Subgroups — #550×D₁₃₇ #31794265

Order 17536.

1736 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×D₁₃₇	C₂	1	2	yes	maximal, complemented
2	#22578075	C₂	1	1	yes	maximal, characteristic, complemented
3	#22578074	C₂	1	3	yes	maximal, complemented
4	#22578073	C₂	1	3	yes	maximal, complemented
5	#22578072	C₂	1	2	yes	maximal, complemented
6	#22578071	C₂	1	6	yes	maximal, complemented
7	#127×D₁₃₇	C₂	1	3	yes	maximal, complemented
8	#121×D₁₃₇	C₂	1	6	yes	maximal, complemented
9	C₄×Q₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₄×D₁₃₇	C₂	1	3	yes	maximal, complemented
11	#550×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14555511	C₂²	1	2	yes	complemented
13	C₂×#14555510	C₂²	1	6	yes	complemented
14	C₂×#14555509	C₂²	1	6	yes	complemented
15	#41×D₁₃₇	—	2	16	no	—
16	C₂×Q₈×D₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×D₄×D₁₃₇	C₂²	1	3	yes	—
18	C₂×D₄×D₁₃₇	C₂²	1	6	yes	—
19	C₂×D₄×D₁₃₇	C₂²	1	6	yes	—
20	C₂²×C₄×D₁₃₇	C₂²	1	6	yes	—
21	#17984085	C₂²	1	3	yes	complemented
22	#17984084	C₂²	1	6	yes	complemented
23	#17984084	C₂²	1	6	yes	complemented
24	#17984084	C₂²	1	6	yes	complemented
25	#17984063	C₂²	1	6	yes	complemented
26	#17984062	C₂²	1	6	yes	complemented
27	#17984061	C₂²	1	12	yes	complemented
28	#17984060	C₂²	1	12	yes	complemented
29	#17984059	C₂²	1	6	yes	complemented
30	Q₈×Dic₁₃₇	C₂²	1	1	yes	characteristic, complemented
31	C₄×Dic₂₇₄	C₂²	1	3	yes	complemented
32	#17984053	C₂²	1	3	yes	complemented
33	C₄×D₅₄₈	C₂²	1	3	yes	complemented
34	D₄×Dic₁₃₇	C₂²	1	3	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	C₄⋊C₄×D₁₃₇	C₂²	1	1	yes	characteristic
39	C₄⋊C₄×D₁₃₇	C₂²	1	3	yes	—
40	C₂²⋊C₄×D₁₃₇	C₂²	1	6	yes	—
41	C₄²×D₁₃₇	C₂²	1	3	yes	—
42	C₂×#41×C₁₃₇	C₂²	1	2	yes	complemented
43	#127×C₁₃₇	C₂²	1	3	yes	complemented
44	#121×C₁₃₇	C₂²	1	6	yes	complemented
45	C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
46	C₄×D₄×C₁₃₇	C₂²	1	3	yes	complemented
47	C₂³×D₁₃₇	C₂³	1	6	yes	—
48	#14555511	—	2	8	no	—
49	#14555510	—	2	24	no	—
50	#14555509	—	2	24	no	—
51	C₂×Dic₂₇₄	C₂³	1	3	yes	—
52	Q₈×D₁₃₇	D₄	1	8	yes	complemented
53	C₂×D₅₄₈	C₂³	1	3	yes	—
54	C₂×D₅₄₈	C₂³	1	6	yes	—
55	C₂×D₅₄₈	C₂³	1	6	yes	—
56	C₂×C₂²⋊D₁₃₇	C₂³	1	6	yes	—
57	C₂×C₂²⋊D₁₃₇	C₂³	1	12	yes	—
