Subgroups — C₁₂₇×#13193 #44576866

Order 32512.

446 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#2931	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#350×C₁₂₇	C₂	1	4	yes	maximal, complemented
4	C₄×M₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#131×C₁₂₇	C₂²	1	2	yes	—
6	#532×C₁₂₇	C₄	1	1	yes	characteristic
7	C₂×M₃₂×C₁₂₇	C₂²	1	2	yes	—
8	C₂²×C₁₆×C₁₂₇	C₂²	1	2	yes	—
9	#436×C₁₂₇	—	2	1	no	—
10	#436×C₁₂₇	—	2	1	no	—
11	C₈×D₄×C₁₂₇	—	2	1	no	—
12	C₈×D₄×C₁₂₇	—	2	1	no	—
13	#423×C₁₂₇	C₄	1	1	yes	characteristic
14	#408×C₁₂₇	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆×C₁₂₇	—	2	1	no	—
17	C₄⋊C₁₆×C₁₂₇	—	2	1	no	—
18	#350×C₁₂₇	—	2	8	no	—
19	#346×C₁₂₇	—	2	1	no	—
20	C₄×C₁₆×C₁₂₇	—	2	1	no	—
21	C₂×#41×C₁₂₇	C₂×C₄	1	2	yes	—
22	#131×C₁₂₇	—	2	4	no	—
23	#131×C₁₂₇	—	2	4	no	—
24	C₂×M₁₆×C₁₂₇	—	2	2	no	—
25	C₂×M₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₂₇	—	2	2	no	—
27	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
29	C₄⋊Q₈×C₁₂₇	C₈	1	1	yes	characteristic
30	#127×C₁₂₇	C₈	1	1	yes	characteristic
31	#124×C₁₂₇	C₈	1	2	yes	—
32	#121×C₁₂₇	—	2	1	no	—
33	#121×C₁₂₇	—	2	1	no	—
34	C₄×D₄×C₁₂₇	—	2	1	no	—
35	C₄×D₄×C₁₂₇	—	2	1	no	—
36	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
37	M₃₂×C₁₂₇	—	2	2	no	—
38	M₃₂×C₁₂₇	D₄	1	4	yes	complemented
39	C₂×C₁₆×C₁₂₇	—	2	2	no	—
40	C₂×C₁₆×C₁₂₇	—	2	4	no	—
41	C₂×C₁₆×C₁₂₇	—	2	2	no	—
42	C₄⋊C₈×C₁₂₇	—	2	1	no	—
43	C₄⋊C₈×C₁₂₇	—	2	1	no	—
44	C₂²⋊C₈×C₁₂₇	—	2	2	no	—
45	C₂²⋊C₈×C₁₂₇	—	2	2	no	—
46	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
47	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
48	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
49	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
50	#41×C₁₂₇	—	2	4	no	—
51	#41×C₁₂₇	—	2	4	no	—
52	C₂×Q₈×C₁₂₇	C₂×C₈	1	2	yes	—
53	C₂×D₄×C₁₂₇	—	2	2	no	—
54	C₂×D₄×C₁₂₇	—	2	1	no	—
55	C₂×D₄×C₁₂₇	—	2	1	no	—
56	C₂×D₄×C₁₂₇	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₁₂₇	—	2	2	no	—
58	C₂²×C₄×C₁₂₇	M₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₁₂₇	M₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
61	M₁₆×C₁₂₇	—	2	4	no	—
62	M₁₆×C₁₂₇	—	4	2	no	—
63	M₁₆×C₁₂₇	—	4	2	no	—
64	C₂×C₈×C₁₂₇	—	2	2	no	—
65	C₂×C₈×C₁₂₇	—	2	2	no	—
66	C₂×C₈×C₁₂₇	—	4	4	no	—
67	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₁₂₇	#41	1	2	yes	—
73	C₄⋊C₄×C₁₂₇	—	2	1	no	—
74	C₄⋊C₄×C₁₂₇	—	2	1	no	—
75	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
76	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
77	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₄²×C₁₂₇	C₂×C₈	1	1	yes	characteristic
80	C₄²×C₁₂₇	C₂×C₈	1	1	yes	characteristic
81	C₁₆×C₁₂₇	—	2	4	no	—
82	C₁₆×C₁₂₇	—	4	2	no	—
83	C₂³×C₁₂₇	—	2	2	no	—
84	C₂³×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
85	Q₈×C₁₂₇	—	2	4	no	—
86	D₄×C₁₂₇	—	2	4	no	—
87	D₄×C₁₂₇	—	4	2	no	—
88	D₄×C₁₂₇	—	4	4	no	—
89	D₄×C₁₂₇	—	4	2	no	—
90	C₂×C₄×C₁₂₇	—	2	2	no	—
91	C₂×C₄×C₁₂₇	—	2	2	no	—
92	C₂×C₄×C₁₂₇	—	4	1	no	—
93	C₂×C₄×C₁₂₇	—	4	2	no	—
94	C₂×C₄×C₁₂₇	—	4	2	no	—
95	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
