Subgroups — D₄×#14555511 #31794281

Order 17536.

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×#14555511	C₂	1	2	yes	maximal, complemented
2	#22578075	C₂	1	1	yes	maximal, characteristic, complemented
3	#22578064	C₂	1	2	yes	maximal, complemented
4	#22578056	C₂	1	3	yes	maximal, complemented
5	#22578044	C₂	1	6	yes	maximal, complemented
6	D₄×D₅₄₈	C₂	1	3	yes	maximal, complemented
7	#22577954	C₂	1	3	yes	maximal, complemented
8	#22577916	C₂	1	6	yes	maximal, complemented
9	C₄×#14555511	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₄×D₁₃₇	C₂	1	3	yes	maximal, complemented
11	D₄×Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14555511	—	2	16	no	—
13	C₂×#14555511	C₂²	1	1	yes	characteristic
14	C₂×#14555511	C₂²	1	6	yes	complemented
15	C₂×#14555511	C₂²	1	6	yes	complemented
16	C₂×#14555511	C₂²	1	2	yes	complemented
17	C₂²×D₅₄₈	C₂²	1	6	yes	complemented
18	C₂×D₄×D₁₃₇	C₂²	1	3	yes	—
19	C₂²×C₄×D₁₃₇	C₂²	1	6	yes	complemented
20	#17984085	C₂²	1	3	yes	complemented
21	#17984077	C₂²	1	6	yes	complemented
22	#17984073	C₂²	1	12	yes	complemented
23	#17984064	C₂²	1	6	yes	complemented
24	#17984063	C₂²	1	6	yes	complemented
25	#17984063	C₂²	1	6	yes	complemented
26	#17984063	C₂²	1	6	yes	complemented
27	#17984062	C₂²	1	6	yes	complemented
28	#17984058	C₂²	1	12	yes	complemented
29	Q₈×Dic₁₃₇	C₂²	1	1	yes	characteristic, complemented
30	C₄×D₅₄₈	C₂²	1	3	yes	complemented
31	C₄×D₅₄₈	C₂²	1	3	yes	complemented
32	D₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
33	#17984050	C₂²	1	3	yes	complemented
34	#17984050	C₂²	1	6	yes	complemented
35	#17984050	C₂²	1	3	yes	complemented
36	C₄×C₂²⋊D₁₃₇	C₂²	1	6	yes	complemented
37	#17984048	C₂²	1	6	yes	complemented
38	#17984046	C₂²	1	3	yes	complemented
39	C₄⋊C₄×D₁₃₇	C₂²	1	3	yes	complemented
40	C₂²⋊C₄×D₁₃₇	C₂²	1	6	yes	complemented
41	C₄²×D₁₃₇	C₂²	1	3	yes	complemented
42	C₂²×Q₈×C₁₃₇	C₂²	1	2	yes	complemented
43	C₄⋊Q₈×C₁₃₇	C₂²	1	3	yes	complemented
44	C₂²⋊Q₈×C₁₃₇	C₂²	1	6	yes	complemented
45	C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
46	C₄×D₄×C₁₃₇	C₂²	1	3	yes	—
47	C₂³×D₁₃₇	C₂³	1	6	yes	—
48	#14555511	—	2	24	no	—
49	#14555511	—	2	24	no	—
50	#14555511	—	2	8	no	—
51	#14555511	D₄	1	8	yes	direct product factor
