Subgroups — C₂×C₁₂₇×#1332 #44575263

Order 32512.

406 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#335×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1332	C₂	1	4	yes	maximal, direct product factor
5	C₂²×Q₁₆×C₁₂₇	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₁₂₇	—	2	1	no	—
7	C₂²×C₁₆×C₁₂₇	—	2	1	no	—
8	C₄×Q₁₆×C₁₂₇	—	2	1	no	—
9	C₄×Q₁₆×C₁₂₇	—	2	1	no	—
10	C₄×Q₁₆×C₁₂₇	C₄	1	2	yes	—
11	C₄×Q₁₆×C₁₂₇	C₂²	1	2	yes	—
12	C₂×#107×C₁₂₇	C₄	1	1	yes	characteristic
13	C₂×C₄⋊C₈×C₁₂₇	—	2	1	no	—
14	C₂×#103×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	#335×C₁₂₇	—	2	1	no	—
19	#335×C₁₂₇	C₂²	1	2	yes	—
20	C₂²×Q₈×C₁₂₇	—	2	1	no	—
21	C₂×Q₁₆×C₁₂₇	—	2	1	no	—
22	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
23	C₂×Q₁₆×C₁₂₇	C₈	1	4	yes	complemented
24	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₂₇	—	4	1	no	—
26	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
28	C₄×Q₈×C₁₂₇	—	2	2	no	—
29	C₄×Q₈×C₁₂₇	—	2	2	no	—
30	C₄×Q₈×C₁₂₇	—	4	1	no	—
31	C₄×Q₈×C₁₂₇	—	4	1	no	—
32	C₂×C₄⋊C₄×C₁₂₇	—	2	1	no	—
33	C₂×C₄⋊C₄×C₁₂₇	—	4	1	no	—
34	C₂×C₄²×C₁₂₇	—	4	1	no	—
35	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
36	C₂×C₁₆×C₁₂₇	—	2	4	no	—
37	C₂×C₁₆×C₁₂₇	—	2	2	no	—
38	#107×C₁₂₇	—	2	1	no	—
39	#107×C₁₂₇	C₂×C₄	1	2	yes	—
40	C₄⋊C₈×C₁₂₇	—	2	2	no	—
41	C₄⋊C₈×C₁₂₇	—	4	1	no	—
42	#103×C₁₂₇	—	2	2	no	—
43	#103×C₁₂₇	—	2	2	no	—
44	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
46	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
47	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
48	C₂×Q₈×C₁₂₇	—	2	4	no	—
49	C₂×Q₈×C₁₂₇	—	2	2	no	—
50	C₂×Q₈×C₁₂₇	—	4	1	no	—
51	C₂×Q₈×C₁₂₇	—	4	2	no	—
52	C₂²×C₄×C₁₂₇	—	4	1	no	—
53	C₂²×C₄×C₁₂₇	—	4	1	no	—
54	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
57	Q₁₆×C₁₂₇	—	2	4	no	—
58	Q₁₆×C₁₂₇	—	2	2	no	—
59	Q₁₆×C₁₂₇	C₂×C₈	1	4	yes	complemented
60	C₂×C₈×C₁₂₇	—	2	2	no	—
61	C₂×C₈×C₁₂₇	—	4	2	no	—
62	C₂×C₈×C₁₂₇	—	8	2	no	—
63	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₂₇	C₂×C₈	1	2	yes	—
66	C₂×C₈×C₁₂₇	M₁₆	1	2	yes	—
67	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
69	C₄⋊C₄×C₁₂₇	—	2	2	no	—
70	C₄⋊C₄×C₁₂₇	—	4	1	no	—
71	C₄⋊C₄×C₁₂₇	—	4	2	no	—
72	C₄⋊C₄×C₁₂₇	—	4	2	no	—
73	C₄²×C₁₂₇	—	4	2	no	—
74	C₄²×C₁₂₇	—	4	2	no	—
75	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₁₂₇	D₈	1	1	yes	characteristic
77	C₄²×C₁₂₇	SD₁₆	1	1	yes	characteristic
78	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
79	C₁₆×C₁₂₇	—	2	4	no	—
80	C₂³×C₁₂₇	#102	1	1	yes	characteristic
81	Q₈×C₁₂₇	—	2	4	no	—
82	Q₈×C₁₂₇	—	4	4	no	—
83	Q₈×C₁₂₇	—	4	2	no	—
84	C₂×C₄×C₁₂₇	—	2	2	no	—
85	C₂×C₄×C₁₂₇	—	4	4	no	—
86	C₂×C₄×C₁₂₇	—	4	2	no	—
87	C₂×C₄×C₁₂₇	—	4	2	no	—
88	C₂×C₄×C₁₂₇	—	8	2	no	—
89	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	2	yes	—
90	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
91	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
92	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
93	C₂×C₄×C₁₂₇	#104	1	2	yes	—
94	C₂×C₄×C₁₂₇	SD₃₂	1	2	yes	—
95	C₂×C₄×C₁₂₇	Q₃₂	1	2	yes	—
