Subgroups — #31797352

Order 17536.

564 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₃₇⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984167	C₂	1	1	yes	maximal, characteristic
3	#22578369	C₂	1	4	yes	maximal, complemented
4	#22578366	C₂	1	4	yes	maximal, complemented
5	M₁₆×#10587739	C₂	1	4	yes	maximal, complemented
6	C₂×M₁₆×D₁₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#14555520	C₄	1	2	yes	complemented
8	D₁₃₇⋊M₁₆	C₄	1	2	yes	complemented
9	D₁₃₇⋊M₁₆	C₄	1	2	yes	complemented
10	D₁₃₇⋊M₁₆	C₂²	1	2	yes	—
11	D₁₃₇⋊M₁₆	C₂²	1	2	yes	—
12	C₂×#14555519	C₄	1	2	yes	complemented
13	C₂×#14555518	C₂²	1	2	yes	—
14	#17984167	C₄	1	2	yes	—
15	#17984167	C₄	1	2	yes	—
16	#17984167	C₂²	1	2	yes	—
17	#17984167	C₂²	1	2	yes	—
18	C₂×C₄⋊#10587739	C₄	1	2	yes	—
19	C₂×C₂²⋊#10587739	C₄	1	2	yes	—
20	C₂×C₄×#10587739	C₂²	1	2	yes	—
21	#17984142	C₂²	1	8	yes	complemented
22	C₈×#10587739	C₂²	1	8	yes	complemented
23	C₂²×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
24	C₂×#14555494	C₄	1	1	yes	characteristic, complemented
25	C₂×#14555493	C₄	1	2	yes	complemented
26	M₁₆×D₁₃₇	C₄	1	2	yes	complemented
27	M₁₆×D₁₃₇	C₄	1	2	yes	complemented
28	M₁₆×D₁₃₇	C₂²	1	2	yes	—
29	M₁₆×D₁₃₇	C₂²	1	2	yes	—
30	C₂×C₈×D₁₃₇	C₂²	1	2	yes	—
31	C₂×M₁₆×C₁₃₇	C₄	1	1	yes	characteristic, complemented
32	C₂²×#10587739	C₂×C₄	1	4	yes	—
33	#14555520	C₂×C₄	1	4	yes	—
34	#14555520	C₂×C₄	1	4	yes	complemented
35	#14555519	C₂×C₄	1	4	yes	—
36	#14555519	C₂×C₄	1	4	yes	complemented
37	#14555518	C₂×C₄	1	4	yes	complemented
38	#14555518	C₂³	1	4	yes	—
39	C₂×#11764047	C₂×C₄	1	4	yes	complemented
40	C₄⋊#10587739	C₂×C₄	1	4	yes	—
41	C₄⋊#10587739	C₂×C₄	1	4	yes	—
42	C₂²⋊#10587739	C₂×C₄	1	4	yes	—
43	C₂²⋊#10587739	C₂×C₄	1	4	yes	—
44	C₄×#10587739	C₂×C₄	1	4	yes	—
45	C₄×#10587739	C₂³	1	4	yes	—
46	C₂³×D₁₃₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
58	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
59	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
60	#14555494	C₂×C₄	1	2	yes	—
61	#14555494	C₂×C₄	1	2	yes	complemented
62	#14555493	C₂×C₄	1	4	yes	—
63	#14555493	C₂×C₄	1	4	yes	complemented
64	C₂×#11764040	C₂×C₄	1	2	yes	complemented
65	C₈×D₁₃₇	C₂×C₄	1	4	yes	complemented
66	C₈×D₁₃₇	C₂³	1	4	yes	—
67	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
68	M₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
69	M₁₆×C₁₃₇	C₂×C₄	1	2	yes	complemented
70	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	complemented
71	C₂×#10587739	—	4	4	no	—
72	C₂×#10587739	C₂²×C₄	1	8	yes	—
73	#11764047	C₂²×C₄	1	8	yes	—
74	C₂²×D₁₃₇	—	4	2	no	—
75	C₂²×D₁₃₇	C₄²	1	1	yes	characteristic