52	C₂×D₅₄₈	—	2	24	no	—
53	C₂×D₅₄₈	C₂³	1	3	yes	—
54	C₂×D₅₄₈	C₂³	1	12	yes	—
55	C₂×D₅₄₈	C₂³	1	6	yes	—
56	C₂×D₅₄₈	C₂³	1	6	yes	—
57	C₂×D₅₄₈	C₂³	1	3	yes	—
58	C₂×C₂²⋊D₁₃₇	C₂³	1	6	yes	—
59	D₄×D₁₃₇	—	2	12	no	—
60	C₂²×Dic₁₃₇	C₂³	1	2	yes	—
61	C₂×C₄×D₁₃₇	—	2	24	no	—
62	C₂×C₄×D₁₃₇	C₂³	1	3	yes	—
63	C₂×C₄×D₁₃₇	C₂³	1	6	yes	—
64	C₂×C₄×D₁₃₇	C₂³	1	6	yes	—
65	C₂×C₄×D₁₃₇	C₂³	1	6	yes	—
66	C₂×C₄×D₁₃₇	C₂³	1	3	yes	—
67	C₂×C₄×D₁₃₇	C₂³	1	3	yes	—
68	C₄⋊Dic₁₃₇	C₂³	1	1	yes	characteristic
69	C₄⋊Dic₁₃₇	C₂³	1	3	yes	—
70	#14555489	C₂³	1	6	yes	—
71	C₂²⋊Dic₁₃₇	C₂³	1	2	yes	—
72	#14555487	C₂³	1	6	yes	—
73	#14555487	C₂³	1	6	yes	—
74	#14555487	C₂³	1	6	yes	—
75	#14555487	C₂³	1	12	yes	—
76	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
77	C₄×Dic₁₃₇	C₂³	1	3	yes	—
78	C₄×Dic₁₃₇	C₂³	1	3	yes	—
79	C₂×Q₈×C₁₃₇	—	2	8	no	—
80	C₂×Q₈×C₁₃₇	C₂³	1	6	yes	—
81	C₂×Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
82	C₂×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
83	C₂²×C₄×C₁₃₇	C₂³	1	6	yes	—
84	C₄⋊C₄×C₁₃₇	C₂³	1	3	yes	—
85	C₄⋊C₄×C₁₃₇	C₂³	1	3	yes	—
86	C₄⋊C₄×C₁₃₇	C₂³	1	6	yes	—
87	C₂²⋊C₄×C₁₃₇	C₂³	1	6	yes	—
88	C₄²×C₁₃₇	C₂³	1	3	yes	—
89	C₂²×D₁₃₇	—	2	12	no	—
90	C₂²×D₁₃₇	—	2	12	no	—
91	C₂²×D₁₃₇	—	4	6	no	—
92	C₂²×D₁₃₇	C₂⁴	1	6	yes	—
93	C₂²×D₁₃₇	C₂⁴	1	3	yes	—
94	D₅₄₈	—	2	24	no	—
95	D₅₄₈	—	2	12	no	—
96	D₅₄₈	—	2	6	no	—
97	D₅₄₈	—	4	6	no	—
98	D₅₄₈	C₂×D₄	1	12	yes	complemented
99	C₂²⋊D₁₃₇	—	2	12	no	—
100	C₂×Dic₁₃₇	—	2	4	no	—
101	C₂×Dic₁₃₇	C₂⁴	1	1	yes	characteristic
102	C₂×Dic₁₃₇	C₂⁴	1	3	yes	—
103	C₂×Dic₁₃₇	C₂⁴	1	2	yes	—
104	C₂×Dic₁₃₇	C₂⁴	1	1	yes	characteristic
105	C₄×D₁₃₇	—	2	12	no	—
106	C₄×D₁₃₇	—	2	12	no	—
107	C₄×D₁₃₇	—	2	6	no	—
108	C₄×D₁₃₇	—	2	12	no	—
109	C₄×D₁₃₇	—	4	3	no	—
110	C₄×D₁₃₇	C₂×D₄	1	12	yes	complemented
111	C₂³×C₁₃₇	C₂⁴	1	2	yes	—
112	Q₈×C₁₃₇	—	2	12	no	—
113	Q₈×C₁₃₇	C₂×D₄	1	4	yes	complemented
114	D₄×C₁₃₇	#41	1	4	yes	complemented
115	C₂×C₄×C₁₃₇	—	2	12	no	—
116	C₂×C₄×C₁₃₇	C₂⁴	1	3	yes	—
117	C₂×C₄×C₁₃₇	C₂⁴	1	6	yes	—
118	C₂×C₄×C₁₃₇	C₂⁴	1	3	yes	—
119	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
120	C₂²×C₁₃₇	—	2	2	no	—
121	C₂²×C₁₃₇	—	2	2	no	—
122	C₂²×C₁₃₇	C₂×#41	1	4	yes	complemented
123	C₂²×C₁₃₇	C₂⁵	1	1	yes	characteristic
124	C₂×D₁₃₇	—	2	3	no	—
125	C₂×D₁₃₇	—	2	3	no	—
