Subgroups — C₁₂₇×#9598 #44573271

Order 32512.

378 conjugacy classes of subgroups.

The subgroup lattice has 234 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×#107×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×#107×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	#107×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	1	no	—
45	C₂×C₈×C₁₂₇	—	2	1	no	—
46	C₂×C₈×C₁₂₇	—	2	2	no	—
47	C₂×C₈×C₁₂₇	—	2	2	no	—
48	C₂×C₈×C₁₂₇	—	2	1	no	—
49	C₂×C₈×C₁₂₇	—	2	1	no	—
50	C₂×C₈×C₁₂₇	—	2	2	no	—
51	C₂×C₈×C₁₂₇	—	4	2	no	—
52	C₂×C₈×C₁₂₇	—	4	2	no	—
53	C₂×C₈×C₁₂₇	—	4	2	no	—
54	C₂×C₈×C₁₂₇	—	4	2	no	—
55	C₂×C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₁₂₇	D₈	1	2	yes	—
58	C₂×C₈×C₁₂₇	Q₁₆	1	2	yes	—
59	C₄⋊C₄×C₁₂₇	—	2	2	no	—
60	C₄⋊C₄×C₁₂₇	—	2	2	no	—
61	C₄⋊C₄×C₁₂₇	—	2	2	no	—
62	C₄⋊C₄×C₁₂₇	M₁₆	1	4	yes	—
63	C₄²×C₁₂₇	—	4	2	no	—
64	C₄²×C₁₂₇	—	4	2	no	—
65	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
68	C₄²×C₁₂₇	#41	1	1	yes	characteristic
69	C₂³×C₁₂₇	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₁₂₇	—	2	2	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	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×C₄×C₁₂₇	#135	1	1	yes	characteristic
88	C₂×C₄×C₁₂₇	#135	1	1	yes	characteristic
89	C₈×C₁₂₇	—	2	2	no	—
90	C₈×C₁₂₇	—	2	2	no	—
91	C₈×C₁₂₇	—	2	2	no	—
92	C₈×C₁₂₇	—	2	2	no	—
93	C₈×C₁₂₇	—	4	2	no	—
94	C₈×C₁₂₇	—	8	2	no	—
95	C₂²×C₁₂₇	#389	1	1	yes	characteristic
96	C₂²×C₁₂₇	#435	1	2	yes	—
97	C₂²×C₁₂₇	C₄×D₈	1	1	yes	characteristic
98	C₂²×C₁₂₇	C₄×Q₁₆	1	1	yes	characteristic
99	C₂²×C₁₂₇	#444	1	2	yes	—
100	C₄×C₁₂₇	—	2	1	no	—
101	C₄×C₁₂₇	—	2	1	no	—
102	C₄×C₁₂₇	—	4	4	no	—
103	C₄×C₁₂₇	—	8	2	no	—
104	C₄×C₁₂₇	#423	1	1	yes	characteristic
105	C₄×C₁₂₇	#424	1	1	yes	characteristic
106	C₄×C₁₂₇	#446	1	2	yes	—
107	C₄×C₁₂₇	#466	1	1	yes	characteristic
108	C₄×C₁₂₇	#467	1	1	yes	characteristic
109	C₄×C₁₂₇	#499	1	1	yes	characteristic
110	C₄×C₁₂₇	#500	1	1	yes	characteristic
111	#9598	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂×C₁₂₇	#1581	1	2	yes	—
113	C₂×C₁₂₇	#1584	1	2	yes	—
114	C₂×C₁₂₇	#1940	1	1	yes	characteristic
115	C₂×C₁₂₇	#1942	1	1	yes	characteristic
116	C₂×C₁₂₇	#1946	1	1	yes	characteristic
117	#1939	C₂×C₁₂₇	1	2	yes	—
118	C₄×#107	C₂×C₁₂₇	1	1	yes	characteristic
119	C₂×#335	C₂×C₁₂₇	1	1	yes	characteristic
120	C₂×#322	C₂×C₁₂₇	1	1	yes	characteristic
121	#1279	C₂×C₁₂₇	1	2	yes	—
122	C₁₂₇	#9598	1	1	yes	characteristic, direct product factor, Sylow
123	C₂×#107	C₄×C₁₂₇	1	2	yes	—
124	C₂×C₄⋊C₈	—	2	2	no	—
125	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
126	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
127	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
128	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
129	C₄×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
130	#340	C₄×C₁₂₇	1	1	yes	characteristic
131	#340	C₄×C₁₂₇	1	1	yes	characteristic
132	#336	—	2	2	no	—
133	#335	—	2	2	no	—
134	#322	—	2	2	no	—
135	C₂²×C₈	—	2	1	no	—
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
138	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
