Subgroups — #31789621

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×#17984114	C₂	1	1	yes	maximal, characteristic
2	#22577883	C₂	1	1	yes	maximal, characteristic, complemented
3	#22577513	C₂	1	2	yes	maximal, complemented
4	#22577503	C₂	1	2	yes	maximal, complemented
5	C₄×#14555498	C₂	1	2	yes	maximal, complemented
6	C₄×#14555495	C₂	1	2	yes	maximal, complemented
7	#22577446	C₂	1	1	yes	maximal, characteristic, complemented
8	#22577397	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#17984009	C₂	1	1	yes	maximal, characteristic, complemented
10	#22577359	C₂	1	1	yes	maximal, characteristic, complemented
11	#518×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14555510	C₂²	1	1	yes	characteristic
13	#17984114	C₄	1	8	yes	complemented
14	C₂×#14555498	C₂²	1	2	yes	—
15	C₂×#14555495	C₂²	1	2	yes	—
16	C₂×#14555494	C₂²	1	1	yes	characteristic
17	C₂²×#11764040	C₂²	1	1	yes	characteristic
18	C₄×Dic₂₇₄	C₂²	1	2	yes	complemented
19	C₄×D₅₄₈	C₂²	1	2	yes	complemented
20	#17984048	—	2	2	no	—
21	#17984046	—	2	1	no	—
22	C₂×C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
23	C₄⋊C₄×D₁₃₇	—	2	1	no	—
24	#17984021	C₂²	1	2	yes	complemented
25	#17984019	C₂²	1	2	yes	complemented
26	#17984011	C₂²	1	1	yes	characteristic, complemented
27	#17984011	C₂²	1	1	yes	characteristic, complemented
28	#17984009	C₂²	1	2	yes	—
29	#17984009	C₂²	1	2	yes	—
30	#17984008	C₂²	1	1	yes	characteristic, complemented
31	#17984008	C₂²	1	1	yes	characteristic, complemented
32	#17983999	C₂²	1	2	yes	complemented
33	C₄×#11764040	C₂²	1	2	yes	complemented
34	C₂²×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
35	C₄×D₄×C₁₃₇	—	2	1	no	—
36	C₄×D₄×C₁₃₇	—	2	1	no	—
37	C₄×D₄×C₁₃₇	C₂²	1	2	yes	—
38	C₄×D₄×C₁₃₇	C₂²	1	2	yes	—
39	#117×C₁₃₇	C₂²	1	1	yes	characteristic
40	#117×C₁₃₇	C₂²	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
42	#14555510	—	2	2	no	—
43	#14555510	C₂×C₄	1	4	yes	complemented
44	C₂×Dic₂₇₄	C₂³	1	1	yes	characteristic
45	C₂×D₅₄₈	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₃₇	—	2	1	no	—
47	C₂²×Dic₁₃₇	—	2	1	no	—
48	C₂×C₄×D₁₃₇	—	2	1	no	—
49	C₂×C₄×D₁₃₇	—	2	2	no	—
50	#14555498	C₂×C₄	1	8	yes	complemented
51	#14555495	C₂×C₄	1	8	yes	complemented
52	#14555494	C₂×C₄	1	4	yes	complemented
53	C₂×#11764040	—	2	2	no	—
54	C₂×#11764040	C₂³	1	1	yes	characteristic
55	C₂×#11764040	C₂³	1	1	yes	characteristic
56	C₂×#11764040	C₂³	1	2	yes	—
57	C₄⋊Dic₁₃₇	—	2	1	no	—
58	C₄⋊Dic₁₃₇	C₂³	1	2	yes	—
59	#14555489	—	2	2	no	—
60	#14555487	—	2	2	no	—
61	C₄×Dic₁₃₇	—	2	2	no	—
62	C₂⁴×C₁₃₇	—	2	1	no	—
63	C₂×D₄×C₁₃₇	—	2	1	no	—
64	C₂×D₄×C₁₃₇	—	2	2	no	—
65	C₂×D₄×C₁₃₇	C₂×C₄	1	4	yes	complemented
66	C₂×D₄×C₁₃₇	C₂³	1	2	yes	—
67	C₂²×C₄×C₁₃₇	—	2	2	no	—
68	C₂²×C₄×C₁₃₇	—	2	2	no	—
69	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
71	C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
74	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
75	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
76	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
77	C₂²⋊C₄×C₁₃₇	D₄	1	2	yes	—
78	C₄²×C₁₃₇	—	2	1	no	—
79	C₄²×C₁₃₇	C₂³	1	2	yes	—