126	C₂×D₁₃₇	—	2	6	no	—
127	C₂×D₁₃₇	—	2	6	no	—
128	C₂×D₁₃₇	—	2	6	no	—
129	C₂×D₁₃₇	—	4	24	no	—
130	C₂×D₁₃₇	C₂²×D₄	1	6	yes	—
131	C₄×C₁₃₇	—	2	6	no	—
132	C₄×C₁₃₇	—	2	3	no	—
133	C₄×C₁₃₇	C₂²×D₄	1	6	yes	—
134	C₄×C₁₃₇	C₂×#41	1	2	yes	complemented
135	Dic₁₃₇	—	2	2	no	—
136	Dic₁₃₇	—	2	3	no	—
137	Dic₁₃₇	—	2	1	no	—
138	Dic₁₃₇	C₂²×D₄	1	2	yes	—
139	C₂×C₁₃₇	—	2	4	no	—
140	C₂×C₁₃₇	C₂³×D₄	1	1	yes	characteristic
141	C₂×C₁₃₇	C₂²×#41	1	1	yes	characteristic
142	C₂×C₁₃₇	#585	1	1	yes	characteristic
143	D₁₃₇	—	2	6	no	—
144	D₁₃₇	—	4	6	no	—
145	C₁₃₇	D₄×#41	1	1	yes	characteristic, complemented, Sylow
146	D₄×#41	—	137	1	no	maximal, Sylow
147	C₂²×#41	—	137	2	no	—
148	#550	—	137	1	no	—
149	D₄×Q₈	D₁₃₇	1	1	yes	characteristic, complemented
150	#548	—	137	2	no	—
151	#547	—	137	3	no	—
152	#546	—	137	6	no	—
153	D₄²	—	137	3	no	—
154	#532	—	137	3	no	—
155	#525	—	137	6	no	—
156	C₄×#41	—	137	1	no	—
157	C₂×C₄×D₄	—	137	3	no	—
158	C₂×#41	—	274	16	no	—
159	C₂×#41	—	137	6	no	—
160	C₂×#41	—	137	6	no	—
161	C₂×#41	—	137	2	no	—
162	C₂×#41	—	137	1	no	—
163	C₂²×Q₈	C₂×D₁₃₇	1	2	yes	complemented
164	C₂²×D₄	—	137	3	no	—
165	C₂²×D₄	—	137	6	no	—
166	C₂³×C₄	—	137	6	no	—
167	C₄⋊Q₈	C₂×D₁₃₇	1	3	yes	complemented
168	#127	—	137	3	no	—
169	#124	—	137	6	no	—
170	#123	—	137	12	no	—
171	C₂²⋊Q₈	—	137	6	no	—
172	C₂²⋊Q₈	C₂×D₁₃₇	1	6	yes	complemented
173	#121	—	137	6	no	—
174	#121	—	137	6	no	—
175	#121	—	137	6	no	—
176	#121	—	137	6	no	—
177	#120	—	137	12	no	—
178	C₄×Q₈	—	137	1	no	—
179	C₄×Q₈	C₂×D₁₃₇	1	1	yes	characteristic, complemented
180	C₄×D₄	—	137	1	no	—
181	C₄×D₄	—	137	6	no	—
182	C₄×D₄	—	137	3	no	—
183	C₄×D₄	—	137	3	no	—
184	C₄×D₄	—	137	6	no	—
185	C₄×D₄	—	137	6	no	—
186	C₄×D₄	—	137	3	no	—
187	C₄×D₄	—	137	3	no	—
188	C₄×D₄	C₂×D₁₃₇	1	3	yes	—
189	#117	—	137	3	no	—
190	C₂×C₄⋊C₄	—	137	3	no	—
191	C₂×C₂²⋊C₄	—	137	6	no	—
192	C₂×C₄²	—	137	3	no	—
193	C₂⁴	—	137	6	no	—
194	#41	—	274	8	no	—
195	#41	—	274	24	no	—
196	#41	—	274	24	no	—
197	#41	—	137	8	no	—
198	C₂×Q₈	—	2	8	no	—
199	C₂×Q₈	C₂²×D₁₃₇	1	6	yes	—
200	C₂×Q₈	C₂²×D₁₃₇	1	1	yes	characteristic
201	C₂×D₄	—	274	24	no	—
202	C₂×D₄	—	274	12	no	—
203	C₂×D₄	—	137	6	no	—
204	C₂×D₄	—	137	3	no	—
205	C₂×D₄	—	137	12	no	—
206	C₂×D₄	—	137	6	no	—
