Subgroups — D₁₃₇⋊#444 #31797560

Order 17536.

464 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₃₇⋊#131	C₂	1	2	yes	maximal, complemented
2	#22578450	C₂	1	2	yes	maximal, complemented
3	#22578449	C₂	1	2	yes	maximal, complemented
4	#22578448	C₂	1	4	yes	maximal, complemented
5	D₁₃₇⋊#108	C₂	1	1	yes	maximal, characteristic, complemented
6	D₁₃₇⋊#104	C₂	1	2	yes	maximal, complemented
7	C₂×#17984142	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₁₃₇	C₂	1	1	yes	maximal, characteristic
9	#17984176	C₂²	1	4	yes	complemented
10	#17984175	—	2	4	no	—
11	#17984175	C₂²	1	4	yes	complemented
12	D₁₃₇⋊M₁₆	—	2	2	no	—
13	D₁₃₇⋊M₁₆	C₂²	1	2	yes	—
14	C₂×#14555518	—	2	2	no	—
15	C₂×#14555518	C₂²	1	1	yes	characteristic
16	C₂×C₄×#10587739	C₂²	1	1	yes	characteristic
17	#17984159	C₂²	1	2	yes	complemented
18	#17984158	C₂²	1	2	yes	complemented
19	#17984153	C₂²	1	4	yes	complemented
20	#17984152	C₂²	1	4	yes	complemented
21	#17984142	C₂²	1	2	yes	complemented
22	#17984142	C₂²	1	2	yes	complemented
23	#41×D₁₃₇	C₂²	1	2	yes	—
24	#17984111	C₂²	1	1	yes	characteristic
25	#17984110	C₄	1	1	yes	characteristic, complemented
26	#17984109	C₂²	1	2	yes	—
27	#17984108	C₄	1	2	yes	complemented
28	#17984107	C₂²	1	1	yes	characteristic
29	Q₁₆×D₁₃₇	C₄	1	1	yes	characteristic, complemented
30	SD₁₆×D₁₃₇	C₄	1	2	yes	complemented
31	D₈×D₁₃₇	C₄	1	1	yes	characteristic, complemented
32	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
33	#135×C₁₃₇	C₄	1	1	yes	characteristic, complemented
34	C₂²×#10587739	—	2	1	no	—
35	#14555520	—	2	4	no	—
36	#14555520	C₂³	1	4	yes	—
37	#14555519	—	2	4	no	—
38	#14555519	—	2	2	no	—
39	#14555518	—	2	4	no	—
40	#14555518	—	2	2	no	—
41	#14555518	D₄	1	2	yes	—
42	#14555518	C₂³	1	2	yes	—
43	C₂×#11764047	—	2	4	no	—
44	C₂×#11764047	D₄	1	2	yes	—
45	C₄×#10587739	—	2	1	no	—
46	C₄×#10587739	C₂³	1	2	yes	—
47	#14555511	C₂³	1	2	yes	—
48	#14555510	—	2	2	no	—
49	#14555510	C₂×C₄	1	2	yes	—
50	#14555509	—	2	2	no	—
51	#14555509	C₂³	1	2	yes	—
52	Q₈×D₁₃₇	C₂×C₄	1	2	yes	—
53	D₄×D₁₃₇	—	2	2	no	—
54	D₄×D₁₃₇	C₂×C₄	1	2	yes	—
55	C₂×C₄×D₁₃₇	—	2	2	no	—
56	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
57	Dic₅₄₈	C₂×C₄	1	1	yes	characteristic, complemented
58	#14555500	C₂×C₄	1	2	yes	complemented
59	#14555499	C₂×C₄	1	2	yes	complemented
60	#14555498	C₂×C₄	1	2	yes	complemented
61	#14555497	C₂×C₄	1	2	yes	complemented
62	D₁₀₉₆	C₂×C₄	1	1	yes	characteristic, complemented
