Subgroups — C₁₂₇×#20730 #44584403

Order 32512.

1166 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#519×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#2921	C₂	1	2	yes	maximal
3	C₁₂₇×#1822	C₂	1	2	yes	maximal
4	C₁₂₇×#1801	C₂	1	4	yes	maximal, complemented
5	C₁₂₇×#1784	C₂	1	2	yes	maximal
6	C₄×#101×C₁₂₇	C₂	1	4	yes	maximal, complemented
7	C₂³×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
8	#519×C₁₂₇	—	2	8	no	—
9	C₂×C₄×Q₈×C₁₂₇	C₄	1	4	yes	—
10	C₂×C₄×Q₈×C₁₂₇	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₁₂₇	C₂²	1	2	yes	—
12	C₂×#117×C₁₂₇	C₄	1	4	yes	—
13	C₂×#117×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×#117×C₁₂₇	C₂²	1	1	yes	characteristic
15	#433×C₁₂₇	—	2	8	no	—
16	#423×C₁₂₇	C₂²	1	4	yes	—
17	C₂×#101×C₁₂₇	C₂²	1	8	yes	—
18	#407×C₁₂₇	C₂²	1	4	yes	—
19	C₄×M₁₆×C₁₂₇	C₂²	1	4	yes	—
20	#343×C₁₂₇	C₂²	1	8	yes	—
21	C₂²×Q₈×C₁₂₇	—	2	8	no	—
22	C₂²×Q₈×C₁₂₇	—	2	4	no	—
23	C₂²×Q₈×C₁₂₇	C₂×C₄	1	2	yes	—
24	C₂²×Q₈×C₁₂₇	C₂³	1	2	yes	—
25	C₂³×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
27	C₂×M₁₆×C₁₂₇	C₂³	1	8	yes	—
28	C₄×Q₈×C₁₂₇	—	2	16	no	—
29	C₄×Q₈×C₁₂₇	—	2	8	no	—
30	C₄×Q₈×C₁₂₇	—	2	8	no	—
31	#117×C₁₂₇	—	2	4	no	—
32	#117×C₁₂₇	—	2	16	no	—
33	#117×C₁₂₇	D₄	1	8	yes	—
34	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
36	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
38	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
39	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₁₂₇	—	2	8	no	—
46	#101×C₁₂₇	—	2	16	no	—
47	C₂²⋊C₈×C₁₂₇	—	2	8	no	—
48	#97×C₁₂₇	—	2	4	no	—
49	C₄×C₈×C₁₂₇	—	2	4	no	—
50	C₂⁴×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×Q₈×C₁₂₇	—	2	32	no	—
52	C₂×Q₈×C₁₂₇	—	2	4	no	—
53	C₂×Q₈×C₁₂₇	—	2	8	no	—
54	C₂×Q₈×C₁₂₇	—	2	4	no	—
55	C₂×Q₈×C₁₂₇	—	2	4	no	—
56	C₂×Q₈×C₁₂₇	#41	1	8	yes	—
57	C₂²×C₄×C₁₂₇	—	2	4	no	—
58	C₂²×C₄×C₁₂₇	—	2	8	no	—
59	C₂²×C₄×C₁₂₇	—	2	4	no	—
60	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
66	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
67	M₁₆×C₁₂₇	—	2	16	no	—
68	C₂×C₈×C₁₂₇	—	2	8	no	—
69	C₄⋊C₄×C₁₂₇	—	2	8	no	—
70	C₄⋊C₄×C₁₂₇	—	2	8	no	—
71	C₄⋊C₄×C₁₂₇	—	2	4	no	—
72	C₄⋊C₄×C₁₂₇	C₂²⋊C₄	1	8	yes	—
73	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
74	C₂²⋊C₄×C₁₂₇	C₂²⋊C₄	1	8	yes	—
75	C₄²×C₁₂₇	—	2	8	no	—
76	C₄²×C₁₂₇	—	2	8	no	—
77	C₄²×C₁₂₇	—	2	4	no	—
78	C₄²×C₁₂₇	C₂×D₄	1	4	yes	—
79	C₄²×C₁₂₇	C₂×D₄	1	4	yes	—
80	C₂³×C₁₂₇	—	2	2	no	—
81	C₂³×C₁₂₇	—	4	2	no	—
82	C₂³×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
83	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
84	Q₈×C₁₂₇	—	2	16	no	—
85	Q₈×C₁₂₇	—	2	16	no	—
86	C₂×C₄×C₁₂₇	—	2	16	no	—
87	C₂×C₄×C₁₂₇	—	2	4	no	—
88	C₂×C₄×C₁₂₇	—	2	2	no	—
89	C₂×C₄×C₁₂₇	—	2	4	no	—
90	C₂×C₄×C₁₂₇	—	2	8	no	—
91	C₂×C₄×C₁₂₇	—	2	4	no	—
