Subgroups — #31789704

Order 17536.

452 conjugacy classes of subgroups.

The subgroup lattice has 282 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	#22578169	C₂	1	1	yes	maximal, characteristic, complemented
3	#22577523	C₂	1	4	yes	maximal, complemented
4	#22577518	C₂	1	4	yes	maximal, complemented
5	C₂×#17984023	C₂	1	1	yes	maximal, characteristic, complemented
6	#22577420	C₂	1	1	yes	maximal, characteristic, complemented
7	#22577419	C₂	1	1	yes	maximal, characteristic, complemented
8	#22577361	C₂	1	1	yes	maximal, characteristic, complemented
9	#568×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14555510	C₂²	1	1	yes	characteristic
11	#17984108	C₄	1	2	yes	complemented
12	#17984108	C₄	1	2	yes	complemented
13	#17984108	C₂²	1	2	yes	—
14	#17984108	C₂²	1	2	yes	—
15	C₂×#14555500	C₄	1	1	yes	characteristic, complemented
16	C₂×#14555498	C₄	1	1	yes	characteristic, complemented
17	C₂×#14555497	C₄	1	1	yes	characteristic, complemented
18	C₂×#14555495	C₄	1	1	yes	characteristic, complemented
19	#17984096	—	2	2	no	—
20	#17984094	C₂²	1	4	yes	complemented
21	C₂×#14555494	C₂²	1	1	yes	characteristic
22	M₁₆×D₁₃₇	—	2	2	no	—
23	C₂²×#11764040	C₂²	1	1	yes	characteristic
24	#17984047	C₂²	1	1	yes	characteristic
25	#17984023	C₂²	1	2	yes	complemented
26	#17984023	C₂²	1	2	yes	complemented
27	#17984015	C₂²	1	4	yes	complemented
28	#17984014	C₂²	1	4	yes	complemented
29	#17983998	C₂²	1	2	yes	complemented
30	C₈×Dic₁₃₇	C₂²	1	2	yes	complemented
31	C₂×#41×C₁₃₇	C₂²	1	1	yes	characteristic
32	#131×C₁₃₇	—	2	2	no	—
33	#131×C₁₃₇	C₂²	1	4	yes	complemented
34	C₂×M₁₆×C₁₃₇	—	2	2	no	—
35	C₂×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
36	C₂×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
37	#14555510	—	2	1	no	—
38	#14555510	—	2	1	no	—
39	#14555510	C₂×C₄	1	2	yes	—
40	#14555510	C₂³	1	2	yes	—
41	C₂×Dic₂₇₄	C₂×C₄	1	1	yes	characteristic
42	C₂×D₅₄₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₁₃₇	—	2	1	no	—
44	C₂×C₄×D₁₃₇	—	2	1	no	—
45	#14555500	C₂×C₄	1	2	yes	—
46	#14555500	C₂×C₄	1	2	yes	—
47	#14555498	C₂×C₄	1	2	yes	—
48	#14555498	C₂×C₄	1	2	yes	—
49	#14555497	C₂×C₄	1	2	yes	—
50	#14555497	C₂×C₄	1	2	yes	—
51	#14555495	C₂×C₄	1	2	yes	—
52	#14555495	C₂×C₄	1	2	yes	—
53	#14555494	—	2	1	no	—
54	#14555494	C₂³	1	2	yes	—
55	#14555493	—	2	4	no	—
56	C₂×#11764040	—	2	1	no	—
57	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
58	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
59	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
60	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
61	C₂×#11764040	C₂³	1	1	yes	characteristic
62	C₂×#11764040	C₂³	1	1	yes	characteristic
63	C₈×D₁₃₇	—	2	4	no	—
64	C₄⋊Dic₁₃₇	—	2	1	no	—
65	C₂²⋊Dic₁₃₇	—	2	1	no	—
66	C₄×Dic₁₃₇	C₂³	1	2	yes	—
67	#41×C₁₃₇	—	2	1	no	—
68	#41×C₁₃₇	—	2	1	no	—
69	#41×C₁₃₇	C₂×C₄	1	2	yes	—
70	#41×C₁₃₇	C₂³	1	2	yes	—
71	C₂×Q₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
72	C₂×D₄×C₁₃₇	—	2	1	no	—
73	C₂×D₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃₇	—	2	1	no	—
75	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
76	M₁₆×C₁₃₇	—	2	1	no	—
77	M₁₆×C₁₃₇	—	2	4	no	—
78	M₁₆×C₁₃₇	D₄	1	2	yes	—
79	M₁₆×C₁₃₇	D₄	1	2	yes	—
80	M₁₆×C₁₃₇	C₂³	1	2	yes	—
81	C₂×C₈×C₁₃₇	—	2	1	no	—
82	C₂×C₈×C₁₃₇	—	2	4	no	—
83	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
84	C₂²×D₁₃₇	—	2	1	no	—
85	Dic₂₇₄	—	2	1	no	—
86	Dic₂₇₄	C₂²×C₄	1	2	yes	—
87	D₅₄₈	—	2	1	no	—
88	D₅₄₈	C₂²×C₄	1	2	yes	—
89	C₂²⋊D₁₃₇	—	2	2	no	—
90	C₂²⋊D₁₃₇	—	2	2	no	—
91	C₂×Dic₁₃₇	—	2	1	no	—
92	C₂×Dic₁₃₇	—	2	2	no	—
93	C₄×D₁₃₇	—	2	2	no	—
94	C₄×D₁₃₇	—	2	2	no	—
95	#11764040	—	2	2	no	—
96	#11764040	C₂²×C₄	1	1	yes	characteristic
97	#11764040	C₂²×C₄	1	1	yes	characteristic
98	#11764040	C₂²×C₄	1	1	yes	characteristic
99	#11764040	C₂²×C₄	1	1	yes	characteristic
100	C₂³×C₁₃₇	—	2	1	no	—
101	C₂³×C₁₃₇	#41	1	1	yes	characteristic
102	Q₈×C₁₃₇	—	2	1	no	—
103	Q₈×C₁₃₇	C₂²×C₄	1	2	yes	—
104	D₄×C₁₃₇	—	2	2	no	—
105	D₄×C₁₃₇	—	2	2	no	—
106	D₄×C₁₃₇	—	2	1	no	—
107	D₄×C₁₃₇	C₂²×C₄	1	2	yes	—
108	C₂×C₄×C₁₃₇	—	2	1	no	—
109	C₂×C₄×C₁₃₇	—	2	2	no	—
110	C₂×C₄×C₁₃₇	—	2	2	no	—
111	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃₇	#41	1	1	yes	characteristic
116	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
117	C₈×C₁₃₇	—	2	2	no	—
118	C₈×C₁₃₇	C₂×D₄	1	4	yes	—
119	C₂²×C₁₃₇	—	2	2	no	—
120	C₂²×C₁₃₇	—	4	1	no	—
121	C₂²×C₁₃₇	—	8	1	no	—
122	C₂²×C₁₃₇	C₄×D₄	1	1	yes	characteristic
123	C₂²×C₁₃₇	C₄×D₄	1	1	yes	characteristic
124	C₂²×C₁₃₇	C₂×#41	1	1	yes	characteristic
125	C₂×D₁₃₇	—	2	2	no	—
126	C₂×D₁₃₇	—	8	1	no	—
127	C₄×C₁₃₇	—	2	2	no	—
