Subgroups — #31789498

Order 17536.

500 conjugacy classes of subgroups.

The subgroup lattice has 276 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	2	yes	maximal, complemented
2	#22577950	C₂	1	1	yes	maximal, characteristic
3	#22577482	C₂	1	1	yes	maximal, characteristic, complemented
4	#22577479	C₂	1	2	yes	maximal, complemented
5	D₄×#11764040	C₂	1	1	yes	maximal, characteristic, complemented
6	#22577468	C₂	1	1	yes	maximal, characteristic, complemented
7	#22577466	C₂	1	2	yes	maximal, complemented
8	#22577425	C₂	1	1	yes	maximal, characteristic, complemented
9	#22577368	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#436×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14555510	C₂²	1	2	yes	—
13	#17984096	—	2	8	no	—
14	C₂×#14555493	C₂²	1	2	yes	—
15	C₂×#14555493	C₂²	1	2	yes	—
16	C₂²×#11764040	C₂²	1	2	yes	—
17	C₂×C₈×D₁₃₇	C₂²	1	2	yes	—
18	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
20	#17984087	C₄	1	1	yes	characteristic
21	#17984084	C₄	1	1	yes	characteristic
22	#17984074	C₄	1	2	yes	—
23	#17984063	C₄	1	1	yes	characteristic
24	#17984061	C₄	1	2	yes	—
25	#17984059	C₄	1	1	yes	characteristic
26	C₄×D₅₄₈	C₂²	1	1	yes	characteristic
27	C₄×C₂²⋊D₁₃₇	C₂²	1	2	yes	—
28	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
29	C₄⋊#11764040	C₂²	1	1	yes	characteristic
30	#17984017	C₂²	1	2	yes	—
31	C₂²⋊#11764040	C₂²	1	2	yes	—
32	#17984000	C₂²	1	2	yes	—
33	#17984000	C₂²	1	2	yes	—
34	C₄×#11764040	C₂²	1	1	yes	characteristic
35	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
36	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
37	C₂×M₁₆×C₁₃₇	C₂²	1	2	yes	—
38	C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
40	C₂²⋊C₈×C₁₃₇	C₂²	1	2	yes	—
41	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
42	#14555510	—	2	4	no	—
43	#14555510	—	2	4	no	—
44	C₂×Dic₂₇₄	C₂×C₄	1	2	yes	—
45	C₂×D₅₄₈	C₂×C₄	1	1	yes	characteristic
46	C₂×D₅₄₈	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊D₁₃₇	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊D₁₃₇	C₂×C₄	1	2	yes	—
49	C₂×C₂²⋊D₁₃₇	C₂×C₄	1	2	yes	—
50	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
53	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
54	#14555493	—	2	4	no	—
55	#14555493	—	2	4	no	—
56	C₂×#11764040	—	2	4	no	—
57	C₂×#11764040	C₂³	1	1	yes	characteristic
58	C₂×#11764040	C₂³	1	2	yes	—
59	C₂×#11764040	C₂³	1	1	yes	characteristic
60	C₈×D₁₃₇	—	2	2	no	—
61	C₈×D₁₃₇	—	2	4	no	—
62	C₈×D₁₃₇	D₄	1	4	yes	complemented
63	C₄⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
64	#14555489	C₂×C₄	1	2	yes	—
65	C₂²⋊Dic₁₃₇	C₂×C₄	1	2	yes	—
66	#14555487	C₂×C₄	1	2	yes	—
67	#14555487	C₂×C₄	1	2	yes	—
68	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
71	C₂×D₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
73	M₁₆×C₁₃₇	—	2	4	no	—
74	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
76	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
78	C₂²⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
79	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
80	C₂²×D₁₃₇	C₂²×C₄	1	2	yes	—
