Subgroups — C₄×#16812137 #28345273

Order 13696.

548 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812137	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133818	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₄₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#20292740	C₂	1	2	yes	maximal, complemented
5	#20292729	C₂	1	2	yes	maximal, complemented
6	C₄×#13133807	C₂	1	2	yes	maximal, complemented
7	C₄×D₈₅₆	C₂	1	2	yes	maximal, complemented
8	#20292669	C₂	1	1	yes	maximal, characteristic, complemented
9	#20292633	C₂	1	1	yes	maximal, characteristic, complemented
10	#20292629	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×M₁₆×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133818	C₂²	1	1	yes	characteristic
13	C₂²×D₄₂₈	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₁₀₇	—	2	1	no	—
15	#16812137	C₄	1	8	yes	direct product factor
16	C₂×#13133807	C₂²	1	2	yes	—
17	C₂×D₈₅₆	C₂²	1	2	yes	—
18	C₄×Dic₂₁₄	C₂²	1	1	yes	characteristic, complemented
19	C₄×Dic₂₁₄	C₂²	1	1	yes	characteristic, complemented
20	C₄×D₄₂₈	—	2	1	no	—
21	C₄×D₄₂₈	—	2	1	no	—
22	C₄×D₄₂₈	C₂²	1	2	yes	—
23	C₄×D₄₂₈	C₂²	1	2	yes	—
24	C₄×D₄₂₈	C₂²	1	1	yes	characteristic, complemented
25	C₄×D₄₂₈	C₂²	1	1	yes	characteristic, complemented
26	C₄×C₂²⋊D₁₀₇	—	2	1	no	—
27	C₄×C₂²⋊D₁₀₇	—	2	1	no	—
28	#16812068	C₂²	1	1	yes	characteristic
29	#16812066	—	2	1	no	—
30	C₂×C₄⋊Dic₁₀₇	C₂²	1	1	yes	characteristic
31	C₂×(C₂×D₁₀₇)⋊C₄	—	2	1	no	—
32	C₄²×D₁₀₇	—	2	1	no	—
33	#16812043	C₂²	1	2	yes	complemented
34	#16812041	C₂²	1	2	yes	complemented
35	#16812034	C₂²	1	2	yes	complemented
36	#16812031	C₂²	1	2	yes	—
37	#16812031	C₂²	1	2	yes	—
38	#16812031	C₂²	1	2	yes	complemented
39	C₂×M₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
40	C₂×M₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
41	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
42	#97×C₁₀₇	C₂²	1	2	yes	complemented
43	C₄×C₈×C₁₀₇	C₂²	1	2	yes	complemented
44	C₂³×D₁₀₇	—	2	1	no	—
45	#13133818	—	2	2	no	—
46	#13133818	C₂×C₄	1	4	yes	complemented
47	C₂×Dic₂₁₄	C₂³	1	1	yes	characteristic
48	C₂×D₄₂₈	—	2	1	no	—
49	C₂×D₄₂₈	—	2	2	no	—
50	C₂×D₄₂₈	C₂×C₄	1	4	yes	complemented
51	C₂×D₄₂₈	C₂³	1	2	yes	—
52	C₂×D₄₂₈	C₂³	1	1	yes	characteristic
53	C₂×C₂²⋊D₁₀₇	—	2	1	no	—
54	C₂²×Dic₁₀₇	—	2	1	no	—
55	C₂×C₄×D₁₀₇	—	2	1	no	—
56	C₂×C₄×D₁₀₇	—	2	1	no	—
57	C₂×C₄×D₁₀₇	—	2	1	no	—
58	C₂×C₄×D₁₀₇	—	2	2	no	—
59	C₂×C₄×D₁₀₇	—	2	2	no	—
60	C₂×C₄×D₁₀₇	—	4	4	no	—
61	#13133807	C₂×C₄	1	8	yes	complemented
62	D₈₅₆	C₂×C₄	1	8	yes	complemented
63	C₄⋊Dic₁₀₇	—	2	1	no	—
64	C₄⋊Dic₁₀₇	C₂³	1	2	yes	—
65	C₄⋊Dic₁₀₇	C₂³	1	1	yes	characteristic
66	C₄⋊Dic₁₀₇	C₂³	1	1	yes	characteristic
67	Dic₁₀₇⋊C₄	—	2	1	no	—
68	Dic₁₀₇⋊C₄	—	2	1	no	—
69	C₂²⋊Dic₁₀₇	—	2	1	no	—
70	(C₂×D₁₀₇)⋊C₄	—	2	2	no	—
71	(C₂×D₁₀₇)⋊C₄	—	2	2	no	—
