Subgroups — #31789388

Order 17536.

476 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	C₂×#17984096	C₂	1	1	yes	maximal, characteristic
2	#22577884	C₂	1	1	yes	maximal, characteristic
3	#22577883	C₂	1	1	yes	maximal, characteristic
4	#22577445	C₂	1	1	yes	maximal, characteristic, complemented
5	#22577444	C₂	1	2	yes	maximal, complemented
6	#22577442	C₂	1	2	yes	maximal, complemented
7	#22577439	C₂	1	1	yes	maximal, characteristic, complemented
8	#22577438	C₂	1	2	yes	maximal, complemented
9	C₂×#17984021	C₂	1	1	yes	maximal, characteristic, complemented
10	#107×D₁₃₇	C₂	1	2	yes	maximal, complemented
11	#428×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14555510	C₂²	1	1	yes	characteristic
13	#17984096	C₄	1	8	yes	complemented
14	C₂×#14555493	C₂²	1	2	yes	—
15	C₂²×#11764040	C₂²	1	1	yes	characteristic
16	C₂×C₈×D₁₃₇	C₂²	1	2	yes	—
17	C₄×Dic₂₇₄	—	2	1	no	—
18	#17984053	—	2	1	no	—
19	C₄×D₅₄₈	—	2	1	no	—
20	#17984050	—	2	1	no	—
21	C₄×C₂²⋊D₁₃₇	—	2	2	no	—
22	#17984048	—	2	2	no	—
23	#17984047	C₂²	1	1	yes	characteristic
24	#17984046	C₂²	1	2	yes	—
25	#17984046	C₂²	1	2	yes	—
26	C₂×C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×D₁₃₇	C₂²	1	2	yes	—
28	C₄⋊C₄×D₁₃₇	C₂²	1	2	yes	—
29	#17984022	C₂²	1	2	yes	—
30	#17984021	C₂²	1	1	yes	characteristic
31	#17984021	C₂²	1	2	yes	—
32	#17984021	C₂²	1	2	yes	—
33	#17984021	C₂²	1	1	yes	characteristic
34	#17984020	C₂²	1	2	yes	—
35	#17984019	C₂²	1	1	yes	characteristic
36	#17984019	C₂²	1	1	yes	characteristic
37	C₂×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
38	#117×C₁₃₇	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
40	#107×C₁₃₇	C₂²	1	2	yes	—
41	#106×C₁₃₇	C₂²	1	2	yes	—
42	#14555510	C₂×C₄	1	4	yes	—
43	#14555510	Q₈	1	4	yes	—
44	C₂×Dic₂₇₄	D₄	1	1	yes	characteristic
45	C₂×D₅₄₈	D₄	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₃₇	D₄	1	2	yes	—
47	C₂²×Dic₁₃₇	—	2	1	no	—
48	C₂×C₄×D₁₃₇	—	2	2	no	—
49	C₂×C₄×D₁₃₇	—	2	2	no	—
50	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
51	#14555493	C₂×C₄	1	8	yes	complemented
52	C₂×#11764040	C₂×C₄	1	4	yes	complemented
53	C₂×#11764040	C₂³	1	1	yes	characteristic
54	C₂×#11764040	C₂³	1	1	yes	characteristic
55	C₈×D₁₃₇	C₂×C₄	1	8	yes	complemented
56	C₄⋊Dic₁₃₇	—	2	1	no	—
57	C₄⋊Dic₁₃₇	C₂³	1	1	yes	characteristic
58	C₄⋊Dic₁₃₇	C₂³	1	1	yes	characteristic
59	C₄⋊Dic₁₃₇	C₂³	1	2	yes	—
60	#14555489	—	2	2	no	—
61	#14555489	—	2	2	no	—
62	C₂²⋊Dic₁₃₇	—	2	1	no	—
63	#14555487	—	2	2	no	—
64	#14555487	—	2	2	no	—
65	C₄×Dic₁₃₇	—	2	2	no	—
66	C₄×Dic₁₃₇	—	2	2	no	—
67	C₄×Dic₁₃₇	—	2	1	no	—
68	C₂²×C₄×C₁₃₇	—	2	1	no	—
69	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
70	M₁₆×C₁₃₇	C₂×C₄	1	4	yes	complemented
71	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
72	C₄⋊C₄×C₁₃₇	—	2	1	no	—
73	C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
76	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
77	C₄²×C₁₃₇	—	2	1	no	—
78	C₂²×D₁₃₇	C₂×D₄	1	2	yes	—
79	Dic₂₇₄	C₄⋊C₄	1	4	yes	—
