Subgroups — #31789662

Order 17536.

484 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984108	C₂	1	1	yes	maximal, characteristic
2	#22577883	C₂	1	1	yes	maximal, characteristic, complemented
3	#22577503	C₂	1	2	yes	maximal, complemented
4	#22577501	C₂	1	2	yes	maximal, complemented
5	C₄×#14555500	C₂	1	2	yes	maximal, complemented
6	C₄×#14555495	C₂	1	2	yes	maximal, complemented
7	C₂×#17984021	C₂	1	1	yes	maximal, characteristic, complemented
8	#22577404	C₂	1	1	yes	maximal, characteristic, complemented
9	#22577402	C₂	1	1	yes	maximal, characteristic, complemented
10	#22577359	C₂	1	1	yes	maximal, characteristic, complemented
11	#520×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14555510	C₂²	1	1	yes	characteristic
13	#17984108	C₄	1	8	yes	complemented
14	C₂×#14555500	C₂²	1	1	yes	characteristic
15	C₂×#14555498	C₂²	1	1	yes	characteristic
16	C₂×#14555497	C₂²	1	1	yes	characteristic
17	C₂×#14555495	C₂²	1	1	yes	characteristic
18	C₂×#14555494	C₂²	1	1	yes	characteristic
19	C₂²×#11764040	C₂²	1	1	yes	characteristic
20	C₄×Dic₂₇₄	C₂²	1	2	yes	—
21	C₄×D₅₄₈	C₂²	1	2	yes	—
22	#17984048	—	2	2	no	—
23	#17984046	—	2	1	no	—
24	C₂×C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
25	C₄⋊C₄×D₁₃₇	—	2	1	no	—
26	#17984021	C₂²	1	2	yes	—
27	#17984021	C₂²	1	2	yes	—
28	#17984012	C₂²	1	2	yes	—
29	#17984011	C₂²	1	2	yes	—
30	#17984009	C₂²	1	2	yes	—
31	#17984008	C₂²	1	2	yes	—
32	#17983999	C₂²	1	2	yes	—
33	C₄×#11764040	C₂²	1	2	yes	—
34	C₂×#41×C₁₃₇	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₁₃₇	C₂²	1	2	yes	—
36	C₄×D₄×C₁₃₇	—	2	2	no	—
37	C₄×D₄×C₁₃₇	C₂²	1	2	yes	—
38	#117×C₁₃₇	—	2	1	no	—
39	#117×C₁₃₇	C₂²	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
41	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
42	#14555510	—	2	2	no	—
43	#14555510	C₂×C₄	1	4	yes	complemented
44	C₂×Dic₂₇₄	C₂³	1	1	yes	characteristic
45	C₂×D₅₄₈	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₃₇	—	2	1	no	—
47	C₂²×Dic₁₃₇	—	2	1	no	—
48	C₂×C₄×D₁₃₇	—	2	1	no	—
49	C₂×C₄×D₁₃₇	—	2	2	no	—
50	#14555500	C₂×C₄	1	4	yes	complemented
51	#14555498	C₂×C₄	1	4	yes	complemented
52	#14555497	C₂×C₄	1	4	yes	complemented
53	#14555495	C₂×C₄	1	4	yes	complemented
54	#14555494	—	2	2	no	—
55	C₂×#11764040	C₂×C₄	1	4	yes	complemented
56	C₂×#11764040	C₂³	1	2	yes	—
57	C₂×#11764040	C₂³	1	1	yes	characteristic
58	C₂×#11764040	C₂³	1	1	yes	characteristic
59	C₄⋊Dic₁₃₇	—	2	1	no	—
60	C₄⋊Dic₁₃₇	C₂³	1	2	yes	—
61	#14555489	—	2	2	no	—
62	#14555487	—	2	2	no	—
63	C₄×Dic₁₃₇	—	2	2	no	—
64	#41×C₁₃₇	—	2	2	no	—
65	#41×C₁₃₇	C₂×C₄	1	4	yes	complemented
66	C₂×Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
67	C₂×D₄×C₁₃₇	—	2	1	no	—
68	C₂×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₃₇	—	2	1	no	—
70	C₂²×C₄×C₁₃₇	—	2	1	no	—
71	C₂²×C₄×C₁₃₇	—	2	2	no	—
72	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₃₇	—	2	1	no	—
74	C₄⋊C₄×C₁₃₇	—	2	2	no	—
75	C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
76	C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
77	C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
78	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
79	C₂²⋊C₄×C₁₃₇	D₄	1	2	yes	—
80	C₄²×C₁₃₇	—	2	2	no	—
81	C₄²×C₁₃₇	C₂³	1	2	yes	—
82	C₂²×D₁₃₇	—	2	1	no	—
83	Dic₂₇₄	—	2	1	no	—
84	Dic₂₇₄	C₂²×C₄	1	2	yes	—
85	D₅₄₈	—	2	1	no	—
86	D₅₄₈	C₂²×C₄	1	2	yes	—