96	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
97	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
98	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
99	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
101	C₂×C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₁₂₇	#131	1	2	yes	—
103	C₈×C₁₂₇	—	2	1	no	—
104	C₈×C₁₂₇	—	2	1	no	—
105	C₈×C₁₂₇	—	4	2	no	—
106	C₈×C₁₂₇	#121	1	2	yes	—
107	C₈×C₁₂₇	#124	1	1	yes	characteristic
108	C₈×C₁₂₇	#127	1	1	yes	characteristic
109	C₂²×C₁₂₇	—	2	1	no	—
110	C₂²×C₁₂₇	—	2	1	no	—
111	C₂²×C₁₂₇	—	4	2	no	—
112	C₂²×C₁₂₇	—	4	2	no	—
113	C₂²×C₁₂₇	—	4	2	no	—
114	C₂²×C₁₂₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
115	C₂²×C₁₂₇	#436	1	2	yes	—
116	C₄×C₁₂₇	—	2	1	no	—
117	C₄×C₁₂₇	—	2	1	no	—
118	C₄×C₁₂₇	—	2	1	no	—
119	C₄×C₁₂₇	—	4	2	no	—
120	C₄×C₁₂₇	#350	1	2	yes	complemented
121	C₄×C₁₂₇	#390	1	1	yes	characteristic
122	C₄×C₁₂₇	#408	1	1	yes	characteristic
123	C₄×C₁₂₇	C₈×D₄	1	2	yes	—
124	#13193	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂×C₁₂₇	—	2	2	no	—
126	C₂×C₁₂₇	—	8	2	no	—
127	C₂×C₁₂₇	#1954	1	1	yes	characteristic
128	C₂×C₁₂₇	C₂×#350	1	1	yes	characteristic
129	C₂×C₁₂₇	#2117	1	1	yes	characteristic
130	#2931	C₂×C₁₂₇	1	1	yes	characteristic
131	C₂×C₄⋊C₁₆	C₂×C₁₂₇	1	1	yes	characteristic, complemented
132	C₂×#350	C₂×C₁₂₇	1	4	yes	complemented
133	C₄×M₃₂	C₂×C₁₂₇	1	1	yes	characteristic, complemented
134	C₁₂₇	#13193	1	1	yes	characteristic, direct product factor, Sylow
135	C₂×#131	C₂²×C₁₂₇	1	2	yes	—
136	#532	C₄×C₁₂₇	1	1	yes	characteristic
137	C₂×M₃₂	C₂²×C₁₂₇	1	2	yes	—
138	C₂²×C₁₆	C₂²×C₁₂₇	1	2	yes	—
139	#436	—	2	1	no	—
140	#436	—	2	1	no	—
141	C₈×D₄	—	2	1	no	—
142	C₈×D₄	—	2	1	no	—
143	#423	C₄×C₁₂₇	1	1	yes	characteristic
144	#408	C₄×C₁₂₇	1	2	yes	—
145	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
146	C₄⋊C₁₆	—	2	1	no	—
147	C₄⋊C₁₆	—	2	1	no	—
148	#350	—	2	8	no	—
149	#346	—	2	1	no	—
150	C₄×C₁₆	—	2	1	no	—
151	C₂×#41	C₂×C₄×C₁₂₇	1	2	yes	—
152	#131	—	2	4	no	—
153	#131	—	2	4	no	—
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
158	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
159	C₄⋊Q₈	C₈×C₁₂₇	1	1	yes	characteristic
160	#127	C₈×C₁₂₇	1	1	yes	characteristic
161	#124	C₈×C₁₂₇	1	2	yes	—
162	#121	—	2	1	no	—
163	#121	—	2	1	no	—
164	C₄×D₄	—	2	1	no	—
165	C₄×D₄	—	2	1	no	—
166	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
167	M₃₂	—	2	2	no	—
168	M₃₂	D₄×C₁₂₇	1	4	yes	complemented
169	C₂×C₁₆	—	2	2	no	—
170	C₂×C₁₆	—	2	4	no	—
171	C₂×C₁₆	—	2	2	no	—
172	C₄⋊C₈	—	2	1	no	—
173	C₄⋊C₈	—	2	1	no	—
174	C₂²⋊C₈	—	2	2	no	—
175	C₂²⋊C₈	—	2	2	no	—
176	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
177	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
178	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
179	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
180	#41	—	2	4	no	—
181	#41	—	2	4	no	—
182	C₂×Q₈	C₂×C₈×C₁₂₇	1	2	yes	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	2	1	no	—
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	C₂×C₈×C₁₂₇	1	2	yes	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	M₁₆×C₁₂₇	1	1	yes	characteristic
