Subgroups — D₄×#11758189 #21454550

Order 7808.

544 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#11758189	C₂	1	2	yes	maximal, complemented
2	C₄×D₄×D₆₁	C₂	1	1	yes	maximal, characteristic
3	#16839841	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839839	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839833	C₂	1	2	yes	maximal, complemented
6	#16839830	C₂	1	2	yes	maximal, complemented
7	D₄×#92418	C₂	1	1	yes	maximal, characteristic, complemented
8	#16839784	C₂	1	1	yes	maximal, characteristic, complemented
9	#16839724	C₂	1	2	yes	maximal, complemented
10	C₄×#11758189	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₆₁	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₆₁	C₂²	1	2	yes	—
14	C₂×#11758189	—	2	8	no	—
15	C₂×#11758189	C₂²	1	1	yes	characteristic
16	C₂×#11758189	C₂²	1	2	yes	—
17	C₂×#11758189	C₂²	1	1	yes	characteristic
18	C₂×#11758189	C₂²	1	2	yes	complemented
19	C₂×#11758189	C₂²	1	2	yes	complemented
20	C₂²×#92418	C₂²	1	2	yes	complemented
21	C₄×D₂₄₄	C₂²	1	1	yes	characteristic
22	D₄×Dic₆₁	C₄	1	1	yes	characteristic
23	#14519739	C₄	1	1	yes	characteristic
24	C₄×C₂²⋊D₆₁	C₂²	1	2	yes	—
25	#14519737	C₄	1	2	yes	—
26	C₄⋊C₄×D₆₁	C₄	1	1	yes	characteristic
27	C₂²⋊C₄×D₆₁	C₄	1	2	yes	—
28	C₄²×D₆₁	C₂²	1	1	yes	characteristic
29	C₄⋊#92418	C₂²	1	1	yes	characteristic, complemented
30	#14519706	C₂²	1	2	yes	complemented
31	C₂²⋊#92418	C₂²	1	2	yes	complemented
32	#14519689	C₂²	1	2	yes	complemented
33	#14519689	C₂²	1	2	yes	complemented
34	#14519687	C₂²	1	1	yes	characteristic, complemented
35	#14519687	C₂²	1	1	yes	characteristic, complemented
36	C₄×#92418	C₂²	1	1	yes	characteristic, complemented
37	C₂²×C₈×C₆₁	C₂²	1	2	yes	complemented
38	C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic, complemented
40	C₂²⋊C₈×C₆₁	C₂²	1	2	yes	complemented
41	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic, complemented
42	C₂³×D₆₁	C₂×C₄	1	2	yes	—
43	C₂×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
45	C₂×C₂²⋊D₆₁	C₂×C₄	1	2	yes	—
46	C₂²×Dic₆₁	C₂×C₄	1	2	yes	—
47	D₄×D₆₁	—	2	4	no	—
48	C₂×D₂₄₄	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₆₁	—	2	4	no	—
50	C₂×C₄×D₆₁	—	2	4	no	—
51	C₂×C₄×D₆₁	C₂×C₄	1	2	yes	—
52	C₂×C₄×D₆₁	C₂×C₄	1	2	yes	—
53	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₆₁	C₂³	1	2	yes	—
56	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₆₁	—	2	4	no	—
58	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₆₁	C₂³	1	2	yes	—
60	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
62	C₂²⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
