Subgroups — C₂×C₁₂₇×#1363 #44575606

Order 32512.

318 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#499×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#334×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1363	C₂	1	4	yes	maximal, direct product factor
5	C₂×#125×C₁₂₇	—	2	1	no	—
6	C₂²×C₁₆×C₁₂₇	—	2	1	no	—
7	#499×C₁₂₇	—	2	1	no	—
8	#499×C₁₂₇	—	2	1	no	—
9	#499×C₁₂₇	C₄	1	2	yes	—
10	#499×C₁₂₇	C₂²	1	2	yes	—
11	C₂×#107×C₁₂₇	C₄	1	1	yes	characteristic
12	C₂×#107×C₁₂₇	C₄	1	1	yes	characteristic
13	C₂×#106×C₁₂₇	—	2	1	no	—
14	C₂×C₄⋊C₈×C₁₂₇	—	2	1	no	—
15	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆×C₁₂₇	—	2	1	no	—
17	C₄⋊C₁₆×C₁₂₇	C₂²	1	2	yes	—
18	#334×C₁₂₇	—	2	1	no	—
19	#334×C₁₂₇	C₂²	1	2	yes	—
20	C₂²×C₈×C₁₂₇	—	4	1	no	—
21	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
23	#125×C₁₂₇	—	2	2	no	—
24	#125×C₁₂₇	—	2	2	no	—
25	#125×C₁₂₇	—	4	1	no	—
26	#125×C₁₂₇	—	4	1	no	—
27	C₂×C₄⋊C₄×C₁₂₇	—	2	1	no	—
28	C₂×C₄⋊C₄×C₁₂₇	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₂₇	—	4	1	no	—
30	C₂×C₄⋊C₄×C₁₂₇	—	4	1	no	—
31	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
32	C₂×C₁₆×C₁₂₇	—	2	2	no	—
33	C₂×C₁₆×C₁₂₇	—	4	2	no	—
34	#107×C₁₂₇	—	2	1	no	—
35	#107×C₁₂₇	—	2	1	no	—
36	#107×C₁₂₇	C₂×C₄	1	2	yes	—
37	#107×C₁₂₇	C₂×C₄	1	2	yes	—
38	#106×C₁₂₇	—	2	2	no	—
39	#106×C₁₂₇	—	2	2	no	—
40	C₄⋊C₈×C₁₂₇	—	2	2	no	—
41	C₄⋊C₈×C₁₂₇	—	4	1	no	—
42	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
45	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₂₇	—	4	1	no	—
47	C₂²×C₄×C₁₂₇	—	4	1	no	—
48	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
51	C₂×C₈×C₁₂₇	—	2	2	no	—
52	C₂×C₈×C₁₂₇	—	4	2	no	—
53	C₂×C₈×C₁₂₇	—	8	2	no	—
54	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₂₇	SD₁₆	1	2	yes	—
57	C₂×C₈×C₁₂₇	Q₁₆	1	2	yes	—
58	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
60	C₄⋊C₄×C₁₂₇	—	2	2	no	—
61	C₄⋊C₄×C₁₂₇	—	2	2	no	—
62	C₄⋊C₄×C₁₂₇	—	4	1	no	—
63	C₄⋊C₄×C₁₂₇	—	4	1	no	—
64	C₄⋊C₄×C₁₂₇	—	4	2	no	—
65	C₄⋊C₄×C₁₂₇	—	4	2	no	—
66	C₄⋊C₄×C₁₂₇	—	4	2	no	—
67	C₄⋊C₄×C₁₂₇	—	4	2	no	—
68	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₄²×C₁₂₇	D₈	1	1	yes	characteristic
70	C₄²×C₁₂₇	SD₁₆	1	1	yes	characteristic
71	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
72	C₁₆×C₁₂₇	—	4	2	no	—
73	C₂³×C₁₂₇	#102	1	1	yes	characteristic
74	C₂×C₄×C₁₂₇	—	2	2	no	—
75	C₂×C₄×C₁₂₇	—	2	2	no	—
76	C₂×C₄×C₁₂₇	—	4	2	no	—
77	C₂×C₄×C₁₂₇	—	4	2	no	—
78	C₂×C₄×C₁₂₇	—	8	2	no	—
79	C₂×C₄×C₁₂₇	—	8	2	no	—
80	C₂×C₄×C₁₂₇	#101	1	2	yes	—
81	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
82	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
83	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
84	C₂×C₄×C₁₂₇	#103	1	2	yes	—
85	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
88	C₈×C₁₂₇	—	2	1	no	—
89	C₈×C₁₂₇	—	2	1	no	—
90	C₈×C₁₂₇	—	8	2	no	—
91	C₈×C₁₂₇	#103	1	2	yes	—
92	C₈×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
93	C₈×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
94	C₂²×C₁₂₇	#332	1	2	yes	—
95	C₂²×C₁₂₇	#359	1	2	yes	—
96	C₂²×C₁₂₇	#360	1	2	yes	—
97	C₂²×C₁₂₇	C₂×#102	1	1	yes	characteristic
98	C₄×C₁₂₇	—	2	1	no	—
99	C₄×C₁₂₇	—	2	1	no	—
100	C₄×C₁₂₇	—	2	1	no	—
101	C₄×C₁₂₇	—	2	1	no	—
102	C₄×C₁₂₇	—	8	2	no	—
103	C₄×C₁₂₇	—	8	2	no	—
104	C₄×C₁₂₇	#332	1	2	yes	—
105	C₄×C₁₂₇	C₂×#101	1	1	yes	characteristic
106	C₄×C₁₂₇	C₂×#103	1	1	yes	characteristic
107	C₂×#1363	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂×C₁₂₇	#1363	1	4	yes	direct product factor
109	C₂×C₁₂₇	C₂×#332	1	1	yes	characteristic
110	C₂×C₁₂₇	C₂×#359	1	1	yes	characteristic
111	C₂×C₁₂₇	C₂×#360	1	1	yes	characteristic
112	C₂×#499	C₂×C₁₂₇	1	1	yes	characteristic
113	C₂×C₄⋊C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
114	C₂×#334	C₂×C₁₂₇	1	1	yes	characteristic
115	#1363	C₂×C₁₂₇	1	4	yes	direct product factor
116	C₁₂₇	C₂×#1363	1	1	yes	characteristic, direct product factor, Sylow