63	#14555495	C₂×C₄	1	2	yes	complemented
64	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
65	C₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic
66	C₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic
67	C₈×D₁₃₇	C₂³	1	1	yes	characteristic
68	C₈×D₁₃₇	C₂³	1	1	yes	characteristic
69	#41×C₁₃₇	C₂×C₄	1	2	yes	—
70	Q₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic, complemented
71	SD₁₆×C₁₃₇	C₂×C₄	1	2	yes	complemented
72	D₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic, complemented
73	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
74	C₂×#10587739	—	2	2	no	—
75	C₂×#10587739	—	4	2	no	—
76	#11764047	—	2	4	no	—
77	#11764047	C₂×D₄	1	4	yes	—
78	C₂²×D₁₃₇	—	2	2	no	—
79	C₂²×D₁₃₇	#41	1	1	yes	characteristic
80	Dic₂₇₄	—	2	2	no	—
81	Dic₂₇₄	C₂²×C₄	1	2	yes	—
82	D₅₄₈	—	2	2	no	—
83	D₅₄₈	C₂²×C₄	1	2	yes	—
84	C₂²⋊D₁₃₇	—	2	2	no	—
85	C₂²⋊D₁₃₇	—	2	2	no	—
86	C₂²⋊D₁₃₇	—	2	2	no	—
87	C₂×Dic₁₃₇	—	2	2	no	—
88	C₂×Dic₁₃₇	C₂×D₄	1	1	yes	characteristic
89	C₄×D₁₃₇	—	2	2	no	—
90	C₄×D₁₃₇	—	2	2	no	—
91	C₄×D₁₃₇	—	2	2	no	—
92	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
93	C₄×D₁₃₇	C₂×D₄	1	1	yes	characteristic
94	C₄×D₁₃₇	#41	1	1	yes	characteristic
95	C₄×D₁₃₇	C₂⁴	1	1	yes	characteristic
96	#11764040	C₂²×C₄	1	1	yes	characteristic
97	#11764040	C₂²×C₄	1	1	yes	characteristic
98	Q₈×C₁₃₇	C₂²×C₄	1	2	yes	—
99	D₄×C₁₃₇	—	2	2	no	—
100	D₄×C₁₃₇	C₂²×C₄	1	2	yes	—
101	C₂×C₄×C₁₃₇	—	2	2	no	—
102	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
103	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
104	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₁₃₇	—	2	2	no	—
106	C₂²×C₁₃₇	C₄×D₄	1	1	yes	characteristic
107	C₂×D₁₃₇	—	2	2	no	—
108	C₂×D₁₃₇	—	2	2	no	—
109	C₂×D₁₃₇	—	4	4	no	—
110	C₂×D₁₃₇	C₄×D₄	1	1	yes	characteristic
111	C₂×D₁₃₇	C₂×#41	1	1	yes	characteristic
112	#10587739	—	4	2	no	—
113	C₄×C₁₃₇	—	2	2	no	—
114	C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
115	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
116	Dic₁₃₇	—	2	2	no	—
117	Dic₁₃₇	C₄×D₄	1	1	yes	characteristic
118	Dic₁₃₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₁₃₇	—	2	1	no	—
120	C₂×C₁₃₇	—	4	2	no	—
121	C₂×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
122	D₁₃₇	—	2	1	no	—
123	D₁₃₇	—	4	2	no	—
124	D₁₃₇	#444	1	2	yes	complemented
125	C₁₃₇	C₂×#444	1	1	yes	characteristic, complemented, Sylow
126	C₂×#444	—	137	1	no	maximal, Sylow
127	C₂×#135	—	137	1	no	—
