Subgroups — C₁₂₇×#9697 #44573370

Order 32512.

498 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#2862	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#1842	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#333×C₁₂₇	C₂	1	2	yes	maximal
5	C₂×#332×C₁₂₇	C₂	1	2	yes	maximal
6	C₂×C₄⋊Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₈×C₁₂₇	—	2	2	no	—
9	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₂₇	—	2	1	no	—
12	#391×C₁₂₇	—	2	1	no	—
13	#391×C₁₂₇	—	2	1	no	—
14	#391×C₁₂₇	C₄	1	4	yes	—
15	#382×C₁₂₇	—	2	2	no	—
16	C₄³×C₁₂₇	C₂²	1	1	yes	characteristic
17	#336×C₁₂₇	—	2	1	no	—
18	#333×C₁₂₇	—	2	4	no	—
19	#332×C₁₂₇	—	2	4	no	—
20	C₂²×Q₈×C₁₂₇	—	2	1	no	—
21	C₂²×Q₈×C₁₂₇	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₂₇	—	2	2	no	—
23	C₂²×C₈×C₁₂₇	—	2	2	no	—
24	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
25	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
26	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
27	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₂₇	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₂₇	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₂₇	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₂₇	D₄	1	2	yes	—
36	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₁₂₇	—	2	2	no	—
38	C₄⋊C₈×C₁₂₇	—	2	2	no	—
39	C₄⋊C₈×C₁₂₇	—	2	2	no	—
40	C₄⋊C₈×C₁₂₇	—	4	4	no	—
41	C₄⋊C₈×C₁₂₇	D₄	1	4	yes	—
42	C₄×C₈×C₁₂₇	—	4	2	no	—
43	#95×C₁₂₇	—	2	4	no	—
44	#95×C₁₂₇	—	2	4	no	—
45	C₂×Q₈×C₁₂₇	—	2	4	no	—
46	C₂×Q₈×C₁₂₇	—	4	4	no	—
47	C₂×Q₈×C₁₂₇	—	4	2	no	—
48	C₂×Q₈×C₁₂₇	—	4	2	no	—
49	C₂×Q₈×C₁₂₇	—	4	1	no	—
50	C₂×Q₈×C₁₂₇	—	4	1	no	—
51	C₂²×C₄×C₁₂₇	—	2	2	no	—
52	C₂²×C₄×C₁₂₇	—	2	2	no	—
53	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₁₂₇	—	2	4	no	—
59	C₂×C₈×C₁₂₇	—	4	2	no	—
60	C₂×C₈×C₁₂₇	—	4	2	no	—
61	C₂×C₈×C₁₂₇	—	4	2	no	—
62	C₂×C₈×C₁₂₇	—	4	4	no	—
63	C₄⋊C₄×C₁₂₇	—	2	4	no	—
64	C₄⋊C₄×C₁₂₇	—	4	2	no	—
65	C₄⋊C₄×C₁₂₇	—	4	1	no	—
66	C₄⋊C₄×C₁₂₇	—	4	1	no	—
67	C₄⋊C₄×C₁₂₇	—	4	1	no	—
68	C₄⋊C₄×C₁₂₇	—	4	1	no	—
69	C₄²×C₁₂₇	—	2	2	no	—
70	C₄²×C₁₂₇	—	2	1	no	—
71	C₄²×C₁₂₇	—	2	1	no	—
72	C₄²×C₁₂₇	—	4	2	no	—
73	C₄²×C₁₂₇	SD₁₆	1	4	yes	—
74	C₄²×C₁₂₇	Q₁₆	1	4	yes	—
75	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
76	C₄²×C₁₂₇	#41	1	2	yes	—
77	C₂³×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	Q₈×C₁₂₇	—	8	4	no	—
79	Q₈×C₁₂₇	—	8	2	no	—
80	Q₈×C₁₂₇	—	8	2	no	—
81	C₂×C₄×C₁₂₇	—	2	2	no	—
82	C₂×C₄×C₁₂₇	—	2	2	no	—
83	C₂×C₄×C₁₂₇	—	2	2	no	—
84	C₂×C₄×C₁₂₇	—	2	2	no	—
85	C₂×C₄×C₁₂₇	—	2	2	no	—
86	C₂×C₄×C₁₂₇	—	4	2	no	—
87	C₂×C₄×C₁₂₇	—	4	2	no	—
88	C₂×C₄×C₁₂₇	—	4	4	no	—
89	C₂×C₄×C₁₂₇	—	4	2	no	—
90	C₂×C₄×C₁₂₇	—	4	2	no	—
91	C₂×C₄×C₁₂₇	#103	1	4	yes	—