92	C₂×C₄×C₁₂₇	—	4	2	no	—
93	C₂×C₄×C₁₂₇	—	4	2	no	—
94	C₂×C₄×C₁₂₇	—	4	4	no	—
95	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
96	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
97	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂×C₄×C₁₂₇	#117	1	8	yes	—
99	C₂×C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
100	C₂×C₄×C₁₂₇	C₂×#41	1	4	yes	—
101	C₈×C₁₂₇	—	4	8	no	—
102	C₂²×C₁₂₇	—	2	2	no	—
103	C₂²×C₁₂₇	—	2	4	no	—
104	C₂²×C₁₂₇	—	2	2	no	—
105	C₂²×C₁₂₇	—	4	4	no	—
106	C₂²×C₁₂₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
107	C₂²×C₁₂₇	C₂×#117	1	2	yes	—
108	C₄×C₁₂₇	—	2	4	no	—
109	C₄×C₁₂₇	—	2	4	no	—
110	C₄×C₁₂₇	—	2	2	no	—
111	C₄×C₁₂₇	—	2	8	no	—
112	C₄×C₁₂₇	—	4	4	no	—
113	C₄×C₁₂₇	#525	1	4	yes	—
114	#20730	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂×C₁₂₇	—	2	2	no	—
116	C₂×C₁₂₇	—	4	2	no	—
117	C₂×C₁₂₇	#2280	1	1	yes	characteristic
118	C₂×C₁₂₇	#2889	1	2	yes	—
119	C₂×#519	C₂×C₁₂₇	1	1	yes	characteristic
120	#2921	C₂×C₁₂₇	1	2	yes	—
121	#1822	C₂×C₁₂₇	1	2	yes	—
122	#1801	C₂×C₁₂₇	1	4	yes	complemented
123	#1784	C₂×C₁₂₇	1	2	yes	—
124	C₄×#101	C₂×C₁₂₇	1	4	yes	complemented
125	C₁₂₇	#20730	1	1	yes	characteristic, direct product factor, Sylow
126	C₂³×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
127	#519	—	2	8	no	—
128	C₂×C₄×Q₈	C₄×C₁₂₇	1	4	yes	—
129	C₂×C₄×Q₈	C₂²×C₁₂₇	1	2	yes	—
130	C₂×C₄×Q₈	C₂²×C₁₂₇	1	2	yes	—
131	C₂×#117	C₄×C₁₂₇	1	4	yes	—
132	C₂×#117	C₂²×C₁₂₇	1	1	yes	characteristic
133	C₂×#117	C₂²×C₁₂₇	1	1	yes	characteristic
134	#433	—	2	8	no	—
135	#423	C₂²×C₁₂₇	1	4	yes	—
136	C₂×#101	C₂²×C₁₂₇	1	8	yes	—
137	#407	C₂²×C₁₂₇	1	4	yes	—
138	C₄×M₁₆	C₂²×C₁₂₇	1	4	yes	—
139	#343	C₂²×C₁₂₇	1	8	yes	—
140	C₂²×Q₈	—	2	8	no	—
141	C₂²×Q₈	—	2	4	no	—
142	C₂²×Q₈	C₂×C₄×C₁₂₇	1	2	yes	—
143	C₂²×Q₈	C₂³×C₁₂₇	1	2	yes	—
144	C₂³×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
145	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
146	C₂×M₁₆	C₂³×C₁₂₇	1	8	yes	—
147	C₄×Q₈	—	2	16	no	—
148	C₄×Q₈	—	2	8	no	—
149	C₄×Q₈	—	2	8	no	—
150	#117	—	2	4	no	—
151	#117	—	2	16	no	—
152	#117	D₄×C₁₂₇	1	8	yes	—
153	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
154	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
155	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
156	C₂×C₄⋊C₄	C₂³×C₁₂₇	1	2	yes	—
157	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
158	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
159	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
160	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
161	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
162	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
163	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
164	C₄⋊C₈	—	2	8	no	—
