Subgroups — C₄×#430×C₁₂₇ #44584131

Order 32512.

1034 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#430×C₁₂₇	C₂	1	2	yes	maximal
2	C₂×C₄×M₁₆×C₁₂₇	C₂	1	2	yes	maximal
3	C₂×C₄×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1778	C₂	1	4	yes	maximal, complemented
5	C₄×#108×C₁₂₇	C₂	1	4	yes	maximal, complemented
6	C₁₂₇×#1738	C₂	1	2	yes	maximal
7	C₂²×M₁₆×C₁₂₇	C₂²	1	4	yes	—
8	C₂²×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
11	#430×C₁₂₇	C₄	1	16	yes	direct product factor
12	C₂×#108×C₁₂₇	C₂²	1	8	yes	—
13	C₄×M₁₆×C₁₂₇	—	2	2	no	—
14	C₄×M₁₆×C₁₂₇	—	2	2	no	—
15	C₄×M₁₆×C₁₂₇	C₄	1	8	yes	—
16	C₄×M₁₆×C₁₂₇	C₂²	1	4	yes	—
17	C₄×M₁₆×C₁₂₇	C₂²	1	4	yes	—
18	C₂×#97×C₁₂₇	—	2	2	no	—
19	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₁₂₇	—	2	2	no	—
21	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
22	#341×C₁₂₇	C₂²	1	8	yes	—
23	C₂³×C₄×C₁₂₇	Q₈	1	2	yes	—
24	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
25	C₂×M₁₆×C₁₂₇	—	2	4	no	—
26	C₂×M₁₆×C₁₂₇	—	2	8	no	—
27	C₂×M₁₆×C₁₂₇	—	2	4	no	—
28	C₂×M₁₆×C₁₂₇	C₂×C₄	1	16	yes	—
29	C₂×M₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
30	C₂×M₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
31	C₂×M₁₆×C₁₂₇	C₂×C₄	1	8	yes	—
32	C₂×M₁₆×C₁₂₇	C₂³	1	8	yes	—
33	C₂²×C₈×C₁₂₇	—	2	4	no	—
34	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
36	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₁₂₇	D₄	1	2	yes	—
38	C₂×C₄²×C₁₂₇	D₄	1	2	yes	—
39	C₂×C₄²×C₁₂₇	Q₈	1	2	yes	—
40	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
42	#108×C₁₂₇	C₂×C₄	1	32	yes	complemented
43	#97×C₁₂₇	—	2	4	no	—
44	#97×C₁₂₇	—	2	4	no	—
45	#97×C₁₂₇	C₂×C₄	1	8	yes	—
46	C₄×C₈×C₁₂₇	—	2	4	no	—
47	C₄×C₈×C₁₂₇	—	2	4	no	—
48	C₄×C₈×C₁₂₇	C₂×C₄	1	8	yes	—
49	C₂⁴×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₂₇	—	4	4	no	—
51	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
52	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
53	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
55	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
58	C₂²×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₂₇	#41	1	4	yes	—
60	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
61	M₁₆×C₁₂₇	—	2	8	no	—
62	M₁₆×C₁₂₇	—	2	16	no	—
63	M₁₆×C₁₂₇	—	2	8	no	—
64	M₁₆×C₁₂₇	C₄²	1	16	yes	—
65	M₁₆×C₁₂₇	C₂²×C₄	1	16	yes	—
66	C₂×C₈×C₁₂₇	—	2	16	no	—
67	C₂×C₈×C₁₂₇	—	2	8	no	—
68	C₂×C₈×C₁₂₇	—	2	4	no	—
69	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
70	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
71	C₂×C₈×C₁₂₇	C₂²×C₄	1	8	yes	—
72	C₄²×C₁₂₇	C₄⋊C₄	1	4	yes	—
73	C₄²×C₁₂₇	C₄⋊C₄	1	4	yes	—
74	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
75	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
76	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
77	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
79	C₂³×C₁₂₇	—	4	2	no	—
80	C₂³×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
81	C₂³×C₁₂₇	C₄×Q₈	1	4	yes	—
82	C₂³×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₁₂₇	—	2	8	no	—
84	C₂×C₄×C₁₂₇	—	2	4	no	—
85	C₂×C₄×C₁₂₇	—	4	4	no	—
86	C₂×C₄×C₁₂₇	—	4	2	no	—
87	C₂×C₄×C₁₂₇	C₂×C₄²	1	4	yes	—
88	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
89	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
90	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₁₂₇	#117	1	4	yes	—
93	C₂×C₄×C₁₂₇	C₄×D₄	1	8	yes	—
94	C₂×C₄×C₁₂₇	C₄×Q₈	1	4	yes	—
95	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₂₇	C₂³×C₄	1	2	yes	—
97	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₂₇	C₂×#41	1	2	yes	—
99	C₈×C₁₂₇	—	2	4	no	—
100	C₈×C₁₂₇	—	2	16	no	—
101	C₈×C₁₂₇	C₂×C₄²	1	8	yes	—
102	C₂²×C₁₂₇	—	2	4	no	—
103	C₂²×C₁₂₇	—	2	2	no	—
104	C₂²×C₁₂₇	—	4	4	no	—
105	C₂²×C₁₂₇	C₄×C₄⋊C₄	1	4	yes	—
106	C₂²×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₁₂₇	C₂×C₄×Q₈	1	2	yes	—
108	C₄×C₁₂₇	—	2	4	no	—
109	C₄×C₁₂₇	—	2	2	no	—
110	C₄×C₁₂₇	C₄×C₄⋊C₄	1	4	yes	—
111	C₄×C₁₂₇	#430	1	4	yes	direct product factor
112	C₄×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
113	C₄×C₁₂₇	C₂×#117	1	1	yes	characteristic
114	C₄×C₁₂₇	C₂×C₄×D₄	1	2	yes	—
115	C₄×#430	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂×C₁₂₇	—	2	2	no	—
117	C₂×C₁₂₇	—	2	4	no	—
118	C₂×C₁₂₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₁₂₇	C₂×#430	1	2	yes	—
120	C₂×#430	C₂×C₁₂₇	1	2	yes	—
121	C₂×C₄×M₁₆	C₂×C₁₂₇	1	2	yes	—
122	C₂×C₄×M₁₆	C₂×C₁₂₇	1	1	yes	characteristic
123	#1778	C₂×C₁₂₇	1	4	yes	complemented
124	C₄×#108	C₂×C₁₂₇	1	4	yes	complemented
125	#1738	C₂×C₁₂₇	1	2	yes	—
