Subgroups — C₂×C₄×SD₁₆×C₁₂₇ #30635889

Order 16256.

552 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×SD₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₄×SD₁₆×C₁₂₇	C₂	1	8	yes	maximal, direct product factor
5	C₂×#106×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#103×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#102×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂²×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₂₇	—	2	1	no	—
12	C₂×SD₁₆×C₁₂₇	C₄	1	8	yes	direct product factor
13	C₂×SD₁₆×C₁₂₇	C₂²	1	4	yes	—
14	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₄×Q₈×C₁₂₇	—	2	2	no	—
17	C₄×Q₈×C₁₂₇	C₂²	1	4	yes	complemented
18	C₄×D₄×C₁₂₇	—	2	2	no	—
19	C₄×D₄×C₁₂₇	C₂²	1	4	yes	—
20	C₂×C₄⋊C₄×C₁₂₇	—	2	1	no	—
21	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₄×C₁₂₇	—	2	1	no	—
24	C₂×C₄²×C₁₂₇	—	2	1	no	—
25	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
26	#106×C₁₂₇	C₂²	1	4	yes	complemented
27	#103×C₁₂₇	C₂²	1	4	yes	complemented
28	#102×C₁₂₇	C₂²	1	4	yes	—
29	C₄×C₈×C₁₂₇	C₂²	1	4	yes	complemented
30	C₂⁴×C₁₂₇	—	2	1	no	—
31	C₂×Q₈×C₁₂₇	—	2	1	no	—
32	C₂×Q₈×C₁₂₇	—	2	2	no	—
33	C₂×Q₈×C₁₂₇	C₂×C₄	1	4	yes	complemented
34	C₂×Q₈×C₁₂₇	C₂³	1	2	yes	—
35	C₂×D₄×C₁₂₇	—	2	1	no	—
36	C₂×D₄×C₁₂₇	—	2	2	no	—
37	C₂×D₄×C₁₂₇	C₂×C₄	1	4	yes	—
38	C₂×D₄×C₁₂₇	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₂₇	—	2	1	no	—
40	C₂²×C₄×C₁₂₇	—	2	1	no	—
41	C₂²×C₄×C₁₂₇	—	2	1	no	—
42	C₂²×C₄×C₁₂₇	—	2	4	no	—
43	C₂²×C₄×C₁₂₇	—	4	4	no	—
44	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
47	SD₁₆×C₁₂₇	C₂×C₄	1	16	yes	direct product factor
48	C₂×C₈×C₁₂₇	—	2	2	no	—
49	C₂×C₈×C₁₂₇	C₂×C₄	1	4	yes	complemented
50	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
51	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
52	C₄⋊C₄×C₁₂₇	—	2	4	no	—
53	C₄⋊C₄×C₁₂₇	—	2	1	no	—
54	C₄⋊C₄×C₁₂₇	—	2	1	no	—
55	C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
56	C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
57	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
58	C₄²×C₁₂₇	—	2	4	no	—
59	C₄²×C₁₂₇	D₄	1	2	yes	—
60	C₄²×C₁₂₇	C₂³	1	2	yes	—
61	C₂³×C₁₂₇	—	2	4	no	—
62	C₂³×C₁₂₇	—	2	2	no	—
63	C₂³×C₁₂₇	—	4	2	no	—
64	C₂³×C₁₂₇	—	4	2	no	—
65	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
66	Q₈×C₁₂₇	—	2	4	no	—
67	Q₈×C₁₂₇	—	2	2	no	—
68	Q₈×C₁₂₇	C₂²×C₄	1	4	yes	complemented
69	D₄×C₁₂₇	—	2	4	no	—
70	D₄×C₁₂₇	—	2	2	no	—
71	D₄×C₁₂₇	C₂²×C₄	1	4	yes	—
72	C₂×C₄×C₁₂₇	—	2	4	no	—
73	C₂×C₄×C₁₂₇	—	2	2	no	—
74	C₂×C₄×C₁₂₇	—	2	2	no	—
75	C₂×C₄×C₁₂₇	—	2	2	no	—
76	C₂×C₄×C₁₂₇	—	2	2	no	—
77	C₂×C₄×C₁₂₇	—	4	2	no	—
78	C₂×C₄×C₁₂₇	—	4	2	no	—
79	C₂×C₄×C₁₂₇	—	4	8	no	—
80	C₂×C₄×C₁₂₇	SD₁₆	1	4	yes	direct product factor
81	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
82	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
83	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₁₂₇	#41	1	2	yes	—
86	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
87	C₈×C₁₂₇	—	2	2	no	—
88	C₈×C₁₂₇	C₂²×C₄	1	4	yes	complemented
89	C₂²×C₁₂₇	—	2	4	no	—
90	C₂²×C₁₂₇	—	4	2	no	—
91	C₂²×C₁₂₇	—	4	2	no	—
92	C₂²×C₁₂₇	—	4	8	no	—