128	C₂×#131	—	137	2	no	—
129	#444	—	137	4	no	—
130	#444	—	137	2	no	—
131	#444	—	137	2	no	—
132	C₂×#108	—	137	1	no	—
133	C₂×#104	—	137	2	no	—
134	C₂×#97	—	137	1	no	—
135	C₂×#41	—	137	2	no	—
136	#135	—	137	2	no	—
137	#135	—	137	1	no	—
138	#135	—	137	2	no	—
139	#135	—	137	1	no	—
140	#135	—	137	1	no	—
141	#135	#10587739	1	1	yes	characteristic, complemented
142	C₂×Q₁₆	—	137	1	no	—
143	C₂×SD₁₆	—	137	2	no	—
144	C₂×D₈	—	137	1	no	—
145	#131	—	274	4	no	—
146	#131	—	137	4	no	—
147	#131	—	137	4	no	—
148	C₂×M₁₆	—	274	2	no	—
149	C₂×M₁₆	—	137	2	no	—
150	C₂²×C₈	—	274	2	no	—
151	C₂²×C₈	—	137	1	no	—
152	C₂²×C₈	—	137	1	no	—
153	C₂×C₄²	—	137	1	no	—
154	#108	—	137	2	no	—
155	#108	—	137	2	no	—
156	#104	—	137	4	no	—
157	#104	—	137	4	no	—
158	#97	—	137	2	no	—
159	#97	—	137	2	no	—
160	#41	—	274	2	no	—
161	#41	—	274	2	no	—
162	#41	—	137	2	no	—
163	#41	—	137	2	no	—
164	#41	—	137	2	no	—
165	#41	C₂×#10587739	1	2	yes	—
166	C₂×Q₈	—	137	2	no	—
167	C₂×D₄	—	274	2	no	—
168	C₂×D₄	—	137	2	no	—
169	C₂²×C₄	—	274	2	no	—
170	C₂²×C₄	—	274	1	no	—
171	C₂²×C₄	—	137	1	no	—
172	Q₁₆	—	137	2	no	—
173	Q₁₆	—	137	1	no	—
174	Q₁₆	C₂×#10587739	1	1	yes	characteristic, complemented
175	SD₁₆	—	137	2	no	—
176	SD₁₆	—	137	2	no	—
177	SD₁₆	—	137	2	no	—
178	SD₁₆	C₂×#10587739	1	2	yes	complemented
179	D₈	—	137	2	no	—
180	D₈	—	137	1	no	—
181	D₈	C₂×#10587739	1	1	yes	characteristic, complemented
182	M₁₆	—	274	2	no	—
183	M₁₆	—	274	4	no	—
184	M₁₆	—	274	4	no	—
185	M₁₆	—	137	4	no	—
186	C₂×C₈	—	274	2	no	—
187	C₂×C₈	—	274	4	no	—
188	C₂×C₈	—	274	4	no	—
189	C₂×C₈	—	137	2	no	—
190	C₂×C₈	—	137	2	no	—
191	C₂×C₈	—	137	2	no	—
192	C₂×C₈	—	137	1	no	—
193	C₂×C₈	—	137	1	no	—
194	C₂×C₈	—	137	1	no	—
195	C₂×C₈	—	137	1	no	—
196	C₂×C₈	—	137	1	no	—
197	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
198	C₄²	—	274	1	no	—
199	C₄²	—	137	2	no	—
200	C₂³	—	274	2	no	—
201	C₂³	—	137	1	no	—
202	Q₈	—	274	2	no	—
203	Q₈	—	137	2	no	—
204	Q₈	C₂²×#10587739	1	2	yes	—
205	D₄	—	2	2	no	—
206	D₄	—	274	2	no	—
207	D₄	—	274	2	no	—
208	D₄	—	274	2	no	—
209	D₄	—	274	2	no	—
210	D₄	—	137	2	no	—
211	D₄	C₂²×#10587739	1	2	yes	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	274	2	no	—
214	C₂×C₄	—	274	2	no	—
215	C₂×C₄	—	274	2	no	—
216	C₂×C₄	—	274	2	no	—
217	C₂×C₄	—	274	2	no	—
218	C₂×C₄	—	548	2	no	—
219	C₂×C₄	—	137	1	no	—
220	C₂×C₄	—	137	1	no	—
221	C₂×C₄	—	137	1	no	—
222	C₂×C₄	—	137	1	no	—
223	C₂×C₄	—	137	1	no	—
