Subgroups — C₁₂₇×#9733 #44573406

Order 32512.

518 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×#494×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#1988	C₂	1	2	yes	maximal
3	C₈×C₄⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#335×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#332×C₁₂₇	C₂	1	2	yes	maximal
6	C₂²×Q₁₆×C₁₂₇	—	2	1	no	—
7	C₂²×Q₁₆×C₁₂₇	—	2	1	no	—
8	C₂×C₄⋊Q₈×C₁₂₇	C₂²	1	2	yes	—
9	#494×C₁₂₇	—	2	2	no	—
10	#494×C₁₂₇	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₂₇	—	2	1	no	—
14	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
16	#340×C₁₂₇	—	2	4	no	—
17	#336×C₁₂₇	—	2	1	no	—
18	#335×C₁₂₇	—	2	2	no	—
19	#332×C₁₂₇	—	2	4	no	—
20	C₂²×Q₈×C₁₂₇	—	2	2	no	—
21	C₂×Q₁₆×C₁₂₇	—	2	4	no	—
22	C₂×Q₁₆×C₁₂₇	—	2	4	no	—
23	C₂×Q₁₆×C₁₂₇	—	4	2	no	—
24	C₂×Q₁₆×C₁₂₇	—	4	2	no	—
25	C₂×Q₁₆×C₁₂₇	—	4	2	no	—
26	C₂×Q₁₆×C₁₂₇	—	4	2	no	—
27	C₂²×C₈×C₁₂₇	—	2	1	no	—
28	C₂²×C₈×C₁₂₇	—	2	2	no	—
29	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
31	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
32	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
33	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
34	C₄⋊Q₈×C₁₂₇	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₁₂₇	D₄	1	2	yes	—
38	C₂×C₄²×C₁₂₇	—	2	1	no	—
39	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₁₂₇	—	2	2	no	—
41	C₄⋊C₈×C₁₂₇	—	2	2	no	—
42	C₄⋊C₈×C₁₂₇	—	2	1	no	—
43	C₄⋊C₈×C₁₂₇	D₄	1	2	yes	—
44	C₄×C₈×C₁₂₇	—	4	2	no	—
45	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
47	#95×C₁₂₇	—	2	1	no	—
48	C₂×Q₈×C₁₂₇	—	2	4	no	—
49	C₂×Q₈×C₁₂₇	—	4	2	no	—
50	C₂×Q₈×C₁₂₇	—	4	2	no	—
51	C₂×Q₈×C₁₂₇	—	4	2	no	—
52	C₂×Q₈×C₁₂₇	—	4	2	no	—
53	C₂×Q₈×C₁₂₇	—	4	2	no	—
54	C₂²×C₄×C₁₂₇	—	2	2	no	—
55	C₂²×C₄×C₁₂₇	—	4	2	no	—
56	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₂₇	#41	1	1	yes	characteristic
59	Q₁₆×C₁₂₇	—	4	8	no	—
60	Q₁₆×C₁₂₇	—	4	8	no	—
61	C₂×C₈×C₁₂₇	—	2	1	no	—
62	C₂×C₈×C₁₂₇	—	2	1	no	—
63	C₂×C₈×C₁₂₇	—	2	2	no	—
64	C₂×C₈×C₁₂₇	—	4	1	no	—
65	C₂×C₈×C₁₂₇	—	4	1	no	—
66	C₂×C₈×C₁₂₇	—	4	2	no	—
67	C₂×C₈×C₁₂₇	—	4	4	no	—
68	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	2	yes	—
69	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈×C₁₂₇	D₈	1	2	yes	—
71	C₂×C₈×C₁₂₇	SD₁₆	1	2	yes	—
72	C₄⋊C₄×C₁₂₇	—	2	2	no	—
73	C₄⋊C₄×C₁₂₇	—	2	4	no	—
74	C₄⋊C₄×C₁₂₇	—	2	4	no	—
75	C₄⋊C₄×C₁₂₇	—	4	2	no	—
76	C₄⋊C₄×C₁₂₇	—	4	2	no	—
77	C₄⋊C₄×C₁₂₇	Q₁₆	1	4	yes	complemented
78	C₄²×C₁₂₇	—	4	1	no	—
79	C₄²×C₁₂₇	—	4	1	no	—
80	C₄²×C₁₂₇	C₂²×C₄	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	#41	1	1	yes	characteristic
84	C₂³×C₁₂₇	#121	1	1	yes	characteristic
85	Q₈×C₁₂₇	—	4	4	no	—
86	Q₈×C₁₂₇	—	4	4	no	—
87	Q₈×C₁₂₇	—	8	2	no	—
88	Q₈×C₁₂₇	—	8	2	no	—
89	C₂×C₄×C₁₂₇	—	2	2	no	—
90	C₂×C₄×C₁₂₇	—	2	4	no	—
91	C₂×C₄×C₁₂₇	—	4	2	no	—
92	C₂×C₄×C₁₂₇	—	4	2	no	—
93	C₂×C₄×C₁₂₇	—	4	4	no	—
94	C₂×C₄×C₁₂₇	—	8	2	no	—
95	C₂×C₄×C₁₂₇	—	8	2	no	—
96	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₂₇	#121	1	1	yes	characteristic
