Subgroups — #31797552

Order 17536.

484 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578472	C₂	1	2	yes	maximal, complemented
2	#22578444	C₂	1	2	yes	maximal, complemented
3	#22578443	C₂	1	2	yes	maximal, complemented
4	SD₁₆×#10587739	C₂	1	4	yes	maximal, complemented
5	C₂×#17984156	C₂	1	1	yes	maximal, characteristic, complemented
6	#22578394	C₂	1	2	yes	maximal, complemented
7	#22578366	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₁₃₇	C₂	1	1	yes	maximal, characteristic
9	D₁₃₇⋊M₁₆	C₂²	1	1	yes	characteristic
10	Q₈×#10587739	C₂²	1	4	yes	—
11	D₄×#10587739	—	2	4	no	—
12	D₄×#10587739	C₂²	1	4	yes	—
13	#17984167	—	2	2	no	—
14	#17984167	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#10587739	—	2	2	no	—
16	C₂×C₄⋊#10587739	C₂²	1	2	yes	—
17	#17984156	C₂²	1	2	yes	—
18	#17984156	C₂²	1	2	yes	—
19	#17984151	C₂²	1	4	yes	—
20	#17984150	C₂²	1	4	yes	—
21	#17984142	C₂²	1	2	yes	—
22	C₈×#10587739	C₂²	1	2	yes	—
23	#41×D₁₃₇	C₂²	1	2	yes	—
24	#17984111	C₄	1	1	yes	characteristic, complemented
25	#17984110	C₄	1	1	yes	characteristic, complemented
26	#17984109	C₄	1	2	yes	complemented
27	#17984108	C₄	1	2	yes	complemented
28	#17984107	C₄	1	1	yes	characteristic, complemented
29	Q₁₆×D₁₃₇	C₂²	1	1	yes	characteristic
30	SD₁₆×D₁₃₇	C₂²	1	2	yes	—
31	D₈×D₁₃₇	C₂²	1	1	yes	characteristic
32	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
33	#135×C₁₃₇	C₄	1	1	yes	characteristic, complemented
34	C₂²×#10587739	—	2	4	no	—
35	C₂²×#10587739	—	2	2	no	—
36	#14555520	—	2	1	no	—
37	#14555519	—	2	1	no	—
38	#14555518	C₂³	1	2	yes	—
39	C₄⋊#10587739	—	2	4	no	—
40	C₄⋊#10587739	—	2	2	no	—
41	C₄⋊#10587739	D₄	1	2	yes	—
42	C₄⋊#10587739	C₂³	1	4	yes	—
43	C₂²⋊#10587739	—	2	4	no	—
44	C₂²⋊#10587739	D₄	1	2	yes	—
45	C₄×#10587739	—	2	4	no	—
46	C₄×#10587739	C₂³	1	2	yes	—
47	#14555511	C₂×C₄	1	2	yes	complemented
48	#14555510	—	2	2	no	—
49	#14555510	C₂×C₄	1	2	yes	complemented
50	#14555509	—	2	2	no	—
51	#14555509	C₂×C₄	1	2	yes	complemented
52	Q₈×D₁₃₇	C₂³	1	2	yes	—
53	D₄×D₁₃₇	—	2	2	no	—
54	D₄×D₁₃₇	C₂³	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, complemented
65	C₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic, complemented
66	C₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic, complemented
67	C₈×D₁₃₇	C₂³	1	1	yes	characteristic
68	C₈×D₁₃₇	C₂³	1	1	yes	characteristic
69	#41×C₁₃₇	C₂×C₄	1	2	yes	complemented
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, complemented
74	C₂×#10587739	—	2	4	no	—
75	C₂×#10587739	—	8	4	no	—
76	C₂×#10587739	—	8	2	no	—
77	C₂×#10587739	C₂×D₄	1	4	yes	—
78	#11764047	—	2	2	no	—
79	C₂²×D₁₃₇	—	2	2	no	—
80	C₂²×D₁₃₇	C₂×D₄	1	1	yes	characteristic
81	Dic₂₇₄	—	2	2	no	—
82	Dic₂₇₄	C₂²×C₄	1	2	yes	—
83	D₅₄₈	—	2	2	no	—
84	D₅₄₈	C₂²×C₄	1	2	yes	—
85	C₂²⋊D₁₃₇	—	2	2	no	—
86	C₂²⋊D₁₃₇	—	2	2	no	—
87	C₂²⋊D₁₃₇	—	2	2	no	—
88	C₂×Dic₁₃₇	—	2	2	no	—
89	C₂×Dic₁₃₇	#41	1	1	yes	characteristic
90	C₄×D₁₃₇	—	2	2	no	—
91	C₄×D₁₃₇	—	2	2	no	—
92	C₄×D₁₃₇	—	2	2	no	—