76	C₂²×D₁₃₇	C₄²	1	1	yes	characteristic
77	C₂²×D₁₃₇	C₄²	1	1	yes	characteristic
78	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₁₃₇	C₄²	1	1	yes	characteristic
82	C₂×Dic₁₃₇	C₄²	1	1	yes	characteristic
83	C₂×Dic₁₃₇	C₄²	1	1	yes	characteristic
84	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
87	C₄×D₁₃₇	C₄²	1	1	yes	characteristic
88	C₄×D₁₃₇	C₄²	1	1	yes	characteristic
89	C₄×D₁₃₇	C₄²	1	1	yes	characteristic
90	C₄×D₁₃₇	C₄²	1	1	yes	characteristic
91	C₄×D₁₃₇	C₄²	1	1	yes	characteristic
92	C₄×D₁₃₇	C₄²	1	1	yes	characteristic
93	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
94	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
95	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
96	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
97	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
98	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
99	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
100	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
101	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
102	C₄×D₁₃₇	C₂⁴	1	1	yes	characteristic
103	#11764040	C₂²×C₄	1	4	yes	—
104	C₂³×C₁₃₇	C₄²	1	1	yes	characteristic
105	C₂×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
106	C₂×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
107	C₂×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
108	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
111	C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
112	C₂²×C₁₃₇	—	4	1	no	—
113	C₂²×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
114	C₂²×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
115	C₂²×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
116	C₂×D₁₃₇	—	2	2	no	—
117	C₂×D₁₃₇	—	2	2	no	—
118	C₂×D₁₃₇	—	2	2	no	—
119	C₂×D₁₃₇	—	4	1	no	—
120	C₂×D₁₃₇	—	4	1	no	—
121	C₂×D₁₃₇	—	4	1	no	—
122	C₂×D₁₃₇	C₂×C₄²	1	1	yes	characteristic
123	C₂×D₁₃₇	C₂×C₄²	1	1	yes	characteristic
124	C₂×D₁₃₇	C₂×C₄²	1	1	yes	characteristic
125	C₂×D₁₃₇	C₂³×C₄	1	1	yes	characteristic
126	#10587739	—	2	8	no	—
127	C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
128	C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
129	C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
130	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
131	Dic₁₃₇	C₂×C₄²	1	1	yes	characteristic
132	Dic₁₃₇	C₂×C₄²	1	1	yes	characteristic
133	Dic₁₃₇	C₂×C₄²	1	1	yes	characteristic
134	Dic₁₃₇	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₁₃₇	—	2	1	no	—
136	C₂×C₁₃₇	—	2	1	no	—
137	C₂×C₁₃₇	—	2	1	no	—
138	C₂×C₁₃₇	C₂²×C₄²	1	1	yes	characteristic
139	D₁₃₇	—	2	1	no	—
