Subgroups — C₂×C₁₂₇×Q₈⋊C₁₆ #44575278

Order 32512.

382 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×C₈×Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×Q₈⋊C₁₆	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₄×Q₈×C₁₂₇	C₄	1	1	yes	characteristic
6	C₂²×C₁₆×C₁₂₇	—	2	1	no	—
7	C₂²×C₁₆×C₁₂₇	—	2	1	no	—
8	C₈×Q₈×C₁₂₇	—	2	1	no	—
9	C₈×Q₈×C₁₂₇	—	2	1	no	—
10	C₈×Q₈×C₁₂₇	C₄	1	2	yes	—
11	C₈×Q₈×C₁₂₇	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₁₂₇	—	2	1	no	—
13	C₂×C₄⋊C₈×C₁₂₇	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₂₇	—	2	1	no	—
15	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆×C₁₂₇	—	2	1	no	—
17	C₄⋊C₁₆×C₁₂₇	C₂²	1	2	yes	—
18	C₄×C₁₆×C₁₂₇	—	2	1	no	—
19	C₄×C₁₆×C₁₂₇	C₂²	1	2	yes	—
20	C₂²×Q₈×C₁₂₇	C₈	1	1	yes	characteristic
21	C₂²×C₈×C₁₂₇	—	2	1	no	—
22	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
24	C₄×Q₈×C₁₂₇	—	2	1	no	—
25	C₄×Q₈×C₁₂₇	—	2	1	no	—
26	C₄×Q₈×C₁₂₇	C₈	1	2	yes	—
27	C₄×Q₈×C₁₂₇	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₁₂₇	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₂₇	C₈	1	1	yes	characteristic
30	C₂×C₄²×C₁₂₇	—	2	1	no	—
31	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₁₂₇	—	2	4	no	—
33	C₂×C₁₆×C₁₂₇	—	2	2	no	—
34	C₂×C₁₆×C₁₂₇	—	2	2	no	—
35	C₂×C₁₆×C₁₂₇	—	4	2	no	—
36	C₄⋊C₈×C₁₂₇	—	2	2	no	—
37	C₄⋊C₈×C₁₂₇	—	2	2	no	—
38	C₄⋊C₈×C₁₂₇	—	2	1	no	—
39	C₄⋊C₈×C₁₂₇	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₁₂₇	—	2	2	no	—
41	C₄×C₈×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	1	no	—
47	C₂×Q₈×C₁₂₇	—	2	2	no	—
48	C₂×Q₈×C₁₂₇	C₁₆	1	4	yes	complemented
49	C₂×Q₈×C₁₂₇	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₁₂₇	—	2	1	no	—
51	C₂²×C₄×C₁₂₇	—	2	1	no	—
52	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₂₇	M₁₆	1	1	yes	characteristic
55	C₂×C₈×C₁₂₇	—	2	2	no	—
56	C₂×C₈×C₁₂₇	—	2	2	no	—
57	C₂×C₈×C₁₂₇	—	4	2	no	—
58	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₁₂₇	SD₁₆	1	2	yes	—
61	C₂×C₈×C₁₂₇	Q₁₆	1	2	yes	—
62	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
64	C₄⋊C₄×C₁₂₇	—	2	2	no	—
65	C₄⋊C₄×C₁₂₇	—	2	1	no	—
66	C₄⋊C₄×C₁₂₇	—	2	2	no	—
67	C₄⋊C₄×C₁₂₇	C₂×C₈	1	2	yes	—
68	C₄²×C₁₂₇	—	2	2	no	—
69	C₄²×C₁₂₇	—	2	2	no	—
70	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₁₂₇	C₂×C₈	1	1	yes	characteristic
72	C₄²×C₁₂₇	M₁₆	1	1	yes	characteristic
73	C₄²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
74	C₁₆×C₁₂₇	—	2	4	no	—
75	C₁₆×C₁₂₇	—	4	2	no	—
76	C₂³×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
77	Q₈×C₁₂₇	—	2	4	no	—
78	Q₈×C₁₂₇	—	2	2	no	—
79	Q₈×C₁₂₇	C₂×C₁₆	1	4	yes	complemented
80	C₂×C₄×C₁₂₇	—	2	4	no	—
81	C₂×C₄×C₁₂₇	—	2	2	no	—
82	C₂×C₄×C₁₂₇	—	2	2	no	—
83	C₂×C₄×C₁₂₇	—	2	2	no	—
84	C₂×C₄×C₁₂₇	—	4	2	no	—
85	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
87	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
88	C₂×C₄×C₁₂₇	#103	1	2	yes	—
89	C₂×C₄×C₁₂₇	#104	1	2	yes	—
90	C₂×C₄×C₁₂₇	C₂×C₁₆	1	2	yes	—