81	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
82	Dic₂₇₄	—	2	4	no	—
83	D₅₄₈	—	2	4	no	—
84	D₅₄₈	—	2	2	no	—
85	C₂²⋊D₁₃₇	—	2	4	no	—
86	C₂²⋊D₁₃₇	—	2	4	no	—
87	C₂²⋊D₁₃₇	—	2	4	no	—
88	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
90	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
92	C₄×D₁₃₇	—	2	2	no	—
93	C₄×D₁₃₇	—	2	2	no	—
94	C₄×D₁₃₇	—	2	2	no	—
95	C₄×D₁₃₇	—	2	2	no	—
96	C₄×D₁₃₇	C₂×D₄	1	2	yes	—
97	C₄×D₁₃₇	#41	1	2	yes	—
98	#11764040	—	2	2	no	—
99	#11764040	—	2	1	no	—
100	#11764040	C₂×D₄	1	2	yes	—
101	C₂³×C₁₃₇	C₂²×C₄	1	2	yes	—
102	D₄×C₁₃₇	—	2	2	no	—
103	C₂×C₄×C₁₃₇	—	2	2	no	—
104	C₂×C₄×C₁₃₇	—	2	2	no	—
105	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
109	C₈×C₁₃₇	—	2	2	no	—
110	C₈×C₁₃₇	—	2	1	no	—
111	C₈×C₁₃₇	C₂×D₄	1	2	yes	—
112	C₂²×C₁₃₇	—	2	2	no	—
113	C₂²×C₁₃₇	—	2	2	no	—
114	C₂²×C₁₃₇	—	2	2	no	—
115	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
116	C₂×D₁₃₇	—	2	1	no	—
117	C₂×D₁₃₇	—	2	1	no	—
118	C₂×D₁₃₇	—	2	2	no	—
119	C₂×D₁₃₇	—	2	2	no	—
120	C₂×D₁₃₇	—	2	2	no	—
121	C₂×D₁₃₇	C₄×D₄	1	2	yes	—
122	C₄×C₁₃₇	—	2	2	no	—
123	C₄×C₁₃₇	—	2	1	no	—
124	C₄×C₁₃₇	#131	1	2	yes	complemented
125	C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
126	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
127	Dic₁₃₇	—	2	2	no	—
128	Dic₁₃₇	—	2	1	no	—
129	Dic₁₃₇	C₄×D₄	1	2	yes	—
130	Dic₁₃₇	#131	1	2	yes	complemented
131	C₂×C₁₃₇	—	4	2	no	—
132	C₂×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
133	C₂×C₁₃₇	C₂×#131	1	1	yes	characteristic
134	C₂×C₁₃₇	C₂×#131	1	1	yes	characteristic
135	D₁₃₇	—	2	2	no	—
136	D₁₃₇	—	4	2	no	—
137	C₁₃₇	#2931	1	1	yes	characteristic, complemented, Sylow
138	#2931	—	137	1	no	maximal, Sylow
139	C₂×#131	—	137	2	no	—
140	#532	—	137	1	no	—
141	#436	—	137	2	no	—
142	#436	—	137	1	no	—
143	#436	D₁₃₇	1	1	yes	characteristic, complemented
144	C₈×D₄	—	137	1	no	—
145	C₈×D₄	—	137	1	no	—
146	C₈×D₄	—	137	2	no	—
147	#423	—	137	1	no	—
148	#408	—	137	2	no	—
149	C₂×C₄×C₈	—	137	1	no	—
150	C₂×#41	—	137	2	no	—
151	#131	—	274	8	no	—
152	C₂×M₁₆	—	137	2	no	—
153	C₂×M₁₆	—	137	2	no	—
154	C₂×M₁₆	C₂×D₁₃₇	1	2	yes	—
155	C₂²×C₈	—	137	2	no	—
156	C₂²×C₈	—	137	2	no	—
157	C₂²×C₈	—	137	1	no	—
158	C₂²×C₈	—	137	1	no	—
159	C₄⋊Q₈	—	137	1	no	—
160	#127	—	137	1	no	—
161	#124	—	137	2	no	—
162	#121	—	137	1	no	—
163	#121	—	137	1	no	—
164	#121	—	137	2	no	—
165	C₄×D₄	—	137	2	no	—
166	C₄×D₄	—	137	1	no	—
167	C₄×D₄	C₂×D₁₃₇	1	1	yes	characteristic
168	C₂×C₄²	—	137	1	no	—
169	C₄⋊C₈	—	137	2	no	—
170	C₄⋊C₈	—	137	1	no	—
171	C₄⋊C₈	C₂×D₁₃₇	1	1	yes	characteristic
172	C₂²⋊C₈	—	137	2	no	—
173	C₂²⋊C₈	—	137	2	no	—
174	C₂²⋊C₈	—	137	2	no	—
175	C₂²⋊C₈	C₂×D₁₃₇	1	2	yes	—
176	C₄×C₈	—	137	1	no	—
177	C₄×C₈	—	137	1	no	—
178	C₄×C₈	—	137	1	no	—
179	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
180	#41	—	274	4	no	—
181	#41	—	274	4	no	—
182	C₂×Q₈	—	137	2	no	—
183	C₂×D₄	—	137	2	no	—
184	C₂×D₄	—	137	2	no	—
185	C₂×D₄	—	137	2	no	—
186	C₂×D₄	—	137	1	no	—
187	C₂×D₄	—	137	2	no	—
188	C₂×D₄	C₄×D₁₃₇	1	1	yes	characteristic
189	C₂²×C₄	—	137	1	no	—
190	C₂²×C₄	—	137	1	no	—
191	C₂²×C₄	—	137	2	no	—
192	C₂²×C₄	—	137	1	no	—