87	C₂²⋊D₁₃₇	—	2	4	no	—
88	C₂×Dic₁₃₇	—	2	1	no	—
89	C₂×Dic₁₃₇	—	2	2	no	—
90	C₂×Dic₁₃₇	—	8	1	no	—
91	C₄×D₁₃₇	—	2	4	no	—
92	C₄×D₁₃₇	—	8	2	no	—
93	#11764040	—	2	2	no	—
94	#11764040	C₂²×C₄	1	2	yes	—
95	#11764040	C₂²×C₄	1	2	yes	—
96	C₂³×C₁₃₇	—	2	1	no	—
97	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
98	Q₈×C₁₃₇	—	2	1	no	—
99	Q₈×C₁₃₇	C₂²×C₄	1	2	yes	—
100	D₄×C₁₃₇	—	2	4	no	—
101	D₄×C₁₃₇	—	2	1	no	—
102	D₄×C₁₃₇	C₂²×C₄	1	2	yes	—
103	C₂×C₄×C₁₃₇	—	2	1	no	—
104	C₂×C₄×C₁₃₇	—	2	4	no	—
105	C₂×C₄×C₁₃₇	—	2	2	no	—
106	C₂×C₄×C₁₃₇	—	8	2	no	—
107	C₂×C₄×C₁₃₇	—	8	1	no	—
108	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
109	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
111	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃₇	#41	1	2	yes	—
113	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
114	C₂²×C₁₃₇	—	2	1	no	—
115	C₂²×C₁₃₇	—	2	1	no	—
116	C₂²×C₁₃₇	—	2	2	no	—
117	C₂²×C₁₃₇	—	4	1	no	—
118	C₂²×C₁₃₇	—	4	1	no	—
119	C₂²×C₁₃₇	C₄×D₄	1	2	yes	—
120	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×D₁₃₇	—	2	2	no	—
122	C₂×D₁₃₇	—	4	1	no	—
123	C₂×D₁₃₇	—	4	1	no	—
124	C₄×C₁₃₇	—	2	2	no	—
125	C₄×C₁₃₇	—	2	2	no	—
126	C₄×C₁₃₇	—	2	2	no	—
127	C₄×C₁₃₇	—	4	2	no	—
128	C₄×C₁₃₇	C₄×D₄	1	2	yes	—
129	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
130	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
131	Dic₁₃₇	—	2	2	no	—
132	Dic₁₃₇	—	4	2	no	—
133	C₂×C₁₃₇	—	2	2	no	—
134	C₂×C₁₃₇	—	4	2	no	—
135	C₂×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂×C₁₃₇	#576	1	1	yes	characteristic
137	C₂×C₁₃₇	#578	1	1	yes	characteristic
138	D₁₃₇	—	4	2	no	—
139	C₁₃₇	#2951	1	1	yes	characteristic, complemented, Sylow
140	#2951	—	137	1	no	maximal, Sylow
141	C₂×#135	—	137	1	no	—
142	#520	—	137	1	no	—
143	#520	D₁₃₇	1	1	yes	characteristic, complemented
144	#440	—	137	2	no	—
145	#440	—	137	2	no	—
146	C₄×Q₁₆	—	137	2	no	—
147	C₄×D₈	—	137	2	no	—
148	C₂×#107	—	137	1	no	—
149	#417	—	137	1	no	—
150	#417	—	137	1	no	—
151	#405	—	137	1	no	—
152	C₂×#41	—	137	1	no	—
153	C₂×#41	C₂×D₁₃₇	1	1	yes	characteristic
154	#135	—	137	8	no	—
155	C₂×Q₁₆	—	137	1	no	—
156	C₂×SD₁₆	—	137	1	no	—
157	C₂×SD₁₆	—	137	1	no	—
158	C₂×D₈	—	137	1	no	—
159	C₂×M₁₆	—	137	1	no	—
160	C₂²×C₈	—	137	1	no	—
161	C₄×Q₈	—	137	2	no	—
162	C₄×Q₈	C₂×D₁₃₇	1	2	yes	—
163	C₄×D₄	—	2	2	no	—
164	C₄×D₄	—	274	2	no	—
165	C₄×D₄	—	137	2	no	—
166	C₄×D₄	C₂×D₁₃₇	1	2	yes	—
167	#117	—	2	1	no	—
168	#117	—	274	1	no	—
169	#117	C₂×D₁₃₇	1	1	yes	characteristic
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄⋊C₄	—	274	1	no	—
172	C₂×C₄⋊C₄	—	137	1	no	—
173	C₂×C₄⋊C₄	C₂×D₁₃₇	1	1	yes	characteristic
174	#107	—	137	2	no	—
175	#107	—	137	2	no	—
176	#103	—	137	2	no	—
177	#103	—	137	2	no	—
178	#102	—	137	2	no	—
179	#102	—	137	2	no	—
180	#97	—	137	2	no	—
181	C₄×C₈	—	137	2	no	—
182	#41	—	2	2	no	—
183	#41	—	274	2	no	—
184	#41	—	137	4	no	—
185	#41	C₄×D₁₃₇	1	4	yes	complemented
186	C₂×Q₈	—	137	1	no	—
187	C₂×Q₈	C₂²×D₁₃₇	1	1	yes	characteristic
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	274	1	no	—
190	C₂×D₄	—	137	1	no	—
191	C₂×D₄	C₂²×D₁₃₇	1	1	yes	characteristic
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	274	1	no	—
196	C₂²×C₄	—	274	1	no	—
197	C₂²×C₄	—	274	2	no	—
198	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
199	Q₁₆	—	137	4	no	—
200	SD₁₆	—	137	4	no	—
201	SD₁₆	—	137	4	no	—
