Subgroups — C₂×C₁₂₇×#1707 #44583267

Order 32512.

814 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#498×C₁₂₇	C₂	1	2	yes	maximal
3	C₂×#494×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#487×C₁₂₇	C₂	1	2	yes	maximal
5	C₁₂₇×#1707	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₈²×C₁₂₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₁₂₇	—	2	2	no	—
8	C₂×C₄⋊Q₈×C₁₂₇	C₂²	1	2	yes	—
9	C₂×C₄⋊Q₈×C₁₂₇	C₂²	1	2	yes	—
10	#500×C₁₂₇	—	2	2	no	—
11	#500×C₁₂₇	C₂²	1	4	yes	—
12	#498×C₁₂₇	—	2	4	no	—
13	#498×C₁₂₇	C₂²	1	8	yes	—
14	#494×C₁₂₇	—	2	2	no	—
15	#494×C₁₂₇	C₂²	1	4	yes	—
16	#487×C₁₂₇	—	2	4	no	—
17	#487×C₁₂₇	C₂²	1	8	yes	—
18	C₂×#107×C₁₂₇	—	2	2	no	—
19	C₂×#106×C₁₂₇	—	2	4	no	—
20	C₂×#103×C₁₂₇	—	2	4	no	—
21	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
24	C₈²×C₁₂₇	C₂²	1	4	yes	—
25	C₂²×Q₈×C₁₂₇	—	2	2	no	—
26	C₂²×Q₈×C₁₂₇	—	2	2	no	—
27	C₂×Q₁₆×C₁₂₇	—	2	8	no	—
28	C₂×Q₁₆×C₁₂₇	—	4	8	no	—
29	C₂²×C₈×C₁₂₇	D₄	1	2	yes	—
30	C₂²×C₈×C₁₂₇	D₄	1	2	yes	—
31	C₂²×C₈×C₁₂₇	D₄	1	2	yes	—
32	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
33	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
34	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
35	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
36	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
37	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
38	C₄⋊Q₈×C₁₂₇	C₂³	1	4	yes	—
39	C₄⋊Q₈×C₁₂₇	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
44	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
45	#107×C₁₂₇	—	2	8	no	—
46	#106×C₁₂₇	—	2	16	no	—
47	#103×C₁₂₇	—	2	16	no	—
48	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
49	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
50	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
51	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
52	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
53	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
54	C₂×Q₈×C₁₂₇	—	2	4	no	—
55	C₂×Q₈×C₁₂₇	—	2	4	no	—
56	C₂×Q₈×C₁₂₇	—	4	2	no	—
57	C₂×Q₈×C₁₂₇	—	4	2	no	—
58	C₂×Q₈×C₁₂₇	—	4	8	no	—
59	C₂×Q₈×C₁₂₇	—	4	8	no	—
60	C₂²×C₄×C₁₂₇	—	4	2	no	—
61	C₂²×C₄×C₁₂₇	—	4	2	no	—
62	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
65	Q₁₆×C₁₂₇	—	4	16	no	—
66	C₂×C₈×C₁₂₇	—	2	4	no	—
67	C₂×C₈×C₁₂₇	SD₁₆	1	8	yes	—
68	C₂×C₈×C₁₂₇	Q₁₆	1	8	yes	complemented
69	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
70	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
71	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
72	C₄⋊C₄×C₁₂₇	—	2	4	no	—
73	C₄⋊C₄×C₁₂₇	—	2	4	no	—
74	C₄⋊C₄×C₁₂₇	—	2	4	no	—
75	C₄⋊C₄×C₁₂₇	—	2	4	no	—
76	C₄⋊C₄×C₁₂₇	—	4	2	no	—
77	C₄⋊C₄×C₁₂₇	—	4	2	no	—
78	C₄⋊C₄×C₁₂₇	—	4	2	no	—
79	C₄⋊C₄×C₁₂₇	—	4	2	no	—
80	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₁₂₇	C₂⁴	1	1	yes	characteristic
84	C₂³×C₁₂₇	#127	1	1	yes	characteristic
85	Q₈×C₁₂₇	—	4	8	no	—