96	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
99	C₈×C₁₂₇	—	2	1	no	—
100	C₈×C₁₂₇	—	2	1	no	—
101	C₈×C₁₂₇	—	8	2	no	—
102	C₈×C₁₂₇	C₂²⋊C₈	1	2	yes	—
103	C₈×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
104	C₈×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
105	C₂²×C₁₂₇	D₄⋊C₈	1	2	yes	—
106	C₂²×C₁₂₇	#358	1	2	yes	—
107	C₂²×C₁₂₇	#359	1	2	yes	—
108	C₂²×C₁₂₇	C₂×#102	1	1	yes	characteristic
109	C₄×C₁₂₇	—	2	1	no	—
110	C₄×C₁₂₇	—	2	1	no	—
111	C₄×C₁₂₇	—	4	4	no	—
112	C₄×C₁₂₇	—	8	2	no	—
113	C₄×C₁₂₇	D₄⋊C₈	1	2	yes	—
114	C₄×C₁₂₇	#358	1	2	yes	—
115	C₄×C₁₂₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
116	C₄×C₁₂₇	C₂×#104	1	1	yes	characteristic
117	C₄×C₁₂₇	C₂×SD₃₂	1	1	yes	characteristic
118	C₄×C₁₂₇	C₂×Q₃₂	1	1	yes	characteristic
119	C₂×#1332	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂×C₁₂₇	#1332	1	4	yes	direct product factor
121	C₂×C₁₂₇	C₂×D₄⋊C₈	1	1	yes	characteristic
122	C₂×C₁₂₇	C₂×#358	1	1	yes	characteristic
123	C₂×C₁₂₇	C₂×#359	1	1	yes	characteristic
124	C₂×C₄×Q₁₆	C₂×C₁₂₇	1	1	yes	characteristic
125	C₂×C₄×C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
126	C₂×#335	C₂×C₁₂₇	1	1	yes	characteristic
127	#1332	C₂×C₁₂₇	1	4	yes	direct product factor
128	C₁₂₇	C₂×#1332	1	1	yes	characteristic, direct product factor, Sylow
129	C₂²×Q₁₆	C₄×C₁₂₇	1	1	yes	characteristic
130	C₂×C₄×Q₈	—	2	1	no	—
131	C₂²×C₁₆	—	2	1	no	—
132	C₄×Q₁₆	—	2	1	no	—
133	C₄×Q₁₆	—	2	1	no	—
134	C₄×Q₁₆	C₄×C₁₂₇	1	2	yes	—
135	C₄×Q₁₆	C₂²×C₁₂₇	1	2	yes	—
136	C₂×#107	C₄×C₁₂₇	1	1	yes	characteristic
137	C₂×C₄⋊C₈	—	2	1	no	—
138	C₂×#103	—	2	1	no	—
139	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
140	C₄×C₁₆	—	2	1	no	—
141	C₄×C₁₆	C₂²×C₁₂₇	1	2	yes	—
142	#335	—	2	1	no	—
143	#335	C₂²×C₁₂₇	1	2	yes	—
144	C₂²×Q₈	—	2	1	no	—
145	C₂×Q₁₆	—	2	1	no	—
146	C₂×Q₁₆	—	2	2	no	—
147	C₂×Q₁₆	C₈×C₁₂₇	1	4	yes	complemented
148	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
149	C₂²×C₈	—	4	1	no	—
150	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
151	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
152	C₄×Q₈	—	2	2	no	—
153	C₄×Q₈	—	2	2	no	—
154	C₄×Q₈	—	4	1	no	—
155	C₄×Q₈	—	4	1	no	—
156	C₂×C₄⋊C₄	—	2	1	no	—
157	C₂×C₄⋊C₄	—	4	1	no	—
158	C₂×C₄²	—	4	1	no	—
159	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
160	C₂×C₁₆	—	2	4	no	—
161	C₂×C₁₆	—	2	2	no	—
162	#107	—	2	1	no	—
163	#107	C₂×C₄×C₁₂₇	1	2	yes	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	—	4	1	no	—
166	#103	—	2	2	no	—
167	#103	—	2	2	no	—
168	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
169	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
170	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
171	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
172	C₂×Q₈	—	2	4	no	—
173	C₂×Q₈	—	2	2	no	—
174	C₂×Q₈	—	4	1	no	—
175	C₂×Q₈	—	4	2	no	—
176	C₂²×C₄	—	4	1	no	—
177	C₂²×C₄	—	4	1	no	—
178	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
179	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
180	C₂²×C₄	SD₁₆×C₁₂₇	1	1	yes	characteristic
