Subgroups — #555×D₁₃₇ #31794325

Order 17536.

1240 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578119	C₂	1	1	yes	maximal, characteristic, complemented
2	#22578118	C₂	1	2	yes	maximal, complemented
3	#22578117	C₂	1	4	yes	maximal, complemented
4	#22578116	C₂	1	4	yes	maximal, complemented
5	#22578115	C₂	1	4	yes	maximal, complemented
6	C₄⋊Q₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#124×D₁₃₇	C₂	1	4	yes	maximal, complemented
8	#121×D₁₃₇	C₂	1	4	yes	maximal, complemented
9	C₄×D₄×D₁₃₇	C₂	1	2	yes	maximal, complemented
10	#117×D₁₃₇	C₂	1	4	yes	maximal, complemented
11	#555×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₁₃₇	C₂²	1	2	yes	—
13	C₂×D₄×D₁₃₇	C₂²	1	2	yes	—
14	C₂×D₄×D₁₃₇	C₂²	1	4	yes	—
15	C₂²×C₄×D₁₃₇	C₂²	1	4	yes	—
16	#17984088	C₂²	1	1	yes	characteristic, complemented
17	#17984087	C₂²	1	4	yes	complemented
18	#17984086	C₂²	1	2	yes	complemented
19	#17984077	C₂²	1	4	yes	complemented
20	#17984076	C₂²	1	4	yes	complemented
21	#17984075	C₂²	1	4	yes	complemented
22	#17984074	C₂²	1	8	yes	complemented
23	#17984074	C₂²	1	8	yes	complemented
24	#17984063	C₂²	1	4	yes	complemented
25	#17984062	C₂²	1	4	yes	complemented
26	#17984061	C₂²	1	8	yes	complemented
27	#17984060	C₂²	1	8	yes	complemented
28	#17984059	C₂²	1	4	yes	complemented
29	C₄×D₅₄₈	C₂²	1	2	yes	complemented
30	D₄×Dic₁₃₇	C₂²	1	2	yes	complemented
31	#17984050	C₂²	1	2	yes	complemented
32	C₄×C₂²⋊D₁₃₇	C₂²	1	4	yes	complemented
33	#17984048	C₂²	1	4	yes	complemented
34	#17984047	C₂²	1	4	yes	complemented
35	#17984046	C₂²	1	4	yes	complemented
36	#17984046	C₂²	1	4	yes	complemented
37	#17984045	C₂²	1	8	yes	complemented
38	#17984044	C₂²	1	8	yes	complemented
39	C₄⋊C₄×D₁₃₇	C₂²	1	4	yes	—
40	C₄⋊C₄×D₁₃₇	C₂²	1	2	yes	—
41	C₂²⋊C₄×D₁₃₇	C₂²	1	8	yes	—
42	C₂²⋊C₄×D₁₃₇	C₂²	1	4	yes	—
43	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
44	C₄²×D₁₃₇	C₂²	1	4	yes	—
45	C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
46	#124×C₁₃₇	C₂²	1	4	yes	complemented
47	#121×C₁₃₇	C₂²	1	4	yes	complemented
48	C₄×D₄×C₁₃₇	C₂²	1	2	yes	complemented
49	#117×C₁₃₇	C₂²	1	4	yes	complemented
50	C₂³×D₁₃₇	C₂³	1	4	yes	—
51	C₂×Dic₂₇₄	C₂³	1	2	yes	—