202	D₈	—	137	4	no	—
203	M₁₆	—	274	2	no	—
204	C₂×C₈	—	137	2	no	—
205	C₂×C₈	—	137	4	no	—
206	C₂×C₈	—	137	1	no	—
207	C₂×C₈	—	137	1	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	274	2	no	—
211	C₄⋊C₄	—	274	1	no	—
212	C₄⋊C₄	—	137	2	no	—
213	C₄⋊C₄	C₂²⋊D₁₃₇	1	1	yes	characteristic
214	C₄⋊C₄	C₂²⋊D₁₃₇	1	1	yes	characteristic
215	C₄⋊C₄	C₂²×D₁₃₇	1	2	yes	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	274	2	no	—
218	C₂²⋊C₄	C₂²⋊D₁₃₇	1	2	yes	—
219	C₄²	—	2	2	no	—
220	C₄²	—	274	2	no	—
221	C₄²	C₂²×D₁₃₇	1	2	yes	—
222	C₂³	—	2	1	no	—
223	C₂³	—	274	1	no	—
224	C₂³	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
225	Q₈	—	2	1	no	—
226	Q₈	—	274	1	no	—
227	Q₈	—	137	2	no	—
228	Q₈	C₂×C₄×D₁₃₇	1	2	yes	—
229	D₄	—	2	4	no	—
230	D₄	—	2	1	no	—
231	D₄	—	274	4	no	—
232	D₄	—	274	1	no	—
233	D₄	—	137	2	no	—
234	D₄	C₂×C₄×D₁₃₇	1	2	yes	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	—	8	1	no	—
240	C₂×C₄	—	274	1	no	—
241	C₂×C₄	—	274	4	no	—
242	C₂×C₄	—	274	2	no	—
243	C₂×C₄	—	1096	2	no	—
244	C₂×C₄	—	1096	1	no	—
245	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
246	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	2	yes	—
247	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	2	yes	—
248	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
249	C₂×C₄	#14555510	1	2	yes	—
250	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
251	C₈	—	274	2	no	—
252	C₈	—	137	2	no	—
253	C₈	—	137	2	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	4	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	274	2	no	—
260	C₂²	—	548	1	no	—
261	C₂²	—	548	1	no	—
262	C₂²	C₄×C₂²⋊D₁₃₇	1	2	yes	—
263	C₂²	C₂²×C₂²⋊D₁₃₇	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	4	2	no	—
268	C₄	—	274	2	no	—
269	C₄	—	548	2	no	—
270	C₄	C₄×C₂²⋊D₁₃₇	1	2	yes	—
271	C₄	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
272	C₄	C₂×#14555510	1	1	yes	characteristic
273	C₂	—	2	2	no	—
274	C₂	—	4	2	no	—
275	C₂	—	548	2	no	—
276	C₂	C₂×C₄×C₂²⋊D₁₃₇	1	1	yes	characteristic
277	C₂	#22578206	1	1	yes	characteristic
278	C₂	#22578218	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₃₇
commutator	C₄×C₁₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	#520×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂²×D₄
prime radical	C₂×#17984108	C₂
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	#520	D₁₃₇
hypocenter	C₁₃₇	#2951
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #520

Complements

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

N	H ≅ G/N	type
2. #22577883	C₂	semidirect
3. #22577503	C₂	semidirect
4. #22577501	C₂	semidirect
5. C₄×#14555500	C₂	semidirect
6. C₄×#14555495	C₂	semidirect
7. C₂×#17984021	C₂	semidirect
8. #22577404	C₂	semidirect
9. #22577402	C₂	semidirect
10. #22577359	C₂	semidirect
11. #520×C₁₃₇	C₂	semidirect
13. #17984108	C₄	semidirect
43. #14555510	C₂×C₄	semidirect
50. #14555500	C₂×C₄	semidirect
51. #14555498	C₂×C₄	semidirect
52. #14555497	C₂×C₄	semidirect
53. #14555495	C₂×C₄	semidirect
55. C₂×#11764040	C₂×C₄	semidirect
65. #41×C₁₃₇	C₂×C₄	semidirect
139. C₁₃₇	#2951	semidirect
143. #520	D₁₃₇	semidirect
185. #41	C₄×D₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984108	2
#22577883	2
#22577503	2
#22577501	2
C₄×#14555500	2
C₄×#14555495	2
C₂×#17984021	2
#22577404	2
#22577402	2
#22577359	2
#520×C₁₃₇	2
#2951	137