Subgroups — D₄×#14555786 #31819896

Order 17792.

1648 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14555786	C₂	1	2	yes	maximal, complemented
2	#22581625	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581614	C₂	1	2	yes	maximal, complemented
4	#22581606	C₂	1	3	yes	maximal, complemented
5	#22581594	C₂	1	6	yes	maximal, complemented
6	D₄×D₅₅₆	C₂	1	3	yes	maximal, complemented
7	#22581504	C₂	1	3	yes	maximal, complemented
8	#22581466	C₂	1	6	yes	maximal, complemented
9	C₄×#14555786	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₄×D₁₃₉	C₂	1	3	yes	maximal, complemented
11	D₄×Q₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14555786	—	2	16	no	—
13	C₂×#14555786	C₂²	1	1	yes	characteristic
14	C₂×#14555786	C₂²	1	6	yes	complemented
15	C₂×#14555786	C₂²	1	6	yes	complemented
16	C₂×#14555786	C₂²	1	2	yes	complemented
17	C₂²×D₅₅₆	C₂²	1	6	yes	complemented
18	C₂×D₄×D₁₃₉	C₂²	1	3	yes	—
19	C₂²×C₄×D₁₃₉	C₂²	1	6	yes	complemented
20	#17984839	C₂²	1	3	yes	complemented
21	#17984831	C₂²	1	6	yes	complemented
22	#17984827	C₂²	1	12	yes	complemented
23	#17984818	C₂²	1	6	yes	complemented
24	#17984817	C₂²	1	6	yes	complemented
25	#17984817	C₂²	1	6	yes	complemented
26	#17984817	C₂²	1	6	yes	complemented
27	#17984816	C₂²	1	6	yes	complemented
28	#17984812	C₂²	1	12	yes	complemented
29	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic, complemented
30	C₄×D₅₅₆	C₂²	1	3	yes	complemented
31	C₄×D₅₅₆	C₂²	1	3	yes	complemented
32	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
33	#17984804	C₂²	1	3	yes	complemented
34	#17984804	C₂²	1	6	yes	complemented
35	#17984804	C₂²	1	3	yes	complemented
36	C₄×C₂²⋊D₁₃₉	C₂²	1	6	yes	complemented
37	(C₂²⋊D₁₃₉)⋊C₄	C₂²	1	6	yes	complemented
38	#17984800	C₂²	1	3	yes	complemented
39	C₄⋊C₄×D₁₃₉	C₂²	1	3	yes	complemented
40	C₂²⋊C₄×D₁₃₉	C₂²	1	6	yes	complemented
41	C₄²×D₁₃₉	C₂²	1	3	yes	complemented
42	C₂²×Q₈×C₁₃₉	C₂²	1	2	yes	complemented
43	C₄⋊Q₈×C₁₃₉	C₂²	1	3	yes	complemented
44	C₂²⋊Q₈×C₁₃₉	C₂²	1	6	yes	complemented
45	C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
46	C₄×D₄×C₁₃₉	C₂²	1	3	yes	—
47	C₂³×D₁₃₉	C₂³	1	6	yes	—
48	#14555786	—	2	24	no	—
49	#14555786	—	2	24	no	—
50	#14555786	—	2	8	no	—
51	#14555786	D₄	1	8	yes	direct product factor