91	C₂×C₄×C₁₂₇	M₃₂	1	2	yes	—
92	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
95	C₈×C₁₂₇	—	2	1	no	—
96	C₈×C₁₂₇	—	2	1	no	—
97	C₈×C₁₂₇	—	4	2	no	—
98	C₈×C₁₂₇	#103	1	2	yes	—
99	C₈×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
100	C₈×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
101	C₂²×C₁₂₇	Q₈⋊C₈	1	2	yes	—
102	C₂²×C₁₂₇	C₂²⋊C₁₆	1	2	yes	—
103	C₂²×C₁₂₇	#350	1	2	yes	—
104	C₂²×C₁₂₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
105	C₄×C₁₂₇	—	2	4	no	—
106	C₄×C₁₂₇	—	2	1	no	—
107	C₄×C₁₂₇	—	2	1	no	—
108	C₄×C₁₂₇	—	4	2	no	—
109	C₄×C₁₂₇	Q₈⋊C₈	1	2	yes	—
110	C₄×C₁₂₇	C₂²⋊C₁₆	1	2	yes	—
111	C₄×C₁₂₇	C₂×#103	1	1	yes	characteristic
112	C₄×C₁₂₇	C₂×#104	1	1	yes	characteristic
113	C₄×C₁₂₇	C₂²×C₁₆	1	1	yes	characteristic
114	C₄×C₁₂₇	C₂×M₃₂	1	1	yes	characteristic
115	C₂×Q₈⋊C₁₆	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂×C₁₂₇	Q₈⋊C₁₆	1	4	yes	direct product factor
117	C₂×C₁₂₇	C₂×Q₈⋊C₈	1	1	yes	characteristic
118	C₂×C₁₂₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
119	C₂×C₁₂₇	C₂×#350	1	1	yes	characteristic
120	C₂×C₈×Q₈	C₂×C₁₂₇	1	1	yes	characteristic
121	C₂×C₄⋊C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
122	C₂×C₄×C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
123	Q₈⋊C₁₆	C₂×C₁₂₇	1	4	yes	direct product factor
124	C₁₂₇	C₂×Q₈⋊C₁₆	1	1	yes	characteristic, direct product factor, Sylow
125	C₂×C₄×Q₈	C₄×C₁₂₇	1	1	yes	characteristic
126	C₂²×C₁₆	—	2	1	no	—
127	C₂²×C₁₆	—	2	1	no	—
128	C₈×Q₈	—	2	1	no	—
129	C₈×Q₈	—	2	1	no	—
130	C₈×Q₈	C₄×C₁₂₇	1	2	yes	—
131	C₈×Q₈	C₂²×C₁₂₇	1	2	yes	—
132	C₂×C₄⋊C₈	—	2	1	no	—
133	C₂×C₄⋊C₈	C₄×C₁₂₇	1	1	yes	characteristic
134	C₂×C₄×C₈	—	2	1	no	—
135	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
136	C₄⋊C₁₆	—	2	1	no	—
137	C₄⋊C₁₆	C₂²×C₁₂₇	1	2	yes	—
138	C₄×C₁₆	—	2	1	no	—
139	C₄×C₁₆	C₂²×C₁₂₇	1	2	yes	—
140	C₂²×Q₈	C₈×C₁₂₇	1	1	yes	characteristic
141	C₂²×C₈	—	2	1	no	—
142	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
143	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
144	C₄×Q₈	—	2	1	no	—
145	C₄×Q₈	—	2	1	no	—
146	C₄×Q₈	C₈×C₁₂₇	1	2	yes	—
147	C₄×Q₈	C₂×C₄×C₁₂₇	1	2	yes	—
148	C₂×C₄⋊C₄	—	2	1	no	—
149	C₂×C₄⋊C₄	C₈×C₁₂₇	1	1	yes	characteristic
150	C₂×C₄²	—	2	1	no	—
151	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
152	C₂×C₁₆	—	2	4	no	—
153	C₂×C₁₆	—	2	2	no	—
154	C₂×C₁₆	—	2	2	no	—
155	C₂×C₁₆	—	4	2	no	—
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	2	1	no	—
159	C₄⋊C₈	C₂×C₄×C₁₂₇	1	2	yes	—
160	C₄×C₈	—	2	2	no	—
161	C₄×C₈	—	2	2	no	—
162	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
163	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
164	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
165	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
166	C₂×Q₈	—	2	1	no	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	C₁₆×C₁₂₇	1	4	yes	complemented
169	C₂×Q₈	C₂×C₈×C₁₂₇	1	2	yes	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₈×C₁₂₇	1	1	yes	characteristic