165	#101	—	2	16	no	—
166	C₂²⋊C₈	—	2	8	no	—
167	#97	—	2	4	no	—
168	C₄×C₈	—	2	4	no	—
169	C₂⁴	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
170	C₂×Q₈	—	2	32	no	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	—	2	8	no	—
173	C₂×Q₈	—	2	4	no	—
174	C₂×Q₈	—	2	4	no	—
175	C₂×Q₈	#41×C₁₂₇	1	8	yes	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	8	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
180	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
181	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
182	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
183	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
184	C₂²×C₄	C₂×D₄×C₁₂₇	1	4	yes	—
185	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
186	M₁₆	—	2	16	no	—
187	C₂×C₈	—	2	8	no	—
188	C₄⋊C₄	—	2	8	no	—
189	C₄⋊C₄	—	2	8	no	—
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	C₂²⋊C₄×C₁₂₇	1	8	yes	—
192	C₂²⋊C₄	—	2	4	no	—
193	C₂²⋊C₄	C₂²⋊C₄×C₁₂₇	1	8	yes	—
194	C₄²	—	2	8	no	—
195	C₄²	—	2	8	no	—
196	C₄²	—	2	4	no	—
197	C₄²	C₂×D₄×C₁₂₇	1	4	yes	—
198	C₄²	C₂×D₄×C₁₂₇	1	4	yes	—
199	C₂³	—	2	2	no	—
200	C₂³	—	4	2	no	—
201	C₂³	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
202	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
203	Q₈	—	2	16	no	—
204	Q₈	—	2	16	no	—
205	C₂×C₄	—	2	16	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	8	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
215	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
216	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
217	C₂×C₄	#117×C₁₂₇	1	8	yes	—
218	C₂×C₄	C₂²×D₄×C₁₂₇	1	2	yes	—
219	C₂×C₄	C₂×#41×C₁₂₇	1	4	yes	—
220	C₈	—	4	8	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	4	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	4	no	—
225	C₂²	C₂²×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
226	C₂²	C₂×#117×C₁₂₇	1	2	yes	—
227	C₄	—	2	4	no	—
228	C₄	—	2	4	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	8	no	—
231	C₄	—	4	4	no	—
232	C₄	#525×C₁₂₇	1	4	yes	—
233	C₂	—	2	2	no	—
234	C₂	—	4	2	no	—
235	C₂	C₁₂₇×#2280	1	1	yes	characteristic
236	C₂	C₁₂₇×#2889	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
prime radical	C₂⁴×C₁₂₇	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂×#117×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₂₇×#1801	C₂	semidirect
6. C₄×#101×C₁₂₇	C₂	semidirect
114. #20730	C₁₂₇	direct (N × H)
122. #1801	C₂×C₁₂₇	semidirect
124. C₄×#101	C₂×C₁₂₇	semidirect
125. C₁₂₇	#20730	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#519×C₁₂₇	2
C₁₂₇×#2921	2
C₁₂₇×#1822	2
C₁₂₇×#1801	2
C₁₂₇×#1784	2
C₄×#101×C₁₂₇	2
#20730	127

Subgroups — C127×#20730 #44584403

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#20730 #44584403