Subgroups — #31789623

Order 17536.

506 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984116	C₂	1	1	yes	maximal, characteristic
2	#22577883	C₂	1	1	yes	maximal, characteristic, complemented
3	#22577877	C₂	1	1	yes	maximal, characteristic, complemented
4	#22577516	C₂	1	2	yes	maximal, complemented
5	#22577505	C₂	1	2	yes	maximal, complemented
6	C₄×#14555499	C₂	1	2	yes	maximal, complemented
7	C₄×D₁₀₉₆	C₂	1	2	yes	maximal, complemented
8	#22577445	C₂	1	1	yes	maximal, characteristic, complemented
9	#22577400	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#17984010	C₂	1	1	yes	maximal, characteristic, complemented
11	#405×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14555510	C₂²	1	1	yes	characteristic
13	C₂²×D₅₄₈	C₂²	1	1	yes	characteristic
14	#17984116	C₄	1	8	yes	complemented
15	C₂×#14555499	C₂²	1	2	yes	—
16	C₂×D₁₀₉₆	C₂²	1	2	yes	—
17	C₄×Dic₂₇₄	C₂²	1	2	yes	complemented
18	C₄×D₅₄₈	C₂²	1	2	yes	—
19	C₄×D₅₄₈	C₂²	1	2	yes	—
20	C₄×D₅₄₈	C₂²	1	2	yes	complemented
21	#17984050	—	2	1	no	—
22	#17984050	—	2	1	no	—
23	#17984048	—	2	2	no	—
24	#17984047	C₂²	1	1	yes	characteristic
25	#17984046	—	2	1	no	—
26	C₂×C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×D₁₃₇	—	2	1	no	—
28	C₂²⋊C₄×D₁₃₇	—	2	2	no	—
29	#17984022	C₂²	1	2	yes	complemented
30	#17984020	C₂²	1	2	yes	complemented
31	#17984013	C₂²	1	1	yes	characteristic, complemented
32	#17984013	C₂²	1	1	yes	characteristic, complemented
33	#17984010	C₂²	1	2	yes	—
34	#17984010	C₂²	1	2	yes	—
35	#17984010	C₂²	1	1	yes	characteristic, complemented
36	#17984010	C₂²	1	1	yes	characteristic, complemented
37	C₂×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
38	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
39	#117×C₁₃₇	C₂²	1	1	yes	characteristic
40	#97×C₁₃₇	C₂²	1	2	yes	complemented
41	C₄×C₈×C₁₃₇	C₂²	1	2	yes	complemented
42	C₂³×D₁₃₇	—	2	1	no	—
43	#14555510	—	2	2	no	—
44	#14555510	C₂×C₄	1	4	yes	complemented
45	C₂×Dic₂₇₄	C₂³	1	1	yes	characteristic
46	C₂×D₅₄₈	—	2	1	no	—
47	C₂×D₅₄₈	—	2	2	no	—
48	C₂×D₅₄₈	C₂×C₄	1	4	yes	complemented
49	C₂×D₅₄₈	C₂³	1	2	yes	—
50	C₂×D₅₄₈	C₂³	1	1	yes	characteristic
51	C₂×C₂²⋊D₁₃₇	—	2	1	no	—
52	C₂²×Dic₁₃₇	—	2	1	no	—
53	C₂×C₄×D₁₃₇	—	2	1	no	—
54	C₂×C₄×D₁₃₇	—	2	2	no	—
55	C₂×C₄×D₁₃₇	—	2	2	no	—
56	C₂×C₄×D₁₃₇	—	2	2	no	—
57	#14555499	C₂×C₄	1	8	yes	complemented
58	D₁₀₉₆	C₂×C₄	1	8	yes	complemented
59	C₄⋊Dic₁₃₇	—	2	1	no	—
60	C₄⋊Dic₁₃₇	C₂³	1	2	yes	—
61	C₄⋊Dic₁₃₇	C₂³	1	1	yes	characteristic