86	Q₈×C₁₂₇	—	4	8	no	—
87	Q₈×C₁₂₇	—	8	4	no	—
88	Q₈×C₁₂₇	—	8	4	no	—
89	C₂×C₄×C₁₂₇	—	4	4	no	—
90	C₂×C₄×C₁₂₇	—	4	4	no	—
91	C₂×C₄×C₁₂₇	—	8	4	no	—
92	C₂×C₄×C₁₂₇	—	8	4	no	—
93	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
94	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
95	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
96	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	4	yes	—
97	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	4	yes	—
98	C₂×C₄×C₁₂₇	#135	1	4	yes	—
99	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
102	C₈×C₁₂₇	—	2	4	no	—
103	C₈×C₁₂₇	C₂×SD₁₆	1	8	yes	—
104	C₈×C₁₂₇	C₂×Q₁₆	1	8	yes	complemented
105	C₂²×C₁₂₇	#492	1	2	yes	—
106	C₂²×C₁₂₇	#494	1	2	yes	—
107	C₂²×C₁₂₇	#495	1	2	yes	—
108	C₂²×C₁₂₇	C₂×#127	1	1	yes	characteristic
109	C₄×C₁₂₇	—	8	4	no	—
110	C₄×C₁₂₇	—	8	4	no	—
111	C₄×C₁₂₇	#492	1	2	yes	—
112	C₄×C₁₂₇	#494	1	2	yes	—
113	C₄×C₁₂₇	#495	1	2	yes	—
114	C₄×C₁₂₇	C₂²×SD₁₆	1	2	yes	—
115	C₄×C₁₂₇	C₂²×Q₁₆	1	2	yes	—
116	C₄×C₁₂₇	C₂×#135	1	2	yes	—
117	C₂×#1707	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂×C₁₂₇	#1707	1	4	yes	direct product factor
119	C₂×C₁₂₇	C₂×#492	1	1	yes	characteristic
120	C₂×C₁₂₇	C₂×#494	1	1	yes	characteristic
121	C₂×C₁₂₇	C₂×#495	1	1	yes	characteristic
122	C₂×#500	C₂×C₁₂₇	1	1	yes	characteristic
123	C₂×#498	C₂×C₁₂₇	1	2	yes	—
124	C₂×#494	C₂×C₁₂₇	1	1	yes	characteristic
125	C₂×#487	C₂×C₁₂₇	1	2	yes	—
126	#1707	C₂×C₁₂₇	1	8	yes	direct product factor
127	C₂×C₈²	C₂×C₁₂₇	1	1	yes	characteristic
128	C₁₂₇	C₂×#1707	1	1	yes	characteristic, direct product factor, Sylow
129	C₂²×Q₁₆	—	2	2	no	—
130	C₂×C₄⋊Q₈	C₂²×C₁₂₇	1	2	yes	—
131	C₂×C₄⋊Q₈	C₂²×C₁₂₇	1	2	yes	—
132	#500	—	2	2	no	—
133	#500	C₂²×C₁₂₇	1	4	yes	—
134	#498	—	2	4	no	—
135	#498	C₂²×C₁₂₇	1	8	yes	—
136	#494	—	2	2	no	—
137	#494	C₂²×C₁₂₇	1	4	yes	—
138	#487	—	2	4	no	—
139	#487	C₂²×C₁₂₇	1	8	yes	—
140	C₂×#107	—	2	2	no	—
141	C₂×#106	—	2	4	no	—
142	C₂×#103	—	2	4	no	—
143	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
144	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
145	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
146	C₈²	C₂²×C₁₂₇	1	4	yes	—
147	C₂²×Q₈	—	2	2	no	—
148	C₂²×Q₈	—	2	2	no	—
149	C₂×Q₁₆	—	2	8	no	—
150	C₂×Q₁₆	—	4	8	no	—
151	C₂²×C₈	D₄×C₁₂₇	1	2	yes	—
152	C₂²×C₈	D₄×C₁₂₇	1	2	yes	—
153	C₂²×C₈	D₄×C₁₂₇	1	2	yes	—
154	C₄⋊Q₈	—	2	2	no	—
155	C₄⋊Q₈	—	2	2	no	—
156	C₄⋊Q₈	—	2	2	no	—
157	C₄⋊Q₈	—	2	2	no	—
158	C₄⋊Q₈	—	2	2	no	—
159	C₄⋊Q₈	—	2	2	no	—
160	C₄⋊Q₈	C₂³×C₁₂₇	1	4	yes	—
161	C₄⋊Q₈	C₂³×C₁₂₇	1	4	yes	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
167	#107	—	2	8	no	—
168	#106	—	2	16	no	—
169	#103	—	2	16	no	—
170	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
171	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
172	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
173	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