93	C₂²×C₁₂₇	C₄×D₄	1	2	yes	—
94	C₂²×C₁₂₇	C₂×SD₁₆	1	2	yes	—
95	C₂²×C₁₂₇	#135	1	2	yes	—
96	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
97	C₄×C₁₂₇	—	2	4	no	—
98	C₄×C₁₂₇	—	2	2	no	—
99	C₄×C₁₂₇	—	4	2	no	—
100	C₄×C₁₂₇	—	4	2	no	—
101	C₄×C₁₂₇	C₄×D₄	1	2	yes	—
102	C₄×C₁₂₇	C₂×SD₁₆	1	4	yes	direct product factor
103	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
104	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
105	C₂×C₁₂₇	—	4	4	no	—
106	C₂×C₁₂₇	C₄×SD₁₆	1	4	yes	direct product factor
107	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
108	C₂×C₁₂₇	C₂²×SD₁₆	1	1	yes	characteristic
109	C₂×C₁₂₇	C₂×#135	1	1	yes	characteristic
110	C₂×C₄×SD₁₆	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₁₂₇	C₂×C₄×SD₁₆	1	1	yes	characteristic, direct product factor, Sylow
112	C₂²×SD₁₆	C₂×C₁₂₇	1	1	yes	characteristic
113	C₂×C₄×Q₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
114	C₂×C₄×D₄	C₂×C₁₂₇	1	1	yes	characteristic
115	C₄×SD₁₆	C₂×C₁₂₇	1	8	yes	direct product factor
116	C₂×#106	C₂×C₁₂₇	1	1	yes	characteristic, complemented
117	C₂×#103	C₂×C₁₂₇	1	1	yes	characteristic, complemented
118	C₂×#102	C₂×C₁₂₇	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
120	C₂²×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
121	C₂²×D₄	C₂²×C₁₂₇	1	1	yes	characteristic
122	C₂³×C₄	—	2	1	no	—
123	C₂×SD₁₆	C₄×C₁₂₇	1	8	yes	direct product factor
124	C₂×SD₁₆	C₂²×C₁₂₇	1	4	yes	—
125	C₂²×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
126	C₂²×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
127	C₄×Q₈	—	2	2	no	—
128	C₄×Q₈	C₂²×C₁₂₇	1	4	yes	complemented
129	C₄×D₄	—	2	2	no	—
130	C₄×D₄	C₂²×C₁₂₇	1	4	yes	—
131	C₂×C₄⋊C₄	—	2	1	no	—
132	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
133	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
134	C₂×C₂²⋊C₄	—	2	1	no	—
135	C₂×C₄²	—	2	1	no	—
136	C₂×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
137	#106	C₂²×C₁₂₇	1	4	yes	complemented
138	#103	C₂²×C₁₂₇	1	4	yes	complemented
139	#102	C₂²×C₁₂₇	1	4	yes	—
140	C₄×C₈	C₂²×C₁₂₇	1	4	yes	complemented
141	C₂⁴	—	2	1	no	—
142	C₂×Q₈	—	2	1	no	—
143	C₂×Q₈	—	2	2	no	—
144	C₂×Q₈	C₂×C₄×C₁₂₇	1	4	yes	complemented
145	C₂×Q₈	C₂³×C₁₂₇	1	2	yes	—
146	C₂×D₄	—	2	1	no	—
147	C₂×D₄	—	2	2	no	—
148	C₂×D₄	C₂×C₄×C₁₂₇	1	4	yes	—
149	C₂×D₄	C₂³×C₁₂₇	1	2	yes	—
150	C₂²×C₄	—	2	1	no	—
151	C₂²×C₄	—	2	1	no	—
152	C₂²×C₄	—	2	1	no	—
153	C₂²×C₄	—	2	4	no	—
154	C₂²×C₄	—	4	4	no	—
155	C₂²×C₄	D₄×C₁₂₇	1	1	yes	characteristic
156	C₂²×C₄	D₄×C₁₂₇	1	1	yes	characteristic
157	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
158	SD₁₆	C₂×C₄×C₁₂₇	1	16	yes	direct product factor
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	C₂×C₄×C₁₂₇	1	4	yes	complemented
161	C₂×C₈	C₂³×C₁₂₇	1	2	yes	—
162	C₂×C₈	C₂³×C₁₂₇	1	2	yes	—
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	C₂³×C₁₂₇	1	2	yes	—
167	C₄⋊C₄	C₂³×C₁₂₇	1	2	yes	—
168	C₂²⋊C₄	—	2	4	no	—
169	C₄²	—	2	4	no	—
170	C₄²	D₄×C₁₂₇	1	2	yes	—
171	C₄²	C₂³×C₁₂₇	1	2	yes	—
172	C₂³	—	2	4	no	—
173	C₂³	—	2	2	no	—
174	C₂³	—	4	2	no	—
175	C₂³	—	4	2	no	—
176	C₂³	C₂×D₄×C₁₂₇	1	1	yes	characteristic
177	Q₈	—	2	4	no	—
178	Q₈	—	2	2	no	—