62	C₄⋊Dic₁₃₇	C₂³	1	1	yes	characteristic
63	#14555489	—	2	2	no	—
64	C₂²⋊Dic₁₃₇	—	2	1	no	—
65	#14555487	—	2	2	no	—
66	#14555487	—	2	2	no	—
67	#14555487	—	2	2	no	—
68	C₄×Dic₁₃₇	—	2	2	no	—
69	C₄×Dic₁₃₇	—	2	1	no	—
70	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
71	M₁₆×C₁₃₇	C₂×C₄	1	4	yes	complemented
72	C₂×C₈×C₁₃₇	—	2	2	no	—
73	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
76	C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
77	C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
78	C₂²⋊C₄×C₁₃₇	D₄	1	2	yes	—
79	C₄²×C₁₃₇	C₂³	1	2	yes	—
80	C₂²×D₁₃₇	—	2	1	no	—
81	C₂²×D₁₃₇	—	2	2	no	—
82	C₂²×D₁₃₇	—	2	4	no	—
83	C₂²×D₁₃₇	—	8	1	no	—
84	C₂²×D₁₃₇	—	8	1	no	—
85	Dic₂₇₄	—	2	1	no	—
86	Dic₂₇₄	C₂²×C₄	1	2	yes	—
87	D₅₄₈	—	2	4	no	—
88	D₅₄₈	—	2	2	no	—
89	D₅₄₈	—	2	1	no	—
90	D₅₄₈	C₂²×C₄	1	2	yes	—
91	D₅₄₈	C₂²×C₄	1	4	yes	—
92	C₂²⋊D₁₃₇	—	2	4	no	—
93	C₂×Dic₁₃₇	—	2	1	no	—
94	C₂×Dic₁₃₇	—	2	1	no	—
95	C₂×Dic₁₃₇	—	2	1	no	—
96	C₂×Dic₁₃₇	—	2	2	no	—
97	C₂×Dic₁₃₇	—	8	1	no	—
98	C₄×D₁₃₇	—	2	4	no	—
99	C₄×D₁₃₇	—	8	2	no	—
100	C₄×D₁₃₇	—	8	2	no	—
101	C₄×D₁₃₇	—	8	2	no	—
102	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
104	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃₇	#41	1	2	yes	—
108	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
109	C₈×C₁₃₇	—	2	1	no	—
110	C₈×C₁₃₇	—	2	1	no	—
111	C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
112	C₂²×C₁₃₇	—	2	1	no	—
113	C₂²×C₁₃₇	—	2	1	no	—
114	C₂²×C₁₃₇	C₄×D₄	1	2	yes	—
115	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×D₁₃₇	—	2	2	no	—
117	C₂×D₁₃₇	—	2	4	no	—
118	C₂×D₁₃₇	—	4	1	no	—
119	C₂×D₁₃₇	—	4	2	no	—
120	C₂×D₁₃₇	—	4	2	no	—
121	C₂×D₁₃₇	—	4	1	no	—
122	C₂×D₁₃₇	—	8	4	no	—
123	C₄×C₁₃₇	—	2	2	no	—
124	C₄×C₁₃₇	—	2	2	no	—
125	C₄×C₁₃₇	C₄×D₄	1	2	yes	—
126	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
127	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
128	Dic₁₃₇	—	2	2	no	—
129	Dic₁₃₇	—	4	1	no	—
130	Dic₁₃₇	—	4	2	no	—
131	Dic₁₃₇	—	4	1	no	—
132	C₂×C₁₃₇	—	2	2	no	—
133	C₂×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
134	C₂×C₁₃₇	#576	1	1	yes	characteristic
135	C₂×C₁₃₇	#577	1	1	yes	characteristic
136	D₁₃₇	—	4	2	no	—
137	D₁₃₇	—	4	4	no	—
138	C₁₃₇	#2946	1	1	yes	characteristic, complemented, Sylow
139	#2946	—	137	1	no	maximal, Sylow
140	C₂×AGL(1,ℤ/8)	—	137	1	no	—
141	#520	—	137	1	no	—
142	#518	—	137	1	no	—
143	#442	—	137	2	no	—
144	#440	—	137	2	no	—
145	C₄×SD₁₆	—	137	2	no	—
146	C₄×D₈	—	137	2	no	—
147	#428	—	137	1	no	—
148	#416	—	137	1	no	—