100	C₂×C₄×C₁₂₇	#124	1	1	yes	characteristic
101	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
102	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
104	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	2	yes	—
105	C₂×C₄×C₁₂₇	#135	1	2	yes	—
106	C₂×C₄×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
107	C₂×C₄×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₈×C₁₂₇	—	2	2	no	—
109	C₈×C₁₂₇	—	4	2	no	—
110	C₈×C₁₂₇	—	8	2	no	—
111	C₈×C₁₂₇	#102	1	4	yes	—
112	C₂²×C₁₂₇	#390	1	1	yes	characteristic
113	C₂²×C₁₂₇	#459	1	1	yes	characteristic
114	C₂²×C₁₂₇	#460	1	1	yes	characteristic
115	C₂²×C₁₂₇	#467	1	2	yes	—
116	C₂²×C₁₂₇	#473	1	2	yes	—
117	C₄×C₁₂₇	—	2	1	no	—
118	C₄×C₁₂₇	—	2	1	no	—
119	C₄×C₁₂₇	—	4	4	no	—
120	C₄×C₁₂₇	—	8	4	no	—
121	C₄×C₁₂₇	C₂×#102	1	1	yes	characteristic
122	C₄×C₁₂₇	#418	1	1	yes	characteristic
123	C₄×C₁₂₇	C₄×Q₁₆	1	2	yes	—
124	C₄×C₁₂₇	#459	1	1	yes	characteristic
125	C₄×C₁₂₇	#460	1	1	yes	characteristic
126	C₄×C₁₂₇	#494	1	1	yes	characteristic
127	C₄×C₁₂₇	#495	1	1	yes	characteristic
128	#9733	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂×C₁₂₇	#1669	1	2	yes	—
130	C₂×C₁₂₇	#1670	1	2	yes	—
131	C₂×C₁₂₇	#1969	1	1	yes	characteristic
132	C₂×C₁₂₇	#1979	1	1	yes	characteristic
133	C₂×C₁₂₇	#1982	1	1	yes	characteristic
134	C₂×#494	C₂×C₁₂₇	1	1	yes	characteristic
135	#1988	C₂×C₁₂₇	1	2	yes	—
136	C₈×C₄⋊C₄	C₂×C₁₂₇	1	1	yes	characteristic
137	C₂×#335	C₂×C₁₂₇	1	1	yes	characteristic
138	C₂×#332	C₂×C₁₂₇	1	2	yes	—
139	C₁₂₇	#9733	1	1	yes	characteristic, direct product factor, Sylow
140	C₂²×Q₁₆	—	2	1	no	—
141	C₂²×Q₁₆	—	2	1	no	—
142	C₂×C₄⋊Q₈	C₂²×C₁₂₇	1	2	yes	—
143	#494	—	2	2	no	—
144	#494	C₄×C₁₂₇	1	4	yes	complemented
145	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
146	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	1	yes	characteristic
147	C₂×C₄×C₈	—	2	1	no	—
148	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
149	C₄×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
150	#340	—	2	4	no	—
151	#336	—	2	1	no	—
152	#335	—	2	2	no	—
153	#332	—	2	4	no	—
154	C₂²×Q₈	—	2	2	no	—
155	C₂×Q₁₆	—	2	4	no	—
156	C₂×Q₁₆	—	2	4	no	—
157	C₂×Q₁₆	—	4	2	no	—
158	C₂×Q₁₆	—	4	2	no	—
159	C₂×Q₁₆	—	4	2	no	—
160	C₂×Q₁₆	—	4	2	no	—
161	C₂²×C₈	—	2	1	no	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
164	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
165	C₄⋊Q₈	—	2	2	no	—
166	C₄⋊Q₈	—	2	2	no	—
167	C₄⋊Q₈	—	2	2	no	—
168	C₄⋊Q₈	C₂×C₄×C₁₂₇	1	4	yes	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	D₄×C₁₂₇	1	2	yes	—
172	C₂×C₄²	—	2	1	no	—
173	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	—	2	2	no	—
176	C₄⋊C₈	—	2	1	no	—
177	C₄⋊C₈	D₄×C₁₂₇	1	2	yes	—
178	C₄×C₈	—	4	2	no	—
179	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
180	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
181	#95	—	2	1	no	—
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	—	4	2	no	—