93	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
94	C₄×D₁₃₇	C₂×D₄	1	1	yes	characteristic
95	C₄×D₁₃₇	#41	1	1	yes	characteristic
96	C₄×D₁₃₇	C₂⁴	1	1	yes	characteristic
97	#11764040	C₂²×C₄	1	1	yes	characteristic
98	#11764040	C₂²×C₄	1	1	yes	characteristic
99	Q₈×C₁₃₇	C₂²×C₄	1	2	yes	—
100	D₄×C₁₃₇	—	2	2	no	—
101	D₄×C₁₃₇	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₁₃₇	—	2	2	no	—
103	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
104	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
105	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₁₃₇	—	2	2	no	—
107	C₂²×C₁₃₇	C₄×D₄	1	1	yes	characteristic
108	C₂×D₁₃₇	—	2	2	no	—
109	C₂×D₁₃₇	—	2	2	no	—
110	C₂×D₁₃₇	—	4	2	no	—
111	C₂×D₁₃₇	—	4	2	no	—
112	C₂×D₁₃₇	C₄×D₄	1	1	yes	characteristic
113	C₂×D₁₃₇	C₂²×D₄	1	1	yes	characteristic
114	#10587739	—	2	4	no	—
115	#10587739	—	4	4	no	—
116	C₄×C₁₃₇	—	2	2	no	—
117	C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
118	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
119	Dic₁₃₇	—	2	2	no	—
120	Dic₁₃₇	C₄×D₄	1	1	yes	characteristic
121	Dic₁₃₇	C₂×#41	1	1	yes	characteristic
122	C₂×C₁₃₇	—	2	1	no	—
123	C₂×C₁₃₇	—	4	2	no	—
124	C₂×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
125	D₁₃₇	—	2	1	no	—
126	D₁₃₇	—	4	2	no	—
127	D₁₃₇	#577	1	2	yes	—
128	C₁₃₇	#2952	1	1	yes	characteristic, complemented, Sylow
129	#2952	—	137	1	no	maximal, Sylow
130	C₂×#135	—	137	1	no	—
131	#520	—	137	2	no	—
132	#442	—	137	2	no	—
133	#441	—	137	2	no	—
134	C₄×SD₁₆	—	137	4	no	—
135	C₂×#106	—	137	1	no	—
136	#417	—	137	2	no	—
137	#405	—	137	1	no	—
138	C₂×#41	—	137	2	no	—
139	#135	—	137	2	no	—
140	#135	—	137	1	no	—
141	#135	—	137	2	no	—
142	#135	—	137	1	no	—
143	#135	—	137	1	no	—
144	#135	#10587739	1	1	yes	characteristic, complemented
145	C₂×Q₁₆	—	137	1	no	—
146	C₂×SD₁₆	—	137	2	no	—
147	C₂×D₈	—	137	1	no	—
148	C₂×M₁₆	—	137	1	no	—
149	C₂²×C₈	—	137	1	no	—
150	C₄×Q₈	—	137	4	no	—
151	C₄×D₄	—	274	4	no	—
152	C₄×D₄	—	137	4	no	—
153	#117	—	274	2	no	—
154	#117	—	137	1	no	—
155	C₂×C₄⋊C₄	—	274	2	no	—
156	C₂×C₄⋊C₄	—	137	2	no	—
157	#106	—	137	2	no	—
158	#106	—	137	2	no	—
159	#103	—	137	4	no	—
160	#102	—	137	4	no	—
161	#97	—	137	2	no	—
162	C₄×C₈	—	137	2	no	—
163	#41	—	274	2	no	—
164	#41	—	274	2	no	—
165	#41	—	137	2	no	—
166	#41	—	137	2	no	—
167	#41	—	137	2	no	—
168	#41	C₂×#10587739	1	2	yes	complemented
169	C₂×Q₈	—	137	2	no	—
170	C₂×D₄	—	274	2	no	—
171	C₂×D₄	—	137	2	no	—
172	C₂²×C₄	—	274	2	no	—
173	C₂²×C₄	—	274	2	no	—
174	C₂²×C₄	—	274	4	no	—
175	C₂²×C₄	—	137	1	no	—
176	Q₁₆	—	137	2	no	—
177	Q₁₆	—	137	1	no	—
178	Q₁₆	C₂×#10587739	1	1	yes	characteristic, complemented
179	SD₁₆	—	137	2	no	—
180	SD₁₆	—	137	2	no	—
181	SD₁₆	—	137	2	no	—
182	SD₁₆	C₂×#10587739	1	2	yes	complemented
183	D₈	—	137	2	no	—
184	D₈	—	137	1	no	—
185	D₈	C₂×#10587739	1	1	yes	characteristic, complemented
186	M₁₆	—	274	1	no	—
187	M₁₆	—	274	1	no	—
188	C₂×C₈	—	137	2	no	—
189	C₂×C₈	—	137	1	no	—