58	C₂×C₂²⋊D₁₃₇	C₂³	1	12	yes	—
59	D₄×D₁₃₇	—	2	24	no	—
60	D₄×D₁₃₇	—	2	12	no	—
61	D₄×D₁₃₇	—	4	12	no	—
62	C₂²×Dic₁₃₇	C₂³	1	6	yes	—
63	C₂×C₄×D₁₃₇	—	2	24	no	—
64	C₂×C₄×D₁₃₇	C₂³	1	3	yes	—
65	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
66	C₂×C₄×D₁₃₇	C₂³	1	6	yes	—
67	C₂×C₄×D₁₃₇	C₂³	1	6	yes	—
68	C₂×C₄×D₁₃₇	C₂³	1	3	yes	—
69	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₁₃₇	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₁₃₇	C₂³	1	3	yes	—
72	#14555489	C₂³	1	2	yes	—
73	#14555489	C₂³	1	6	yes	—
74	C₂²⋊Dic₁₃₇	C₂³	1	6	yes	—
75	#14555487	C₂³	1	6	yes	—
76	#14555487	C₂³	1	6	yes	—
77	C₄×Dic₁₃₇	C₂³	1	3	yes	—
78	C₄×Dic₁₃₇	C₂³	1	3	yes	—
79	C₄×Dic₁₃₇	C₂³	1	3	yes	—
80	#41×C₁₃₇	—	2	8	no	—
81	C₂×Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
82	C₂×D₄×C₁₃₇	C₂³	1	3	yes	—
83	C₂×D₄×C₁₃₇	C₂³	1	6	yes	—
84	C₂×D₄×C₁₃₇	C₂³	1	6	yes	—
85	C₂²×C₄×C₁₃₇	C₂³	1	6	yes	—
86	C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
87	C₄⋊C₄×C₁₃₇	C₂³	1	3	yes	—
88	C₂²⋊C₄×C₁₃₇	C₂³	1	6	yes	—
89	C₄²×C₁₃₇	C₂³	1	3	yes	—
90	C₂²×D₁₃₇	—	2	12	no	—
91	C₂²×D₁₃₇	—	2	12	no	—
92	C₂²×D₁₃₇	—	2	12	no	—
93	C₂²×D₁₃₇	C₂⁴	1	6	yes	—
94	C₂²×D₁₃₇	C₂⁴	1	1	yes	characteristic
95	Dic₂₇₄	C₂×D₄	1	12	yes	complemented
96	D₅₄₈	—	2	12	no	—
97	D₅₄₈	—	2	6	no	—
98	D₅₄₈	—	4	6	no	—
99	C₂²⋊D₁₃₇	—	2	24	no	—
100	C₂²⋊D₁₃₇	—	2	12	no	—
101	C₂²⋊D₁₃₇	—	4	12	no	—
102	C₂×Dic₁₃₇	—	2	12	no	—
103	C₂×Dic₁₃₇	C₂⁴	1	3	yes	—
104	C₂×Dic₁₃₇	C₂⁴	1	2	yes	—
105	C₂×Dic₁₃₇	C₂⁴	1	3	yes	—
106	C₂×Dic₁₃₇	C₂⁴	1	1	yes	characteristic
107	C₄×D₁₃₇	—	2	4	no	—
108	C₄×D₁₃₇	—	2	6	no	—
109	C₄×D₁₃₇	—	2	12	no	—
110	C₄×D₁₃₇	—	2	12	no	—
111	C₄×D₁₃₇	C₂×D₄	1	12	yes	—
112	C₄×D₁₃₇	#41	1	4	yes	complemented
113	C₂³×C₁₃₇	C₂⁴	1	6	yes	—
114	Q₈×C₁₃₇	C₂×D₄	1	4	yes	complemented
115	D₄×C₁₃₇	—	2	12	no	—
116	D₄×C₁₃₇	—	2	6	no	—
117	D₄×C₁₃₇	—	4	6	no	—
118	C₂×C₄×C₁₃₇	—	2	12	no	—
119	C₂×C₄×C₁₃₇	C₂⁴	1	3	yes	—
120	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
121	C₂×C₄×C₁₃₇	C₂⁴	1	3	yes	—
122	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
123	C₂²×C₁₃₇	—	2	6	no	—
124	C₂²×C₁₃₇	—	2	6	no	—
125	C₂²×C₁₃₇	—	2	6	no	—
126	C₂²×C₁₃₇	C₂⁵	1	1	yes	characteristic
127	C₂×D₁₃₇	—	2	6	no	—
128	C₂×D₁₃₇	—	2	6	no	—
129	C₂×D₁₃₇	—	2	6	no	—