140	D₁₃₇	—	2	1	no	—
141	D₁₃₇	—	2	1	no	—
142	D₁₃₇	#568	1	2	yes	—
143	C₁₃₇	#2873	1	1	yes	characteristic, complemented, Sylow
144	#2873	—	137	1	no	maximal, Sylow
145	C₂²×M₁₆	—	137	1	no	—
146	C₂²×M₁₆	—	137	1	no	—
147	C₂×#117	—	137	1	no	—
148	#405	—	137	4	no	—
149	#405	—	137	4	no	—
150	C₄×M₁₆	—	137	4	no	—
151	C₂³×C₄	—	137	1	no	—
152	C₂×M₁₆	—	137	2	no	—
153	C₂×M₁₆	—	137	2	no	—
154	C₂×M₁₆	—	137	2	no	—
155	C₂×M₁₆	—	137	2	no	—
156	C₂×M₁₆	—	137	2	no	—
157	C₂×M₁₆	—	137	2	no	—
158	C₂×M₁₆	—	137	1	no	—
159	C₂×M₁₆	—	137	2	no	—
160	C₂×M₁₆	—	137	2	no	—
161	C₂×M₁₆	—	137	2	no	—
162	C₂×M₁₆	—	137	2	no	—
163	C₂×M₁₆	—	137	2	no	—
164	C₂×M₁₆	#10587739	1	1	yes	characteristic, complemented
165	C₂²×C₈	—	137	2	no	—
166	C₂²×C₈	—	137	2	no	—
167	#117	—	137	2	no	—
168	#117	—	137	2	no	—
169	#117	—	137	2	no	—
170	#117	—	137	2	no	—
171	C₂×C₄⋊C₄	—	137	2	no	—
172	C₂×C₂²⋊C₄	—	137	2	no	—
173	C₂×C₄²	—	137	2	no	—
174	#97	—	137	8	no	—
175	C₄×C₈	—	137	8	no	—
176	C₂⁴	—	137	1	no	—
177	C₂²×C₄	—	137	4	no	—
178	C₂²×C₄	—	137	1	no	—
179	C₂²×C₄	—	137	1	no	—
180	C₂²×C₄	—	137	1	no	—
181	C₂²×C₄	—	137	1	no	—
182	C₂²×C₄	—	137	1	no	—
183	C₂²×C₄	—	137	1	no	—
184	C₂²×C₄	—	137	1	no	—
185	C₂²×C₄	—	137	1	no	—
186	C₂²×C₄	—	137	1	no	—
187	C₂²×C₄	—	137	1	no	—
188	C₂²×C₄	—	137	1	no	—
189	C₂²×C₄	—	137	1	no	—
190	C₂²×C₄	—	137	1	no	—
191	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
192	M₁₆	—	137	4	no	—
193	M₁₆	—	137	4	no	—
194	M₁₆	—	137	4	no	—
195	M₁₆	—	137	4	no	—
196	M₁₆	—	137	2	no	—
197	M₁₆	—	137	2	no	—
198	M₁₆	—	137	4	no	—
199	M₁₆	—	137	4	no	—
200	M₁₆	C₂×#10587739	1	2	yes	—
201	M₁₆	C₂×#10587739	1	2	yes	complemented
202	C₂×C₈	—	137	4	no	—
203	C₂×C₈	—	137	4	no	—
204	C₂×C₈	—	137	4	no	—
205	C₂×C₈	—	137	2	no	—
206	C₂×C₈	—	137	4	no	—
207	C₂×C₈	—	137	4	no	—
208	C₂×C₈	C₂×#10587739	1	2	yes	complemented
209	C₄⋊C₄	—	137	4	no	—
210	C₄⋊C₄	—	137	4	no	—
211	C₂²⋊C₄	—	137	4	no	—
212	C₂²⋊C₄	—	137	4	no	—
213	C₄²	—	137	4	no	—
214	C₄²	—	137	4	no	—
215	C₂³	—	548	2	no	—
216	C₂³	—	137	1	no	—
217	C₂³	—	137	1	no	—
218	C₂³	—	137	1	no	—
219	C₂³	—	137	1	no	—
220	C₂³	—	137	1	no	—
221	C₂³	—	137	1	no	—
222	C₂³	C₄×#10587739	1	1	yes	characteristic
223	C₂×C₄	—	548	4	no	—
224	C₂×C₄	—	137	8	no	—
225	C₂×C₄	—	137	1	no	—
226	C₂×C₄	—	137	1	no	—
227	C₂×C₄	—	137	1	no	—
228	C₂×C₄	—	137	1	no	—
229	C₂×C₄	—	137	1	no	—
230	C₂×C₄	—	137	1	no	—
231	C₂×C₄	—	137	1	no	—
232	C₂×C₄	—	137	1	no	—
233	C₂×C₄	—	137	1	no	—
234	C₂×C₄	—	137	1	no	—
235	C₂×C₄	—	137	1	no	—
236	C₂×C₄	—	137	1	no	—
237	C₂×C₄	—	137	1	no	—
238	C₂×C₄	—	137	1	no	—
239	C₂×C₄	—	137	1	no	—
240	C₂×C₄	—	137	1	no	—
241	C₂×C₄	—	137	1	no	—