128	C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
129	C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
130	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
131	C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
132	Dic₁₃₇	—	2	2	no	—
133	Dic₁₃₇	—	4	2	no	—
134	C₂×C₁₃₇	—	2	1	no	—
135	C₂×C₁₃₇	—	2	1	no	—
136	C₂×C₁₃₇	—	2	1	no	—
137	C₂×C₁₃₇	—	4	2	no	—
138	C₂×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
139	D₁₃₇	—	4	2	no	—
140	C₁₃₇	#2957	1	1	yes	characteristic, complemented, Sylow
141	#2957	—	137	1	no	maximal, Sylow
142	C₂×#135	—	137	1	no	—
143	#568	—	137	1	no	—
144	#568	D₁₃₇	1	1	yes	characteristic, complemented
145	#444	—	137	4	no	—
146	#443	—	137	4	no	—
147	C₂×#108	—	137	1	no	—
148	#421	—	137	1	no	—
149	#421	—	137	1	no	—
150	#405	—	137	1	no	—
151	C₂×#41	—	137	1	no	—
152	C₂×#41	C₂×D₁₃₇	1	1	yes	characteristic
153	#135	—	137	2	no	—
154	#135	—	137	2	no	—
155	#135	—	137	2	no	—
156	#135	—	137	2	no	—
157	C₂×Q₁₆	—	137	1	no	—
158	C₂×SD₁₆	—	137	1	no	—
159	C₂×SD₁₆	—	137	1	no	—
160	C₂×D₈	—	137	1	no	—
161	#131	—	2	2	no	—
162	#131	—	274	2	no	—
163	#131	—	137	4	no	—
164	#131	C₂×D₁₃₇	1	4	yes	complemented
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	274	2	no	—
167	C₂×M₁₆	—	137	1	no	—
168	C₂×M₁₆	C₂×D₁₃₇	1	1	yes	characteristic
169	C₂×M₁₆	C₂×D₁₃₇	1	1	yes	characteristic
170	C₂²×C₈	—	137	1	no	—
171	#117	—	137	1	no	—
172	#108	—	137	2	no	—
173	#108	—	137	2	no	—
174	#104	—	137	4	no	—
175	#104	—	137	4	no	—
176	#97	—	137	2	no	—
177	C₄×C₈	—	137	2	no	—
178	#41	—	2	1	no	—
179	#41	—	2	1	no	—
180	#41	—	274	1	no	—
181	#41	—	274	1	no	—
182	#41	—	137	2	no	—
183	#41	—	137	2	no	—
184	#41	C₄×D₁₃₇	1	2	yes	—
185	#41	C₂²×D₁₃₇	1	2	yes	—
186	C₂×Q₈	—	137	1	no	—
187	C₂×Q₈	C₄×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	C₄×D₁₃₇	1	1	yes	characteristic
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	274	1	no	—
194	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
195	Q₁₆	—	137	2	no	—
196	Q₁₆	—	137	2	no	—
197	SD₁₆	—	137	2	no	—
198	SD₁₆	—	137	2	no	—
199	SD₁₆	—	137	2	no	—
200	SD₁₆	—	137	2	no	—
201	D₈	—	137	2	no	—
202	D₈	—	137	2	no	—
203	M₁₆	—	2	1	no	—
204	M₁₆	—	2	4	no	—
205	M₁₆	—	274	1	no	—
206	M₁₆	—	274	4	no	—
207	M₁₆	—	137	2	no	—
208	M₁₆	C₂²⋊D₁₃₇	1	2	yes	—
209	M₁₆	C₂²⋊D₁₃₇	1	2	yes	—
210	M₁₆	C₂²×D₁₃₇	1	2	yes	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	274	1	no	—
214	C₂×C₈	—	274	4	no	—
215	C₂×C₈	—	137	1	no	—
216	C₂×C₈	—	137	1	no	—
217	C₂×C₈	—	137	1	no	—
218	C₂×C₈	—	137	1	no	—
219	C₂×C₈	—	137	1	no	—
220	C₂×C₈	—	137	1	no	—
221	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
222	C₄⋊C₄	—	274	1	no	—
223	C₂²⋊C₄	—	274	1	no	—
224	C₄²	—	137	2	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	274	1	no	—
227	C₂³	#14555510	1	1	yes	characteristic
228	Q₈	—	2	1	no	—
229	Q₈	—	274	1	no	—
230	Q₈	—	137	2	no	—
231	Q₈	C₂×C₄×D₁₃₇	1	2	yes	—
232	D₄	—	2	2	no	—
233	D₄	—	2	2	no	—
234	D₄	—	2	1	no	—
235	D₄	—	274	2	no	—
236	D₄	—	274	2	no	—
237	D₄	—	274	1	no	—
238	D₄	—	137	2	no	—
239	D₄	C₂×C₄×D₁₃₇	1	2	yes	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	274	1	no	—
244	C₂×C₄	—	274	2	no	—
245	C₂×C₄	—	274	2	no	—
246	C₂×C₄	—	274	2	no	—
247	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
249	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
250	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
251	C₂×C₄	#14555510	1	1	yes	characteristic
252	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
253	C₈	—	2	2	no	—
254	C₈	—	274	2	no	—
255	C₈	—	137	1	no	—
256	C₈	—	137	1	no	—
257	C₈	—	137	1	no	—
258	C₈	—	137	1	no	—
259	C₈	C₂×C₂²⋊D₁₃₇	1	4	yes	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	8	1	no	—
263	C₂²	—	274	2	no	—
264	C₂²	—	1096	1	no	—
265	C₂²	C₄×C₂²⋊D₁₃₇	1	1	yes	characteristic
266	C₂²	C₄×C₂²⋊D₁₃₇	1	1	yes	characteristic
267	C₂²	C₂×#14555510	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	274	2	no	—
270	C₄	—	548	2	no	—
271	C₄	C₄×C₂²⋊D₁₃₇	1	1	yes	characteristic
272	C₄	C₄×C₂²⋊D₁₃₇	1	1	yes	characteristic
273	C₄	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
274	C₄	C₂²×C₂²⋊D₁₃₇	1	1	yes	characteristic
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	4	2	no	—
279	C₂	—	548	2	no	—
280	C₂	C₂×C₄×C₂²⋊D₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₂²⋊D₁₃₇
commutator	C₄×C₁₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	#568×C₁₃₇	C₂
socle	C₂×C₁₃₇	C₂×C₄×D₄
prime radical	C₂×#17984108	C₂
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	#568	D₁₃₇
hypocenter	C₁₃₇	#2957
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #568