52	C₂×D₅₅₆	—	2	24	no	—
53	C₂×D₅₅₆	C₂³	1	3	yes	—
54	C₂×D₅₅₆	C₂³	1	12	yes	—
55	C₂×D₅₅₆	C₂³	1	6	yes	—
56	C₂×D₅₅₆	C₂³	1	6	yes	—
57	C₂×D₅₅₆	C₂³	1	3	yes	—
58	C₂×C₂²⋊D₁₃₉	C₂³	1	6	yes	—
59	D₄×D₁₃₉	—	2	12	no	—
60	C₂²×Dic₁₃₉	C₂³	1	2	yes	—
61	C₂×C₄×D₁₃₉	—	2	24	no	—
62	C₂×C₄×D₁₃₉	C₂³	1	3	yes	—
63	C₂×C₄×D₁₃₉	C₂³	1	6	yes	—
64	C₂×C₄×D₁₃₉	C₂³	1	6	yes	—
65	C₂×C₄×D₁₃₉	C₂³	1	6	yes	—
66	C₂×C₄×D₁₃₉	C₂³	1	3	yes	—
67	C₂×C₄×D₁₃₉	C₂³	1	3	yes	—
68	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
69	C₄⋊Dic₁₃₉	C₂³	1	3	yes	—
70	Dic₁₃₉⋊C₄	C₂³	1	6	yes	—
71	C₂²⋊Dic₁₃₉	C₂³	1	2	yes	—
72	(C₂×D₁₃₉)⋊C₄	C₂³	1	6	yes	—
73	(C₂×D₁₃₉)⋊C₄	C₂³	1	6	yes	—
74	(C₂×D₁₃₉)⋊C₄	C₂³	1	6	yes	—
75	(C₂×D₁₃₉)⋊C₄	C₂³	1	12	yes	—
76	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
77	C₄×Dic₁₃₉	C₂³	1	3	yes	—
78	C₄×Dic₁₃₉	C₂³	1	3	yes	—
79	C₂×Q₈×C₁₃₉	—	2	8	no	—
80	C₂×Q₈×C₁₃₉	C₂³	1	6	yes	—
81	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
82	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
83	C₂²×C₄×C₁₃₉	C₂³	1	6	yes	—
84	C₄⋊C₄×C₁₃₉	C₂³	1	3	yes	—
85	C₄⋊C₄×C₁₃₉	C₂³	1	3	yes	—
86	C₄⋊C₄×C₁₃₉	C₂³	1	6	yes	—
87	C₂²⋊C₄×C₁₃₉	C₂³	1	6	yes	—
88	C₄²×C₁₃₉	C₂³	1	3	yes	—
89	C₂²×D₁₃₉	—	2	12	no	—
90	C₂²×D₁₃₉	—	2	12	no	—
91	C₂²×D₁₃₉	—	4	6	no	—
92	C₂²×D₁₃₉	C₂⁴	1	6	yes	—
93	C₂²×D₁₃₉	C₂⁴	1	3	yes	—
94	D₅₅₆	—	2	24	no	—
95	D₅₅₆	—	2	12	no	—
96	D₅₅₆	—	2	6	no	—
97	D₅₅₆	—	4	6	no	—
98	D₅₅₆	C₂×D₄	1	12	yes	complemented
99	C₂²⋊D₁₃₉	—	2	12	no	—
100	C₂×Dic₁₃₉	—	2	4	no	—
101	C₂×Dic₁₃₉	C₂⁴	1	1	yes	characteristic
102	C₂×Dic₁₃₉	C₂⁴	1	3	yes	—
103	C₂×Dic₁₃₉	C₂⁴	1	2	yes	—
104	C₂×Dic₁₃₉	C₂⁴	1	1	yes	characteristic
105	C₄×D₁₃₉	—	2	12	no	—
106	C₄×D₁₃₉	—	2	12	no	—
107	C₄×D₁₃₉	—	2	6	no	—
108	C₄×D₁₃₉	—	2	12	no	—
109	C₄×D₁₃₉	—	4	3	no	—
110	C₄×D₁₃₉	C₂×D₄	1	12	yes	complemented
111	C₂³×C₁₃₉	C₂⁴	1	2	yes	—
112	Q₈×C₁₃₉	—	2	12	no	—
113	Q₈×C₁₃₉	C₂×D₄	1	4	yes	complemented
114	D₄×C₁₃₉	#41	1	4	yes	complemented
115	C₂×C₄×C₁₃₉	—	2	12	no	—
116	C₂×C₄×C₁₃₉	C₂⁴	1	3	yes	—
117	C₂×C₄×C₁₃₉	C₂⁴	1	6	yes	—
118	C₂×C₄×C₁₃₉	C₂⁴	1	3	yes	—
119	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
120	C₂²×C₁₃₉	—	2	2	no	—
121	C₂²×C₁₃₉	—	2	2	no	—
122	C₂²×C₁₃₉	C₂×#41	1	4	yes	complemented
123	C₂²×C₁₃₉	C₂⁵	1	1	yes	characteristic
124	C₂×D₁₃₉	—	2	3	no	—
