Subgroups — C₁₂₇×#13209 #44576882

Order 32512.

482 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×#3147	C₂	1	1	yes	maximal, characteristic
2	C₂×#359×C₁₂₇	C₂	1	4	yes	maximal, complemented
3	C₂×#346×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₂₇×#1848	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#135×C₁₂₇	—	2	1	no	—
6	C₂×#135×C₁₂₇	C₂²	1	2	yes	—
7	#532×C₁₂₇	—	2	1	no	—
8	C₂²×C₁₆×C₁₂₇	C₂²	1	2	yes	—
9	#495×C₁₂₇	—	2	1	no	—
10	#495×C₁₂₇	C₄	1	2	yes	—
11	#494×C₁₂₇	C₄	1	1	yes	characteristic
12	#493×C₁₂₇	C₄	1	1	yes	characteristic
13	#492×C₁₂₇	—	2	1	no	—
14	C₂×#108×C₁₂₇	—	2	1	no	—
15	C₂×#108×C₁₂₇	C₂²	1	2	yes	—
16	#423×C₁₂₇	—	2	1	no	—
17	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
18	#359×C₁₂₇	—	2	8	no	—
19	#346×C₁₂₇	—	2	1	no	—
20	#346×C₁₂₇	—	2	1	no	—
21	C₂×#41×C₁₂₇	—	2	2	no	—
22	#135×C₁₂₇	—	2	4	no	—
23	#135×C₁₂₇	—	2	4	no	—
24	#135×C₁₂₇	—	4	2	no	—
25	#135×C₁₂₇	—	4	2	no	—
26	C₂×Q₁₆×C₁₂₇	—	2	1	no	—
27	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
28	C₂×SD₁₆×C₁₂₇	—	2	2	no	—
29	C₂×SD₁₆×C₁₂₇	—	2	2	no	—
30	C₂×D₈×C₁₂₇	—	2	1	no	—
31	C₂×D₈×C₁₂₇	C₂×C₄	1	2	yes	—
32	C₂×M₁₆×C₁₂₇	—	2	2	no	—
33	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
35	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
36	#127×C₁₂₇	—	2	1	no	—
37	#124×C₁₂₇	—	2	2	no	—
38	#121×C₁₂₇	—	4	2	no	—
39	C₄×D₄×C₁₂₇	—	4	2	no	—
40	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
41	C₂×C₁₆×C₁₂₇	—	2	2	no	—
42	C₂×C₁₆×C₁₂₇	—	2	2	no	—
43	C₂×C₁₆×C₁₂₇	D₄	1	4	yes	complemented
44	#108×C₁₂₇	—	2	4	no	—
45	#108×C₁₂₇	—	4	2	no	—
46	C₄⋊C₈×C₁₂₇	—	4	2	no	—
47	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
50	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
51	#41×C₁₂₇	—	4	2	no	—
52	#41×C₁₂₇	—	4	2	no	—
53	#41×C₁₂₇	—	4	2	no	—
54	#41×C₁₂₇	—	4	2	no	—
55	C₂×Q₈×C₁₂₇	—	2	2	no	—
56	C₂×D₄×C₁₂₇	—	2	2	no	—
57	C₂×D₄×C₁₂₇	—	4	2	no	—
58	C₂×D₄×C₁₂₇	—	4	2	no	—
59	C₂²×C₄×C₁₂₇	—	4	2	no	—
60	C₂²×C₄×C₁₂₇	SD₁₆	1	2	yes	—
61	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
62	Q₁₆×C₁₂₇	—	2	4	no	—
63	Q₁₆×C₁₂₇	—	4	2	no	—
64	SD₁₆×C₁₂₇	—	4	4	no	—
65	SD₁₆×C₁₂₇	—	4	4	no	—
66	D₈×C₁₂₇	—	2	4	no	—
67	D₈×C₁₂₇	—	4	2	no	—
68	M₁₆×C₁₂₇	—	4	2	no	—
69	M₁₆×C₁₂₇	—	4	2	no	—
70	C₂×C₈×C₁₂₇	—	4	2	no	—
71	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₂₇	D₈	1	2	yes	—
74	C₂×C₈×C₁₂₇	SD₁₆	1	2	yes	—
75	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₁₂₇	#41	1	2	yes	—
79	C₄⋊C₄×C₁₂₇	—	4	2	no	—
80	C₂²⋊C₄×C₁₂₇	—	4	4	no	—
81	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₄²×C₁₂₇	D₈	1	2	yes	—
84	C₁₆×C₁₂₇	—	2	4	no	—
85	C₂³×C₁₂₇	—	4	2	no	—
86	C₂³×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
87	Q₈×C₁₂₇	—	4	2	no	—
88	Q₈×C₁₂₇	—	4	2	no	—
89	D₄×C₁₂₇	—	4	2	no	—
90	D₄×C₁₂₇	—	4	2	no	—
91	D₄×C₁₂₇	—	8	4	no	—
92	D₄×C₁₂₇	—	8	2	no	—
93	D₄×C₁₂₇	—	8	2	no	—
94	C₂×C₄×C₁₂₇	—	4	2	no	—
95	C₂×C₄×C₁₂₇	—	4	2	no	—
96	C₂×C₄×C₁₂₇	—	8	2	no	—