130	C₂×D₁₃₇	—	4	24	no	—
131	C₂×D₁₃₇	C₂²×D₄	1	2	yes	—
132	C₂×D₁₃₇	C₂×#41	1	2	yes	—
133	C₂×D₁₃₇	H(5,𝔽₂)	1	2	yes	—
134	C₄×C₁₃₇	—	2	2	no	—
135	C₄×C₁₃₇	—	2	3	no	—
136	C₄×C₁₃₇	C₂²×D₄	1	6	yes	—
137	C₄×C₁₃₇	C₂×#41	1	2	yes	complemented
138	Dic₁₃₇	—	2	2	no	—
139	Dic₁₃₇	—	2	3	no	—
140	Dic₁₃₇	C₂²×D₄	1	6	yes	—
141	Dic₁₃₇	C₂×#41	1	2	yes	complemented
142	C₂×C₁₃₇	—	4	6	no	—
143	C₂×C₁₃₇	C₂³×D₄	1	1	yes	characteristic
144	C₂×C₁₃₇	C₂²×#41	1	1	yes	characteristic
145	C₂×C₁₃₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
146	D₁₃₇	—	4	6	no	—
147	D₁₃₇	#550	1	4	yes	direct product factor
148	C₁₃₇	C₂×#550	1	1	yes	characteristic, complemented, Sylow
149	C₂×#550	—	137	1	no	maximal, Sylow
150	C₂²×#41	—	137	2	no	—
151	#550	—	137	2	no	—
152	#550	—	137	3	no	—
153	#550	—	137	6	no	—
154	#550	—	137	1	no	—
155	#550	—	137	3	no	—
156	#550	D₁₃₇	1	1	yes	characteristic, direct product factor
157	C₂×#127	—	137	3	no	—
158	C₂×#121	—	137	6	no	—
159	C₂×C₄×Q₈	—	137	1	no	—
160	C₂×C₄×D₄	—	137	3	no	—
161	C₂×#41	—	274	16	no	—
162	C₂×#41	—	137	2	no	—
163	C₂×#41	—	137	6	no	—
164	C₂×#41	—	137	6	no	—
165	C₂×#41	C₂×D₁₃₇	1	2	yes	complemented
166	C₂²×Q₈	—	137	1	no	—
167	C₂²×D₄	—	137	3	no	—
168	C₂²×D₄	—	137	6	no	—
169	C₂²×D₄	—	137	6	no	—
170	C₂³×C₄	—	137	6	no	—
171	#127	—	137	6	no	—
172	#127	—	137	6	no	—
173	#127	—	137	6	no	—
174	#127	—	137	3	no	—
175	#127	C₂×D₁₃₇	1	3	yes	complemented
176	#121	—	137	6	no	—
177	#121	—	137	12	no	—
178	#121	—	137	6	no	—
179	#121	—	137	6	no	—
180	#121	—	137	12	no	—
181	#121	C₂×D₁₃₇	1	6	yes	complemented
182	C₄×Q₈	—	137	1	no	—
183	C₄×Q₈	—	137	3	no	—
184	C₄×Q₈	—	137	3	no	—
185	C₄×Q₈	C₂×D₁₃₇	1	1	yes	characteristic, complemented
186	C₄×D₄	—	137	3	no	—
187	C₄×D₄	—	137	6	no	—
188	C₄×D₄	—	137	3	no	—
189	C₄×D₄	—	137	6	no	—
190	C₄×D₄	—	137	3	no	—
191	C₄×D₄	C₂×D₁₃₇	1	3	yes	complemented
192	C₂×C₄⋊C₄	—	137	1	no	—
193	C₂×C₄⋊C₄	—	137	3	no	—
194	C₂×C₂²⋊C₄	—	137	6	no	—
195	C₂×C₄²	—	137	3	no	—
196	C₂⁴	—	137	6	no	—
197	#41	—	2	8	no	—
198	#41	—	274	24	no	—
199	#41	—	274	24	no	—
200	#41	—	274	8	no	—
201	C₂×Q₈	—	137	8	no	—
202	C₂×Q₈	—	137	3	no	—
203	C₂×Q₈	C₂²×D₁₃₇	1	1	yes	characteristic
204	C₂×D₄	—	274	12	no	—
205	C₂×D₄	—	274	24	no	—
206	C₂×D₄	—	548	12	no	—