72	(C₂×D₁₀₇)⋊C₄	—	2	1	no	—
73	(C₂×D₁₀₇)⋊C₄	—	2	1	no	—
74	C₄×Dic₁₀₇	—	2	1	no	—
75	C₄×Dic₁₀₇	—	2	1	no	—
76	C₄×Dic₁₀₇	—	2	1	no	—
77	C₂²×C₄×C₁₀₇	D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₀₇	D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
80	M₁₆×C₁₀₇	—	2	2	no	—
81	M₁₆×C₁₀₇	C₂×C₄	1	4	yes	complemented
82	C₂×C₈×C₁₀₇	C₂³	1	2	yes	—
83	C₂×C₈×C₁₀₇	C₂³	1	2	yes	—
84	C₄²×C₁₀₇	D₄	1	1	yes	characteristic
85	C₄²×C₁₀₇	D₄	1	1	yes	characteristic
86	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
87	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
88	C₂³×C₁₀₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×D₁₀₇	—	2	4	no	—
90	C₂²×D₁₀₇	—	2	1	no	—
91	C₂²×D₁₀₇	—	2	2	no	—
92	C₂²×D₁₀₇	—	4	2	no	—
93	C₂²×D₁₀₇	—	4	2	no	—
94	C₂×C₄×C₁₀₇	—	2	2	no	—
95	C₂×C₄×C₁₀₇	—	4	1	no	—
96	C₂×C₄×C₁₀₇	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₀₇	#41	1	2	yes	—
103	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
104	C₂²⋊D₁₀₇	—	2	4	no	—
105	C₂×Dic₁₀₇	—	2	1	no	—
106	C₂×Dic₁₀₇	—	2	1	no	—
107	C₂×Dic₁₀₇	—	2	1	no	—
108	C₂×Dic₁₀₇	—	2	2	no	—
109	C₂×Dic₁₀₇	—	4	2	no	—
110	D₄₂₈	—	2	4	no	—
111	D₄₂₈	—	2	2	no	—
112	D₄₂₈	—	2	1	no	—
113	D₄₂₈	C₂²×C₄	1	4	yes	—
114	D₄₂₈	C₂²×C₄	1	2	yes	complemented
115	C₄×D₁₀₇	—	2	4	no	—
116	C₄×D₁₀₇	—	4	2	no	—
117	C₄×D₁₀₇	—	4	4	no	—
118	C₄×D₁₀₇	—	4	2	no	—
119	C₄×D₁₀₇	—	4	4	no	—
120	Dic₂₁₄	—	2	1	no	—
121	Dic₂₁₄	C₂²×C₄	1	2	yes	complemented
122	C₈×C₁₀₇	—	2	2	no	—
123	C₈×C₁₀₇	C₂²×C₄	1	4	yes	complemented
124	C₂²×C₁₀₇	—	2	1	no	—
125	C₂²×C₁₀₇	—	2	1	no	—
126	C₂²×C₁₀₇	C₄×D₄	1	2	yes	—
127	C₂²×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×D₁₀₇	—	2	2	no	—
129	C₂×D₁₀₇	—	2	4	no	—
130	C₂×D₁₀₇	—	4	8	no	—
131	C₂×D₁₀₇	—	4	2	no	—
132	C₂×D₁₀₇	—	4	1	no	—
133	C₂×D₁₀₇	—	4	1	no	—
134	C₂×D₁₀₇	—	4	2	no	—
135	C₄×C₁₀₇	—	2	1	no	—
136	C₄×C₁₀₇	—	2	1	no	—
137	C₄×C₁₀₇	—	2	1	no	—
138	C₄×C₁₀₇	C₄×D₄	1	2	yes	—
139	C₄×C₁₀₇	AGL(1,ℤ/8)	1	2	yes	complemented
140	C₄×C₁₀₇	C₂³×C₄	1	1	yes	characteristic
141	C₄×C₁₀₇	C₂×#41	1	1	yes	characteristic
142	Dic₁₀₇	—	2	2	no	—
143	Dic₁₀₇	—	4	2	no	—
144	Dic₁₀₇	—	4	1	no	—
145	Dic₁₀₇	—	4	1	no	—
146	C₂×C₁₀₇	—	2	2	no	—
147	C₂×C₁₀₇	C₂×C₄×D₄	1	1	yes	characteristic
148	C₂×C₁₀₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂×C₁₀₇	#575	1	1	yes	characteristic
150	D₁₀₇	—	4	2	no	—
151	D₁₀₇	—	4	4	no	—
152	C₄×AGL(1,ℤ/8)	—	107	1	no	maximal, Sylow
153	C₁₀₇	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
154	C₂×AGL(1,ℤ/8)	—	107	1	no	—
155	C₄×#41	—	107	1	no	—
156	C₂×C₄×D₄	—	107	1	no	—
157	#442	—	107	2	no	—
158	#440	—	107	2	no	—
159	C₄×SD₁₆	—	107	2	no	—
160	C₄×D₈	—	107	2	no	—
161	#428	—	107	1	no	—
162	#418	—	107	1	no	—
163	#417	—	107	1	no	—
164	C₄×M₁₆	D₁₀₇	1	1	yes	characteristic, complemented
165	C₂×#41	—	107	1	no	—
166	C₂²×D₄	—	107	1	no	—
167	C₂³×C₄	—	214	1	no	—