63	C₄²×C₆₁	C₂³	1	1	yes	characteristic
64	C₂²⋊Dic₆₁	C₂×C₄	1	2	yes	—
65	#11758198	C₂×C₄	1	2	yes	—
66	#11758198	C₂×C₄	1	2	yes	—
67	C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
68	#11758196	C₂×C₄	1	2	yes	—
69	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
72	C₂×#92418	—	2	4	no	—
73	C₂×#92418	C₂³	1	1	yes	characteristic
74	C₂×#92418	C₂³	1	2	yes	—
75	C₂×#92418	C₂³	1	1	yes	characteristic
76	#11758189	—	2	4	no	—
77	#11758189	—	2	2	no	—
78	#11758189	—	2	4	no	—
79	#11758189	—	2	4	no	—
80	#11758189	D₄	1	4	yes	direct product factor
81	C₂³×C₆₁	C₂²×C₄	1	2	yes	—
82	C₂²×D₆₁	—	2	4	no	—
83	C₂²×D₆₁	—	2	4	no	—
84	C₂²×D₆₁	—	4	2	no	—
85	C₂²×D₆₁	C₂²×C₄	1	2	yes	—
86	C₂²×D₆₁	C₂²×C₄	1	1	yes	characteristic
87	D₄×C₆₁	M₁₆	1	4	yes	complemented
88	C₂×C₄×C₆₁	—	2	2	no	—
89	C₂×C₄×C₆₁	—	2	2	no	—
90	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
94	C₂²⋊D₆₁	—	2	4	no	—
95	C₂×Dic₆₁	—	2	4	no	—
96	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
97	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
98	C₂×Dic₆₁	C₂²×C₄	1	2	yes	—
99	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
100	D₂₄₄	—	2	2	no	—
101	C₄×D₆₁	—	2	4	no	—
102	C₄×D₆₁	—	2	2	no	—
103	C₄×D₆₁	—	2	2	no	—
104	C₄×D₆₁	—	2	2	no	—
105	C₄×D₆₁	—	2	4	no	—
106	C₄×D₆₁	—	4	1	no	—
107	C₄×D₆₁	C₂×D₄	1	2	yes	—
108	C₄×D₆₁	#41	1	2	yes	—
109	C₈×C₆₁	—	2	2	no	—
110	C₈×C₆₁	—	2	1	no	—
111	C₈×C₆₁	C₂×D₄	1	2	yes	complemented
112	#92418	—	2	2	no	—
113	#92418	—	2	1	no	—
114	#92418	C₂×D₄	1	2	yes	complemented
115	C₂²×C₆₁	—	2	2	no	—
116	C₂²×C₆₁	—	2	2	no	—
117	C₂²×C₆₁	C₂×M₁₆	1	4	yes	complemented
118	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
119	C₂×D₆₁	—	2	1	no	—
120	C₂×D₆₁	—	2	1	no	—
121	C₂×D₆₁	—	2	2	no	—
122	C₂×D₆₁	—	2	2	no	—
123	C₂×D₆₁	—	2	2	no	—
124	C₂×D₆₁	—	4	8	no	—
125	C₂×D₆₁	C₄×D₄	1	2	yes	—
126	C₄×C₆₁	—	2	2	no	—
127	C₄×C₆₁	—	2	1	no	—
128	C₄×C₆₁	C₂×M₁₆	1	2	yes	complemented
129	C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
130	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
131	Dic₆₁	—	2	2	no	—
132	Dic₆₁	—	2	1	no	—
133	Dic₆₁	—	2	1	no	—
134	Dic₆₁	C₄×D₄	1	2	yes	—
135	D₄×M₁₆	—	61	1	no	maximal, Sylow
136	C₂×C₆₁	—	2	4	no	—
137	C₂×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
138	C₂×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
139	C₂×C₆₁	#568	1	1	yes	characteristic
140	D₆₁	—	2	2	no	—
141	D₆₁	—	4	2	no	—
142	C₂²×M₁₆	—	61	2	no	—
143	C₂×C₄×D₄	—	61	1	no	—
144	#436	—	61	1	no	—
145	#436	—	61	1	no	—
146	#435	—	61	2	no	—
147	#435	—	61	2	no	—
148	C₈×D₄	—	61	1	no	—