126	C₁₂₇	C₄×#430	1	1	yes	characteristic, direct product factor, Sylow
127	C₂²×M₁₆	C₂²×C₁₂₇	1	4	yes	—
128	C₂²×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
129	C₂²×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
130	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
131	#430	C₄×C₁₂₇	1	16	yes	direct product factor
132	C₂×#108	C₂²×C₁₂₇	1	8	yes	—
133	C₄×M₁₆	—	2	2	no	—
134	C₄×M₁₆	—	2	2	no	—
135	C₄×M₁₆	C₄×C₁₂₇	1	8	yes	—
136	C₄×M₁₆	C₂²×C₁₂₇	1	4	yes	—
137	C₄×M₁₆	C₂²×C₁₂₇	1	4	yes	—
138	C₂×#97	—	2	2	no	—
139	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
140	C₂×C₄×C₈	—	2	2	no	—
141	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
142	#341	C₂²×C₁₂₇	1	8	yes	—
143	C₂³×C₄	Q₈×C₁₂₇	1	2	yes	—
144	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
145	C₂×M₁₆	—	2	4	no	—
146	C₂×M₁₆	—	2	8	no	—
147	C₂×M₁₆	—	2	4	no	—
148	C₂×M₁₆	C₂×C₄×C₁₂₇	1	16	yes	—
149	C₂×M₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
150	C₂×M₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
151	C₂×M₁₆	C₂×C₄×C₁₂₇	1	8	yes	—
152	C₂×M₁₆	C₂³×C₁₂₇	1	8	yes	—
153	C₂²×C₈	—	2	4	no	—
154	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
155	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
156	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
157	C₂×C₄²	D₄×C₁₂₇	1	2	yes	—
158	C₂×C₄²	D₄×C₁₂₇	1	2	yes	—
159	C₂×C₄²	Q₈×C₁₂₇	1	2	yes	—
160	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
161	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
162	#108	C₂×C₄×C₁₂₇	1	32	yes	complemented
163	#97	—	2	4	no	—
164	#97	—	2	4	no	—
165	#97	C₂×C₄×C₁₂₇	1	8	yes	—
166	C₄×C₈	—	2	4	no	—
167	C₄×C₈	—	2	4	no	—
168	C₄×C₈	C₂×C₄×C₁₂₇	1	8	yes	—
169	C₂⁴	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
170	C₂²×C₄	—	4	4	no	—
171	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	4	yes	—
172	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	4	yes	—
173	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
174	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
175	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
176	C₂²×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
177	C₂²×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
178	C₂²×C₄	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
179	C₂²×C₄	#41×C₁₂₇	1	4	yes	—
180	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
181	M₁₆	—	2	8	no	—
182	M₁₆	—	2	16	no	—
183	M₁₆	—	2	8	no	—
184	M₁₆	C₄²×C₁₂₇	1	16	yes	—
185	M₁₆	C₂²×C₄×C₁₂₇	1	16	yes	—
186	C₂×C₈	—	2	16	no	—
187	C₂×C₈	—	2	8	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
190	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
191	C₂×C₈	C₂²×C₄×C₁₂₇	1	8	yes	—
192	C₄²	C₄⋊C₄×C₁₂₇	1	4	yes	—
193	C₄²	C₄⋊C₄×C₁₂₇	1	4	yes	—
194	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
195	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
196	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
197	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
198	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
199	C₂³	—	4	2	no	—
200	C₂³	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
201	C₂³	C₄×Q₈×C₁₂₇	1	4	yes	—
202	C₂³	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	4	4	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	C₂×C₄²×C₁₂₇	1	4	yes	—
208	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
209	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	4	yes	—
210	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
211	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
212	C₂×C₄	#117×C₁₂₇	1	4	yes	—
213	C₂×C₄	C₄×D₄×C₁₂₇	1	8	yes	—
214	C₂×C₄	C₄×Q₈×C₁₂₇	1	4	yes	—
215	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
216	C₂×C₄	C₂³×C₄×C₁₂₇	1	2	yes	—
217	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
218	C₂×C₄	C₂×#41×C₁₂₇	1	2	yes	—
219	C₈	—	2	4	no	—
220	C₈	—	2	16	no	—
221	C₈	C₂×C₄²×C₁₂₇	1	8	yes	—
222	C₂²	—	2	4	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	4	no	—
225	C₂²	C₄×C₄⋊C₄×C₁₂₇	1	4	yes	—
226	C₂²	C₂²×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
227	C₂²	C₂×C₄×Q₈×C₁₂₇	1	2	yes	—
228	C₄	—	2	4	no	—
229	C₄	—	2	2	no	—
230	C₄	C₄×C₄⋊C₄×C₁₂₇	1	4	yes	—
231	C₄	#430×C₁₂₇	1	4	yes	direct product factor
232	C₄	C₂²×C₄²×C₁₂₇	1	1	yes	characteristic
233	C₄	C₂×#117×C₁₂₇	1	1	yes	characteristic
234	C₄	C₂×C₄×D₄×C₁₂₇	1	2	yes	—
235	C₂	—	2	2	no	—
236	C₂	—	2	4	no	—
237	C₂	C₂×C₄×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
238	C₂	C₂×#430×C₁₂₇	1	2	yes	—