168	AGL(1,ℤ/8)	—	107	8	no	—
169	C₂×SD₁₆	—	107	2	no	—
170	C₂×D₈	—	107	2	no	—
171	C₂×M₁₆	C₂×D₁₀₇	1	1	yes	characteristic
172	C₂×M₁₆	C₂×D₁₀₇	1	1	yes	characteristic
173	C₄×Q₈	—	107	1	no	—
174	C₄×Q₈	—	107	1	no	—
175	C₄×D₄	—	107	2	no	—
176	C₄×D₄	—	107	2	no	—
177	C₄×D₄	—	107	1	no	—
178	C₄×D₄	—	107	1	no	—
179	C₄×D₄	—	214	1	no	—
180	C₄×D₄	—	214	1	no	—
181	C₄×D₄	—	214	1	no	—
182	C₄×D₄	—	214	1	no	—
183	#117	—	107	1	no	—
184	#117	—	214	1	no	—
185	C₂×C₄⋊C₄	—	107	1	no	—
186	C₂×C₂²⋊C₄	—	214	1	no	—
187	C₂×C₄²	—	214	1	no	—
188	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
189	#107	—	107	2	no	—
190	#106	—	107	2	no	—
191	#103	—	107	2	no	—
192	#102	—	107	2	no	—
193	#102	—	107	2	no	—
194	#102	—	107	2	no	—
195	#97	C₂×D₁₀₇	1	2	yes	complemented
196	C₄×C₈	C₂×D₁₀₇	1	2	yes	complemented
197	C₂⁴	—	214	1	no	—
198	#41	—	107	4	no	—
199	#41	—	214	2	no	—
200	C₂×Q₈	—	107	1	no	—
201	C₂×D₄	—	107	4	no	—
202	C₂×D₄	—	107	1	no	—
203	C₂×D₄	—	107	2	no	—
204	C₂×D₄	—	214	1	no	—
205	C₂×D₄	—	214	1	no	—
206	C₂×D₄	—	214	2	no	—
207	C₂²×C₄	—	428	4	no	—
208	C₂²×C₄	—	214	1	no	—
209	C₂²×C₄	—	214	1	no	—
210	C₂²×C₄	—	214	2	no	—
211	C₂²×C₄	—	214	1	no	—
212	C₂²×C₄	—	214	2	no	—
213	C₂²×C₄	—	214	1	no	—
214	C₂²×C₄	D₄₂₈	1	1	yes	characteristic
215	C₂²×C₄	D₄₂₈	1	1	yes	characteristic
216	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
217	SD₁₆	—	107	8	no	—
218	D₈	—	107	8	no	—
219	M₁₆	—	2	2	no	—
220	M₁₆	C₄×D₁₀₇	1	4	yes	complemented
221	C₂×C₈	C₂²×D₁₀₇	1	2	yes	—
222	C₂×C₈	C₂²×D₁₀₇	1	2	yes	—
223	C₄⋊C₄	—	107	1	no	—
224	C₄⋊C₄	—	107	1	no	—
225	C₄⋊C₄	—	107	2	no	—
226	C₄⋊C₄	—	214	1	no	—
227	C₄⋊C₄	—	214	1	no	—
228	C₄⋊C₄	—	214	1	no	—
229	C₂²⋊C₄	—	214	1	no	—
230	C₂²⋊C₄	—	214	2	no	—
231	C₂²⋊C₄	—	214	2	no	—
232	C₂²⋊C₄	—	214	1	no	—
233	C₂²⋊C₄	—	214	1	no	—
234	C₄²	—	214	1	no	—
235	C₄²	—	214	1	no	—
236	C₄²	—	214	1	no	—
237	C₄²	D₄₂₈	1	1	yes	characteristic
238	C₄²	D₄₂₈	1	1	yes	characteristic
239	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
240	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
241	C₂³	—	428	2	no	—
242	C₂³	—	428	2	no	—
243	C₂³	—	214	4	no	—
244	C₂³	—	214	1	no	—
245	C₂³	—	214	2	no	—
246	C₂³	C₂×D₄₂₈	1	1	yes	characteristic
247	Q₈	—	107	2	no	—
248	Q₈	—	214	1	no	—
249	D₄	—	107	2	no	—
250	D₄	—	107	4	no	—
251	D₄	—	214	4	no	—
252	D₄	—	214	4	no	—
253	D₄	—	214	2	no	—
254	D₄	—	214	1	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	428	4	no	—
258	C₂×C₄	—	428	2	no	—
259	C₂×C₄	—	428	2	no	—
260	C₂×C₄	—	428	4	no	—
261	C₂×C₄	—	428	2	no	—
262	C₂×C₄	—	214	1	no	—
263	C₂×C₄	—	214	1	no	—
264	C₂×C₄	—	214	2	no	—
265	C₂×C₄	—	214	1	no	—
266	C₂×C₄	—	214	4	no	—
267	C₂×C₄	C₂×C₄×D₁₀₇	1	2	yes	—
268	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
269	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