193	C₂²×C₄	C₂²×D₁₃₇	1	2	yes	—
194	M₁₆	—	2	4	no	—
195	M₁₆	—	274	4	no	—
196	M₁₆	—	274	4	no	—
197	C₂×C₈	—	274	2	no	—
198	C₂×C₈	—	274	4	no	—
199	C₂×C₈	—	274	4	no	—
200	C₂×C₈	—	137	1	no	—
201	C₂×C₈	—	137	2	no	—
202	C₂×C₈	—	137	1	no	—
203	C₂×C₈	—	137	4	no	—
204	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
205	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
206	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
207	C₄⋊C₄	—	137	2	no	—
208	C₄⋊C₄	—	137	1	no	—
209	C₄⋊C₄	C₄×D₁₃₇	1	1	yes	characteristic
210	C₂²⋊C₄	—	137	2	no	—
211	C₂²⋊C₄	—	137	2	no	—
212	C₂²⋊C₄	—	137	2	no	—
213	C₂²⋊C₄	C₄×D₁₃₇	1	2	yes	—
214	C₄²	—	137	1	no	—
215	C₄²	—	137	1	no	—
216	C₄²	—	137	1	no	—
217	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
218	C₂³	—	137	2	no	—
219	C₂³	—	137	1	no	—
220	C₂³	C₂×C₄×D₁₃₇	1	2	yes	—
221	Q₈	—	274	4	no	—
222	D₄	—	2	2	no	—
223	D₄	—	274	4	no	—
224	D₄	—	274	4	no	—
225	D₄	—	274	4	no	—
226	D₄	—	274	4	no	—
227	D₄	—	274	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	274	2	no	—
231	C₂×C₄	—	274	2	no	—
232	C₂×C₄	—	274	2	no	—
233	C₂×C₄	—	274	2	no	—
234	C₂×C₄	—	137	1	no	—
235	C₂×C₄	—	137	2	no	—
236	C₂×C₄	—	137	1	no	—
237	C₂×C₄	—	137	1	no	—
238	C₂×C₄	—	137	2	no	—
239	C₂×C₄	—	137	2	no	—
240	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
241	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
242	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
243	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	2	1	no	—
246	C₈	—	274	2	no	—
247	C₈	—	274	1	no	—
248	C₈	—	137	2	no	—
249	C₈	D₄×D₁₃₇	1	2	yes	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	274	1	no	—
254	C₂²	—	274	1	no	—
255	C₂²	—	274	2	no	—
256	C₂²	—	274	2	no	—
257	C₂²	—	274	2	no	—
258	C₂²	—	137	2	no	—
259	C₂²	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	1	no	—
262	C₄	—	274	2	no	—
263	C₄	—	274	1	no	—
264	C₄	—	137	2	no	—
265	C₄	—	137	2	no	—
266	C₄	#17984096	1	2	yes	complemented
267	C₄	C₂×D₄×D₁₃₇	1	1	yes	characteristic
268	C₄	#41×D₁₃₇	1	1	yes	characteristic
269	C₂	—	4	2	no	—
270	C₂	—	274	2	no	—
271	C₂	—	548	2	no	—
272	C₂	C₄×D₄×D₁₃₇	1	1	yes	characteristic
273	C₂	#131×D₁₃₇	1	1	yes	characteristic
274	C₂	C₂×#17984096	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂³×C₄
Fitting	#436×C₁₃₇	C₂
prime radical	#22577950	C₂
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	#436	D₁₃₇
hypocenter	C₁₃₇	#2931
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
1. C₂×#17984096	C₂	semidirect
3. #22577482	C₂	semidirect
4. #22577479	C₂	semidirect
5. D₄×#11764040	C₂	semidirect
6. #22577468	C₂	semidirect
7. #22577466	C₂	semidirect
8. #22577425	C₂	semidirect
9. #22577368	C₂	semidirect
10. C₄×C₈×D₁₃₇	C₂	semidirect
11. #436×C₁₃₇	C₂	semidirect
62. C₈×D₁₃₇	D₄	semidirect
124. C₄×C₁₃₇	#131	semidirect
130. Dic₁₃₇	#131	semidirect
137. C₁₃₇	#2931	semidirect
143. #436	D₁₃₇	semidirect
266. C₄	#17984096	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
#22577950	2
#22577482	2
#22577479	2
D₄×#11764040	2
#22577468	2
#22577466	2
#22577425	2
#22577368	2
C₄×C₈×D₁₃₇	2
#436×C₁₃₇	2
#2931	137