52	C₂×Dic₂₇₄	C₂³	1	4	yes	—
53	Q₈×D₁₃₇	—	4	4	no	—
54	C₂×D₅₄₈	C₂³	1	2	yes	—
55	C₂×D₅₄₈	C₂³	1	4	yes	—
56	C₂×C₂²⋊D₁₃₇	C₂³	1	4	yes	—
57	C₂×C₂²⋊D₁₃₇	C₂³	1	8	yes	—
58	D₄×D₁₃₇	—	2	8	no	—
59	D₄×D₁₃₇	—	4	8	no	—
60	C₂²×Dic₁₃₇	C₂³	1	4	yes	—
61	C₂×C₄×D₁₃₇	—	2	16	no	—
62	C₂×C₄×D₁₃₇	C₂³	1	4	yes	—
63	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₁₃₇	C₂³	1	4	yes	—
65	C₂×C₄×D₁₃₇	C₂³	1	4	yes	—
66	C₂×C₄×D₁₃₇	C₂³	1	4	yes	—
67	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
68	C₄⋊Dic₁₃₇	C₂³	1	2	yes	—
69	C₄⋊Dic₁₃₇	C₂³	1	4	yes	—
70	#14555489	C₂³	1	4	yes	—
71	#14555489	C₂³	1	8	yes	—
72	C₂²⋊Dic₁₃₇	C₂³	1	8	yes	—
73	C₂²⋊Dic₁₃₇	C₂³	1	4	yes	—
74	#14555487	C₂³	1	8	yes	—
75	#14555487	C₂³	1	8	yes	—
76	#14555487	C₂³	1	4	yes	—
77	#14555487	C₂³	1	4	yes	—
78	C₄×Dic₁₃₇	C₂³	1	4	yes	—
79	C₄×Dic₁₃₇	C₂³	1	8	yes	—
80	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
81	C₄×Dic₁₃₇	C₂³	1	2	yes	—
82	C₂×Q₈×C₁₃₇	C₂³	1	2	yes	—
83	C₂×D₄×C₁₃₇	C₂³	1	2	yes	—
84	C₂×D₄×C₁₃₇	C₂³	1	4	yes	—
85	C₂²×C₄×C₁₃₇	C₂³	1	4	yes	—
86	C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
87	C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
88	C₂²⋊C₄×C₁₃₇	C₂³	1	8	yes	—
89	C₂²⋊C₄×C₁₃₇	C₂³	1	4	yes	—
90	C₄²×C₁₃₇	C₂³	1	4	yes	—
91	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
92	C₂²×D₁₃₇	—	2	8	no	—
93	C₂²×D₁₃₇	—	2	8	no	—
94	C₂²×D₁₃₇	—	2	8	no	—
95	C₂²×D₁₃₇	C₂⁴	1	4	yes	—
96	C₂²×D₁₃₇	C₂⁴	1	1	yes	characteristic
97	Dic₂₇₄	—	4	2	no	—
98	Dic₂₇₄	—	4	4	no	—
99	D₅₄₈	—	2	4	no	—
100	D₅₄₈	—	4	4	no	—
101	C₂²⋊D₁₃₇	—	2	8	no	—
102	C₂²⋊D₁₃₇	—	4	8	no	—
103	C₂×Dic₁₃₇	—	2	8	no	—
104	C₂×Dic₁₃₇	C₂⁴	1	4	yes	—
105	C₂×Dic₁₃₇	C₂⁴	1	1	yes	characteristic
106	C₂×Dic₁₃₇	C₂⁴	1	4	yes	—
107	C₂×Dic₁₃₇	C₂⁴	1	2	yes	—
108	C₄×D₁₃₇	—	2	8	no	—
109	C₄×D₁₃₇	—	2	8	no	—
110	C₄×D₁₃₇	—	2	2	no	—
111	C₄×D₁₃₇	—	2	8	no	—
112	C₄×D₁₃₇	—	2	8	no	—
113	C₄×D₁₃₇	#41	1	8	yes	—
114	C₂³×C₁₃₇	C₂⁴	1	4	yes	—
115	Q₈×C₁₃₇	—	4	2	no	—
116	D₄×C₁₃₇	—	2	4	no	—
117	D₄×C₁₃₇	—	4	4	no	—
118	C₂×C₄×C₁₃₇	—	2	8	no	—
119	C₂×C₄×C₁₃₇	C₂⁴	1	4	yes	—
120	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
121	C₂×C₄×C₁₃₇	C₂⁴	1	4	yes	—
122	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