181	Q₁₆	—	2	4	no	—
182	Q₁₆	—	2	2	no	—
183	Q₁₆	C₂×C₈×C₁₂₇	1	4	yes	complemented
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	8	2	no	—
187	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
188	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
189	C₂×C₈	C₂×C₈×C₁₂₇	1	2	yes	—
190	C₂×C₈	M₁₆×C₁₂₇	1	2	yes	—
191	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
192	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	4	1	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄²	—	4	2	no	—
198	C₄²	—	4	2	no	—
199	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
200	C₄²	D₈×C₁₂₇	1	1	yes	characteristic
201	C₄²	SD₁₆×C₁₂₇	1	1	yes	characteristic
202	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
203	C₁₆	—	2	4	no	—
204	C₂³	#102×C₁₂₇	1	1	yes	characteristic
205	Q₈	—	2	4	no	—
206	Q₈	—	4	4	no	—
207	Q₈	—	4	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	8	2	no	—
213	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	2	yes	—
214	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
215	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
216	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
217	C₂×C₄	#104×C₁₂₇	1	2	yes	—
218	C₂×C₄	SD₃₂×C₁₂₇	1	2	yes	—
219	C₂×C₄	Q₃₂×C₁₂₇	1	2	yes	—
220	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
221	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
222	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	8	2	no	—
226	C₈	C₂²⋊C₈×C₁₂₇	1	2	yes	—
227	C₈	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
228	C₈	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
229	C₂²	D₄⋊C₈×C₁₂₇	1	2	yes	—
230	C₂²	#358×C₁₂₇	1	2	yes	—
231	C₂²	#359×C₁₂₇	1	2	yes	—
232	C₂²	C₂×#102×C₁₂₇	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	4	4	no	—
236	C₄	—	8	2	no	—
237	C₄	D₄⋊C₈×C₁₂₇	1	2	yes	—
238	C₄	#358×C₁₂₇	1	2	yes	—
239	C₄	C₂×C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
240	C₄	C₂×#104×C₁₂₇	1	1	yes	characteristic
241	C₄	C₂×SD₃₂×C₁₂₇	1	1	yes	characteristic
242	C₄	C₂×Q₃₂×C₁₂₇	1	1	yes	characteristic
243	C₂	C₁₂₇×#1332	1	4	yes	direct product factor
244	C₂	C₂×D₄⋊C₈×C₁₂₇	1	1	yes	characteristic
245	C₂	C₂×#358×C₁₂₇	1	1	yes	characteristic
246	C₂	C₂×#359×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₂₇	D₈
commutator	C₈	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
socle, prime radical	C₂³×C₁₂₇	#102
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×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×#1332	C₂	direct (N × H)
23. C₂×Q₁₆×C₁₂₇	C₈	semidirect
59. Q₁₆×C₁₂₇	C₂×C₈	semidirect
119. C₂×#1332	C₁₂₇	direct (N × H)
120. C₂×C₁₂₇	#1332	direct (N × H)
127. #1332	C₂×C₁₂₇	direct (N × H)
128. C₁₂₇	C₂×#1332	direct (N × H)
147. C₂×Q₁₆	C₈×C₁₂₇	semidirect
183. Q₁₆	C₂×C₈×C₁₂₇	semidirect
243. C₂	C₁₂₇×#1332	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×C₄×Q₁₆×C₁₂₇	2
C₂×C₄×C₁₆×C₁₂₇	2
C₂×#335×C₁₂₇	2
C₁₂₇×#1332	2
C₂×#1332	127

Subgroups — C2×C127×#1332 #44575263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₂₇×#1332 #44575263