174	C₂²×C₄	M₁₆×C₁₂₇	1	1	yes	characteristic
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
179	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
180	C₂×C₈	SD₁₆×C₁₂₇	1	2	yes	—
181	C₂×C₈	Q₁₆×C₁₂₇	1	2	yes	—
182	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
183	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	C₂×C₈×C₁₂₇	1	2	yes	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
191	C₄²	C₂×C₈×C₁₂₇	1	1	yes	characteristic
192	C₄²	M₁₆×C₁₂₇	1	1	yes	characteristic
193	C₄²	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
194	C₁₆	—	2	4	no	—
195	C₁₆	—	4	2	no	—
196	C₂³	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
197	Q₈	—	2	2	no	—
198	Q₈	—	2	4	no	—
199	Q₈	C₂×C₁₆×C₁₂₇	1	4	yes	complemented
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
206	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
207	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
208	C₂×C₄	#103×C₁₂₇	1	2	yes	—
209	C₂×C₄	#104×C₁₂₇	1	2	yes	—
210	C₂×C₄	C₂×C₁₆×C₁₂₇	1	2	yes	—
211	C₂×C₄	M₃₂×C₁₂₇	1	2	yes	—
212	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
213	C₂×C₄	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
214	C₂×C₄	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	1	no	—
217	C₈	—	4	2	no	—
218	C₈	#103×C₁₂₇	1	2	yes	—
219	C₈	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
220	C₈	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
221	C₂²	Q₈⋊C₈×C₁₂₇	1	2	yes	—
222	C₂²	C₂²⋊C₁₆×C₁₂₇	1	2	yes	—
223	C₂²	#350×C₁₂₇	1	2	yes	—
224	C₂²	C₂×C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
225	C₄	—	2	4	no	—
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	4	2	no	—
229	C₄	Q₈⋊C₈×C₁₂₇	1	2	yes	—
230	C₄	C₂²⋊C₁₆×C₁₂₇	1	2	yes	—
231	C₄	C₂×#103×C₁₂₇	1	1	yes	characteristic
232	C₄	C₂×#104×C₁₂₇	1	1	yes	characteristic
233	C₄	C₂²×C₁₆×C₁₂₇	1	1	yes	characteristic
234	C₄	C₂×M₃₂×C₁₂₇	1	1	yes	characteristic
235	C₂	C₁₂₇×Q₈⋊C₁₆	1	4	yes	direct product factor
236	C₂	C₂×Q₈⋊C₈×C₁₂₇	1	1	yes	characteristic
237	C₂	C₂×C₂²⋊C₁₆×C₁₂₇	1	1	yes	characteristic
238	C₂	C₂×#350×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈×C₁₂₇	D₄
commutator	C₄	C₂²×C₁₆×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
socle, prime radical	C₂³×C₁₂₇	C₂²⋊C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₂₇×Q₈⋊C₁₆	C₂	direct (N × H)
48. C₂×Q₈×C₁₂₇	C₁₆	semidirect
79. Q₈×C₁₂₇	C₂×C₁₆	semidirect
115. C₂×Q₈⋊C₁₆	C₁₂₇	direct (N × H)
116. C₂×C₁₂₇	Q₈⋊C₁₆	direct (N × H)
123. Q₈⋊C₁₆	C₂×C₁₂₇	direct (N × H)
124. C₁₂₇	C₂×Q₈⋊C₁₆	direct (N × H)
168. C₂×Q₈	C₁₆×C₁₂₇	semidirect
199. Q₈	C₂×C₁₆×C₁₂₇	semidirect
235. C₂	C₁₂₇×Q₈⋊C₁₆	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×C₈×Q₈×C₁₂₇	2
C₂×C₄⋊C₁₆×C₁₂₇	2
C₂×C₄×C₁₆×C₁₂₇	2
C₁₂₇×Q₈⋊C₁₆	2
C₂×Q₈⋊C₁₆	127

Subgroups — C2×C127×Q8⋊C16 #44575278

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₂₇×Q₈⋊C₁₆ #44575278