125	C₂×D₁₃₉	—	2	3	no	—
126	C₂×D₁₃₉	—	2	6	no	—
127	C₂×D₁₃₉	—	2	6	no	—
128	C₂×D₁₃₉	—	2	6	no	—
129	C₂×D₁₃₉	—	4	24	no	—
130	C₂×D₁₃₉	C₂²×D₄	1	6	yes	—
131	C₄×C₁₃₉	—	2	6	no	—
132	C₄×C₁₃₉	—	2	3	no	—
133	C₄×C₁₃₉	C₂²×D₄	1	6	yes	—
134	C₄×C₁₃₉	C₂×#41	1	2	yes	complemented
135	Dic₁₃₉	—	2	2	no	—
136	Dic₁₃₉	—	2	3	no	—
137	Dic₁₃₉	—	2	1	no	—
138	Dic₁₃₉	C₂²×D₄	1	2	yes	—
139	C₂×C₁₃₉	—	2	4	no	—
140	C₂×C₁₃₉	C₂³×D₄	1	1	yes	characteristic
141	C₂×C₁₃₉	C₂²×#41	1	1	yes	characteristic
142	C₂×C₁₃₉	#585	1	1	yes	characteristic
143	D₁₃₉	—	2	6	no	—
144	D₁₃₉	—	4	6	no	—
145	C₁₃₉	D₄×#41	1	1	yes	characteristic, complemented, Sylow
146	D₄×#41	—	139	1	no	maximal, Sylow
147	C₂²×#41	—	139	2	no	—
148	#550	—	139	1	no	—
149	D₄×Q₈	D₁₃₉	1	1	yes	characteristic, complemented
150	#548	—	139	2	no	—
151	#547	—	139	3	no	—
152	#546	—	139	6	no	—
153	D₄²	—	139	3	no	—
154	#532	—	139	3	no	—
155	#525	—	139	6	no	—
156	C₄×#41	—	139	1	no	—
157	C₂×C₄×D₄	—	139	3	no	—
158	C₂×#41	—	278	16	no	—
159	C₂×#41	—	139	6	no	—
160	C₂×#41	—	139	6	no	—
161	C₂×#41	—	139	2	no	—
162	C₂×#41	—	139	1	no	—
163	C₂²×Q₈	C₂×D₁₃₉	1	2	yes	complemented
164	C₂²×D₄	—	139	3	no	—
165	C₂²×D₄	—	139	6	no	—
166	C₂³×C₄	—	139	6	no	—
167	C₄⋊Q₈	C₂×D₁₃₉	1	3	yes	complemented
168	#127	—	139	3	no	—
169	#124	—	139	6	no	—
170	#123	—	139	12	no	—
171	C₂²⋊Q₈	—	139	6	no	—
172	C₂²⋊Q₈	C₂×D₁₃₉	1	6	yes	complemented
173	#121	—	139	6	no	—
174	#121	—	139	6	no	—
175	#121	—	139	6	no	—
176	#121	—	139	6	no	—
177	#120	—	139	12	no	—
178	C₄×Q₈	—	139	1	no	—
179	C₄×Q₈	C₂×D₁₃₉	1	1	yes	characteristic, complemented
180	C₄×D₄	—	139	1	no	—
181	C₄×D₄	—	139	6	no	—
182	C₄×D₄	—	139	3	no	—
183	C₄×D₄	—	139	3	no	—
184	C₄×D₄	—	139	6	no	—
185	C₄×D₄	—	139	6	no	—
186	C₄×D₄	—	139	3	no	—
187	C₄×D₄	—	139	3	no	—
188	C₄×D₄	C₂×D₁₃₉	1	3	yes	—
189	#117	—	139	3	no	—
190	C₂×C₄⋊C₄	—	139	3	no	—
191	C₂×C₂²⋊C₄	—	139	6	no	—
192	C₂×C₄²	—	139	3	no	—
193	C₂⁴	—	139	6	no	—
194	#41	—	278	8	no	—
195	#41	—	278	24	no	—
196	#41	—	278	24	no	—
197	#41	—	139	8	no	—
198	C₂×Q₈	—	2	8	no	—
199	C₂×Q₈	C₂²×D₁₃₉	1	6	yes	—
200	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
201	C₂×D₄	—	278	24	no	—
202	C₂×D₄	—	278	12	no	—
203	C₂×D₄	—	139	6	no	—
204	C₂×D₄	—	139	3	no	—
205	C₂×D₄	—	139	12	no	—
206	C₂×D₄	—	139	6	no	—
