
89	C₂²×C₁₂₇	#424	1	8	yes	—
90	C₂²×C₁₂₇	#424	1	4	yes	—
91	C₂²×C₁₂₇	#433	1	8	yes	—
92	C₄×C₁₂₇	—	2	8	no	—
93	C₄×C₁₂₇	—	4	8	no	—
94	C₄×C₁₂₇	#389	1	4	yes	—
95	C₄×C₁₂₇	#390	1	2	yes	—
96	C₄×C₁₂₇	#391	1	2	yes	—
97	#10833	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂×C₁₂₇	C₄⋊#95	1	1	yes	characteristic
99	C₂×C₁₂₇	#1729	1	2	yes	—
100	C₂×C₁₂₇	#1768	1	4	yes	—
101	C₂×C₁₂₇	#1798	1	4	yes	—
102	C₂×C₁₂₇	#1832	1	2	yes	—
103	C₂×C₁₂₇	#1843	1	2	yes	—
104	C₂²×C₄⋊C₈	C₂×C₁₂₇	1	2	yes	—
105	C₂×#376	C₂×C₁₂₇	1	1	yes	characteristic
106	C₂×#336	C₂×C₁₂₇	1	4	yes	—
107	C₁₂₇	#10833	1	1	yes	characteristic, direct product factor, Sylow
108	C₂³×C₈	C₂²×C₁₂₇	1	4	yes	—
109	C₂²×C₄²	C₂²×C₁₂₇	1	2	yes	—
110	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
111	C₂×C₄⋊C₈	—	2	4	no	—
112	C₂×C₄⋊C₈	—	2	4	no	—
113	C₂×C₄⋊C₈	C₄×C₁₂₇	1	8	yes	complemented
114	#376	—	2	4	no	—
115	C₂×#95	C₄×C₁₂₇	1	4	yes	—
116	#336	—	2	8	no	—
117	C₂³×C₄	C₂×C₄×C₁₂₇	1	4	yes	—
118	C₂³×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
119	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
120	C₂²×C₈	—	2	8	no	—
121	C₂²×C₈	—	2	4	no	—
122	C₂²×C₈	—	2	4	no	—
123	C₂²×C₈	—	2	8	no	—
124	C₂²×C₈	C₂×C₄×C₁₂₇	1	8	yes	—
125	C₂×C₄²	—	2	4	no	—
126	C₂×C₄²	—	2	4	no	—
127	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
128	C₂×C₄²	D₄×C₁₂₇	1	4	yes	—
129	C₂×C₄²	D₄×C₁₂₇	1	2	yes	—
130	C₂×C₄²	D₄×C₁₂₇	1	4	yes	—
131	C₂×C₄²	Q₈×C₁₂₇	1	4	yes	—
132	C₂×C₄²	Q₈×C₁₂₇	1	2	yes	—
133	C₄⋊C₈	—	2	16	no	—
134	#95	—	2	8	no	—
135	C₂⁴	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
136	C₂²×C₄	—	2	8	no	—
137	C₂²×C₄	—	2	4	no	—
138	C₂²×C₄	—	2	4	no	—
139	C₂²×C₄	—	2	16	no	—
140	C₂²×C₄	C₄²×C₁₂₇	1	4	yes	—
141	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	4	yes	—
142	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	8	yes	—
143	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	4	yes	—
144	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
145	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
146	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
147	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
148	C₂²×C₄	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
149	C₂²×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
150	C₂²×C₄	#41×C₁₂₇	1	4	yes	—
151	C₂×C₈	—	2	8	no	—
152	C₂×C₈	—	2	16	no	—
153	C₂×C₈	—	4	8	no	—
154	C₂×C₈	—	4	4	no	—
155	C₂×C₈	—	4	4	no	—
156	C₄²	—	2	8	no	—
157	C₄²	—	2	8	no	—
158	C₄²	C₂²⋊C₄×C₁₂₇	1	8	yes	—
159	C₄²	C₄⋊C₄×C₁₂₇	1	8	yes	—
160	C₂³	C₂×C₄²×C₁₂₇	1	1	yes	characteristic
161	C₂³	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
162	C₂³	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
163	C₂³	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
164	C₂³	#117×C₁₂₇	1	2	yes	—
165	C₂³	#131×C₁₂₇	1	8	yes	—
166	C₂×C₄	—	2	8	no	—
167	C₂×C₄	—	2	8	no	—
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	—	2	8	no	—
170	C₂×C₄	—	2	16	no	—
171	C₂×C₄	—	4	8	no	—
172	C₂×C₄	—	4	8	no	—
173	C₂×C₄	#95×C₁₂₇	1	8	yes	—
174	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
175	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
176	C₂×C₄	C₄×D₄×C₁₂₇	1	4	yes	—
177	C₂×C₄	C₄×Q₈×C₁₂₇	1	4	yes	—
178	C₂×C₄	#121×C₁₂₇	1	4	yes	—
179	C₂×C₄	C₂²⋊Q₈×C₁₂₇	1	4	yes	—
180	C₂×C₄	#124×C₁₂₇	1	2	yes	—
181	C₂×C₄	#125×C₁₂₇	1	2	yes	—
182	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
183	C₂×C₄	C₄⋊Q₈×C₁₂₇	1	2	yes	—
184	C₂×C₄	C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
185	C₈	—	4	8	no	—
186	C₂²	C₂×#95×C₁₂₇	1	1	yes	characteristic
187	C₂²	C₄×C₄⋊C₄×C₁₂₇	1	2	yes	—
188	C₂²	#380×C₁₂₇	1	2	yes	—
189	C₂²	#382×C₁₂₇	1	1	yes	characteristic
190	C₂²	#384×C₁₂₇	1	1	yes	characteristic
191	C₂²	#405×C₁₂₇	1	4	yes	—
192	C₂²	#408×C₁₂₇	1	4	yes	—
193	C₂²	#424×C₁₂₇	1	4	yes	—
194	C₂²	#424×C₁₂₇	1	8	yes	—
195	C₂²	#433×C₁₂₇	1	8	yes	—
196	C₄	—	2	8	no	—
197	C₄	—	4	8	no	—
198	C₄	#389×C₁₂₇	1	4	yes	—
199	C₄	#390×C₁₂₇	1	2	yes	—
200	C₄	#391×C₁₂₇	1	2	yes	—
201	C₂	C₁₂₇×C₄⋊#95	1	1	yes	characteristic
202	C₂	C₁₂₇×#1729	1	2	yes	—
203	C₂	C₁₂₇×#1768	1	4	yes	—
204	C₂	C₁₂₇×#1798	1	4	yes	—
205	C₂	C₁₂₇×#1832	1	2	yes	—
206	C₂	C₁₂₇×#1843	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₁₂₇	C₂³
commutator	C₂³	C₂×C₄²×C₁₂₇
Frattini, cosocle	C₂³×C₄	C₂³×C₁₂₇
socle, prime radical	C₂⁴×C₁₂₇	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₂×C₄⋊C₈×C₁₂₇	C₄	semidirect
97. #10833	C₁₂₇	direct (N × H)
107. C₁₂₇	#10833	direct (N × H)
113. C₂×C₄⋊C₈	C₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊C₈×C₁₂₇	2
C₂×#376×C₁₂₇	2
C₂×#336×C₁₂₇	2
#10833	127

Subgroups — C127×#10833 #44574506

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#10833 #44574506