80	C₂²×D₁₃₇	—	2	1	no	—
81	Dic₂₇₄	—	2	1	no	—
82	Dic₂₇₄	C₂²×C₄	1	2	yes	—
83	D₅₄₈	—	2	1	no	—
84	D₅₄₈	C₂²×C₄	1	2	yes	—
85	C₂²⋊D₁₃₇	—	2	4	no	—
86	C₂×Dic₁₃₇	—	2	1	no	—
87	C₂×Dic₁₃₇	—	2	2	no	—
88	C₂×Dic₁₃₇	—	8	1	no	—
89	C₄×D₁₃₇	—	2	4	no	—
90	C₄×D₁₃₇	—	8	2	no	—
91	#11764040	—	2	1	no	—
92	#11764040	—	2	1	no	—
93	#11764040	C₂²×C₄	1	4	yes	—
94	C₂³×C₁₃₇	—	2	2	no	—
95	C₂³×C₁₃₇	—	2	4	no	—
96	C₂³×C₁₃₇	—	8	1	no	—
97	C₂³×C₁₃₇	—	8	1	no	—
98	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
99	D₄×C₁₃₇	—	2	4	no	—
100	D₄×C₁₃₇	—	2	2	no	—
101	D₄×C₁₃₇	C₂²×C₄	1	4	yes	—
102	C₂×C₄×C₁₃₇	—	2	1	no	—
103	C₂×C₄×C₁₃₇	—	2	1	no	—
104	C₂×C₄×C₁₃₇	—	8	2	no	—
105	C₂×C₄×C₁₃₇	—	8	2	no	—
106	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
108	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃₇	#41	1	2	yes	—
111	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
112	C₂²×C₁₃₇	—	2	1	no	—
113	C₂²×C₁₃₇	—	2	1	no	—
114	C₂²×C₁₃₇	—	2	4	no	—
115	C₂²×C₁₃₇	—	4	2	no	—
116	C₂²×C₁₃₇	—	4	2	no	—
117	C₂²×C₁₃₇	—	8	4	no	—
118	C₂²×C₁₃₇	C₄×D₄	1	2	yes	—
119	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×D₁₃₇	—	2	2	no	—
121	C₂×D₁₃₇	—	4	1	no	—
122	C₂×D₁₃₇	—	4	1	no	—
123	C₄×C₁₃₇	—	2	2	no	—
124	C₄×C₁₃₇	—	2	2	no	—
125	C₄×C₁₃₇	—	4	1	no	—
126	C₄×C₁₃₇	—	4	1	no	—
127	C₄×C₁₃₇	C₄×D₄	1	2	yes	—
128	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
129	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
130	Dic₁₃₇	—	2	2	no	—
131	Dic₁₃₇	—	4	2	no	—
132	C₂×C₁₃₇	—	2	2	no	—
133	C₂×C₁₃₇	—	4	4	no	—
134	C₂×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
135	C₂×C₁₃₇	#576	1	1	yes	characteristic
136	C₂×C₁₃₇	#577	1	1	yes	characteristic
137	D₁₃₇	—	4	2	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	D₁₃₇	1	1	yes	characteristic, complemented
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	—	137	1	no	—
151	C₂×#41	—	137	1	no	—
152	C₂²×D₄	C₂×D₁₃₇	1	1	yes	characteristic
153	AGL(1,ℤ/8)	—	137	8	no	—
154	C₂×SD₁₆	—	137	2	no	—
155	C₂×D₈	—	137	2	no	—
156	C₂×M₁₆	—	137	1	no	—
157	C₂²×C₈	—	137	1	no	—
158	C₄×Q₈	—	137	2	no	—
159	C₄×D₄	—	2	1	no	—
160	C₄×D₄	—	2	1	no	—
161	C₄×D₄	—	274	2	no	—
162	C₄×D₄	—	137	2	no	—
163	C₄×D₄	C₂×D₁₃₇	1	2	yes	—
164	C₄×D₄	C₂×D₁₃₇	1	2	yes	—
165	#117	—	274	1	no	—
166	#117	C₂×D₁₃₇	1	1	yes	characteristic
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	—	2	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	—	137	2	no	—
180	C₄×C₈	—	137	2	no	—
181	C₂⁴	—	2	1	no	—
182	#41	—	274	2	no	—
183	#41	—	137	4	no	—
184	C₂×Q₈	—	137	1	no	—
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	274	1	no	—
188	C₂×D₄	—	137	1	no	—
189	C₂×D₄	C₄×D₁₃₇	1	4	yes	complemented
190	C₂×D₄	C₂²×D₁₃₇	1	2	yes	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	274	1	no	—
194	C₂²×C₄	—	274	1	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	—	137	4	no	—
200	C₂×C₈	—	274	2	no	—
201	C₂×C₈	—	137	1	no	—
202	C₂×C₈	—	137	1	no	—
203	C₂×C₈	—	137	2	no	—
204	C₄⋊C₄	—	274	2	no	—
205	C₄⋊C₄	—	274	1	no	—
206	C₄⋊C₄	—	137	2	no	—
207	C₄⋊C₄	C₂²⋊D₁₃₇	1	1	yes	characteristic
208	C₄⋊C₄	C₂²⋊D₁₃₇	1	1	yes	characteristic
