Subgroups — C₂×#22549647 #30638472

Order 16256.

388 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523232	C₂	1	1	yes	maximal, characteristic
2	C₂×#16861168	C₂	1	1	yes	maximal, characteristic
3	#22549647	C₂	1	4	yes	maximal, direct product factor
4	C₂²×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523232	—	2	1	no	—
6	C₂×#14523232	—	2	1	no	—
7	C₂×#14523232	—	2	1	no	—
8	C₂×#14523232	C₄	1	2	yes	complemented
9	C₂×#14523232	C₄	1	2	yes	complemented
10	C₂×#14523232	C₂²	1	2	yes	—
11	C₂²×C₁₂₇⋊C₈	—	2	1	no	—
12	#16861168	—	2	1	no	—
13	#16861168	—	2	1	no	—
14	#16861168	C₄	1	2	yes	—
15	#16861168	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₁₂₇	—	2	1	no	—
17	C₂×C₂²⋊Dic₁₂₇	C₄	1	2	yes	—
18	C₂×C₄×Dic₁₂₇	—	2	1	no	—
19	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₁₂₇	—	2	1	no	—
21	C₂×M₁₆×C₁₂₇	—	2	1	no	—
22	C₂×M₁₆×C₁₂₇	—	2	1	no	—
23	C₂×M₁₆×C₁₂₇	—	2	1	no	—
24	C₂×M₁₆×C₁₂₇	C₄	1	2	yes	complemented
25	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
26	C₂²×C₈×C₁₂₇	C₄	1	2	yes	complemented
27	C₂²×Dic₁₂₇	—	2	2	no	—
28	#14523232	—	2	2	no	—
29	#14523232	—	2	2	no	—
30	#14523232	—	2	2	no	—
31	#14523232	C₂×C₄	1	4	yes	complemented
32	C₂×C₁₂₇⋊C₈	—	2	2	no	—
33	C₂×C₁₂₇⋊C₈	—	2	2	no	—
34	C₂×C₁₂₇⋊C₈	—	2	2	no	—
35	C₄⋊Dic₁₂₇	—	2	2	no	—
36	C₄⋊Dic₁₂₇	—	2	2	no	—
37	C₂²⋊Dic₁₂₇	—	2	2	no	—
38	C₂²⋊Dic₁₂₇	C₂×C₄	1	4	yes	—
39	C₄×Dic₁₂₇	—	2	2	no	—
40	C₄×Dic₁₂₇	—	2	2	no	—
41	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₂₇	Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₂₇	Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
56	M₁₆×C₁₂₇	—	2	2	no	—
57	M₁₆×C₁₂₇	—	2	2	no	—
58	M₁₆×C₁₂₇	—	2	2	no	—
59	M₁₆×C₁₂₇	—	2	2	no	—
60	C₂×C₈×C₁₂₇	—	2	2	no	—
61	C₂×C₈×C₁₂₇	—	2	2	no	—
62	C₂×C₈×C₁₂₇	C₂×C₄	1	4	yes	complemented
63	C₂×Dic₁₂₇	—	2	4	no	—
64	C₂×Dic₁₂₇	—	4	4	no	—
65	C₁₂₇⋊C₈	—	2	4	no	—
66	C₂³×C₁₂₇	—	4	2	no	—
67	C₂³×C₁₂₇	C₄²	1	1	yes	characteristic
68	C₂³×C₁₂₇	C₄²	1	1	yes	characteristic
69	C₂³×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂³×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂³×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
72	C₂³×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
73	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₂₇	C₄²	1	1	yes	characteristic
75	C₂×C₄×C₁₂₇	C₄²	1	1	yes	characteristic
76	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
102	C₈×C₁₂₇	—	2	4	no	—
103	C₂²×C₁₂₇	—	2	2	no	—
104	C₂²×C₁₂₇	—	2	2	no	—
105	C₂²×C₁₂₇	—	2	2	no	—
106	C₂²×C₁₂₇	—	4	1	no	—
107	C₂²×C₁₂₇	—	4	1	no	—
108	C₂²×C₁₂₇	—	4	1	no	—
109	C₂²×C₁₂₇	—	4	1	no	—
110	C₂²×C₁₂₇	#95	1	1	yes	characteristic
111	C₂²×C₁₂₇	#95	1	1	yes	characteristic
112	C₂²×C₁₂₇	#95	1	1	yes	characteristic
113	C₂²×C₁₂₇	#95	1	1	yes	characteristic
114	C₂²×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
115	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₄×C₁₂₇	#95	1	1	yes	characteristic
118	C₄×C₁₂₇	#95	1	1	yes	characteristic
119	C₄×C₁₂₇	#95	1	1	yes	characteristic
120	C₄×C₁₂₇	#95	1	1	yes	characteristic
121	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
125	Dic₁₂₇	—	4	4	no	—
126	C₂×C₁₂₇	—	2	1	no	—
127	C₂×C₁₂₇	—	2	1	no	—