179	Q₈	C₂²×C₄×C₁₂₇	1	4	yes	complemented
180	D₄	—	2	4	no	—
181	D₄	—	2	2	no	—
182	D₄	C₂²×C₄×C₁₂₇	1	4	yes	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	8	no	—
191	C₂×C₄	SD₁₆×C₁₂₇	1	4	yes	direct product factor
192	C₂×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
193	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
194	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
195	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
196	C₂×C₄	#41×C₁₂₇	1	2	yes	—
197	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	C₂²×C₄×C₁₂₇	1	4	yes	complemented
200	C₂²	—	2	4	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	4	2	no	—
203	C₂²	—	4	8	no	—
204	C₂²	C₄×D₄×C₁₂₇	1	2	yes	—
205	C₂²	C₂×SD₁₆×C₁₂₇	1	2	yes	—
206	C₂²	#135×C₁₂₇	1	2	yes	—
207	C₂²	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
208	C₄	—	2	4	no	—
209	C₄	—	2	2	no	—
210	C₄	—	4	2	no	—
211	C₄	—	4	2	no	—
212	C₄	C₄×D₄×C₁₂₇	1	2	yes	—
213	C₄	C₂×SD₁₆×C₁₂₇	1	4	yes	direct product factor
214	C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
215	C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
216	C₂	—	4	4	no	—
217	C₂	C₄×SD₁₆×C₁₂₇	1	4	yes	direct product factor
218	C₂	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
219	C₂	C₂²×SD₁₆×C₁₂₇	1	1	yes	characteristic
220	C₂	C₂×#135×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₂₇	D₄
commutator	C₄	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₂₇
socle	C₂³×C₁₂₇	C₂×D₄
prime radical	C₂²×D₄×C₁₂₇	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Q₈×C₁₂₇	C₂	semidirect
4. C₄×SD₁₆×C₁₂₇	C₂	direct (N × H)
5. C₂×#106×C₁₂₇	C₂	semidirect
6. C₂×#103×C₁₂₇	C₂	semidirect
8. C₂×C₄×C₈×C₁₂₇	C₂	semidirect
12. C₂×SD₁₆×C₁₂₇	C₄	direct (N × H)
17. C₄×Q₈×C₁₂₇	C₂²	semidirect
26. #106×C₁₂₇	C₂²	semidirect
27. #103×C₁₂₇	C₂²	semidirect
29. C₄×C₈×C₁₂₇	C₂²	semidirect
33. C₂×Q₈×C₁₂₇	C₂×C₄	semidirect
47. SD₁₆×C₁₂₇	C₂×C₄	direct (N × H)
49. C₂×C₈×C₁₂₇	C₂×C₄	semidirect
68. Q₈×C₁₂₇	C₂²×C₄	semidirect
80. C₂×C₄×C₁₂₇	SD₁₆	direct (N × H)
88. C₈×C₁₂₇	C₂²×C₄	semidirect
102. C₄×C₁₂₇	C₂×SD₁₆	direct (N × H)
106. C₂×C₁₂₇	C₄×SD₁₆	direct (N × H)
110. C₂×C₄×SD₁₆	C₁₂₇	direct (N × H)
111. C₁₂₇	C₂×C₄×SD₁₆	direct (N × H)
113. C₂×C₄×Q₈	C₂×C₁₂₇	semidirect
115. C₄×SD₁₆	C₂×C₁₂₇	direct (N × H)
116. C₂×#106	C₂×C₁₂₇	semidirect
117. C₂×#103	C₂×C₁₂₇	semidirect
119. C₂×C₄×C₈	C₂×C₁₂₇	semidirect
123. C₂×SD₁₆	C₄×C₁₂₇	direct (N × H)
128. C₄×Q₈	C₂²×C₁₂₇	semidirect
137. #106	C₂²×C₁₂₇	semidirect
138. #103	C₂²×C₁₂₇	semidirect
140. C₄×C₈	C₂²×C₁₂₇	semidirect
144. C₂×Q₈	C₂×C₄×C₁₂₇	semidirect
158. SD₁₆	C₂×C₄×C₁₂₇	direct (N × H)
160. C₂×C₈	C₂×C₄×C₁₂₇	semidirect
179. Q₈	C₂²×C₄×C₁₂₇	semidirect
191. C₂×C₄	SD₁₆×C₁₂₇	direct (N × H)
199. C₈	C₂²×C₄×C₁₂₇	semidirect
213. C₄	C₂×SD₁₆×C₁₂₇	direct (N × H)
217. C₂	C₄×SD₁₆×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
C₂²×SD₁₆×C₁₂₇	2
C₂×C₄×Q₈×C₁₂₇	2
C₂×C₄×D₄×C₁₂₇	2
C₄×SD₁₆×C₁₂₇	2
C₂×#106×C₁₂₇	2
C₂×#103×C₁₂₇	2
C₂×#102×C₁₂₇	2
C₂×C₄×C₈×C₁₂₇	2
C₂×C₄×SD₁₆	127

Subgroups — C2×C4×SD16×C127 #30635889

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×SD₁₆×C₁₂₇ #30635889