Subgroups — C₁₂₇×#9600 #44573273

Order 32512.

378 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#1939	C₂	1	2	yes	maximal
2	C₄×#106×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#335×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#322×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₁₂₇×#1279	C₂	1	2	yes	maximal
6	C₂×#106×C₁₂₇	C₄	1	2	yes	—
7	C₂×C₄⋊C₈×C₁₂₇	—	2	2	no	—
8	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
13	#340×C₁₂₇	C₄	1	1	yes	characteristic
14	#340×C₁₂₇	C₄	1	1	yes	characteristic
15	#336×C₁₂₇	—	2	2	no	—
16	#335×C₁₂₇	—	2	2	no	—
17	#322×C₁₂₇	—	2	2	no	—
18	C₂²×C₈×C₁₂₇	—	2	1	no	—
19	C₂²×C₈×C₁₂₇	—	2	2	no	—
20	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
25	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₁₂₇	—	2	2	no	—
27	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
28	#106×C₁₂₇	—	2	4	no	—
29	C₄⋊C₈×C₁₂₇	—	2	2	no	—
30	C₄⋊C₈×C₁₂₇	—	2	2	no	—
31	C₄⋊C₈×C₁₂₇	—	4	4	no	—
32	C₄×C₈×C₁₂₇	—	2	1	no	—
33	C₄×C₈×C₁₂₇	—	2	1	no	—
34	C₄×C₈×C₁₂₇	—	2	1	no	—
35	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
36	C₄×C₈×C₁₂₇	Q₈	1	2	yes	—
37	C₄×C₈×C₁₂₇	Q₈	1	2	yes	—
38	#95×C₁₂₇	—	2	2	no	—
39	C₂²×C₄×C₁₂₇	—	2	2	no	—
40	C₂²×C₄×C₁₂₇	—	2	2	no	—
41	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₂₇	#41	1	1	yes	characteristic
44	C₂×C₈×C₁₂₇	—	2	2	no	—
45	C₂×C₈×C₁₂₇	—	2	2	no	—
46	C₂×C₈×C₁₂₇	—	2	2	no	—
47	C₂×C₈×C₁₂₇	—	2	2	no	—
48	C₂×C₈×C₁₂₇	—	2	2	no	—
49	C₂×C₈×C₁₂₇	—	2	1	no	—
50	C₂×C₈×C₁₂₇	—	2	1	no	—
51	C₂×C₈×C₁₂₇	—	2	2	no	—
52	C₂×C₈×C₁₂₇	—	2	2	no	—
53	C₂×C₈×C₁₂₇	—	4	2	no	—
54	C₂×C₈×C₁₂₇	—	4	2	no	—
55	C₂×C₈×C₁₂₇	—	4	2	no	—
56	C₂×C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
58	C₄⋊C₄×C₁₂₇	—	2	2	no	—
59	C₄⋊C₄×C₁₂₇	—	2	2	no	—
60	C₄⋊C₄×C₁₂₇	—	2	2	no	—
61	C₄⋊C₄×C₁₂₇	M₁₆	1	4	yes	—
62	C₄²×C₁₂₇	—	4	2	no	—
63	C₄²×C₁₂₇	—	4	2	no	—
64	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
66	C₄²×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₁₂₇	#41	1	1	yes	characteristic
68	C₂³×C₁₂₇	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₂₇	—	2	1	no	—
70	C₂×C₄×C₁₂₇	—	2	1	no	—
71	C₂×C₄×C₁₂₇	—	2	4	no	—
72	C₂×C₄×C₁₂₇	—	4	2	no	—
73	C₂×C₄×C₁₂₇	—	4	2	no	—
74	C₂×C₄×C₁₂₇	—	4	2	no	—
75	C₂×C₄×C₁₂₇	—	4	2	no	—
76	C₂×C₄×C₁₂₇	—	4	2	no	—
77	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₁₂₇	#121	1	1	yes	characteristic
80	C₂×C₄×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₁₂₇	#125	1	1	yes	characteristic
82	C₂×C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₁₂₇	C₂×M₁₆	1	2	yes	—
84	C₂×C₄×C₁₂₇	#131	1	2	yes	—
85	C₂×C₄×C₁₂₇	AGL(1,ℤ/8)	1	2	yes	—
86	C₂×C₄×C₁₂₇	#137	1	2	yes	—
87	C₈×C₁₂₇	—	2	4	no	—
88	C₈×C₁₂₇	—	2	2	no	—
89	C₈×C₁₂₇	—	2	2	no	—
90	C₈×C₁₂₇	—	4	1	no	—
91	C₈×C₁₂₇	—	4	1	no	—
92	C₈×C₁₂₇	—	8	2	no	—
93	C₂²×C₁₂₇	#389	1	1	yes	characteristic
94	C₂²×C₁₂₇	#435	1	2	yes	—
95	C₂²×C₁₂₇	#440	1	2	yes	—
96	C₂²×C₁₂₇	#443	1	1	yes	characteristic
97	C₂²×C₁₂₇	#443	1	1	yes	characteristic
98	C₄×C₁₂₇	—	2	1	no	—
99	C₄×C₁₂₇	—	2	1	no	—
100	C₄×C₁₂₇	—	4	4	no	—
101	C₄×C₁₂₇	—	8	2	no	—
102	C₄×C₁₂₇	#423	1	1	yes	characteristic
103	C₄×C₁₂₇	#424	1	1	yes	characteristic
104	C₄×C₁₂₇	#446	1	2	yes	—
105	C₄×C₁₂₇	#469	1	1	yes	characteristic
106	C₄×C₁₂₇	#470	1	1	yes	characteristic
107	C₄×C₁₂₇	#501	1	2	yes	—
108	#9600	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂×C₁₂₇	#1585	1	2	yes	—
110	C₂×C₁₂₇	#1588	1	2	yes	—
111	C₂×C₁₂₇	#1940	1	1	yes	characteristic
112	C₂×C₁₂₇	#1944	1	1	yes	characteristic
113	C₂×C₁₂₇	#1945	1	1	yes	characteristic
114	#1939	C₂×C₁₂₇	1	2	yes	—
115	C₄×#106	C₂×C₁₂₇	1	1	yes	characteristic
116	C₂×#335	C₂×C₁₂₇	1	1	yes	characteristic
117	C₂×#322	C₂×C₁₂₇	1	1	yes	characteristic