128	C₂×C₁₂₇	—	2	1	no	—
129	C₂×C₁₂₇	—	2	1	no	—
130	C₂×C₁₂₇	—	2	1	no	—
131	C₂×C₁₂₇	—	2	1	no	—
132	C₂×C₁₂₇	#344	1	2	yes	complemented
133	C₂×C₁₂₇	C₂×#95	1	1	yes	characteristic
134	C₂×#344	—	127	1	no	maximal, Sylow
135	C₁₂₇	C₂×#344	1	1	yes	characteristic, complemented, Sylow
136	C₂²×M₁₆	—	127	1	no	—
137	C₂²×M₁₆	D₁₂₇	1	1	yes	characteristic
138	C₂×#117	—	127	1	no	—
139	#344	—	127	4	no	—
140	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
141	C₂×M₁₆	—	2	1	no	—
142	C₂×M₁₆	—	2	1	no	—
143	C₂×M₁₆	—	2	1	no	—
144	C₂×M₁₆	—	2	1	no	—
145	C₂×M₁₆	—	127	2	no	—
146	C₂×M₁₆	—	127	2	no	—
147	C₂×M₁₆	—	127	2	no	—
148	C₂×M₁₆	—	254	1	no	—
149	C₂×M₁₆	—	254	1	no	—
150	C₂×M₁₆	—	254	1	no	—
151	C₂×M₁₆	Dic₁₂₇	1	2	yes	complemented
152	C₂×M₁₆	C₂×D₁₂₇	1	2	yes	—
153	C₂²×C₈	—	254	1	no	—
154	C₂²×C₈	Dic₁₂₇	1	2	yes	complemented
155	#117	—	127	2	no	—
156	#117	—	127	2	no	—
157	#117	—	254	1	no	—
158	#117	—	254	1	no	—
159	C₂×C₄⋊C₄	—	254	1	no	—
160	C₂×C₂²⋊C₄	—	127	2	no	—
161	C₂×C₄²	—	254	1	no	—
162	C₂⁴	C₄×D₁₂₇	1	1	yes	characteristic
163	C₂²×C₄	—	254	2	no	—
164	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
165	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
166	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
167	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
168	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
169	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
170	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
171	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
172	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
173	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
174	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
175	C₂²×C₄	Dic₂₅₄	1	1	yes	characteristic
176	C₂²×C₄	Dic₂₅₄	1	1	yes	characteristic
177	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
178	M₁₆	—	2	2	no	—
179	M₁₆	—	2	2	no	—
180	M₁₆	—	2	2	no	—
181	M₁₆	—	2	2	no	—
182	M₁₆	—	127	4	no	—
183	M₁₆	—	254	2	no	—
184	M₁₆	—	254	2	no	—
185	M₁₆	—	254	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	254	2	no	—
189	C₂×C₈	—	254	2	no	—
190	C₂×C₈	—	254	2	no	—
191	C₂×C₈	C₂×Dic₁₂₇	1	4	yes	complemented
192	C₄⋊C₄	—	254	2	no	—
193	C₄⋊C₄	—	254	2	no	—
194	C₂²⋊C₄	—	127	4	no	—
195	C₂²⋊C₄	—	254	2	no	—
196	C₄²	—	254	2	no	—
197	C₄²	—	254	2	no	—
198	C₂³	—	4	2	no	—
199	C₂³	C₄×Dic₁₂₇	1	1	yes	characteristic
200	C₂³	C₄×Dic₁₂₇	1	1	yes	characteristic
201	C₂³	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
202	C₂³	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
203	C₂³	Dic₁₂₇⋊C₄	1	1	yes	characteristic
204	C₂³	Dic₁₂₇⋊C₄	1	1	yes	characteristic
205	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
206	C₂×C₄	—	508	4	no	—
207	C₂×C₄	—	254	4	no	—
208	C₂×C₄	C₄×Dic₁₂₇	1	1	yes	characteristic
209	C₂×C₄	C₄×Dic₁₂₇	1	1	yes	characteristic
210	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
211	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
212	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
213	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
214	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