207	C₂×D₄	—	137	6	no	—
208	C₂×D₄	—	137	12	no	—
209	C₂×D₄	—	137	12	no	—
210	C₂×D₄	—	137	3	no	—
211	C₂×D₄	—	137	6	no	—
212	C₂×D₄	—	137	6	no	—
213	C₂×D₄	C₂²×D₁₃₇	1	3	yes	—
214	C₂×D₄	C₂²×D₁₃₇	1	6	yes	—
215	C₂×D₄	C₂²×D₁₃₇	1	6	yes	—
216	C₂²×C₄	—	274	24	no	—
217	C₂²×C₄	—	137	6	no	—
218	C₂²×C₄	—	137	3	no	—
219	C₂²×C₄	—	137	2	no	—
220	C₂²×C₄	—	137	6	no	—
221	C₂²×C₄	—	137	6	no	—
222	C₂²×C₄	—	137	1	no	—
223	C₂²×C₄	—	137	3	no	—
224	C₂²×C₄	C₂²×D₁₃₇	1	6	yes	—
225	C₄⋊C₄	—	137	2	no	—
226	C₄⋊C₄	—	137	6	no	—
227	C₄⋊C₄	—	137	1	no	—
228	C₄⋊C₄	—	137	3	no	—
229	C₄⋊C₄	C₂²×D₁₃₇	1	1	yes	characteristic
230	C₄⋊C₄	C₂²×D₁₃₇	1	3	yes	—
231	C₂²⋊C₄	—	137	6	no	—
232	C₂²⋊C₄	—	137	6	no	—
233	C₂²⋊C₄	—	137	6	no	—
234	C₂²⋊C₄	C₂²×D₁₃₇	1	6	yes	—
235	C₄²	—	137	3	no	—
236	C₄²	—	137	3	no	—
237	C₄²	—	137	3	no	—
238	C₄²	C₂²×D₁₃₇	1	3	yes	—
239	C₂³	—	274	12	no	—
240	C₂³	—	274	12	no	—
241	C₂³	—	274	12	no	—
242	C₂³	—	137	6	no	—
243	C₂³	—	137	1	no	—
244	C₂³	C₂³×D₁₃₇	1	6	yes	—
245	Q₈	—	137	12	no	—
246	Q₈	D₄×D₁₃₇	1	4	yes	complemented
247	D₄	—	2	12	no	—
248	D₄	—	2	6	no	—
249	D₄	—	4	6	no	—
250	D₄	—	274	24	no	—
251	D₄	—	274	12	no	—
252	D₄	—	274	12	no	—
253	D₄	—	274	6	no	—
254	D₄	—	548	6	no	—
255	D₄	—	548	12	no	—
256	C₂×C₄	—	2	12	no	—
257	C₂×C₄	—	274	4	no	—
258	C₂×C₄	—	274	6	no	—
259	C₂×C₄	—	274	12	no	—
260	C₂×C₄	—	274	12	no	—
261	C₂×C₄	—	274	12	no	—
262	C₂×C₄	—	137	3	no	—
263	C₂×C₄	—	137	2	no	—
264	C₂×C₄	—	137	1	no	—
265	C₂×C₄	—	137	3	no	—
266	C₂×C₄	—	137	12	no	—
267	C₂×C₄	—	137	4	no	—
268	C₂×C₄	C₂³×D₁₃₇	1	3	yes	—
269	C₂×C₄	C₂³×D₁₃₇	1	2	yes	—
270	C₂×C₄	C₂³×D₁₃₇	1	3	yes	—
271	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
272	C₂²	—	2	6	no	—
273	C₂²	—	2	6	no	—
274	C₂²	—	2	6	no	—
275	C₂²	—	274	6	no	—
276	C₂²	—	274	6	no	—
277	C₂²	—	274	6	no	—
278	C₂²	—	548	24	no	—
279	C₂²	—	137	2	no	—
280	C₂²	—	137	2	no	—
281	C₂²	—	137	2	no	—
282	C₂²	C₂⁴×D₁₃₇	1	1	yes	characteristic
283	C₄	—	2	2	no	—
284	C₄	—	2	3	no	—
285	C₄	—	274	2	no	—
286	C₄	—	274	3	no	—
287	C₄	—	137	6	no	—
288	C₄	—	137	2	no	—
289	C₄	C₂×D₄×D₁₃₇	1	6	yes	—
290	C₄	#41×D₁₃₇	1	2	yes	complemented
291	C₂	—	4	6	no	—
292	C₂	—	548	6	no	—
293	C₂	—	137	4	no	—
294	C₂	C₂²×D₄×D₁₃₇	1	1	yes	characteristic