149	C₂×#102	—	137	1	no	—
150	#405	D₁₃₇	1	1	yes	characteristic, complemented
151	C₂×#41	—	137	1	no	—
152	C₂²×D₄	—	137	1	no	—
153	AGL(1,ℤ/8)	—	137	8	no	—
154	C₂×SD₁₆	—	137	2	no	—
155	C₂×D₈	—	137	2	no	—
156	C₂×M₁₆	C₂×D₁₃₇	1	1	yes	characteristic
157	C₂²×C₈	C₂×D₁₃₇	1	1	yes	characteristic
158	C₄×Q₈	—	137	2	no	—
159	C₄×D₄	—	274	1	no	—
160	C₄×D₄	—	274	1	no	—
161	C₄×D₄	—	274	2	no	—
162	C₄×D₄	—	137	2	no	—
163	C₄×D₄	—	137	2	no	—
164	C₄×D₄	—	137	2	no	—
165	#117	—	274	1	no	—
166	#117	—	137	1	no	—
167	#117	C₂×D₁₃₇	1	1	yes	characteristic
168	C₂×C₄⋊C₄	—	274	1	no	—
169	C₂×C₄⋊C₄	—	137	1	no	—
170	C₂×C₂²⋊C₄	—	274	2	no	—
171	#107	—	137	2	no	—
172	#106	—	137	2	no	—
173	#103	—	137	1	no	—
174	#103	—	137	1	no	—
175	#102	—	137	2	no	—
176	#102	—	137	2	no	—
177	#102	—	137	1	no	—
178	#102	—	137	1	no	—
179	#97	C₂×D₁₃₇	1	2	yes	complemented
180	C₄×C₈	C₂×D₁₃₇	1	2	yes	complemented
181	C₂⁴	—	274	1	no	—
182	#41	—	274	2	no	—
183	#41	—	137	4	no	—
184	C₂×Q₈	—	137	1	no	—
185	C₂×D₄	—	274	1	no	—
186	C₂×D₄	—	274	1	no	—
187	C₂×D₄	—	274	2	no	—
188	C₂×D₄	—	137	4	no	—
189	C₂×D₄	—	137	1	no	—
190	C₂×D₄	—	137	2	no	—
191	C₂²×C₄	—	274	1	no	—
192	C₂²×C₄	—	274	1	no	—
193	C₂²×C₄	—	274	2	no	—
194	C₂²×C₄	—	274	2	no	—
195	C₂²×C₄	—	274	2	no	—
196	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
197	SD₁₆	—	137	8	no	—
198	D₈	—	137	8	no	—
199	M₁₆	C₄×D₁₃₇	1	4	yes	complemented
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
202	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
203	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
204	C₄⋊C₄	—	274	2	no	—
205	C₄⋊C₄	—	274	1	no	—
206	C₄⋊C₄	—	137	2	no	—
207	C₄⋊C₄	—	137	1	no	—
208	C₄⋊C₄	—	137	1	no	—
209	C₄⋊C₄	D₅₄₈	1	1	yes	characteristic
210	C₄⋊C₄	D₅₄₈	1	1	yes	characteristic
211	C₂²⋊C₄	—	274	1	no	—
212	C₂²⋊C₄	—	274	2	no	—
213	C₂²⋊C₄	—	274	2	no	—
214	C₂²⋊C₄	—	274	2	no	—
215	C₂²⋊C₄	D₅₄₈	1	2	yes	—
216	C₄²	—	274	2	no	—
217	C₄²	—	274	1	no	—
218	C₄²	C₂²×D₁₃₇	1	2	yes	—
219	C₂³	—	274	1	no	—
220	C₂³	—	274	2	no	—
221	C₂³	—	274	4	no	—
222	C₂³	—	1096	1	no	—
223	C₂³	—	1096	1	no	—
224	C₂³	C₂×D₅₄₈	1	1	yes	characteristic
225	Q₈	—	274	1	no	—
226	Q₈	—	137	2	no	—
227	D₄	—	274	4	no	—
228	D₄	—	274	4	no	—
229	D₄	—	274	2	no	—
230	D₄	—	274	1	no	—
231	D₄	—	137	2	no	—
232	D₄	—	137	4	no	—
233	C₂×C₄	—	274	1	no	—
234	C₂×C₄	—	274	4	no	—
235	C₂×C₄	—	274	1	no	—
236	C₂×C₄	—	274	1	no	—
237	C₂×C₄	—	274	2	no	—
238	C₂×C₄	—	1096	2	no	—
239	C₂×C₄	—	1096	2	no	—
240	C₂×C₄	—	1096	2	no	—
241	C₂×C₄	—	1096	1	no	—