139	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
140	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
141	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
142	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
143	C₂×C₄²	—	2	2	no	—
144	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
145	#107	—	2	4	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	—	4	4	no	—
149	C₄×C₈	—	2	1	no	—
150	C₄×C₈	—	2	1	no	—
151	C₄×C₈	—	2	1	no	—
152	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
153	C₄×C₈	Q₈×C₁₂₇	1	2	yes	—
154	C₄×C₈	Q₈×C₁₂₇	1	2	yes	—
155	#95	—	2	2	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
159	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
160	C₂²×C₄	#41×C₁₂₇	1	1	yes	characteristic
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	—	4	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	C₄⋊C₄×C₁₂₇	1	2	yes	—
173	C₂×C₈	C₄⋊C₄×C₁₂₇	1	2	yes	—
174	C₂×C₈	D₈×C₁₂₇	1	2	yes	—
175	C₂×C₈	Q₁₆×C₁₂₇	1	2	yes	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	M₁₆×C₁₂₇	1	4	yes	—
180	C₄²	—	4	2	no	—
181	C₄²	—	4	2	no	—
182	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
183	C₄²	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
184	C₄²	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
185	C₄²	#41×C₁₂₇	1	1	yes	characteristic
186	C₂³	C₄×D₄×C₁₂₇	1	1	yes	characteristic
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
195	C₂×C₄	C₄×Q₈×C₁₂₇	1	1	yes	characteristic
196	C₂×C₄	#121×C₁₂₇	1	1	yes	characteristic
197	C₂×C₄	C₂²⋊Q₈×C₁₂₇	1	1	yes	characteristic
198	C₂×C₄	#125×C₁₂₇	1	1	yes	characteristic
199	C₂×C₄	C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
200	C₂×C₄	C₂×M₁₆×C₁₂₇	1	2	yes	—
201	C₂×C₄	#131×C₁₂₇	1	2	yes	—
202	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
203	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
204	C₂×C₄	#135×C₁₂₇	1	1	yes	characteristic
205	C₂×C₄	#135×C₁₂₇	1	1	yes	characteristic
206	C₈	—	2	2	no	—
207	C₈	—	2	2	no	—
208	C₈	—	2	2	no	—
209	C₈	—	2	2	no	—
210	C₈	—	4	2	no	—
211	C₈	—	8	2	no	—
212	C₂²	#389×C₁₂₇	1	1	yes	characteristic
213	C₂²	#435×C₁₂₇	1	2	yes	—
214	C₂²	C₄×D₈×C₁₂₇	1	1	yes	characteristic
215	C₂²	C₄×Q₁₆×C₁₂₇	1	1	yes	characteristic
216	C₂²	#444×C₁₂₇	1	2	yes	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	4	4	no	—
220	C₄	—	8	2	no	—
221	C₄	#423×C₁₂₇	1	1	yes	characteristic
222	C₄	#424×C₁₂₇	1	1	yes	characteristic
223	C₄	#446×C₁₂₇	1	2	yes	—
224	C₄	#466×C₁₂₇	1	1	yes	characteristic
225	C₄	#467×C₁₂₇	1	1	yes	characteristic
226	C₄	#499×C₁₂₇	1	1	yes	characteristic
227	C₄	#500×C₁₂₇	1	1	yes	characteristic
228	C₂	C₁₂₇×#1581	1	2	yes	—
229	C₂	C₁₂₇×#1584	1	2	yes	—
230	C₂	C₁₂₇×#1940	1	1	yes	characteristic
231	C₂	C₁₂₇×#1942	1	1	yes	characteristic
232	C₂	C₁₂₇×#1946	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
111. #9598	C₁₂₇	direct (N × H)
122. C₁₂₇	#9598	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×#107×C₁₂₇	2
C₂×#335×C₁₂₇	2
C₂×#322×C₁₂₇	2
C₁₂₇×#1279	2
#9598	127

Subgroups — C127×#9598 #44573271

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#9598 #44573271