149	C₈×D₄	D₆₁	1	1	yes	characteristic, complemented
150	#423	—	61	1	no	—
151	#407	—	61	2	no	—
152	C₄×M₁₆	—	61	1	no	—
153	C₆₁	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
154	C₂²×D₄	—	61	1	no	—
155	C₂³×C₄	—	61	2	no	—
156	C₂×M₁₆	—	61	1	no	—
157	C₂×M₁₆	—	61	2	no	—
158	C₂×M₁₆	—	61	2	no	—
159	C₂×M₁₆	—	61	2	no	—
160	C₂×M₁₆	—	61	1	no	—
161	C₂×M₁₆	—	122	8	no	—
162	C₂²×C₈	—	61	2	no	—
163	C₂²×C₈	C₂×D₆₁	1	2	yes	complemented
164	C₄×D₄	—	61	1	no	—
165	C₄×D₄	—	61	2	no	—
166	C₄×D₄	—	61	1	no	—
167	C₄×D₄	—	61	2	no	—
168	C₄×D₄	—	61	1	no	—
169	C₄×D₄	C₂×D₆₁	1	1	yes	characteristic
170	C₂×C₄⋊C₄	—	61	1	no	—
171	C₂×C₂²⋊C₄	—	61	2	no	—
172	C₂×C₄²	—	61	1	no	—
173	C₄⋊C₈	—	61	2	no	—
174	C₄⋊C₈	—	61	1	no	—
175	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic, complemented
176	C₂²⋊C₈	—	61	2	no	—
177	C₂²⋊C₈	—	61	2	no	—
178	C₂²⋊C₈	—	61	2	no	—
179	C₂²⋊C₈	C₂×D₆₁	1	2	yes	complemented
180	#97	—	61	1	no	—
181	#97	—	61	1	no	—
182	C₄×C₈	—	61	1	no	—
183	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic, complemented
184	C₂⁴	—	61	2	no	—
185	C₂×D₄	—	61	2	no	—
186	C₂×D₄	—	61	1	no	—
187	C₂×D₄	—	122	4	no	—
188	C₂×D₄	C₄×D₆₁	1	1	yes	characteristic
189	C₂²×C₄	—	61	1	no	—
190	C₂²×C₄	—	61	2	no	—
191	C₂²×C₄	—	61	1	no	—
192	C₂²×C₄	—	61	2	no	—
193	C₂²×C₄	—	61	2	no	—
194	C₂²×C₄	—	61	2	no	—
195	C₂²×C₄	—	61	1	no	—
196	C₂²×C₄	—	122	4	no	—
197	C₂²×C₄	—	122	4	no	—
198	C₂²×C₄	C₂²×D₆₁	1	2	yes	—
199	M₁₆	—	61	4	no	—
200	M₁₆	—	122	4	no	—
201	M₁₆	—	122	2	no	—
202	M₁₆	—	122	4	no	—
203	M₁₆	—	122	4	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	61	1	no	—
206	C₂×C₈	—	61	1	no	—
207	C₂×C₈	—	61	2	no	—
208	C₂×C₈	—	122	4	no	—
209	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
210	C₂×C₈	C₂²×D₆₁	1	2	yes	—
211	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
212	C₄⋊C₄	—	61	2	no	—
213	C₄⋊C₄	—	61	1	no	—
214	C₄⋊C₄	C₄×D₆₁	1	1	yes	characteristic
215	C₂²⋊C₄	—	61	2	no	—
216	C₂²⋊C₄	—	61	2	no	—
217	C₂²⋊C₄	—	61	2	no	—
218	C₂²⋊C₄	C₄×D₆₁	1	2	yes	—
219	C₄²	—	61	1	no	—
220	C₄²	—	61	1	no	—
221	C₄²	—	61	1	no	—
222	C₄²	C₂²×D₆₁	1	1	yes	characteristic
223	C₂³	—	61	1	no	—
224	C₂³	—	61	2	no	—
225	C₂³	—	122	4	no	—
226	C₂³	—	122	4	no	—
227	C₂³	—	244	2	no	—
228	C₂³	C₂×C₄×D₆₁	1	2	yes	—
229	D₄	—	122	4	no	—
230	D₄	—	122	2	no	—
231	D₄	#11758189	1	4	yes	direct product factor
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	61	1	no	—
235	C₂×C₄	—	61	2	no	—
236	C₂×C₄	—	61	2	no	—
237	C₂×C₄	—	61	1	no	—
238	C₂×C₄	—	61	1	no	—
239	C₂×C₄	—	61	2	no	—
240	C₂×C₄	—	122	4	no	—
241	C₂×C₄	—	122	4	no	—
242	C₂×C₄	—	122	4	no	—
243	C₂×C₄	—	122	2	no	—