97	C₂×C₄×C₁₂₇	—	8	2	no	—
98	C₂×C₄×C₁₂₇	#102	1	2	yes	—
99	C₂×C₄×C₁₂₇	#102	1	2	yes	—
100	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₂₇	#124	1	1	yes	characteristic
102	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
103	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
106	C₈×C₁₂₇	—	8	2	no	—
107	C₈×C₁₂₇	#102	1	2	yes	—
108	C₈×C₁₂₇	#102	1	2	yes	—
109	C₈×C₁₂₇	C₄×D₄	1	2	yes	—
110	C₈×C₁₂₇	#121	1	2	yes	—
111	C₂²×C₁₂₇	—	2	2	no	—
112	C₂²×C₁₂₇	—	8	4	no	—
113	C₂²×C₁₂₇	—	8	2	no	—
114	C₂²×C₁₂₇	C₂×#102	1	1	yes	characteristic
115	C₂²×C₁₂₇	#486	1	1	yes	characteristic
116	C₂²×C₁₂₇	#492	1	1	yes	characteristic
117	C₄×C₁₂₇	—	2	2	no	—
118	C₄×C₁₂₇	—	2	2	no	—
119	C₄×C₁₂₇	—	8	2	no	—
120	C₄×C₁₂₇	#390	1	1	yes	characteristic
121	C₄×C₁₂₇	C₂×#102	1	1	yes	characteristic
122	C₄×C₁₂₇	#486	1	1	yes	characteristic
123	C₄×C₁₂₇	#493	1	1	yes	characteristic
124	#13209	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂×C₁₂₇	—	2	1	no	—
126	C₂×C₁₂₇	—	2	1	no	—
127	C₂×C₁₂₇	—	16	2	no	—
128	C₂×C₁₂₇	#1957	1	1	yes	characteristic
129	C₂×C₁₂₇	#2143	1	2	yes	—
130	#3147	C₂×C₁₂₇	1	1	yes	characteristic
131	C₂×#359	C₂×C₁₂₇	1	4	yes	complemented
132	C₂×#346	C₂×C₁₂₇	1	1	yes	characteristic, complemented
133	#1848	C₂×C₁₂₇	1	1	yes	characteristic, complemented
134	C₁₂₇	#13209	1	1	yes	characteristic, direct product factor, Sylow
135	C₂×#135	—	2	1	no	—
136	C₂×#135	C₂²×C₁₂₇	1	2	yes	—
137	#532	—	2	1	no	—
138	C₂²×C₁₆	C₂²×C₁₂₇	1	2	yes	—
139	#495	—	2	1	no	—
140	#495	C₄×C₁₂₇	1	2	yes	—
141	#494	C₄×C₁₂₇	1	1	yes	characteristic
142	#493	C₄×C₁₂₇	1	1	yes	characteristic
143	#492	—	2	1	no	—
144	C₂×#108	—	2	1	no	—
145	C₂×#108	C₂²×C₁₂₇	1	2	yes	—
146	#423	—	2	1	no	—
147	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
148	#359	—	2	8	no	—
149	#346	—	2	1	no	—
150	#346	—	2	1	no	—
151	C₂×#41	—	2	2	no	—
152	#135	—	2	4	no	—
153	#135	—	2	4	no	—
154	#135	—	4	2	no	—
155	#135	—	4	2	no	—
156	C₂×Q₁₆	—	2	1	no	—
157	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
158	C₂×SD₁₆	—	2	2	no	—
159	C₂×SD₁₆	—	2	2	no	—
160	C₂×D₈	—	2	1	no	—
161	C₂×D₈	C₂×C₄×C₁₂₇	1	2	yes	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
164	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
165	C₄⋊Q₈	—	2	1	no	—
166	#127	—	2	1	no	—
167	#124	—	2	2	no	—
168	#121	—	4	2	no	—
169	C₄×D₄	—	4	2	no	—
170	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
171	C₂×C₁₆	—	2	2	no	—
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	D₄×C₁₂₇	1	4	yes	complemented
174	#108	—	2	4	no	—
175	#108	—	4	2	no	—
176	C₄⋊C₈	—	4	2	no	—
177	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
178	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
179	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
180	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
181	#41	—	4	2	no	—
182	#41	—	4	2	no	—
183	#41	—	4	2	no	—
184	#41	—	4	2	no	—
185	C₂×Q₈	—	2	2	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	SD₁₆×C₁₂₇	1	2	yes	—