207	C₂×D₄	—	137	6	no	—
208	C₂×D₄	—	137	3	no	—
209	C₂×D₄	C₂²×D₁₃₇	1	1	yes	characteristic
210	C₂²×C₄	—	274	24	no	—
211	C₂²×C₄	—	137	2	no	—
212	C₂²×C₄	—	137	3	no	—
213	C₂²×C₄	—	137	6	no	—
214	C₂²×C₄	—	137	6	no	—
215	C₂²×C₄	—	137	6	no	—
216	C₂²×C₄	—	137	3	no	—
217	C₂²×C₄	—	137	3	no	—
218	C₂²×C₄	C₂²×D₁₃₇	1	6	yes	—
219	C₄⋊C₄	—	137	6	no	—
220	C₄⋊C₄	—	137	1	no	—
221	C₄⋊C₄	—	137	3	no	—
222	C₄⋊C₄	C₂²×D₁₃₇	1	3	yes	—
223	C₄⋊C₄	C₂²×D₁₃₇	1	3	yes	—
224	C₄⋊C₄	C₂²×D₁₃₇	1	6	yes	—
225	C₂²⋊C₄	—	137	2	no	—
226	C₂²⋊C₄	—	137	6	no	—
227	C₂²⋊C₄	—	137	6	no	—
228	C₂²⋊C₄	—	137	6	no	—
229	C₂²⋊C₄	—	137	12	no	—
230	C₂²⋊C₄	C₂²×D₁₃₇	1	6	yes	—
231	C₄²	—	137	1	no	—
232	C₄²	—	137	3	no	—
233	C₄²	—	137	3	no	—
234	C₄²	C₂²×D₁₃₇	1	3	yes	—
235	C₂³	—	274	12	no	—
236	C₂³	—	274	12	no	—
237	C₂³	—	548	6	no	—
238	C₂³	—	137	6	no	—
239	C₂³	—	137	3	no	—
240	C₂³	C₂³×D₁₃₇	1	2	yes	—
241	Q₈	—	2	12	no	—
242	Q₈	D₄×D₁₃₇	1	4	yes	complemented
243	D₄	—	274	24	no	—
244	D₄	—	274	12	no	—
245	D₄	—	274	12	no	—
246	D₄	—	274	6	no	—
247	D₄	—	548	6	no	—
248	D₄	—	137	12	no	—
249	D₄	#14555511	1	4	yes	direct product factor
250	C₂×C₄	—	2	12	no	—
251	C₂×C₄	—	274	12	no	—
252	C₂×C₄	—	274	12	no	—
253	C₂×C₄	—	274	4	no	—
254	C₂×C₄	—	274	12	no	—
255	C₂×C₄	—	274	6	no	—
256	C₂×C₄	—	548	3	no	—
257	C₂×C₄	—	137	1	no	—
258	C₂×C₄	—	137	3	no	—
259	C₂×C₄	—	137	2	no	—
260	C₂×C₄	—	137	1	no	—
261	C₂×C₄	—	137	12	no	—
262	C₂×C₄	C₂³×D₁₃₇	1	3	yes	—
263	C₂×C₄	C₂³×D₁₃₇	1	6	yes	—
264	C₂×C₄	C₂³×D₁₃₇	1	3	yes	—
265	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	274	3	no	—
269	C₂²	—	274	3	no	—
270	C₂²	—	274	6	no	—
271	C₂²	—	274	6	no	—
272	C₂²	—	274	6	no	—
273	C₂²	—	548	24	no	—
274	C₂²	—	137	6	no	—
275	C₂²	C₂×#14555511	1	4	yes	complemented
276	C₂²	C₂⁴×D₁₃₇	1	1	yes	characteristic
277	C₄	—	2	6	no	—
278	C₄	—	2	3	no	—
279	C₄	—	274	2	no	—
280	C₄	—	274	3	no	—
281	C₄	—	274	1	no	—
282	C₄	—	137	2	no	—
283	C₄	C₂×D₄×D₁₃₇	1	6	yes	—
284	C₄	C₂×#14555511	1	2	yes	complemented
285	C₂	—	2	4	no	—
286	C₂	—	274	6	no	—
287	C₂	—	548	6	no	—
288	C₂	C₂²×D₄×D₁₃₇	1	1	yes	characteristic
289	C₂	C₂²×#14555511	1	1	yes	characteristic
290	C₂	#22578240	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃₇
commutator, socle, cosocle	C₂²×C₁₃₇	C₂⁵
Fitting	D₄×Q₈×C₁₃₇	C₂