123	C₂²×C₁₃₇	—	2	4	no	—
124	C₂²×C₁₃₇	—	2	4	no	—
125	C₂²×C₁₃₇	—	2	4	no	—
126	C₂²×C₁₃₇	C₂⁵	1	1	yes	characteristic
127	C₂×D₁₃₇	—	2	4	no	—
128	C₂×D₁₃₇	—	2	4	no	—
129	C₂×D₁₃₇	—	2	4	no	—
130	C₂×D₁₃₇	—	4	16	no	—
131	C₂×D₁₃₇	C₂×#41	1	4	yes	—
132	C₂×D₁₃₇	H(5,𝔽₂)	1	2	yes	—
133	C₄×C₁₃₇	—	2	4	no	—
134	C₄×C₁₃₇	—	2	4	no	—
135	C₄×C₁₃₇	—	2	1	no	—
136	C₄×C₁₃₇	C₂×#41	1	4	yes	—
137	Dic₁₃₇	—	2	4	no	—
138	Dic₁₃₇	—	2	4	no	—
139	Dic₁₃₇	—	2	1	no	—
140	Dic₁₃₇	C₂×#41	1	4	yes	—
141	C₂×C₁₃₇	—	4	4	no	—
142	C₂×C₁₃₇	C₂²×#41	1	2	yes	—
143	C₂×C₁₃₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
144	D₁₃₇	—	4	4	no	—
145	D₁₃₇	#555	1	4	yes	direct product factor
146	C₁₃₇	C₂×#555	1	1	yes	characteristic, complemented, Sylow
147	C₂×#555	—	137	1	no	maximal, Sylow
148	#555	—	137	1	no	—
149	#555	—	137	4	no	—
150	#555	—	137	4	no	—
151	#555	—	137	2	no	—
152	#555	—	137	4	no	—
153	#555	D₁₃₇	1	1	yes	characteristic, direct product factor
154	C₂×C₄⋊Q₈	—	137	1	no	—
155	C₂×#124	—	137	4	no	—
156	C₂×#121	—	137	4	no	—
157	C₂×C₄×D₄	—	137	2	no	—
158	C₂×#117	—	137	4	no	—
159	C₂²×Q₈	—	137	2	no	—
160	C₂²×D₄	—	137	2	no	—
161	C₂²×D₄	—	137	4	no	—
162	C₂³×C₄	—	137	4	no	—
163	C₄⋊Q₈	—	137	2	no	—
164	C₄⋊Q₈	—	137	4	no	—
165	C₄⋊Q₈	—	137	1	no	—
166	C₄⋊Q₈	C₂×D₁₃₇	1	1	yes	characteristic, complemented
167	#124	—	137	4	no	—
168	#124	—	137	4	no	—
169	#124	—	137	8	no	—
170	#124	—	137	8	no	—
171	#124	—	137	4	no	—
172	#124	C₂×D₁₃₇	1	4	yes	complemented
173	#121	—	137	4	no	—
174	#121	—	137	8	no	—
175	#121	—	137	4	no	—
176	#121	—	137	4	no	—
177	#121	—	137	8	no	—
178	#121	C₂×D₁₃₇	1	4	yes	complemented
179	C₄×D₄	—	137	2	no	—
180	C₄×D₄	—	137	4	no	—
181	C₄×D₄	—	137	2	no	—
182	C₄×D₄	—	137	4	no	—
183	C₄×D₄	—	137	2	no	—
184	C₄×D₄	C₂×D₁₃₇	1	2	yes	complemented
185	#117	—	137	4	no	—
186	#117	—	137	4	no	—
187	#117	—	137	4	no	—
188	#117	—	137	8	no	—
189	#117	—	137	8	no	—
190	#117	C₂×D₁₃₇	1	4	yes	complemented
191	C₂×C₄⋊C₄	—	137	2	no	—
192	C₂×C₄⋊C₄	—	137	4	no	—
193	C₂×C₂²⋊C₄	—	137	8	no	—
194	C₂×C₂²⋊C₄	—	137	4	no	—
195	C₂×C₄²	—	137	4	no	—
196	C₂×C₄²	—	137	1	no	—
197	C₂⁴	—	137	4	no	—
198	C₂×Q₈	—	548	4	no	—