190	C₂×C₈	—	137	1	no	—
191	C₂×C₈	—	137	1	no	—
192	C₂×C₈	—	137	1	no	—
193	C₂×C₈	—	137	1	no	—
194	C₂×C₈	C₂×#10587739	1	1	yes	characteristic, complemented
195	C₄⋊C₄	—	274	2	no	—
196	C₄⋊C₄	—	274	4	no	—
197	C₄⋊C₄	—	137	4	no	—
198	C₄⋊C₄	—	137	2	no	—
199	C₂²⋊C₄	—	274	4	no	—
200	C₂²⋊C₄	—	137	2	no	—
201	C₄²	—	274	4	no	—
202	C₄²	—	137	2	no	—
203	C₂³	—	274	2	no	—
204	C₂³	—	137	1	no	—
205	Q₈	—	274	2	no	—
206	Q₈	—	137	2	no	—
207	Q₈	C₂²×#10587739	1	2	yes	—
208	D₄	—	2	2	no	—
209	D₄	—	274	2	no	—
210	D₄	—	274	2	no	—
211	D₄	—	274	2	no	—
212	D₄	—	274	2	no	—
213	D₄	—	137	2	no	—
214	D₄	C₂²×#10587739	1	2	yes	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	274	2	no	—
217	C₂×C₄	—	274	2	no	—
218	C₂×C₄	—	274	2	no	—
219	C₂×C₄	—	274	2	no	—
220	C₂×C₄	—	274	4	no	—
221	C₂×C₄	—	1096	4	no	—
222	C₂×C₄	—	1096	2	no	—
223	C₂×C₄	—	137	4	no	—
224	C₂×C₄	—	137	1	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	C₂²×#10587739	1	1	yes	characteristic
230	C₈	—	274	2	no	—
231	C₈	—	137	1	no	—
232	C₈	—	137	1	no	—
233	C₈	C₂²×#10587739	1	1	yes	characteristic
234	C₈	C₂²×#10587739	1	1	yes	characteristic
235	C₂²	—	2	2	no	—
236	C₂²	—	274	2	no	—
237	C₂²	—	274	2	no	—
238	C₂²	—	548	2	no	—
239	C₂²	—	548	2	no	—
240	C₂²	—	137	1	no	—
241	C₂²	—	137	1	no	—
242	C₂²	D₄×#10587739	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	274	4	no	—
245	C₄	—	274	2	no	—
246	C₄	—	548	4	no	—
247	C₄	—	137	1	no	—
248	C₄	—	137	1	no	—
249	C₄	D₄×#10587739	1	1	yes	characteristic
250	C₄	C₂³×#10587739	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	4	2	no	—
253	C₂	—	274	1	no	—
254	C₂	—	548	2	no	—
255	C₂	—	137	2	no	—
256	C₂	C₂×D₄×#10587739	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×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	#2952
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #22578472	C₂	semidirect
2. #22578444	C₂	semidirect
3. #22578443	C₂	semidirect
4. SD₁₆×#10587739	C₂	semidirect
5. C₂×#17984156	C₂	semidirect
6. #22578394	C₂	semidirect
7. #22578366	C₂	semidirect
24. #17984111	C₄	semidirect
25. #17984110	C₄	semidirect
26. #17984109	C₄	semidirect
27. #17984108	C₄	semidirect
28. #17984107	C₄	semidirect
33. #135×C₁₃₇	C₄	semidirect
47. #14555511	C₂×C₄	semidirect
49. #14555510	C₂×C₄	semidirect
51. #14555509	C₂×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
64. C₂×#11764040	C₂×C₄	semidirect
65. C₈×D₁₃₇	C₂×C₄	semidirect
66. C₈×D₁₃₇	C₂×C₄	semidirect
69. #41×C₁₃₇	C₂×C₄	semidirect
70. Q₁₆×C₁₃₇	C₂×C₄	semidirect
71. SD₁₆×C₁₃₇	C₂×C₄	semidirect
72. D₈×C₁₃₇	C₂×C₄	semidirect
73. C₂×C₈×C₁₃₇	C₂×C₄	semidirect
128. C₁₃₇	#2952	semidirect
144. #135	#10587739	semidirect
168. #41	C₂×#10587739	semidirect
178. Q₁₆	C₂×#10587739	semidirect
182. SD₁₆	C₂×#10587739	semidirect
185. D₈	C₂×#10587739	semidirect
194. 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
#22578472	2
#22578444	2
#22578443	2
SD₁₆×#10587739	2
C₂×#17984156	2
#22578394	2
#22578366	2
#135×D₁₃₇	2
#2952	137