80	D₅₄₈	C₄⋊C₄	1	4	yes	—
81	C₂²⋊D₁₃₇	C₄⋊C₄	1	8	yes	—
82	C₂×Dic₁₃₇	—	2	1	no	—
83	C₂×Dic₁₃₇	—	2	2	no	—
84	C₂×Dic₁₃₇	—	2	1	no	—
85	C₂×Dic₁₃₇	—	8	1	no	—
86	C₂×Dic₁₃₇	C₂×D₄	1	2	yes	—
87	C₄×D₁₃₇	—	8	2	no	—
88	C₄×D₁₃₇	—	8	2	no	—
89	C₄×D₁₃₇	C₂²×C₄	1	4	yes	—
90	C₄×D₁₃₇	C₂×Q₈	1	4	yes	—
91	#11764040	C₂²×C₄	1	2	yes	—
92	#11764040	C₂²×C₄	1	2	yes	—
93	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃₇	—	2	1	no	—
95	C₂×C₄×C₁₃₇	—	2	1	no	—
96	C₂×C₄×C₁₃₇	—	2	2	no	—
97	C₂×C₄×C₁₃₇	—	8	1	no	—
98	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
101	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
102	C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
103	C₂²×C₁₃₇	—	2	1	no	—
104	C₂²×C₁₃₇	—	2	1	no	—
105	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
106	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
107	C₂×D₁₃₇	—	2	2	no	—
108	C₂×D₁₃₇	—	2	2	no	—
109	C₂×D₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
110	C₄×C₁₃₇	—	4	2	no	—
111	C₄×C₁₃₇	—	4	1	no	—
112	C₄×C₁₃₇	—	4	1	no	—
113	C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
114	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
115	C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
116	Dic₁₃₇	—	4	1	no	—
117	Dic₁₃₇	—	4	2	no	—
118	Dic₁₃₇	—	4	1	no	—
119	Dic₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
120	C₂×C₁₃₇	—	2	2	no	—
121	C₂×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₁₃₇	#576	1	1	yes	characteristic
123	C₂×C₁₃₇	#578	1	1	yes	characteristic
124	D₁₃₇	—	2	4	no	—
125	C₁₃₇	#2913	1	1	yes	characteristic, complemented, Sylow
126	#2913	—	137	1	no	maximal, Sylow
127	C₂×#131	—	137	1	no	—
128	#520	—	137	1	no	—
129	#520	—	137	1	no	—
130	#428	—	137	1	no	—
131	#428	—	137	2	no	—
132	#428	—	137	2	no	—
133	#428	D₁₃₇	1	1	yes	characteristic, complemented
134	#427	—	137	1	no	—
135	#427	—	137	2	no	—
136	C₂×#107	—	137	1	no	—
137	C₂×#107	—	137	2	no	—
138	C₂×#41	—	137	1	no	—
139	#131	—	137	8	no	—
140	C₂×M₁₆	—	137	2	no	—
141	C₂×M₁₆	C₂×D₁₃₇	1	1	yes	characteristic
142	C₂²×C₈	—	137	1	no	—
143	C₂²×C₈	—	137	2	no	—
144	C₄×Q₈	—	274	1	no	—
145	C₄×Q₈	—	274	1	no	—
146	C₄×D₄	—	274	1	no	—
147	C₄×D₄	—	274	2	no	—
148	C₄×D₄	—	274	1	no	—
149	C₄×D₄	—	274	2	no	—
150	#117	—	137	1	no	—
151	#117	—	137	2	no	—
152	#117	—	137	2	no	—
153	#117	C₂×D₁₃₇	1	1	yes	characteristic
154	C₂×C₄⋊C₄	—	137	1	no	—
155	C₂×C₄⋊C₄	—	137	2	no	—
156	C₂×C₄⋊C₄	—	137	2	no	—
157	C₂×C₄⋊C₄	C₂×D₁₃₇	1	1	yes	characteristic
158	#107	—	137	2	no	—
159	#107	—	137	1	no	—
160	#107	—	137	2	no	—
161	#107	—	137	2	no	—
162	#107	—	137	1	no	—
163	#107	C₂×D₁₃₇	1	2	yes	—
164	#106	—	137	2	no	—
165	#106	—	137	1	no	—
166	#106	—	137	1	no	—
167	#106	C₂×D₁₃₇	1	2	yes	—
168	#41	—	137	4	no	—
169	#41	—	137	4	no	—
170	C₂×Q₈	—	137	1	no	—
171	C₂×D₄	—	137	2	no	—
172	C₂×D₄	—	137	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	274	2	no	—
175	C₂²×C₄	—	274	2	no	—
176	C₂²×C₄	—	274	1	no	—
177	C₂²×C₄	—	137	2	no	—
178	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