174	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
175	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	—	4	2	no	—
179	C₂×Q₈	—	4	2	no	—
180	C₂×Q₈	—	4	8	no	—
181	C₂×Q₈	—	4	8	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
185	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
187	Q₁₆	—	4	16	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	SD₁₆×C₁₂₇	1	8	yes	—
190	C₂×C₈	Q₁₆×C₁₂₇	1	8	yes	complemented
191	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
192	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
193	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	2	4	no	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
203	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
204	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
205	C₄²	C₂⁴×C₁₂₇	1	1	yes	characteristic
206	C₂³	#127×C₁₂₇	1	1	yes	characteristic
207	Q₈	—	4	8	no	—
208	Q₈	—	4	8	no	—
209	Q₈	—	8	4	no	—
210	Q₈	—	8	4	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	8	4	no	—
214	C₂×C₄	—	8	4	no	—
215	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
216	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
217	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
218	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	4	yes	—
219	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	4	yes	—
220	C₂×C₄	#135×C₁₂₇	1	4	yes	—
221	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
222	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
223	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
224	C₈	—	2	4	no	—
225	C₈	C₂×SD₁₆×C₁₂₇	1	8	yes	—
226	C₈	C₂×Q₁₆×C₁₂₇	1	8	yes	complemented
227	C₂²	#492×C₁₂₇	1	2	yes	—
228	C₂²	#494×C₁₂₇	1	2	yes	—
229	C₂²	#495×C₁₂₇	1	2	yes	—
230	C₂²	C₂×#127×C₁₂₇	1	1	yes	characteristic
231	C₄	—	8	4	no	—
232	C₄	—	8	4	no	—
233	C₄	#492×C₁₂₇	1	2	yes	—
234	C₄	#494×C₁₂₇	1	2	yes	—
235	C₄	#495×C₁₂₇	1	2	yes	—
236	C₄	C₂²×SD₁₆×C₁₂₇	1	2	yes	—
237	C₄	C₂²×Q₁₆×C₁₂₇	1	2	yes	—
238	C₄	C₂×#135×C₁₂₇	1	2	yes	—
239	C₂	C₁₂₇×#1707	1	4	yes	direct product factor
240	C₂	C₂×#492×C₁₂₇	1	1	yes	characteristic
241	C₂	C₂×#494×C₁₂₇	1	1	yes	characteristic
242	C₂	C₂×#495×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₂₇	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₂₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₁₂₇×#1707	C₂	direct (N × H)
68. C₂×C₈×C₁₂₇	Q₁₆	semidirect
104. C₈×C₁₂₇	C₂×Q₁₆	semidirect
117. C₂×#1707	C₁₂₇	direct (N × H)
118. C₂×C₁₂₇	#1707	direct (N × H)
126. #1707	C₂×C₁₂₇	direct (N × H)
128. C₁₂₇	C₂×#1707	direct (N × H)
190. C₂×C₈	Q₁₆×C₁₂₇	semidirect
226. C₈	C₂×Q₁₆×C₁₂₇	semidirect
239. C₂	C₁₂₇×#1707	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×#500×C₁₂₇	2
C₂×#498×C₁₂₇	2
C₂×#494×C₁₂₇	2
C₂×#487×C₁₂₇	2
C₁₂₇×#1707	2
C₂×C₈²×C₁₂₇	2
C₂×#1707	127

Subgroups — C2×C127×#1707 #44583267

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₂₇×#1707 #44583267