Subgroups — C₁₂₇×#11664 #44575337

Order 32512.

362 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#2163	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1566	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1334	C₂	1	4	yes	maximal, complemented
5	C₂²×Q₁₆×C₁₂₇	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₁₂₇	—	2	1	no	—
7	C₄×Q₁₆×C₁₂₇	—	2	1	no	—
8	C₄×Q₁₆×C₁₂₇	—	2	1	no	—
9	C₄×Q₁₆×C₁₂₇	C₄	1	2	yes	—
10	C₄×Q₁₆×C₁₂₇	C₂²	1	2	yes	—
11	C₂×#107×C₁₂₇	C₄	1	1	yes	characteristic
12	#423×C₁₂₇	—	2	1	no	—
13	C₂×#103×C₁₂₇	—	2	1	no	—
14	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₁₂₇	—	2	1	no	—
16	C₂²⋊C₁₆×C₁₂₇	—	2	1	no	—
17	#346×C₁₂₇	C₂²	1	2	yes	—
18	#334×C₁₂₇	—	2	1	no	—
19	#334×C₁₂₇	C₂²	1	2	yes	—
20	C₂²×Q₈×C₁₂₇	—	2	1	no	—
21	C₂×Q₁₆×C₁₂₇	—	2	4	no	—
22	C₂×Q₁₆×C₁₂₇	—	2	1	no	—
23	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
24	C₂×M₁₆×C₁₂₇	—	4	1	no	—
25	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
27	C₄×Q₈×C₁₂₇	—	2	2	no	—
28	C₄×Q₈×C₁₂₇	—	2	2	no	—
29	C₄×Q₈×C₁₂₇	—	4	1	no	—
30	C₄×Q₈×C₁₂₇	—	4	1	no	—
31	C₂×C₄⋊C₄×C₁₂₇	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₂₇	—	4	1	no	—
33	C₂×C₄²×C₁₂₇	—	4	1	no	—
34	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₁₂₇	—	2	2	no	—
36	#107×C₁₂₇	—	2	1	no	—
37	#107×C₁₂₇	C₂×C₄	1	2	yes	—
38	C₄⋊C₈×C₁₂₇	—	2	2	no	—
39	C₄⋊C₈×C₁₂₇	—	4	1	no	—
40	#103×C₁₂₇	—	2	2	no	—
41	#103×C₁₂₇	—	2	2	no	—
42	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
45	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
46	C₂×Q₈×C₁₂₇	—	2	4	no	—
47	C₂×Q₈×C₁₂₇	—	2	2	no	—
48	C₂×Q₈×C₁₂₇	—	4	2	no	—
49	C₂×Q₈×C₁₂₇	—	4	1	no	—
50	C₂²×C₄×C₁₂₇	—	4	1	no	—
51	C₂²×C₄×C₁₂₇	—	4	1	no	—
52	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
55	Q₁₆×C₁₂₇	—	2	4	no	—
56	Q₁₆×C₁₂₇	—	2	2	no	—
57	Q₁₆×C₁₂₇	M₁₆	1	4	yes	complemented
58	M₁₆×C₁₂₇	—	8	2	no	—
59	C₂×C₈×C₁₂₇	—	2	2	no	—
60	C₂×C₈×C₁₂₇	—	4	1	no	—
61	C₂×C₈×C₁₂₇	—	4	2	no	—
62	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
66	C₄⋊C₄×C₁₂₇	—	2	2	no	—
67	C₄⋊C₄×C₁₂₇	—	4	1	no	—
68	C₄⋊C₄×C₁₂₇	—	4	2	no	—
69	C₄⋊C₄×C₁₂₇	—	4	2	no	—
70	C₄²×C₁₂₇	—	4	2	no	—
71	C₄²×C₁₂₇	—	4	2	no	—
72	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₁₂₇	D₈	1	1	yes	characteristic
74	C₄²×C₁₂₇	SD₁₆	1	1	yes	characteristic
75	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
76	C₁₆×C₁₂₇	—	4	2	no	—
77	C₂³×C₁₂₇	#102	1	1	yes	characteristic
78	Q₈×C₁₂₇	—	2	4	no	—
79	Q₈×C₁₂₇	—	4	4	no	—
80	Q₈×C₁₂₇	—	4	2	no	—
81	C₂×C₄×C₁₂₇	—	2	2	no	—
82	C₂×C₄×C₁₂₇	—	2	2	no	—
83	C₂×C₄×C₁₂₇	—	2	1	no	—
84	C₂×C₄×C₁₂₇	—	2	1	no	—
85	C₂×C₄×C₁₂₇	—	4	4	no	—
86	C₂×C₄×C₁₂₇	—	4	2	no	—
87	C₂×C₄×C₁₂₇	—	4	2	no	—
88	C₂×C₄×C₁₂₇	—	8	2	no	—
89	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
90	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
91	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
92	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
95	C₈×C₁₂₇	—	2	1	no	—
96	C₈×C₁₂₇	—	2	1	no	—
97	C₈×C₁₂₇	—	2	1	no	—
98	C₈×C₁₂₇	—	8	2	no	—
99	C₈×C₁₂₇	C₂×M₁₆	1	2	yes	—
100	C₂²×C₁₂₇	—	2	1	no	—
101	C₂²×C₁₂₇	—	2	1	no	—
102	C₂²×C₁₂₇	#360	1	2	yes	—
103	C₂²×C₁₂₇	C₂×#102	1	1	yes	characteristic
104	C₄×C₁₂₇	—	2	1	no	—
105	C₄×C₁₂₇	—	2	1	no	—
106	C₄×C₁₂₇	—	2	1	no	—
107	C₄×C₁₂₇	—	2	1	no	—
108	C₄×C₁₂₇	—	4	4	no	—
109	C₄×C₁₂₇	—	8	2	no	—
110	C₄×C₁₂₇	#362	1	2	yes	—
111	C₄×C₁₂₇	#407	1	1	yes	characteristic
112	C₄×C₁₂₇	#421	1	1	yes	characteristic
113	#11664	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂×C₁₂₇	—	2	2	no	—
115	C₂×C₁₂₇	#1486	1	1	yes	characteristic
116	C₂×C₁₂₇	C₂×#360	1	1	yes	characteristic
117	C₂×C₁₂₇	C₂×#362	1	1	yes	characteristic
118	C₂×C₄×Q₁₆	C₂×C₁₂₇	1	1	yes	characteristic
119	#2163	C₂×C₁₂₇	1	1	yes	characteristic
120	#1566	C₂×C₁₂₇	1	1	yes	characteristic
121	#1334	C₂×C₁₂₇	1	4	yes	complemented
122	C₁₂₇	#11664	1	1	yes	characteristic, direct product factor, Sylow
123	C₂²×Q₁₆	C₄×C₁₂₇	1	1	yes	characteristic