295	C₂	C₂×#41×D₁₃₇	1	1	yes	characteristic
296	C₂	H(5,𝔽₂)×D₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃₇
commutator, socle, cosocle	C₂²×C₁₃₇	C₂⁵
Fitting	#550×C₁₃₇	C₂
hypercenter	#550	D₁₃₇
hypocenter	C₁₃₇	C₂×#550
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ #550

Complements

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

N	H ≅ G/N	type
1. C₂×#41×D₁₃₇	C₂	semidirect
2. #22578075	C₂	semidirect
3. #22578074	C₂	semidirect
4. #22578073	C₂	semidirect
5. #22578072	C₂	semidirect
6. #22578071	C₂	semidirect
7. #127×D₁₃₇	C₂	semidirect
8. #121×D₁₃₇	C₂	semidirect
9. C₄×Q₈×D₁₃₇	C₂	semidirect
10. C₄×D₄×D₁₃₇	C₂	semidirect
11. #550×C₁₃₇	C₂	semidirect
12. C₂×#14555511	C₂²	semidirect
13. C₂×#14555510	C₂²	semidirect
14. C₂×#14555509	C₂²	semidirect
21. #17984085	C₂²	semidirect
22. #17984084	C₂²	semidirect
23. #17984084	C₂²	semidirect
24. #17984084	C₂²	semidirect
25. #17984063	C₂²	semidirect
26. #17984062	C₂²	semidirect
27. #17984061	C₂²	semidirect
28. #17984060	C₂²	semidirect
29. #17984059	C₂²	semidirect
30. Q₈×Dic₁₃₇	C₂²	semidirect
31. C₄×Dic₂₇₄	C₂²	semidirect
32. #17984053	C₂²	semidirect
33. C₄×D₅₄₈	C₂²	semidirect
34. D₄×Dic₁₃₇	C₂²	semidirect
35. #17984050	C₂²	semidirect
36. C₄×C₂²⋊D₁₃₇	C₂²	semidirect
37. #17984048	C₂²	semidirect
42. C₂×#41×C₁₃₇	C₂²	semidirect
43. #127×C₁₃₇	C₂²	semidirect
44. #121×C₁₃₇	C₂²	semidirect
45. C₄×Q₈×C₁₃₇	C₂²	semidirect
46. C₄×D₄×C₁₃₇	C₂²	semidirect
52. Q₈×D₁₃₇	D₄	semidirect
95. Dic₂₇₄	C₂×D₄	semidirect
112. C₄×D₁₃₇	#41	semidirect
114. Q₈×C₁₃₇	C₂×D₄	semidirect
137. C₄×C₁₃₇	C₂×#41	semidirect
141. Dic₁₃₇	C₂×#41	semidirect
147. D₁₃₇	#550	direct (N × H)
148. C₁₃₇	C₂×#550	semidirect
156. #550	D₁₃₇	direct (N × H)
165. C₂×#41	C₂×D₁₃₇	semidirect
175. #127	C₂×D₁₃₇	semidirect
181. #121	C₂×D₁₃₇	semidirect
185. C₄×Q₈	C₂×D₁₃₇	semidirect
191. C₄×D₄	C₂×D₁₃₇	semidirect
246. Q₈	D₄×D₁₃₇	semidirect
290. C₄	#41×D₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41×D₁₃₇	2
#22578075	2
#22578074	2
#22578073	2
#22578072	2
#22578071	2
#127×D₁₃₇	2
#121×D₁₃₇	2
C₄×Q₈×D₁₃₇	2
C₄×D₄×D₁₃₇	2
#550×C₁₃₇	2
C₂×#550	137

Subgroups — #550×D137 #31794265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #550×D₁₃₇ #31794265