179	M₁₆	—	137	8	no	—
180	M₁₆	C₄×D₁₃₇	1	4	yes	complemented
181	C₂×C₈	—	137	4	no	—
182	C₂×C₈	—	137	1	no	—
183	C₂×C₈	—	137	1	no	—
184	C₂×C₈	—	137	8	no	—
185	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	274	2	no	—
188	C₄⋊C₄	—	274	2	no	—
189	C₄⋊C₄	—	274	1	no	—
190	C₄⋊C₄	—	137	1	no	—
191	C₄⋊C₄	—	137	1	no	—
192	C₄⋊C₄	—	137	2	no	—
193	C₄⋊C₄	C₂²×D₁₃₇	1	1	yes	characteristic
194	C₄⋊C₄	C₂²×D₁₃₇	1	1	yes	characteristic
195	C₄⋊C₄	C₂²×D₁₃₇	1	2	yes	—
196	C₂²⋊C₄	—	2	1	no	—
197	C₂²⋊C₄	—	274	2	no	—
198	C₂²⋊C₄	—	274	2	no	—
199	C₂²⋊C₄	—	274	1	no	—
200	C₄²	—	2	1	no	—
201	C₄²	—	274	2	no	—
202	C₄²	—	274	2	no	—
203	C₄²	—	274	1	no	—
204	C₂³	—	137	2	no	—
205	C₂³	D₄×D₁₃₇	1	1	yes	characteristic
206	Q₈	—	137	4	no	—
207	D₄	—	137	8	no	—
208	D₄	—	137	4	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	8	1	no	—
213	C₂×C₄	—	274	1	no	—
214	C₂×C₄	—	274	2	no	—
215	C₂×C₄	—	274	1	no	—
216	C₂×C₄	—	1096	2	no	—
217	C₂×C₄	—	1096	2	no	—
218	C₂×C₄	—	1096	1	no	—
219	C₂×C₄	—	137	2	no	—
220	C₂×C₄	—	137	4	no	—
221	C₂×C₄	—	137	4	no	—
222	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
223	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
224	C₂×C₄	Q₈×D₁₃₇	1	2	yes	—
225	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
226	C₈	—	137	2	no	—
227	C₈	—	137	2	no	—
228	C₈	C₂×C₄×D₁₃₇	1	4	yes	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	—	274	2	no	—
232	C₂²	—	274	2	no	—
233	C₂²	—	137	4	no	—
234	C₂²	C₄⋊C₄×D₁₃₇	1	2	yes	—
235	C₂²	C₂×D₄×D₁₃₇	1	1	yes	characteristic
236	C₄	—	4	2	no	—
237	C₄	—	4	1	no	—
238	C₄	—	4	1	no	—
239	C₄	—	548	1	no	—
240	C₄	—	548	2	no	—
241	C₄	—	548	1	no	—
242	C₄	—	137	4	no	—
243	C₄	C₄⋊C₄×D₁₃₇	1	2	yes	—
244	C₄	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
245	C₄	C₂×Q₈×D₁₃₇	1	1	yes	characteristic
246	C₂	—	2	2	no	—
247	C₂	—	274	4	no	—
248	C₂	C₂×C₄⋊C₄×D₁₃₇	1	1	yes	characteristic
249	C₂	#22578207	1	1	yes	characteristic
250	C₂	#22578219	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₇
commutator	C₄×C₁₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	#428×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂²×D₄
prime radical	C₂×#14555510	C₂²
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	#428	D₁₃₇
hypocenter	C₁₃₇	#2913
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
4. #22577445	C₂	semidirect
5. #22577444	C₂	semidirect
6. #22577442	C₂	semidirect
7. #22577439	C₂	semidirect
8. #22577438	C₂	semidirect
9. C₂×#17984021	C₂	semidirect
10. #107×D₁₃₇	C₂	semidirect
11. #428×C₁₃₇	C₂	semidirect
13. #17984096	C₄	semidirect
51. #14555493	C₂×C₄	semidirect
52. C₂×#11764040	C₂×C₄	semidirect
55. C₈×D₁₃₇	C₂×C₄	semidirect
70. M₁₆×C₁₃₇	C₂×C₄	semidirect
125. C₁₃₇	#2913	semidirect
133. #428	D₁₃₇	semidirect
180. 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×#17984096	2
#22577884	2
#22577883	2
#22577445	2
#22577444	2
#22577442	2
#22577439	2
#22577438	2
C₂×#17984021	2
#107×D₁₃₇	2
#428×C₁₃₇	2
#2913	137