187	C₂×Q₈	—	4	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
192	C₂²×C₄	#41×C₁₂₇	1	1	yes	characteristic
193	Q₁₆	—	4	8	no	—
194	Q₁₆	—	4	8	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	4	4	no	—
202	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	2	yes	—
203	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	2	yes	—
204	C₂×C₈	D₈×C₁₂₇	1	2	yes	—
205	C₂×C₈	SD₁₆×C₁₂₇	1	2	yes	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	Q₁₆×C₁₂₇	1	4	yes	complemented
212	C₄²	—	4	1	no	—
213	C₄²	—	4	1	no	—
214	C₄²	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
215	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
216	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
217	C₄²	#41×C₁₂₇	1	1	yes	characteristic
218	C₂³	#121×C₁₂₇	1	1	yes	characteristic
219	Q₈	—	4	4	no	—
220	Q₈	—	4	4	no	—
221	Q₈	—	8	2	no	—
222	Q₈	—	8	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	8	2	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
231	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
232	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
233	C₂×C₄	#121×C₁₂₇	1	1	yes	characteristic
234	C₂×C₄	#124×C₁₂₇	1	1	yes	characteristic
235	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
236	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
237	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
238	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	2	yes	—
239	C₂×C₄	#135×C₁₂₇	1	2	yes	—
240	C₂×C₄	AGL(1,ℤ/8)×C₁₂₇	1	1	yes	characteristic
241	C₂×C₄	AGL(1,ℤ/8)×C₁₂₇	1	1	yes	characteristic
242	C₈	—	2	2	no	—
243	C₈	—	4	2	no	—
244	C₈	—	8	2	no	—
245	C₈	#102×C₁₂₇	1	4	yes	—
246	C₂²	#390×C₁₂₇	1	1	yes	characteristic
247	C₂²	#459×C₁₂₇	1	1	yes	characteristic
248	C₂²	#460×C₁₂₇	1	1	yes	characteristic
249	C₂²	#467×C₁₂₇	1	2	yes	—
250	C₂²	#473×C₁₂₇	1	2	yes	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	4	4	no	—
254	C₄	—	8	4	no	—
255	C₄	C₂×#102×C₁₂₇	1	1	yes	characteristic
256	C₄	#418×C₁₂₇	1	1	yes	characteristic
257	C₄	C₄×Q₁₆×C₁₂₇	1	2	yes	—
258	C₄	#459×C₁₂₇	1	1	yes	characteristic
259	C₄	#460×C₁₂₇	1	1	yes	characteristic
260	C₄	#494×C₁₂₇	1	1	yes	characteristic
261	C₄	#495×C₁₂₇	1	1	yes	characteristic
262	C₂	C₁₂₇×#1669	1	2	yes	—
263	C₂	C₁₂₇×#1670	1	2	yes	—
264	C₂	C₁₂₇×#1969	1	1	yes	characteristic
265	C₂	C₁₂₇×#1979	1	1	yes	characteristic
266	C₂	C₁₂₇×#1982	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₂₇	#121
commutator	C₄²	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂×C₄²	C₂³×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
10. #494×C₁₂₇	C₄	semidirect
77. C₄⋊C₄×C₁₂₇	Q₁₆	semidirect
128. #9733	C₁₂₇	direct (N × H)
139. C₁₂₇	#9733	direct (N × H)
144. #494	C₄×C₁₂₇	semidirect
211. C₄⋊C₄	Q₁₆×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#494×C₁₂₇	2
C₁₂₇×#1988	2
C₈×C₄⋊C₄×C₁₂₇	2
C₂×#335×C₁₂₇	2
C₂×#332×C₁₂₇	2
#9733	127

Subgroups — C127×#9733 #44573406

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#9733 #44573406