118	#1279	C₂×C₁₂₇	1	2	yes	—
119	C₁₂₇	#9600	1	1	yes	characteristic, direct product factor, Sylow
120	C₂×#106	C₄×C₁₂₇	1	2	yes	—
121	C₂×C₄⋊C₈	—	2	2	no	—
122	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
123	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
124	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
125	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
126	C₄×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
127	#340	C₄×C₁₂₇	1	1	yes	characteristic
128	#340	C₄×C₁₂₇	1	1	yes	characteristic
129	#336	—	2	2	no	—
130	#335	—	2	2	no	—
131	#322	—	2	2	no	—
132	C₂²×C₈	—	2	1	no	—
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
135	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
136	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
137	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
138	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
139	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
140	C₂×C₄²	—	2	2	no	—
141	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
142	#106	—	2	4	no	—
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	2	2	no	—
145	C₄⋊C₈	—	4	4	no	—
146	C₄×C₈	—	2	1	no	—
147	C₄×C₈	—	2	1	no	—
148	C₄×C₈	—	2	1	no	—
149	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
150	C₄×C₈	Q₈×C₁₂₇	1	2	yes	—
151	C₄×C₈	Q₈×C₁₂₇	1	2	yes	—
152	#95	—	2	2	no	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
156	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
157	C₂²×C₄	#41×C₁₂₇	1	1	yes	characteristic
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	C₄⋊C₄×C₁₂₇	1	2	yes	—
171	C₂×C₈	C₄⋊C₄×C₁₂₇	1	2	yes	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	M₁₆×C₁₂₇	1	4	yes	—
176	C₄²	—	4	2	no	—
177	C₄²	—	4	2	no	—
178	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
179	C₄²	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
180	C₄²	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
181	C₄²	#41×C₁₂₇	1	1	yes	characteristic
182	C₂³	C₄×D₄×C₁₂₇	1	1	yes	characteristic
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
192	C₂×C₄	C₄×Q₈×C₁₂₇	1	1	yes	characteristic
193	C₂×C₄	#121×C₁₂₇	1	1	yes	characteristic
194	C₂×C₄	C₂²⋊Q₈×C₁₂₇	1	1	yes	characteristic
195	C₂×C₄	#125×C₁₂₇	1	1	yes	characteristic
196	C₂×C₄	C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
197	C₂×C₄	C₂×M₁₆×C₁₂₇	1	2	yes	—
198	C₂×C₄	#131×C₁₂₇	1	2	yes	—
199	C₂×C₄	AGL(1,ℤ/8)×C₁₂₇	1	2	yes	—
200	C₂×C₄	#137×C₁₂₇	1	2	yes	—
201	C₈	—	2	4	no	—
202	C₈	—	2	2	no	—
203	C₈	—	2	2	no	—
204	C₈	—	4	1	no	—
205	C₈	—	4	1	no	—
206	C₈	—	8	2	no	—
207	C₂²	#389×C₁₂₇	1	1	yes	characteristic
208	C₂²	#435×C₁₂₇	1	2	yes	—
209	C₂²	#440×C₁₂₇	1	2	yes	—
210	C₂²	#443×C₁₂₇	1	1	yes	characteristic
211	C₂²	#443×C₁₂₇	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	4	4	no	—
215	C₄	—	8	2	no	—
216	C₄	#423×C₁₂₇	1	1	yes	characteristic
217	C₄	#424×C₁₂₇	1	1	yes	characteristic
218	C₄	#446×C₁₂₇	1	2	yes	—
219	C₄	#469×C₁₂₇	1	1	yes	characteristic
220	C₄	#470×C₁₂₇	1	1	yes	characteristic
221	C₄	#501×C₁₂₇	1	2	yes	—
222	C₂	C₁₂₇×#1585	1	2	yes	—
223	C₂	C₁₂₇×#1588	1	2	yes	—
224	C₂	C₁₂₇×#1940	1	1	yes	characteristic
225	C₂	C₁₂₇×#1944	1	1	yes	characteristic
226	C₂	C₁₂₇×#1945	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₂₇	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₂₇
socle, prime radical	C₂³×C₁₂₇	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
108. #9600	C₁₂₇	direct (N × H)
119. C₁₂₇	#9600	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×#1939	2
C₄×#106×C₁₂₇	2
C₂×#335×C₁₂₇	2
C₂×#322×C₁₂₇	2
C₁₂₇×#1279	2
#9600	127

Subgroups — C127×#9600 #44573273

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#9600 #44573273