215	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
216	C₂×C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
217	C₂×C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
218	C₂×C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
219	C₂×C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
220	C₂×C₄	Dic₁₂₇⋊C₄	1	1	yes	characteristic
221	C₂×C₄	Dic₁₂₇⋊C₄	1	1	yes	characteristic
222	C₂×C₄	Dic₁₂₇⋊C₄	1	1	yes	characteristic
223	C₂×C₄	Dic₁₂₇⋊C₄	1	1	yes	characteristic
224	C₂×C₄	Dic₁₂₇⋊C₄	1	1	yes	characteristic
225	C₂×C₄	Dic₁₂₇⋊C₄	1	1	yes	characteristic
226	C₂×C₄	C₄⋊Dic₁₂₇	1	1	yes	characteristic
227	C₂×C₄	C₄⋊Dic₁₂₇	1	1	yes	characteristic
228	C₂×C₄	C₄⋊Dic₁₂₇	1	1	yes	characteristic
229	C₂×C₄	C₄⋊Dic₁₂₇	1	1	yes	characteristic
230	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
231	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
232	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
233	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
234	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
235	C₂×C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
236	C₈	—	2	4	no	—
237	C₈	—	254	4	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	4	1	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	—	4	1	no	—
245	C₂²	C₁₂₇⋊#95	1	1	yes	characteristic
246	C₂²	C₁₂₇⋊#95	1	1	yes	characteristic
247	C₂²	C₁₂₇⋊#95	1	1	yes	characteristic
248	C₂²	C₁₂₇⋊#95	1	1	yes	characteristic
249	C₂²	C₂×C₄×Dic₁₂₇	1	1	yes	characteristic
250	C₂²	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
251	C₂²	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
252	C₄	—	508	4	no	—
253	C₄	C₁₂₇⋊#95	1	1	yes	characteristic
254	C₄	C₁₂₇⋊#95	1	1	yes	characteristic
255	C₄	C₁₂₇⋊#95	1	1	yes	characteristic
256	C₄	C₁₂₇⋊#95	1	1	yes	characteristic
257	C₄	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
258	C₄	C₂×C₂²⋊Dic₁₂₇	1	1	yes	characteristic
259	C₄	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
260	C₄	C₂×C₄⋊Dic₁₂₇	1	1	yes	characteristic
261	C₂	—	2	1	no	—
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	—	2	1	no	—
267	C₂	#22549647	1	2	yes	direct product factor
268	C₂	C₂×C₁₂₇⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₁₂₇
commutator	C₂²×C₁₂₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂²×M₁₆×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#95
prime radical	C₂⁴×C₁₂₇	C₂×C₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂²×M₁₆	D₁₂₇
hypocenter	C₁₂₇	C₂×#344
LCS	G ▷ C₂²×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #22549647	C₂	direct (N × H)
8. C₂×#14523232	C₄	semidirect
9. C₂×#14523232	C₄	semidirect
24. C₂×M₁₆×C₁₂₇	C₄	semidirect
26. C₂²×C₈×C₁₂₇	C₄	semidirect
31. #14523232	C₂×C₄	semidirect
62. C₂×C₈×C₁₂₇	C₂×C₄	semidirect
132. C₂×C₁₂₇	#344	semidirect
135. C₁₂₇	C₂×#344	semidirect
151. C₂×M₁₆	Dic₁₂₇	semidirect
154. C₂²×C₈	Dic₁₂₇	semidirect
191. C₂×C₈	C₂×Dic₁₂₇	semidirect
267. C₂	#22549647	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×#14523232	2
C₂×#16861168	2
#22549647	2
C₂²×M₁₆×C₁₂₇	2
C₂×#344	127

Subgroups — C2×#22549647 #30638472

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22549647 #30638472