191	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
192	Q₁₆	—	2	4	no	—
193	Q₁₆	—	4	2	no	—
194	SD₁₆	—	4	4	no	—
195	SD₁₆	—	4	4	no	—
196	D₈	—	2	4	no	—
197	D₈	—	4	2	no	—
198	M₁₆	—	4	2	no	—
199	M₁₆	—	4	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
202	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
203	C₂×C₈	D₈×C₁₂₇	1	2	yes	—
204	C₂×C₈	SD₁₆×C₁₂₇	1	2	yes	—
205	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
206	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
207	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
208	C₂×C₈	#41×C₁₂₇	1	2	yes	—
209	C₄⋊C₄	—	4	2	no	—
210	C₂²⋊C₄	—	4	4	no	—
211	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
212	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
213	C₄²	D₈×C₁₂₇	1	2	yes	—
214	C₁₆	—	2	4	no	—
215	C₂³	—	4	2	no	—
216	C₂³	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
217	Q₈	—	4	2	no	—
218	Q₈	—	4	2	no	—
219	D₄	—	4	2	no	—
220	D₄	—	4	2	no	—
221	D₄	—	8	4	no	—
222	D₄	—	8	2	no	—
223	D₄	—	8	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	8	2	no	—
227	C₂×C₄	—	8	2	no	—
228	C₂×C₄	#102×C₁₂₇	1	2	yes	—
229	C₂×C₄	#102×C₁₂₇	1	2	yes	—
230	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
231	C₂×C₄	#124×C₁₂₇	1	1	yes	characteristic
232	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
233	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
234	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
235	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
236	C₈	—	8	2	no	—
237	C₈	#102×C₁₂₇	1	2	yes	—
238	C₈	#102×C₁₂₇	1	2	yes	—
239	C₈	C₄×D₄×C₁₂₇	1	2	yes	—
240	C₈	#121×C₁₂₇	1	2	yes	—
241	C₂²	—	2	2	no	—
242	C₂²	—	8	4	no	—
243	C₂²	—	8	2	no	—
244	C₂²	C₂×#102×C₁₂₇	1	1	yes	characteristic
245	C₂²	#486×C₁₂₇	1	1	yes	characteristic
246	C₂²	#492×C₁₂₇	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	8	2	no	—
250	C₄	#390×C₁₂₇	1	1	yes	characteristic
251	C₄	C₂×#102×C₁₂₇	1	1	yes	characteristic
252	C₄	#486×C₁₂₇	1	1	yes	characteristic
253	C₄	#493×C₁₂₇	1	1	yes	characteristic
254	C₂	—	2	1	no	—
255	C₂	—	2	1	no	—
256	C₂	—	16	2	no	—
257	C₂	C₁₂₇×#1957	1	1	yes	characteristic
258	C₂	C₁₂₇×#2143	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂×#102
prime radical	C₁₂₇×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×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
2. C₂×#359×C₁₂₇	C₂	semidirect
3. C₂×#346×C₁₂₇	C₂	semidirect
4. C₁₂₇×#1848	C₂	semidirect
43. C₂×C₁₆×C₁₂₇	D₄	semidirect
124. #13209	C₁₂₇	direct (N × H)
131. C₂×#359	C₂×C₁₂₇	semidirect
132. C₂×#346	C₂×C₁₂₇	semidirect
133. #1848	C₂×C₁₂₇	semidirect
134. C₁₂₇	#13209	direct (N × H)
173. C₂×C₁₆	D₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#3147	2
C₂×#359×C₁₂₇	2
C₂×#346×C₁₂₇	2
C₁₂₇×#1848	2
#13209	127

Subgroups — C127×#13209 #44576882

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#13209 #44576882