92	C₂×C₄×C₁₂₇	#103	1	4	yes	—
93	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
94	C₂×C₄×C₁₂₇	#121	1	2	yes	—
95	C₂×C₄×C₁₂₇	#124	1	1	yes	characteristic
96	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
97	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	2	yes	—
98	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	2	yes	—
99	C₂×C₄×C₁₂₇	#135	1	2	yes	—
100	C₂×C₄×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
101	C₂×C₄×C₁₂₇	#137	1	1	yes	characteristic
102	C₈×C₁₂₇	—	4	4	no	—
103	C₈×C₁₂₇	—	8	2	no	—
104	C₂²×C₁₂₇	#390	1	1	yes	characteristic
105	C₂²×C₁₂₇	#413	1	2	yes	—
106	C₂²×C₁₂₇	C₂×#103	1	2	yes	—
107	C₂²×C₁₂₇	#416	1	1	yes	characteristic
108	C₂²×C₁₂₇	#417	1	1	yes	characteristic
109	C₄×C₁₂₇	—	2	4	no	—
110	C₄×C₁₂₇	—	2	4	no	—
111	C₄×C₁₂₇	—	2	2	no	—
112	C₄×C₁₂₇	—	8	2	no	—
113	C₄×C₁₂₇	—	8	2	no	—
114	C₄×C₁₂₇	#463	1	2	yes	—
115	C₄×C₁₂₇	#464	1	2	yes	—
116	C₄×C₁₂₇	#487	1	2	yes	—
117	C₄×C₁₂₇	#492	1	1	yes	characteristic
118	C₄×C₁₂₇	#494	1	1	yes	characteristic
119	#9697	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂×C₁₂₇	#1542	1	2	yes	—
121	C₂×C₁₂₇	#1550	1	2	yes	—
122	C₂×C₁₂₇	#1956	1	1	yes	characteristic
123	C₂×C₁₂₇	#1967	1	1	yes	characteristic
124	C₂×C₁₂₇	#1983	1	1	yes	characteristic
125	#2862	C₂×C₁₂₇	1	1	yes	characteristic
126	#1842	C₂×C₁₂₇	1	1	yes	characteristic
127	C₄×C₄⋊C₈	C₂×C₁₂₇	1	1	yes	characteristic
128	C₂×#333	C₂×C₁₂₇	1	2	yes	—
129	C₂×#332	C₂×C₁₂₇	1	2	yes	—
130	C₁₂₇	#9697	1	1	yes	characteristic, direct product factor, Sylow
131	C₂×C₄⋊Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
132	C₂×C₄⋊Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
133	C₂×C₄⋊C₈	—	2	2	no	—
134	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
135	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
136	C₂×C₄×C₈	—	2	1	no	—
137	#391	—	2	1	no	—
138	#391	—	2	1	no	—
139	#391	C₄×C₁₂₇	1	4	yes	—
140	#382	—	2	2	no	—
141	C₄³	C₂²×C₁₂₇	1	1	yes	characteristic
142	#336	—	2	1	no	—
143	#333	—	2	4	no	—
144	#332	—	2	4	no	—
145	C₂²×Q₈	—	2	1	no	—
146	C₂²×Q₈	C₂×C₄×C₁₂₇	1	2	yes	—
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	—	2	2	no	—
149	C₄⋊Q₈	—	2	2	no	—
150	C₄⋊Q₈	—	2	2	no	—
151	C₄⋊Q₈	—	2	2	no	—
152	C₄⋊Q₈	—	2	2	no	—
153	C₂×C₄⋊C₄	—	2	1	no	—
154	C₂×C₄⋊C₄	—	2	1	no	—
155	C₂×C₄⋊C₄	—	2	1	no	—
156	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
157	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
158	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
159	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
160	C₂×C₄²	D₄×C₁₂₇	1	2	yes	—
161	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	—	4	4	no	—
166	C₄⋊C₈	D₄×C₁₂₇	1	4	yes	—