209	C₄⋊C₄	C₂²×D₁₃₇	1	1	yes	characteristic
210	C₄⋊C₄	C₂²×D₁₃₇	1	1	yes	characteristic
211	C₂²⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	—	274	2	no	—
215	C₂²⋊C₄	C₂²⋊D₁₃₇	1	2	yes	—
216	C₄²	—	2	1	no	—
217	C₄²	—	274	2	no	—
218	C₄²	C₂²×D₁₃₇	1	2	yes	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	4	no	—
221	C₂³	—	8	1	no	—
222	C₂³	—	8	1	no	—
223	C₂³	—	274	1	no	—
224	C₂³	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
225	Q₈	—	274	1	no	—
226	Q₈	—	137	2	no	—
227	D₄	—	2	4	no	—
228	D₄	—	2	2	no	—
229	D₄	—	274	4	no	—
230	D₄	—	274	1	no	—
231	D₄	—	137	2	no	—
232	D₄	C₂×C₄×D₁₃₇	1	4	yes	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	274	1	no	—
238	C₂×C₄	—	274	4	no	—
239	C₂×C₄	—	274	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×C₂²⋊D₁₃₇	1	2	yes	—
244	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	2	yes	—
245	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
246	C₂×C₄	#14555510	1	2	yes	—
247	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
248	C₈	—	274	1	no	—
249	C₈	—	274	1	no	—
250	C₈	—	137	4	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	8	4	no	—
257	C₂²	—	274	2	no	—
258	C₂²	—	548	1	no	—
259	C₂²	—	548	1	no	—
260	C₂²	C₄×C₂²⋊D₁₃₇	1	2	yes	—
261	C₂²	C₂²×C₂²⋊D₁₃₇	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	4	1	no	—
266	C₄	—	274	2	no	—
267	C₄	—	548	2	no	—
268	C₄	C₄×C₂²⋊D₁₃₇	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	—	4	4	no	—
273	C₂	—	548	2	no	—
274	C₂	C₂×C₄×C₂²⋊D₁₃₇	1	1	yes	characteristic
275	C₂	#22578206	1	1	yes	characteristic
276	C₂	#22578214	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₃₇
commutator	C₄×C₁₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	#518×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂²×D₄
prime radical	C₂×#17984114	C₂
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	#518	D₁₃₇
hypocenter	C₁₃₇	#2946
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #518

Complements

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

N	H ≅ G/N	type
2. #22577883	C₂	semidirect
3. #22577513	C₂	semidirect
4. #22577503	C₂	semidirect
5. C₄×#14555498	C₂	semidirect
6. C₄×#14555495	C₂	semidirect
7. #22577446	C₂	semidirect
8. #22577397	C₂	semidirect
9. C₂×#17984009	C₂	semidirect
10. #22577359	C₂	semidirect
11. #518×C₁₃₇	C₂	semidirect
13. #17984114	C₄	semidirect
18. C₄×Dic₂₇₄	C₂²	semidirect
19. C₄×D₅₄₈	C₂²	semidirect
24. #17984021	C₂²	semidirect
25. #17984019	C₂²	semidirect
26. #17984011	C₂²	semidirect
27. #17984011	C₂²	semidirect
30. #17984008	C₂²	semidirect
31. #17984008	C₂²	semidirect
32. #17983999	C₂²	semidirect
33. C₄×#11764040	C₂²	semidirect
43. #14555510	C₂×C₄	semidirect
50. #14555498	C₂×C₄	semidirect
51. #14555495	C₂×C₄	semidirect
52. #14555494	C₂×C₄	semidirect
65. C₂×D₄×C₁₃₇	C₂×C₄	semidirect
138. C₁₃₇	#2946	semidirect
142. #518	D₁₃₇	semidirect
189. C₂×D₄	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×#17984114	2
#22577883	2
#22577513	2
#22577503	2
C₄×#14555498	2
C₄×#14555495	2
#22577446	2
#22577397	2
C₂×#17984009	2
#22577359	2
#518×C₁₃₇	2
#2946	137