	subgroup	G/H
center	C₄²×C₁₂₇	C₂×D₄
commutator	C₄	C₂²×C₄²×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle	C₂²×C₁₂₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₁₂₇	C₂×Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₁₂₇ ◁ C₂²×C₄²×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₂₇×#1778	C₂	semidirect
5. C₄×#108×C₁₂₇	C₂	semidirect
11. #430×C₁₂₇	C₄	direct (N × H)
42. #108×C₁₂₇	C₂×C₄	semidirect
111. C₄×C₁₂₇	#430	direct (N × H)
115. C₄×#430	C₁₂₇	direct (N × H)
123. #1778	C₂×C₁₂₇	semidirect
124. C₄×#108	C₂×C₁₂₇	semidirect
126. C₁₂₇	C₄×#430	direct (N × H)
131. #430	C₄×C₁₂₇	direct (N × H)
162. #108	C₂×C₄×C₁₂₇	semidirect
231. C₄	#430×C₁₂₇	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×#430×C₁₂₇	2
C₂×C₄×M₁₆×C₁₂₇	2
C₁₂₇×#1778	2
C₄×#108×C₁₂₇	2
C₁₂₇×#1738	2
C₄×#430	127

Subgroups — C4×#430×C127 #44584131

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#430×C₁₂₇ #44584131