189	C₂²×C₄	M₁₆×C₁₂₇	1	1	yes	characteristic
190	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
191	M₁₆	—	2	4	no	—
192	M₁₆	—	4	2	no	—
193	M₁₆	—	4	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	4	4	no	—
197	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
198	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
199	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
200	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
201	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
202	C₂×C₈	#41×C₁₂₇	1	2	yes	—
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	—	2	2	no	—
206	C₂²⋊C₄	—	2	2	no	—
207	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
208	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
209	C₄²	C₂×C₈×C₁₂₇	1	1	yes	characteristic
210	C₄²	C₂×C₈×C₁₂₇	1	1	yes	characteristic
211	C₁₆	—	2	4	no	—
212	C₁₆	—	4	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
215	Q₈	—	2	4	no	—
216	D₄	—	2	4	no	—
217	D₄	—	4	2	no	—
218	D₄	—	4	4	no	—
219	D₄	—	4	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
226	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
227	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
228	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
229	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
230	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
231	C₂×C₄	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
232	C₂×C₄	#131×C₁₂₇	1	2	yes	—
233	C₈	—	2	1	no	—
234	C₈	—	2	1	no	—
235	C₈	—	4	2	no	—
236	C₈	#121×C₁₂₇	1	2	yes	—
237	C₈	#124×C₁₂₇	1	1	yes	characteristic
238	C₈	#127×C₁₂₇	1	1	yes	characteristic
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	4	2	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	C₂×C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
245	C₂²	#436×C₁₂₇	1	2	yes	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	4	2	no	—
250	C₄	#350×C₁₂₇	1	2	yes	complemented
251	C₄	#390×C₁₂₇	1	1	yes	characteristic
252	C₄	#408×C₁₂₇	1	1	yes	characteristic
253	C₄	C₈×D₄×C₁₂₇	1	2	yes	—
254	C₂	—	2	2	no	—
255	C₂	—	8	2	no	—
256	C₂	C₁₂₇×#1954	1	1	yes	characteristic
257	C₂	C₂×#350×C₁₂₇	1	1	yes	characteristic
258	C₂	C₁₂₇×#2117	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₂₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂×C₂²⋊C₈
prime radical	#532×C₁₂₇	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊C₁₆×C₁₂₇	C₂	semidirect
3. C₂×#350×C₁₂₇	C₂	semidirect
4. C₄×M₃₂×C₁₂₇	C₂	semidirect
38. M₃₂×C₁₂₇	D₄	semidirect
120. C₄×C₁₂₇	#350	semidirect
124. #13193	C₁₂₇	direct (N × H)
131. C₂×C₄⋊C₁₆	C₂×C₁₂₇	semidirect
132. C₂×#350	C₂×C₁₂₇	semidirect
133. C₄×M₃₂	C₂×C₁₂₇	semidirect
134. C₁₂₇	#13193	direct (N × H)
168. M₃₂	D₄×C₁₂₇	semidirect
250. C₄	#350×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#2931	2
C₂×C₄⋊C₁₆×C₁₂₇	2
C₂×#350×C₁₂₇	2
C₄×M₃₂×C₁₂₇	2
#13193	127

Subgroups — C127×#13193 #44576866

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#13193 #44576866