hypercenter	D₄×Q₈	D₁₃₇
hypocenter	C₁₃₇	D₄×#41
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ D₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#14555511	C₂	semidirect
2. #22578075	C₂	semidirect
3. #22578064	C₂	semidirect
4. #22578056	C₂	semidirect
5. #22578044	C₂	semidirect
6. D₄×D₅₄₈	C₂	semidirect
7. #22577954	C₂	semidirect
8. #22577916	C₂	semidirect
9. C₄×#14555511	C₂	semidirect
10. C₄×D₄×D₁₃₇	C₂	semidirect
11. D₄×Q₈×C₁₃₇	C₂	semidirect
14. C₂×#14555511	C₂²	semidirect
15. C₂×#14555511	C₂²	semidirect
16. C₂×#14555511	C₂²	semidirect
17. C₂²×D₅₄₈	C₂²	semidirect
19. C₂²×C₄×D₁₃₇	C₂²	semidirect
20. #17984085	C₂²	semidirect
21. #17984077	C₂²	semidirect
22. #17984073	C₂²	semidirect
23. #17984064	C₂²	semidirect
24. #17984063	C₂²	semidirect
25. #17984063	C₂²	semidirect
26. #17984063	C₂²	semidirect
27. #17984062	C₂²	semidirect
28. #17984058	C₂²	semidirect
29. Q₈×Dic₁₃₇	C₂²	semidirect
30. C₄×D₅₄₈	C₂²	semidirect
31. C₄×D₅₄₈	C₂²	semidirect
33. #17984050	C₂²	semidirect
34. #17984050	C₂²	semidirect
35. #17984050	C₂²	semidirect
36. C₄×C₂²⋊D₁₃₇	C₂²	semidirect
37. #17984048	C₂²	semidirect
38. #17984046	C₂²	semidirect
39. C₄⋊C₄×D₁₃₇	C₂²	semidirect
40. C₂²⋊C₄×D₁₃₇	C₂²	semidirect
41. C₄²×D₁₃₇	C₂²	semidirect
42. C₂²×Q₈×C₁₃₇	C₂²	semidirect
43. C₄⋊Q₈×C₁₃₇	C₂²	semidirect
44. C₂²⋊Q₈×C₁₃₇	C₂²	semidirect
45. C₄×Q₈×C₁₃₇	C₂²	semidirect
51. #14555511	D₄	direct (N × H)
98. D₅₄₈	C₂×D₄	semidirect
110. C₄×D₁₃₇	C₂×D₄	semidirect
113. Q₈×C₁₃₇	C₂×D₄	semidirect
114. D₄×C₁₃₇	#41	semidirect
122. C₂²×C₁₃₇	C₂×#41	semidirect
134. C₄×C₁₃₇	C₂×#41	semidirect
145. C₁₃₇	D₄×#41	semidirect
149. D₄×Q₈	D₁₃₇	semidirect
163. C₂²×Q₈	C₂×D₁₃₇	semidirect
167. C₄⋊Q₈	C₂×D₁₃₇	semidirect
172. C₂²⋊Q₈	C₂×D₁₃₇	semidirect
179. C₄×Q₈	C₂×D₁₃₇	semidirect
242. Q₈	D₄×D₁₃₇	semidirect
249. D₄	#14555511	direct (N × H)
275. C₂²	C₂×#14555511	semidirect
284. C₄	C₂×#14555511	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14555511	2
#22578075	2
#22578064	2
#22578056	2
#22578044	2
D₄×D₅₄₈	2
#22577954	2
#22577916	2
C₄×#14555511	2
C₄×D₄×D₁₃₇	2
D₄×Q₈×C₁₃₇	2
D₄×#41	137

Subgroups — D4×#14555511 #31794281

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#14555511 #31794281