Subgroups — C₁₂₇×#13204 #44576877

Order 32512.

514 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#495×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#358×C₁₂₇	C₂	1	2	yes	maximal, complemented
3	C₂×#357×C₁₂₇	C₂	1	2	yes	maximal
4	C₂×C₄×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₂₇×#1846	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₂²×SD₁₆×C₁₂₇	—	2	1	no	—
8	C₂²×D₈×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₂×#124×C₁₂₇	—	2	1	no	—
10	C₂²×C₁₆×C₁₂₇	C₂²	1	2	yes	—
11	#495×C₁₂₇	—	2	2	no	—
12	#495×C₁₂₇	C₄	1	4	yes	complemented
13	C₂×#107×C₁₂₇	C₂²	1	2	yes	—
14	C₂×#106×C₁₂₇	—	2	1	no	—
15	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
16	#382×C₁₂₇	—	2	1	no	—
17	#358×C₁₂₇	—	2	4	no	—
18	#357×C₁₂₇	—	2	4	no	—
19	C₄×C₁₆×C₁₂₇	—	2	2	no	—
20	C₂²×Q₈×C₁₂₇	—	2	1	no	—
21	C₂²×D₄×C₁₂₇	—	2	1	no	—
22	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
23	C₂×Q₁₆×C₁₂₇	—	2	1	no	—
24	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
25	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
26	C₂×SD₁₆×C₁₂₇	—	2	4	no	—
27	C₂×SD₁₆×C₁₂₇	—	4	2	no	—
28	C₂×SD₁₆×C₁₂₇	—	4	2	no	—
29	C₂×D₈×C₁₂₇	—	2	2	no	—
30	C₂×D₈×C₁₂₇	—	2	1	no	—
31	C₂×D₈×C₁₂₇	—	2	2	no	—
32	C₂×D₈×C₁₂₇	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
35	#124×C₁₂₇	—	2	4	no	—
36	#124×C₁₂₇	—	4	2	no	—
37	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
38	C₂×C₂²⋊C₄×C₁₂₇	—	4	2	no	—
39	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
40	C₂×C₁₆×C₁₂₇	—	2	2	no	—
41	C₂×C₁₆×C₁₂₇	—	2	2	no	—
42	C₂×C₁₆×C₁₂₇	D₄	1	4	yes	—
43	#107×C₁₂₇	—	2	2	no	—
44	#107×C₁₂₇	—	2	2	no	—
45	#106×C₁₂₇	—	4	2	no	—
46	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
48	#95×C₁₂₇	—	4	2	no	—
49	C₂⁴×C₁₂₇	—	4	1	no	—
50	C₂×Q₈×C₁₂₇	—	2	2	no	—
51	C₂×Q₈×C₁₂₇	—	4	1	no	—
52	C₂×Q₈×C₁₂₇	—	4	1	no	—
53	C₂×Q₈×C₁₂₇	—	4	1	no	—
54	C₂×Q₈×C₁₂₇	—	4	1	no	—
55	C₂×Q₈×C₁₂₇	—	4	1	no	—
56	C₂×D₄×C₁₂₇	—	2	2	no	—
57	C₂×D₄×C₁₂₇	—	4	1	no	—
58	C₂×D₄×C₁₂₇	—	4	1	no	—
59	C₂×D₄×C₁₂₇	—	4	1	no	—
60	C₂×D₄×C₁₂₇	—	4	1	no	—
61	C₂×D₄×C₁₂₇	—	4	1	no	—
62	C₂²×C₄×C₁₂₇	—	4	1	no	—
63	C₂²×C₄×C₁₂₇	—	4	2	no	—
64	C₂²×C₄×C₁₂₇	D₈	1	2	yes	—
65	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
66	Q₁₆×C₁₂₇	—	2	4	no	—
67	Q₁₆×C₁₂₇	—	4	2	no	—
68	SD₁₆×C₁₂₇	—	4	8	no	—
69	D₈×C₁₂₇	—	2	4	no	—
70	D₈×C₁₂₇	—	4	2	no	—
71	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₁₂₇	D₈	1	2	yes	—
73	C₂×C₈×C₁₂₇	SD₁₆	1	2	yes	—
74	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₂₇	#41	1	2	yes	—
78	C₄⋊C₄×C₁₂₇	—	4	2	no	—
79	C₄⋊C₄×C₁₂₇	—	4	2	no	—
80	C₂²⋊C₄×C₁₂₇	—	4	8	no	—
81	C₄²×C₁₂₇	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₁₂₇	SD₁₆	1	2	yes	—
83	C₁₆×C₁₂₇	—	2	4	no	—
84	C₂³×C₁₂₇	—	4	2	no	—
85	C₂³×C₁₂₇	—	8	1	no	—
86	C₂³×C₁₂₇	—	8	1	no	—
87	C₂³×C₁₂₇	—	8	4	no	—
88	C₂³×C₁₂₇	C₂×D₈	1	1	yes	characteristic
89	Q₈×C₁₂₇	—	4	2	no	—
90	Q₈×C₁₂₇	—	4	2	no	—
91	Q₈×C₁₂₇	—	8	1	no	—
92	Q₈×C₁₂₇	—	8	1	no	—
93	D₄×C₁₂₇	—	4	2	no	—
94	D₄×C₁₂₇	—	4	2	no	—
95	D₄×C₁₂₇	—	8	1	no	—
96	D₄×C₁₂₇	—	8	1	no	—
97	C₂×C₄×C₁₂₇	—	2	4	no	—
98	C₂×C₄×C₁₂₇	—	4	2	no	—