207	C₂×D₄	—	139	6	no	—
208	C₂×D₄	—	139	3	no	—
209	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
210	C₂²×C₄	—	278	24	no	—
211	C₂²×C₄	—	139	2	no	—
212	C₂²×C₄	—	139	3	no	—
213	C₂²×C₄	—	139	6	no	—
214	C₂²×C₄	—	139	6	no	—
215	C₂²×C₄	—	139	6	no	—
216	C₂²×C₄	—	139	3	no	—
217	C₂²×C₄	—	139	3	no	—
218	C₂²×C₄	C₂²×D₁₃₉	1	6	yes	—
219	C₄⋊C₄	—	139	1	no	—
220	C₄⋊C₄	—	139	6	no	—
221	C₄⋊C₄	—	139	3	no	—
222	C₄⋊C₄	C₂²×D₁₃₉	1	3	yes	—
223	C₄⋊C₄	C₂²×D₁₃₉	1	3	yes	—
224	C₄⋊C₄	C₂²×D₁₃₉	1	6	yes	—
225	C₂²⋊C₄	—	139	2	no	—
226	C₂²⋊C₄	—	139	6	no	—
227	C₂²⋊C₄	—	139	6	no	—
228	C₂²⋊C₄	—	139	6	no	—
229	C₂²⋊C₄	—	139	12	no	—
230	C₂²⋊C₄	C₂²×D₁₃₉	1	6	yes	—
231	C₄²	—	139	1	no	—
232	C₄²	—	139	3	no	—
233	C₄²	—	139	3	no	—
234	C₄²	C₂²×D₁₃₉	1	3	yes	—
235	C₂³	—	278	12	no	—
236	C₂³	—	278	12	no	—
237	C₂³	—	556	6	no	—
238	C₂³	—	139	6	no	—
239	C₂³	—	139	3	no	—
240	C₂³	C₂³×D₁₃₉	1	2	yes	—
241	Q₈	—	2	12	no	—
242	Q₈	D₄×D₁₃₉	1	4	yes	complemented
243	D₄	—	278	24	no	—
244	D₄	—	278	12	no	—
245	D₄	—	278	12	no	—
246	D₄	—	278	6	no	—
247	D₄	—	556	6	no	—
248	D₄	—	139	12	no	—
249	D₄	#14555786	1	4	yes	direct product factor
250	C₂×C₄	—	2	12	no	—
251	C₂×C₄	—	278	12	no	—
252	C₂×C₄	—	278	12	no	—
253	C₂×C₄	—	278	4	no	—
254	C₂×C₄	—	278	12	no	—
255	C₂×C₄	—	278	6	no	—
256	C₂×C₄	—	556	3	no	—
257	C₂×C₄	—	139	1	no	—
258	C₂×C₄	—	139	3	no	—
259	C₂×C₄	—	139	2	no	—
260	C₂×C₄	—	139	1	no	—
261	C₂×C₄	—	139	12	no	—
262	C₂×C₄	C₂³×D₁₃₉	1	3	yes	—
263	C₂×C₄	C₂³×D₁₃₉	1	6	yes	—
264	C₂×C₄	C₂³×D₁₃₉	1	3	yes	—
265	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	278	3	no	—
269	C₂²	—	278	3	no	—
270	C₂²	—	278	6	no	—
271	C₂²	—	278	6	no	—
272	C₂²	—	278	6	no	—
273	C₂²	—	556	24	no	—
274	C₂²	—	139	6	no	—
275	C₂²	C₂×#14555786	1	4	yes	complemented
276	C₂²	C₂⁴×D₁₃₉	1	1	yes	characteristic
277	C₄	—	2	6	no	—
278	C₄	—	2	3	no	—
279	C₄	—	278	2	no	—
280	C₄	—	278	3	no	—
281	C₄	—	278	1	no	—
282	C₄	—	139	2	no	—
283	C₄	C₂×D₄×D₁₃₉	1	6	yes	—
284	C₄	C₂×#14555786	1	2	yes	complemented
285	C₂	—	2	4	no	—
286	C₂	—	278	6	no	—
287	C₂	—	556	6	no	—
288	C₂	C₂²×D₄×D₁₃₉	1	1	yes	characteristic
289	C₂	C₂²×#14555786	1	1	yes	characteristic
290	C₂	#22581790	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃₉
commutator, socle, cosocle	C₂²×C₁₃₉	C₂⁵