244	C₂×C₄	—	122	2	no	—
245	C₂×C₄	—	122	2	no	—
246	C₂×C₄	—	244	1	no	—
247	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄×D₆₁	1	2	yes	—
249	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
250	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	2	1	no	—
253	C₈	—	61	2	no	—
254	C₈	—	122	2	no	—
255	C₈	—	122	1	no	—
256	C₈	D₄×D₆₁	1	2	yes	complemented
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	61	2	no	—
260	C₂²	—	122	1	no	—
261	C₂²	—	122	1	no	—
262	C₂²	—	122	2	no	—
263	C₂²	—	122	2	no	—
264	C₂²	—	122	2	no	—
265	C₂²	—	244	8	no	—
266	C₂²	C₂×#11758189	1	4	yes	complemented
267	C₂²	C₂²×C₄×D₆₁	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	—	61	2	no	—
271	C₄	—	122	2	no	—
272	C₄	—	122	1	no	—
273	C₄	—	122	1	no	—
274	C₄	C₂×#11758189	1	2	yes	complemented
275	C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
276	C₄	#41×D₆₁	1	1	yes	characteristic
277	C₂	—	2	4	no	—
278	C₂	—	122	2	no	—
279	C₂	—	244	2	no	—
280	C₂	C₄×D₄×D₆₁	1	1	yes	characteristic
281	C₂	C₂²×#11758189	1	1	yes	characteristic
282	C₂	#16840527	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	C₈×D₄×C₆₁	C₂
hypercenter	C₈×D₄	D₆₁
hypocenter	C₆₁	D₄×M₁₆
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#11758189	C₂	semidirect
3. #16839841	C₂	semidirect
4. #16839839	C₂	semidirect
5. #16839833	C₂	semidirect
6. #16839830	C₂	semidirect
7. D₄×#92418	C₂	semidirect
8. #16839784	C₂	semidirect
9. #16839724	C₂	semidirect
10. C₄×#11758189	C₂	semidirect
11. C₈×D₄×C₆₁	C₂	semidirect
18. C₂×#11758189	C₂²	semidirect
19. C₂×#11758189	C₂²	semidirect
20. C₂²×#92418	C₂²	semidirect
29. C₄⋊#92418	C₂²	semidirect
30. #14519706	C₂²	semidirect
31. C₂²⋊#92418	C₂²	semidirect
32. #14519689	C₂²	semidirect
33. #14519689	C₂²	semidirect
34. #14519687	C₂²	semidirect
35. #14519687	C₂²	semidirect
36. C₄×#92418	C₂²	semidirect
37. C₂²×C₈×C₆₁	C₂²	semidirect
39. C₄⋊C₈×C₆₁	C₂²	semidirect
40. C₂²⋊C₈×C₆₁	C₂²	semidirect
41. C₄×C₈×C₆₁	C₂²	semidirect
80. #11758189	D₄	direct (N × H)
87. D₄×C₆₁	M₁₆	semidirect
111. C₈×C₆₁	C₂×D₄	semidirect
114. #92418	C₂×D₄	semidirect
117. C₂²×C₆₁	C₂×M₁₆	semidirect
128. C₄×C₆₁	C₂×M₁₆	semidirect
149. C₈×D₄	D₆₁	semidirect
153. C₆₁	D₄×M₁₆	semidirect
163. C₂²×C₈	C₂×D₆₁	semidirect
175. C₄⋊C₈	C₂×D₆₁	semidirect
179. C₂²⋊C₈	C₂×D₆₁	semidirect
183. C₄×C₈	C₂×D₆₁	semidirect
231. D₄	#11758189	direct (N × H)
256. C₈	D₄×D₆₁	semidirect
266. C₂²	C₂×#11758189	semidirect
274. C₄	C₂×#11758189	semidirect

Subgroups — D4×#11758189 #21454550

Conjugacy classes of subgroups

Complements

Subgroups — D₄×#11758189 #21454550