99	C₂×C₄×C₁₂₇	—	8	1	no	—
100	C₂×C₄×C₁₂₇	—	8	1	no	—
101	C₂×C₄×C₁₂₇	—	8	2	no	—
102	C₂×C₄×C₁₂₇	—	8	4	no	—
103	C₂×C₄×C₁₂₇	#102	1	4	yes	—
104	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₂₇	#124	1	1	yes	characteristic
106	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
107	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
110	C₈×C₁₂₇	#102	1	4	yes	—
111	C₈×C₁₂₇	C₄×D₄	1	2	yes	—
112	C₈×C₁₂₇	#121	1	2	yes	—
113	C₂²×C₁₂₇	—	8	4	no	—
114	C₂²×C₁₂₇	—	8	2	no	—
115	C₂²×C₁₂₇	—	16	2	no	—
116	C₂²×C₁₂₇	—	16	2	no	—
117	C₂²×C₁₂₇	C₂×#102	1	1	yes	characteristic
118	C₂²×C₁₂₇	#486	1	1	yes	characteristic
119	C₂²×C₁₂₇	#493	1	1	yes	characteristic
120	C₂²×C₁₂₇	#508	1	4	yes	—
121	C₄×C₁₂₇	—	2	4	no	—
122	C₄×C₁₂₇	—	8	2	no	—
123	C₄×C₁₂₇	—	16	2	no	—
124	C₄×C₁₂₇	#390	1	1	yes	characteristic
125	C₄×C₁₂₇	C₂×#102	1	1	yes	characteristic
126	C₄×C₁₂₇	#486	1	1	yes	characteristic
127	C₄×C₁₂₇	#492	1	1	yes	characteristic
128	#13204	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂×C₁₂₇	—	16	2	no	—
130	C₂×C₁₂₇	#1957	1	1	yes	characteristic
131	C₂×C₁₂₇	#2138	1	2	yes	—
132	C₂×C₁₂₇	#2242	1	2	yes	—
133	C₂×C₁₂₇	#2249	1	2	yes	—
134	C₂×#495	C₂×C₁₂₇	1	1	yes	characteristic
135	C₂×#358	C₂×C₁₂₇	1	2	yes	complemented
136	C₂×#357	C₂×C₁₂₇	1	2	yes	—
137	C₂×C₄×C₁₆	C₂×C₁₂₇	1	1	yes	characteristic, complemented
138	#1846	C₂×C₁₂₇	1	1	yes	characteristic, complemented
139	C₁₂₇	#13204	1	1	yes	characteristic, direct product factor, Sylow
140	C₂²×Q₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
141	C₂²×SD₁₆	—	2	1	no	—
142	C₂²×D₈	C₂²×C₁₂₇	1	1	yes	characteristic
143	C₂×#124	—	2	1	no	—
144	C₂²×C₁₆	C₂²×C₁₂₇	1	2	yes	—
145	#495	—	2	2	no	—
146	#495	C₄×C₁₂₇	1	4	yes	complemented
147	C₂×#107	C₂²×C₁₂₇	1	2	yes	—
148	C₂×#106	—	2	1	no	—
149	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
150	#382	—	2	1	no	—
151	#358	—	2	4	no	—
152	#357	—	2	4	no	—
153	C₄×C₁₆	—	2	2	no	—
154	C₂²×Q₈	—	2	1	no	—
155	C₂²×D₄	—	2	1	no	—
156	C₂×Q₁₆	—	2	2	no	—
157	C₂×Q₁₆	—	2	1	no	—
158	C₂×Q₁₆	—	2	2	no	—
159	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
160	C₂×SD₁₆	—	2	4	no	—
161	C₂×SD₁₆	—	4	2	no	—
162	C₂×SD₁₆	—	4	2	no	—
163	C₂×D₈	—	2	2	no	—
164	C₂×D₈	—	2	1	no	—
165	C₂×D₈	—	2	2	no	—
166	C₂×D₈	C₂×C₄×C₁₂₇	1	2	yes	—
167	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
168	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
169	#124	—	2	4	no	—
170	#124	—	4	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₂²⋊C₄	—	4	2	no	—
173	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
174	C₂×C₁₆	—	2	2	no	—
175	C₂×C₁₆	—	2	2	no	—
176	C₂×C₁₆	D₄×C₁₂₇	1	4	yes	—
177	#107	—	2	2	no	—
178	#107	—	2	2	no	—
179	#106	—	4	2	no	—
180	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
181	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
182	#95	—	4	2	no	—
183	C₂⁴	—	4	1	no	—
184	C₂×Q₈	—	2	2	no	—
185	C₂×Q₈	—	4	1	no	—
186	C₂×Q₈	—	4	1	no	—
187	C₂×Q₈	—	4	1	no	—
188	C₂×Q₈	—	4	1	no	—
189	C₂×Q₈	—	4	1	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	4	1	no	—
192	C₂×D₄	—	4	1	no	—