Fitting	D₄×Q₈×C₁₃₉	C₂
hypercenter	D₄×Q₈	D₁₃₉
hypocenter	C₁₃₉	D₄×#41
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ D₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#14555786	C₂	semidirect
2. #22581625	C₂	semidirect
3. #22581614	C₂	semidirect
4. #22581606	C₂	semidirect
5. #22581594	C₂	semidirect
6. D₄×D₅₅₆	C₂	semidirect
7. #22581504	C₂	semidirect
8. #22581466	C₂	semidirect
9. C₄×#14555786	C₂	semidirect
10. C₄×D₄×D₁₃₉	C₂	semidirect
11. D₄×Q₈×C₁₃₉	C₂	semidirect
14. C₂×#14555786	C₂²	semidirect
15. C₂×#14555786	C₂²	semidirect
16. C₂×#14555786	C₂²	semidirect
17. C₂²×D₅₅₆	C₂²	semidirect
19. C₂²×C₄×D₁₃₉	C₂²	semidirect
20. #17984839	C₂²	semidirect
21. #17984831	C₂²	semidirect
22. #17984827	C₂²	semidirect
23. #17984818	C₂²	semidirect
24. #17984817	C₂²	semidirect
25. #17984817	C₂²	semidirect
26. #17984817	C₂²	semidirect
27. #17984816	C₂²	semidirect
28. #17984812	C₂²	semidirect
29. Q₈×Dic₁₃₉	C₂²	semidirect
30. C₄×D₅₅₆	C₂²	semidirect
31. C₄×D₅₅₆	C₂²	semidirect
33. #17984804	C₂²	semidirect
34. #17984804	C₂²	semidirect
35. #17984804	C₂²	semidirect
36. C₄×C₂²⋊D₁₃₉	C₂²	semidirect
37. (C₂²⋊D₁₃₉)⋊C₄	C₂²	semidirect
38. #17984800	C₂²	semidirect
39. C₄⋊C₄×D₁₃₉	C₂²	semidirect
40. C₂²⋊C₄×D₁₃₉	C₂²	semidirect
41. C₄²×D₁₃₉	C₂²	semidirect
42. C₂²×Q₈×C₁₃₉	C₂²	semidirect
43. C₄⋊Q₈×C₁₃₉	C₂²	semidirect
44. C₂²⋊Q₈×C₁₃₉	C₂²	semidirect
45. C₄×Q₈×C₁₃₉	C₂²	semidirect
51. #14555786	D₄	direct (N × H)
98. D₅₅₆	C₂×D₄	semidirect
110. C₄×D₁₃₉	C₂×D₄	semidirect
113. Q₈×C₁₃₉	C₂×D₄	semidirect
114. D₄×C₁₃₉	#41	semidirect
122. C₂²×C₁₃₉	C₂×#41	semidirect
134. C₄×C₁₃₉	C₂×#41	semidirect
145. C₁₃₉	D₄×#41	semidirect
149. D₄×Q₈	D₁₃₉	semidirect
163. C₂²×Q₈	C₂×D₁₃₉	semidirect
167. C₄⋊Q₈	C₂×D₁₃₉	semidirect
172. C₂²⋊Q₈	C₂×D₁₃₉	semidirect
179. C₄×Q₈	C₂×D₁₃₉	semidirect
242. Q₈	D₄×D₁₃₉	semidirect
249. D₄	#14555786	direct (N × H)
275. C₂²	C₂×#14555786	semidirect
284. C₄	C₂×#14555786	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#14555786	2
#22581625	2
#22581614	2
#22581606	2
#22581594	2
D₄×D₅₅₆	2
#22581504	2
#22581466	2
C₄×#14555786	2
C₄×D₄×D₁₃₉	2
D₄×Q₈×C₁₃₉	2
D₄×#41	139

Subgroups — D4×#14555786 #31819896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#14555786 #31819896