167	C₄×C₈	—	4	2	no	—
168	#95	—	2	4	no	—
169	#95	—	2	4	no	—
170	C₂×Q₈	—	2	4	no	—
171	C₂×Q₈	—	4	4	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	1	no	—
175	C₂×Q₈	—	4	1	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	2	yes	—
179	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	2	yes	—
180	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
181	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	4	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	4	1	no	—
191	C₄⋊C₄	—	4	1	no	—
192	C₄⋊C₄	—	4	1	no	—
193	C₄⋊C₄	—	4	1	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	2	1	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	4	2	no	—
198	C₄²	SD₁₆×C₁₂₇	1	4	yes	—
199	C₄²	Q₁₆×C₁₂₇	1	4	yes	—
200	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
201	C₄²	#41×C₁₂₇	1	2	yes	—
202	C₂³	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
203	Q₈	—	8	4	no	—
204	Q₈	—	8	2	no	—
205	Q₈	—	8	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	#103×C₁₂₇	1	4	yes	—
217	C₂×C₄	#103×C₁₂₇	1	4	yes	—
218	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
219	C₂×C₄	#121×C₁₂₇	1	2	yes	—
220	C₂×C₄	#124×C₁₂₇	1	1	yes	characteristic
221	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
222	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	2	yes	—
223	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	2	yes	—
224	C₂×C₄	#135×C₁₂₇	1	2	yes	—
225	C₂×C₄	AGL(1,ℤ/8)×C₁₂₇	1	1	yes	characteristic
226	C₂×C₄	#137×C₁₂₇	1	1	yes	characteristic
227	C₈	—	4	4	no	—
228	C₈	—	8	2	no	—
229	C₂²	#390×C₁₂₇	1	1	yes	characteristic
230	C₂²	#413×C₁₂₇	1	2	yes	—
231	C₂²	C₂×#103×C₁₂₇	1	2	yes	—
232	C₂²	#416×C₁₂₇	1	1	yes	characteristic
233	C₂²	#417×C₁₂₇	1	1	yes	characteristic
234	C₄	—	2	4	no	—
235	C₄	—	2	2	no	—
236	C₄	—	2	4	no	—
237	C₄	—	8	2	no	—
238	C₄	—	8	2	no	—
239	C₄	#463×C₁₂₇	1	2	yes	—
240	C₄	#464×C₁₂₇	1	2	yes	—
241	C₄	#487×C₁₂₇	1	2	yes	—
242	C₄	#492×C₁₂₇	1	1	yes	characteristic
243	C₄	#494×C₁₂₇	1	1	yes	characteristic
244	C₂	C₁₂₇×#1542	1	2	yes	—
245	C₂	C₁₂₇×#1550	1	2	yes	—
246	C₂	C₁₂₇×#1956	1	1	yes	characteristic
247	C₂	C₁₂₇×#1967	1	1	yes	characteristic
248	C₂	C₁₂₇×#1983	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₂₇	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₂₇
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ C₄³×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
119. #9697	C₁₂₇	direct (N × H)
130. C₁₂₇	#9697	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₂₇×#2862	2
C₁₂₇×#1842	2
C₄×C₄⋊C₈×C₁₂₇	2
C₂×#333×C₁₂₇	2
C₂×#332×C₁₂₇	2
#9697	127

Subgroups — C127×#9697 #44573370

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#9697 #44573370