Complements

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

N	H ≅ G/N	type
2. #22578169	C₂	semidirect
3. #22577523	C₂	semidirect
4. #22577518	C₂	semidirect
5. C₂×#17984023	C₂	semidirect
6. #22577420	C₂	semidirect
7. #22577419	C₂	semidirect
8. #22577361	C₂	semidirect
9. #568×C₁₃₇	C₂	semidirect
11. #17984108	C₄	semidirect
12. #17984108	C₄	semidirect
15. C₂×#14555500	C₄	semidirect
16. C₂×#14555498	C₄	semidirect
17. C₂×#14555497	C₄	semidirect
18. C₂×#14555495	C₄	semidirect
20. #17984094	C₂²	semidirect
25. #17984023	C₂²	semidirect
26. #17984023	C₂²	semidirect
27. #17984015	C₂²	semidirect
28. #17984014	C₂²	semidirect
29. #17983998	C₂²	semidirect
30. C₈×Dic₁₃₇	C₂²	semidirect
33. #131×C₁₃₇	C₂²	semidirect
140. C₁₃₇	#2957	semidirect
144. #568	D₁₃₇	semidirect
164. #131	C₂×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
#22578169	2
#22577523	2
#22577518	2
C₂×#17984023	2
#22577420	2
#22577419	2
#22577361	2
#568×C₁₃₇	2
#2957	137