193	C₂×D₄	—	4	1	no	—
194	C₂×D₄	—	4	1	no	—
195	C₂×D₄	—	4	1	no	—
196	C₂²×C₄	—	4	1	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	D₈×C₁₂₇	1	2	yes	—
199	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
200	Q₁₆	—	2	4	no	—
201	Q₁₆	—	4	2	no	—
202	SD₁₆	—	4	8	no	—
203	D₈	—	2	4	no	—
204	D₈	—	4	2	no	—
205	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	2	yes	—
206	C₂×C₈	D₈×C₁₂₇	1	2	yes	—
207	C₂×C₈	SD₁₆×C₁₂₇	1	2	yes	—
208	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
209	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
210	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
211	C₂×C₈	#41×C₁₂₇	1	2	yes	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₂²⋊C₄	—	4	8	no	—
215	C₄²	C₂²⋊C₄×C₁₂₇	1	2	yes	—
216	C₄²	SD₁₆×C₁₂₇	1	2	yes	—
217	C₁₆	—	2	4	no	—
218	C₂³	—	4	2	no	—
219	C₂³	—	8	4	no	—
220	C₂³	—	8	1	no	—
221	C₂³	—	8	1	no	—
222	C₂³	C₂×D₈×C₁₂₇	1	1	yes	characteristic
223	Q₈	—	4	2	no	—
224	Q₈	—	4	2	no	—
225	Q₈	—	8	1	no	—
226	Q₈	—	8	1	no	—
227	D₄	—	4	2	no	—
228	D₄	—	4	2	no	—
229	D₄	—	8	1	no	—
230	D₄	—	8	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	8	1	no	—
234	C₂×C₄	—	8	1	no	—
235	C₂×C₄	—	8	4	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	#102×C₁₂₇	1	4	yes	—
238	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
239	C₂×C₄	#124×C₁₂₇	1	1	yes	characteristic
240	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
241	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
242	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
243	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
244	C₈	#102×C₁₂₇	1	4	yes	—
245	C₈	C₄×D₄×C₁₂₇	1	2	yes	—
246	C₈	#121×C₁₂₇	1	2	yes	—
247	C₂²	—	8	4	no	—
248	C₂²	—	8	2	no	—
249	C₂²	—	16	2	no	—
250	C₂²	—	16	2	no	—
251	C₂²	C₂×#102×C₁₂₇	1	1	yes	characteristic
252	C₂²	#486×C₁₂₇	1	1	yes	characteristic
253	C₂²	#493×C₁₂₇	1	1	yes	characteristic
254	C₂²	#508×C₁₂₇	1	4	yes	—
255	C₄	—	2	4	no	—
256	C₄	—	8	2	no	—
257	C₄	—	16	2	no	—
258	C₄	#390×C₁₂₇	1	1	yes	characteristic
259	C₄	C₂×#102×C₁₂₇	1	1	yes	characteristic
260	C₄	#486×C₁₂₇	1	1	yes	characteristic
261	C₄	#492×C₁₂₇	1	1	yes	characteristic
262	C₂	—	16	2	no	—
263	C₂	C₁₂₇×#1957	1	1	yes	characteristic
264	C₂	C₁₂₇×#2138	1	2	yes	—
265	C₂	C₁₂₇×#2242	1	2	yes	—
266	C₂	C₁₂₇×#2249	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₂₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
prime radical	C₂²×D₈×C₁₂₇	C₂²
LCS	G ▷ C₂×C₈ ▷ 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
2. C₂×#358×C₁₂₇	C₂	semidirect
4. C₂×C₄×C₁₆×C₁₂₇	C₂	semidirect
5. C₁₂₇×#1846	C₂	semidirect
12. #495×C₁₂₇	C₄	semidirect
128. #13204	C₁₂₇	direct (N × H)
135. C₂×#358	C₂×C₁₂₇	semidirect
137. C₂×C₄×C₁₆	C₂×C₁₂₇	semidirect
138. #1846	C₂×C₁₂₇	semidirect
139. C₁₂₇	#13204	direct (N × H)
146. #495	C₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#495×C₁₂₇	2
C₂×#358×C₁₂₇	2
C₂×#357×C₁₂₇	2
C₂×C₄×C₁₆×C₁₂₇	2
C₁₂₇×#1846	2
#13204	127

Subgroups — C127×#13204 #44576877

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#13204 #44576877