199	C₂×Q₈	—	137	4	no	—
200	C₂×Q₈	—	137	2	no	—
201	C₂×Q₈	C₂²×D₁₃₇	1	2	yes	—
202	C₂×D₄	—	274	8	no	—
203	C₂×D₄	—	548	8	no	—
204	C₂×D₄	—	137	4	no	—
205	C₂×D₄	—	137	8	no	—
206	C₂×D₄	—	137	2	no	—
207	C₂×D₄	—	137	4	no	—
208	C₂×D₄	C₂²×D₁₃₇	1	2	yes	—
209	C₂×D₄	C₂²×D₁₃₇	1	4	yes	—
210	C₂²×C₄	—	274	16	no	—
211	C₂²×C₄	—	137	4	no	—
212	C₂²×C₄	—	137	4	no	—
213	C₂²×C₄	—	137	1	no	—
214	C₂²×C₄	—	137	4	no	—
215	C₂²×C₄	—	137	4	no	—
216	C₂²×C₄	—	137	4	no	—
217	C₂²×C₄	—	137	2	no	—
218	C₂²×C₄	C₂²×D₁₃₇	1	4	yes	—
219	C₄⋊C₄	—	137	4	no	—
220	C₄⋊C₄	—	137	8	no	—
221	C₄⋊C₄	—	137	2	no	—
222	C₄⋊C₄	—	137	4	no	—
223	C₄⋊C₄	C₂²×D₁₃₇	1	2	yes	—
224	C₄⋊C₄	C₂²×D₁₃₇	1	4	yes	—
225	C₂²⋊C₄	—	137	8	no	—
226	C₂²⋊C₄	—	137	4	no	—
227	C₂²⋊C₄	—	137	8	no	—
228	C₂²⋊C₄	—	137	8	no	—
229	C₂²⋊C₄	—	137	4	no	—
230	C₂²⋊C₄	—	137	4	no	—
231	C₂²⋊C₄	C₂²×D₁₃₇	1	8	yes	—
232	C₂²⋊C₄	C₂²×D₁₃₇	1	4	yes	—
233	C₄²	—	137	4	no	—
234	C₄²	—	137	8	no	—
235	C₄²	—	137	2	no	—
236	C₄²	—	137	1	no	—
237	C₄²	C₂²×D₁₃₇	1	4	yes	—
238	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
239	C₂³	—	274	8	no	—
240	C₂³	—	274	8	no	—
241	C₂³	—	274	8	no	—
242	C₂³	—	137	4	no	—
243	C₂³	—	137	1	no	—
244	C₂³	C₂³×D₁₃₇	1	4	yes	—
245	Q₈	—	4	2	no	—
246	Q₈	—	548	4	no	—
247	Q₈	—	548	2	no	—
248	D₄	—	2	4	no	—
249	D₄	—	4	4	no	—
250	D₄	—	274	8	no	—
251	D₄	—	274	4	no	—
252	D₄	—	548	4	no	—
253	D₄	—	548	8	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	274	8	no	—
256	C₂×C₄	—	274	8	no	—
257	C₂×C₄	—	274	2	no	—
258	C₂×C₄	—	274	8	no	—
259	C₂×C₄	—	274	8	no	—
260	C₂×C₄	—	274	8	no	—
261	C₂×C₄	—	137	4	no	—
262	C₂×C₄	—	137	1	no	—
263	C₂×C₄	—	137	4	no	—
264	C₂×C₄	—	137	2	no	—
265	C₂×C₄	—	137	8	no	—
266	C₂×C₄	C₂³×D₁₃₇	1	4	yes	—
267	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
268	C₂×C₄	C₂³×D₁₃₇	1	4	yes	—
269	C₂×C₄	C₂³×D₁₃₇	1	2	yes	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	274	4	no	—
274	C₂²	—	274	4	no	—
275	C₂²	—	274	4	no	—
276	C₂²	—	548	16	no	—
277	C₂²	—	137	4	no	—
278	C₂²	—	137	2	no	—
279	C₂²	C₂⁴×D₁₃₇	1	1	yes	characteristic
280	C₄	—	2	4	no	—
281	C₄	—	2	4	no	—
282	C₄	—	2	1	no	—
283	C₄	—	274	4	no	—
284	C₄	—	274	4	no	—
285	C₄	—	274	1	no	—
286	C₄	—	137	4	no	—