242	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
243	C₂×C₄	C₂×D₅₄₈	1	2	yes	—
244	C₂×C₄	C₂×D₅₄₈	1	2	yes	—
245	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
246	C₂×C₄	#14555510	1	2	yes	—
247	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
248	C₈	—	2	1	no	—
249	C₈	—	2	1	no	—
250	C₈	C₂×C₄×D₁₃₇	1	4	yes	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	274	2	no	—
254	C₂²	—	274	4	no	—
255	C₂²	—	548	1	no	—
256	C₂²	—	548	2	no	—
257	C₂²	—	548	2	no	—
258	C₂²	—	548	1	no	—
259	C₂²	—	1096	4	no	—
260	C₂²	C₄×D₅₄₈	1	2	yes	—
261	C₂²	C₂²×D₅₄₈	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	274	2	no	—
265	C₄	—	548	1	no	—
266	C₄	—	548	2	no	—
267	C₄	—	548	1	no	—
268	C₄	C₄×D₅₄₈	1	2	yes	—
269	C₄	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
270	C₄	C₂×#14555510	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	—	548	4	no	—
273	C₂	—	548	2	no	—
274	C₂	C₂×C₄×D₅₄₈	1	1	yes	characteristic
275	C₂	#22578208	1	1	yes	characteristic
276	C₂	#22578215	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₄₈
commutator	C₄×C₁₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	#405×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂²×D₄
prime radical	C₂×#17984116	C₂
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	#405	D₁₃₇
hypocenter	C₁₃₇	#2946
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
2. #22577883	C₂	semidirect
3. #22577877	C₂	semidirect
4. #22577516	C₂	semidirect
5. #22577505	C₂	semidirect
6. C₄×#14555499	C₂	semidirect
7. C₄×D₁₀₉₆	C₂	semidirect
8. #22577445	C₂	semidirect
9. #22577400	C₂	semidirect
10. C₂×#17984010	C₂	semidirect
11. #405×C₁₃₇	C₂	semidirect
14. #17984116	C₄	semidirect
17. C₄×Dic₂₇₄	C₂²	semidirect
20. C₄×D₅₄₈	C₂²	semidirect
29. #17984022	C₂²	semidirect
30. #17984020	C₂²	semidirect
31. #17984013	C₂²	semidirect
32. #17984013	C₂²	semidirect
35. #17984010	C₂²	semidirect
36. #17984010	C₂²	semidirect
40. #97×C₁₃₇	C₂²	semidirect
41. C₄×C₈×C₁₃₇	C₂²	semidirect
44. #14555510	C₂×C₄	semidirect
48. C₂×D₅₄₈	C₂×C₄	semidirect
57. #14555499	C₂×C₄	semidirect
58. D₁₀₉₆	C₂×C₄	semidirect
71. M₁₆×C₁₃₇	C₂×C₄	semidirect
138. C₁₃₇	#2946	semidirect
150. #405	D₁₃₇	semidirect
179. #97	C₂×D₁₃₇	semidirect
180. C₄×C₈	C₂×D₁₃₇	semidirect
199. M₁₆	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×#17984116	2
#22577883	2
#22577877	2
#22577516	2
#22577505	2
C₄×#14555499	2
C₄×D₁₀₉₆	2
#22577445	2
#22577400	2
C₂×#17984010	2
#405×C₁₃₇	2
#2946	137