117	C₂×#125	—	2	1	no	—
118	C₂²×C₁₆	—	2	1	no	—
119	#499	—	2	1	no	—
120	#499	—	2	1	no	—
121	#499	C₄×C₁₂₇	1	2	yes	—
122	#499	C₂²×C₁₂₇	1	2	yes	—
123	C₂×#107	C₄×C₁₂₇	1	1	yes	characteristic
124	C₂×#107	C₄×C₁₂₇	1	1	yes	characteristic
125	C₂×#106	—	2	1	no	—
126	C₂×C₄⋊C₈	—	2	1	no	—
127	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
128	C₄⋊C₁₆	—	2	1	no	—
129	C₄⋊C₁₆	C₂²×C₁₂₇	1	2	yes	—
130	#334	—	2	1	no	—
131	#334	C₂²×C₁₂₇	1	2	yes	—
132	C₂²×C₈	—	4	1	no	—
133	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
134	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
135	#125	—	2	2	no	—
136	#125	—	2	2	no	—
137	#125	—	4	1	no	—
138	#125	—	4	1	no	—
139	C₂×C₄⋊C₄	—	2	1	no	—
140	C₂×C₄⋊C₄	—	2	1	no	—
141	C₂×C₄⋊C₄	—	4	1	no	—
142	C₂×C₄⋊C₄	—	4	1	no	—
143	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
144	C₂×C₁₆	—	2	2	no	—
145	C₂×C₁₆	—	4	2	no	—
146	#107	—	2	1	no	—
147	#107	—	2	1	no	—
148	#107	C₂×C₄×C₁₂₇	1	2	yes	—
149	#107	C₂×C₄×C₁₂₇	1	2	yes	—
150	#106	—	2	2	no	—
151	#106	—	2	2	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	4	1	no	—
154	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
155	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
156	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
157	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
158	C₂²×C₄	—	4	1	no	—
159	C₂²×C₄	—	4	1	no	—
160	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
161	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
162	C₂²×C₄	SD₁₆×C₁₂₇	1	1	yes	characteristic
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	4	2	no	—
165	C₂×C₈	—	8	2	no	—
166	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
167	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
168	C₂×C₈	SD₁₆×C₁₂₇	1	2	yes	—
169	C₂×C₈	Q₁₆×C₁₂₇	1	2	yes	—
170	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
171	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	4	1	no	—
175	C₄⋊C₄	—	4	1	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
181	C₄²	D₈×C₁₂₇	1	1	yes	characteristic
182	C₄²	SD₁₆×C₁₂₇	1	1	yes	characteristic
183	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
184	C₁₆	—	4	2	no	—
185	C₂³	#102×C₁₂₇	1	1	yes	characteristic
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	#101×C₁₂₇	1	2	yes	—
193	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
194	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
195	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
196	C₂×C₄	#103×C₁₂₇	1	2	yes	—
197	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
198	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
199	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
200	C₈	—	2	1	no	—
201	C₈	—	2	1	no	—
202	C₈	—	8	2	no	—
203	C₈	#103×C₁₂₇	1	2	yes	—
204	C₈	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
205	C₈	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
206	C₂²	#332×C₁₂₇	1	2	yes	—
207	C₂²	#359×C₁₂₇	1	2	yes	—
208	C₂²	#360×C₁₂₇	1	2	yes	—
209	C₂²	C₂×#102×C₁₂₇	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	8	2	no	—
215	C₄	—	8	2	no	—
216	C₄	#332×C₁₂₇	1	2	yes	—
217	C₄	C₂×#101×C₁₂₇	1	1	yes	characteristic
218	C₄	C₂×#103×C₁₂₇	1	1	yes	characteristic
219	C₂	C₁₂₇×#1363	1	4	yes	direct product factor
220	C₂	C₂×#332×C₁₂₇	1	1	yes	characteristic
221	C₂	C₂×#359×C₁₂₇	1	1	yes	characteristic
222	C₂	C₂×#360×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₂₇	#102
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₂₇×#1363	C₂	direct (N × H)
107. C₂×#1363	C₁₂₇	direct (N × H)
108. C₂×C₁₂₇	#1363	direct (N × H)
115. #1363	C₂×C₁₂₇	direct (N × H)
116. C₁₂₇	C₂×#1363	direct (N × H)
219. C₂	C₁₂₇×#1363	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#499×C₁₂₇	2
C₂×C₄⋊C₁₆×C₁₂₇	2
C₂×#334×C₁₂₇	2
C₁₂₇×#1363	2
C₂×#1363	127

Subgroups — C2×C127×#1363 #44575606

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₂₇×#1363 #44575606