242	C₂×C₄	—	137	1	no	—
243	C₂×C₄	—	137	1	no	—
244	C₂×C₄	—	137	1	no	—
245	C₂×C₄	—	137	1	no	—
246	C₂×C₄	—	137	1	no	—
247	C₂×C₄	C₄×#10587739	1	1	yes	characteristic
248	C₂×C₄	C₄×#10587739	1	1	yes	characteristic
249	C₂×C₄	C₄×#10587739	1	1	yes	characteristic
250	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
251	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
252	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
253	C₈	—	137	8	no	—
254	C₈	—	137	4	no	—
255	C₈	C₂²×#10587739	1	4	yes	—
256	C₂²	—	4	1	no	—
257	C₂²	—	274	2	no	—
258	C₂²	—	274	2	no	—
259	C₂²	—	274	2	no	—
260	C₂²	—	548	1	no	—
261	C₂²	—	548	1	no	—
262	C₂²	—	548	1	no	—
263	C₂²	—	137	1	no	—
264	C₂²	—	137	1	no	—
265	C₂²	—	137	1	no	—
266	C₂²	—	137	1	no	—
267	C₂²	C₂×C₄×#10587739	1	1	yes	characteristic
268	C₂²	C₂×C₄×#10587739	1	1	yes	characteristic
269	C₂²	C₂×C₄×#10587739	1	1	yes	characteristic
270	C₄	—	274	8	no	—
271	C₄	—	137	1	no	—
272	C₄	—	137	1	no	—
273	C₄	—	137	1	no	—
274	C₄	—	137	1	no	—
275	C₄	C₂×C₄×#10587739	1	1	yes	characteristic
276	C₄	C₂×C₄×#10587739	1	1	yes	characteristic
277	C₄	C₂×C₄×#10587739	1	1	yes	characteristic
278	C₄	C₂³×#10587739	1	1	yes	characteristic
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	274	1	no	—
283	C₂	—	274	1	no	—
284	C₂	—	274	1	no	—
285	C₂	—	137	2	no	—
286	C₂	C₂²×C₄×#10587739	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂³×#10587739
commutator, socle	C₂×C₁₃₇	C₂²×C₄²
Fitting	C₂×M₁₆×C₁₃₇	C₄
prime radical	C₂³×D₁₃₇	C₂×C₄
cosocle	C₄×D₁₃₇	C₂⁴
hypercenter	C₂×M₁₆	#10587739
hypocenter	C₁₃₇	#2873
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #22578369	C₂	semidirect
4. #22578366	C₂	semidirect
5. M₁₆×#10587739	C₂	semidirect
7. C₂×#14555520	C₄	semidirect
8. D₁₃₇⋊M₁₆	C₄	semidirect
9. D₁₃₇⋊M₁₆	C₄	semidirect
12. C₂×#14555519	C₄	semidirect
21. #17984142	C₂²	semidirect
22. C₈×#10587739	C₂²	semidirect
24. C₂×#14555494	C₄	semidirect
25. C₂×#14555493	C₄	semidirect
26. M₁₆×D₁₃₇	C₄	semidirect
27. M₁₆×D₁₃₇	C₄	semidirect
31. C₂×M₁₆×C₁₃₇	C₄	semidirect
34. #14555520	C₂×C₄	semidirect
36. #14555519	C₂×C₄	semidirect
37. #14555518	C₂×C₄	semidirect
39. C₂×#11764047	C₂×C₄	semidirect
61. #14555494	C₂×C₄	semidirect
63. #14555493	C₂×C₄	semidirect
64. C₂×#11764040	C₂×C₄	semidirect
65. C₈×D₁₃₇	C₂×C₄	semidirect
69. M₁₆×C₁₃₇	C₂×C₄	semidirect
70. C₂×C₈×C₁₃₇	C₂×C₄	semidirect
143. C₁₃₇	#2873	semidirect
164. C₂×M₁₆	#10587739	semidirect
201. M₁₆	C₂×#10587739	semidirect
208. C₂×C₈	C₂×#10587739	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₃₇⋊M₁₆	2
C₂×#17984167	2
#22578369	2
#22578366	2
M₁₆×#10587739	2
C₂×M₁₆×D₁₃₇	2
#2873	137