287	C₄	#41×D₁₃₇	1	4	yes	—
288	C₂	—	4	4	no	—
289	C₂	—	548	4	no	—
290	C₂	—	137	4	no	—
291	C₂	C₂×#41×D₁₃₇	1	2	yes	—
292	C₂	H(5,𝔽₂)×D₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃₇
commutator, socle, cosocle	C₂²×C₁₃₇	C₂⁵
Fitting	#555×C₁₃₇	C₂
hypercenter	#555	D₁₃₇
hypocenter	C₁₃₇	C₂×#555
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ #555

Complements

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

N	H ≅ G/N	type
1. #22578119	C₂	semidirect
2. #22578118	C₂	semidirect
3. #22578117	C₂	semidirect
4. #22578116	C₂	semidirect
5. #22578115	C₂	semidirect
6. C₄⋊Q₈×D₁₃₇	C₂	semidirect
7. #124×D₁₃₇	C₂	semidirect
8. #121×D₁₃₇	C₂	semidirect
9. C₄×D₄×D₁₃₇	C₂	semidirect
10. #117×D₁₃₇	C₂	semidirect
11. #555×C₁₃₇	C₂	semidirect
16. #17984088	C₂²	semidirect
17. #17984087	C₂²	semidirect
18. #17984086	C₂²	semidirect
19. #17984077	C₂²	semidirect
20. #17984076	C₂²	semidirect
21. #17984075	C₂²	semidirect
22. #17984074	C₂²	semidirect
23. #17984074	C₂²	semidirect
24. #17984063	C₂²	semidirect
25. #17984062	C₂²	semidirect
26. #17984061	C₂²	semidirect
27. #17984060	C₂²	semidirect
28. #17984059	C₂²	semidirect
29. C₄×D₅₄₈	C₂²	semidirect
30. D₄×Dic₁₃₇	C₂²	semidirect
31. #17984050	C₂²	semidirect
32. C₄×C₂²⋊D₁₃₇	C₂²	semidirect
33. #17984048	C₂²	semidirect
34. #17984047	C₂²	semidirect
35. #17984046	C₂²	semidirect
36. #17984046	C₂²	semidirect
37. #17984045	C₂²	semidirect
38. #17984044	C₂²	semidirect
45. C₄⋊Q₈×C₁₃₇	C₂²	semidirect
46. #124×C₁₃₇	C₂²	semidirect
47. #121×C₁₃₇	C₂²	semidirect
48. C₄×D₄×C₁₃₇	C₂²	semidirect
49. #117×C₁₃₇	C₂²	semidirect
145. D₁₃₇	#555	direct (N × H)
146. C₁₃₇	C₂×#555	semidirect
153. #555	D₁₃₇	direct (N × H)
166. C₄⋊Q₈	C₂×D₁₃₇	semidirect
172. #124	C₂×D₁₃₇	semidirect
178. #121	C₂×D₁₃₇	semidirect
184. C₄×D₄	C₂×D₁₃₇	semidirect
190. #117	C₂×D₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#22578119	2
#22578118	2
#22578117	2
#22578116	2
#22578115	2
C₄⋊Q₈×D₁₃₇	2
#124×D₁₃₇	2
#121×D₁₃₇	2
C₄×D₄×D₁₃₇	2
#117×D₁₃₇	2
#555×C₁₃₇	2
C₂×#555	137

Subgroups — #555×D137 #31794325

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #555×D₁₃₇ #31794325