124	C₂×C₄×Q₈	—	2	1	no	—
125	C₄×Q₁₆	—	2	1	no	—
126	C₄×Q₁₆	—	2	1	no	—
127	C₄×Q₁₆	C₄×C₁₂₇	1	2	yes	—
128	C₄×Q₁₆	C₂²×C₁₂₇	1	2	yes	—
129	C₂×#107	C₄×C₁₂₇	1	1	yes	characteristic
130	#423	—	2	1	no	—
131	C₂×#103	—	2	1	no	—
132	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
133	C₄⋊C₁₆	—	2	1	no	—
134	C₂²⋊C₁₆	—	2	1	no	—
135	#346	C₂²×C₁₂₇	1	2	yes	—
136	#334	—	2	1	no	—
137	#334	C₂²×C₁₂₇	1	2	yes	—
138	C₂²×Q₈	—	2	1	no	—
139	C₂×Q₁₆	—	2	4	no	—
140	C₂×Q₁₆	—	2	1	no	—
141	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
142	C₂×M₁₆	—	4	1	no	—
143	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
144	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
145	C₄×Q₈	—	2	2	no	—
146	C₄×Q₈	—	2	2	no	—
147	C₄×Q₈	—	4	1	no	—
148	C₄×Q₈	—	4	1	no	—
149	C₂×C₄⋊C₄	—	2	1	no	—
150	C₂×C₄⋊C₄	—	4	1	no	—
151	C₂×C₄²	—	4	1	no	—
152	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
153	C₂×C₁₆	—	2	2	no	—
154	#107	—	2	1	no	—
155	#107	C₂×C₄×C₁₂₇	1	2	yes	—
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	4	1	no	—
158	#103	—	2	2	no	—
159	#103	—	2	2	no	—
160	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
161	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
162	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
163	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
164	C₂×Q₈	—	2	4	no	—
165	C₂×Q₈	—	2	2	no	—
166	C₂×Q₈	—	4	2	no	—
167	C₂×Q₈	—	4	1	no	—
168	C₂²×C₄	—	4	1	no	—
169	C₂²×C₄	—	4	1	no	—
170	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
171	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
172	C₂²×C₄	SD₁₆×C₁₂₇	1	1	yes	characteristic
173	Q₁₆	—	2	4	no	—
174	Q₁₆	—	2	2	no	—
175	Q₁₆	M₁₆×C₁₂₇	1	4	yes	complemented
176	M₁₆	—	8	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	4	1	no	—
179	C₂×C₈	—	4	2	no	—
180	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
181	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
182	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
183	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	4	1	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄²	—	4	2	no	—
189	C₄²	—	4	2	no	—
190	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
191	C₄²	D₈×C₁₂₇	1	1	yes	characteristic
192	C₄²	SD₁₆×C₁₂₇	1	1	yes	characteristic
193	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
194	C₁₆	—	4	2	no	—
195	C₂³	#102×C₁₂₇	1	1	yes	characteristic
196	Q₈	—	2	4	no	—
197	Q₈	—	4	4	no	—
198	Q₈	—	4	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	8	2	no	—
207	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
208	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
209	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
210	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
211	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
212	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
213	C₈	—	2	1	no	—
214	C₈	—	2	1	no	—
215	C₈	—	2	1	no	—
216	C₈	—	8	2	no	—
217	C₈	C₂×M₁₆×C₁₂₇	1	2	yes	—
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	#360×C₁₂₇	1	2	yes	—
221	C₂²	C₂×#102×C₁₂₇	1	1	yes	characteristic
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	4	4	no	—
227	C₄	—	8	2	no	—
228	C₄	#362×C₁₂₇	1	2	yes	—
229	C₄	#407×C₁₂₇	1	1	yes	characteristic
230	C₄	#421×C₁₂₇	1	1	yes	characteristic
231	C₂	—	2	2	no	—
232	C₂	C₁₂₇×#1486	1	1	yes	characteristic
233	C₂	C₂×#360×C₁₂₇	1	1	yes	characteristic
234	C₂	C₂×#362×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
prime radical	C₂³×C₁₂₇	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
4. C₁₂₇×#1334	C₂	semidirect
57. Q₁₆×C₁₂₇	M₁₆	semidirect
113. #11664	C₁₂₇	direct (N × H)
121. #1334	C₂×C₁₂₇	semidirect
122. C₁₂₇	#11664	direct (N × H)
175. Q₁₆	M₁₆×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₁₆×C₁₂₇	2
C₁₂₇×#2163	2
C₁₂₇×#1566	2
C₁₂₇×#1334	2
#11664	127

Subgroups — C127×#11664 #44575337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#11664 #44575337