224	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
225	C₈	—	274	4	no	—
226	C₈	—	137	4	no	—
227	C₈	—	137	1	no	—
228	C₈	—	137	1	no	—
229	C₈	C₂²×#10587739	1	1	yes	characteristic
230	C₈	C₂²×#10587739	1	1	yes	characteristic
231	C₂²	—	2	2	no	—
232	C₂²	—	274	2	no	—
233	C₂²	—	274	2	no	—
234	C₂²	—	548	4	no	—
235	C₂²	—	137	1	no	—
236	C₂²	—	137	1	no	—
237	C₂²	D₄×#10587739	1	1	yes	characteristic
238	C₄	—	2	2	no	—
239	C₄	—	274	2	no	—
240	C₄	—	548	2	no	—
241	C₄	—	137	1	no	—
242	C₄	—	137	1	no	—
243	C₄	D₄×#10587739	1	1	yes	characteristic
244	C₄	C₂³×#10587739	1	1	yes	characteristic
245	C₂	—	2	1	no	—
246	C₂	—	4	2	no	—
247	C₂	—	274	1	no	—
248	C₂	—	548	2	no	—
249	C₂	—	137	2	no	—
250	C₂	C₂×D₄×#10587739	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×#10587739
commutator	C₄×C₁₃₇	C₂³×C₄
Frattini	C₄	C₂³×#10587739
Fitting	#135×C₁₃₇	C₄
socle	C₂×C₁₃₇	C₂×C₄×D₄
prime radical	#135×D₁₃₇	C₂
cosocle	C₄×D₁₃₇	C₂⁴
hypercenter	#135	#10587739
hypocenter	C₁₃₇	C₂×#444
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. D₁₃₇⋊#131	C₂	semidirect
2. #22578450	C₂	semidirect
3. #22578449	C₂	semidirect
4. #22578448	C₂	semidirect
5. D₁₃₇⋊#108	C₂	semidirect
6. D₁₃₇⋊#104	C₂	semidirect
7. C₂×#17984142	C₂	semidirect
9. #17984176	C₂²	semidirect
11. #17984175	C₂²	semidirect
17. #17984159	C₂²	semidirect
18. #17984158	C₂²	semidirect
19. #17984153	C₂²	semidirect
20. #17984152	C₂²	semidirect
21. #17984142	C₂²	semidirect
22. #17984142	C₂²	semidirect
25. #17984110	C₄	semidirect
27. #17984108	C₄	semidirect
29. Q₁₆×D₁₃₇	C₄	semidirect
30. SD₁₆×D₁₃₇	C₄	semidirect
31. D₈×D₁₃₇	C₄	semidirect
33. #135×C₁₃₇	C₄	semidirect
57. Dic₅₄₈	C₂×C₄	semidirect
58. #14555500	C₂×C₄	semidirect
59. #14555499	C₂×C₄	semidirect
60. #14555498	C₂×C₄	semidirect
61. #14555497	C₂×C₄	semidirect
62. D₁₀₉₆	C₂×C₄	semidirect
63. #14555495	C₂×C₄	semidirect
70. Q₁₆×C₁₃₇	C₂×C₄	semidirect
71. SD₁₆×C₁₃₇	C₂×C₄	semidirect
72. D₈×C₁₃₇	C₂×C₄	semidirect
124. D₁₃₇	#444	semidirect
125. C₁₃₇	C₂×#444	semidirect
141. #135	#10587739	semidirect
174. Q₁₆	C₂×#10587739	semidirect
178. SD₁₆	C₂×#10587739	semidirect
181. D₈	C₂×#10587739	semidirect

Maximal subgroups

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

subgroup	index
D₁₃₇⋊#131	2
#22578450	2
#22578449	2
#22578448	2
D₁₃₇⋊#108	2
D₁₃₇⋊#104	2
C₂×#17984142	2
#135×D₁₃₇	2
C₂×#444	137

Subgroups — D137⋊#444 #31797560

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₃₇⋊#444 #31797560