270	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
271	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
272	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
273	C₂×C₄	#13133818	1	2	yes	—
274	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
275	C₈	—	2	2	no	—
276	C₈	C₂×C₄×D₁₀₇	1	4	yes	complemented
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	428	8	no	—
280	C₂²	—	428	2	no	—
281	C₂²	—	428	1	no	—
282	C₂²	—	428	1	no	—
283	C₂²	—	428	2	no	—
284	C₂²	—	214	2	no	—
285	C₂²	—	214	4	no	—
286	C₂²	C₄×D₄₂₈	1	2	yes	—
287	C₂²	C₂²×D₄₂₈	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	428	2	no	—
292	C₄	—	428	1	no	—
293	C₄	—	428	1	no	—
294	C₄	—	214	2	no	—
295	C₄	C₄×D₄₂₈	1	2	yes	—
296	C₄	#16812137	1	2	yes	direct product factor
297	C₄	C₂²×C₄×D₁₀₇	1	1	yes	characteristic
298	C₄	C₂×#13133818	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	428	4	no	—
301	C₂	—	428	2	no	—
302	C₂	C₂×C₄×D₄₂₈	1	1	yes	characteristic
303	C₂	C₂×#16812137	1	1	yes	characteristic
304	C₂	#20293427	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄₂₈
commutator	C₄×C₁₀₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₇
Fitting	C₄×M₁₆×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂²×D₄
prime radical	C₂×#16812137	C₂
cosocle	C₂×C₄×C₁₀₇	C₂⁴
hypercenter	C₄×M₁₆	D₁₀₇
hypocenter	C₁₀₇	C₄×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₄×#13133818	C₂	semidirect
3. C₂×C₄×D₄₂₈	C₂	semidirect
4. #20292740	C₂	semidirect
5. #20292729	C₂	semidirect
6. C₄×#13133807	C₂	semidirect
7. C₄×D₈₅₆	C₂	semidirect
8. #20292669	C₂	semidirect
9. #20292633	C₂	semidirect
10. #20292629	C₂	semidirect
11. C₄×M₁₆×C₁₀₇	C₂	semidirect
15. #16812137	C₄	direct (N × H)
18. C₄×Dic₂₁₄	C₂²	semidirect
19. C₄×Dic₂₁₄	C₂²	semidirect
24. C₄×D₄₂₈	C₂²	semidirect
25. C₄×D₄₂₈	C₂²	semidirect
33. #16812043	C₂²	semidirect
34. #16812041	C₂²	semidirect
35. #16812034	C₂²	semidirect
38. #16812031	C₂²	semidirect
42. #97×C₁₀₇	C₂²	semidirect
43. C₄×C₈×C₁₀₇	C₂²	semidirect
46. #13133818	C₂×C₄	semidirect
50. C₂×D₄₂₈	C₂×C₄	semidirect
61. #13133807	C₂×C₄	semidirect
62. D₈₅₆	C₂×C₄	semidirect
81. M₁₆×C₁₀₇	C₂×C₄	semidirect
114. D₄₂₈	C₂²×C₄	semidirect
121. Dic₂₁₄	C₂²×C₄	semidirect
123. C₈×C₁₀₇	C₂²×C₄	semidirect
139. C₄×C₁₀₇	AGL(1,ℤ/8)	semidirect
153. C₁₀₇	C₄×AGL(1,ℤ/8)	semidirect
164. C₄×M₁₆	D₁₀₇	semidirect
195. #97	C₂×D₁₀₇	semidirect
196. C₄×C₈	C₂×D₁₀₇	semidirect
220. M₁₆	C₄×D₁₀₇	semidirect
276. C₈	C₂×C₄×D₁₀₇	semidirect
296. C₄	#16812137	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×#16812137	2
C₄×#13133818	2
C₂×C₄×D₄₂₈	2
#20292740	2
#20292729	2
C₄×#13133807	2
C₄×D₈₅₆	2
#20292669	2
#20292633	2
#20292629	2
C₄×M₁₆×C₁₀₇	2
C₄×AGL(1,ℤ/8)	107

Subgroups — C4×#16812137 #28345273

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16812137 #28345273