Subgroups — #31789446

Order 17536.

402 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#117×D₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#22577463	C₂	1	2	yes	maximal, complemented
3	#22577463	C₂	1	2	yes	maximal, complemented
4	#22577459	C₂	1	2	yes	maximal, complemented
5	#22577455	C₂	1	2	yes	maximal, complemented
6	#22577430	C₂	1	1	yes	maximal, characteristic, complemented
7	#22577366	C₂	1	1	yes	maximal, characteristic
8	C₂×#17984000	C₂	1	1	yes	maximal, characteristic
9	#22577361	C₂	1	1	yes	maximal, characteristic, complemented
10	#22577358	C₂	1	1	yes	maximal, characteristic, complemented
11	#424×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×#14555494	C₂²	1	1	yes	characteristic
14	C₂²×#11764040	C₂²	1	1	yes	characteristic
15	#17984047	C₂²	1	1	yes	characteristic
16	#17984046	C₄	1	1	yes	characteristic
17	#17984046	C₄	1	1	yes	characteristic
18	#17984045	C₂²	1	2	yes	—
19	#17984044	C₄	1	2	yes	—
20	C₄⋊C₄×D₁₃₇	C₄	1	1	yes	characteristic
21	C₄⋊C₄×D₁₃₇	C₄	1	1	yes	characteristic
22	C₂²⋊C₄×D₁₃₇	C₄	1	2	yes	—
23	C₄²×D₁₃₇	C₂²	1	2	yes	—
24	C₄⋊#11764040	C₂²	1	2	yes	complemented
25	C₄⋊#11764040	C₂²	1	2	yes	complemented
26	#17984000	C₂²	1	1	yes	characteristic
27	#17984000	C₂²	1	2	yes	—
28	#17984000	C₂²	1	1	yes	characteristic
29	#17984000	C₂²	1	1	yes	characteristic
30	#17984000	C₂²	1	1	yes	characteristic
31	#17984000	C₂²	1	2	yes	—
32	#17983998	C₂²	1	2	yes	complemented
33	#17983998	C₂²	1	2	yes	complemented
34	C₈×Dic₁₃₇	C₂²	1	2	yes	complemented
35	C₈×Dic₁₃₇	C₂²	1	2	yes	complemented
36	C₂×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
38	#117×C₁₃₇	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	complemented
40	C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	complemented
41	C₂³×D₁₃₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₃₇	—	2	2	no	—
44	C₂×C₄×D₁₃₇	—	2	2	no	—
45	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
49	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
50	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
51	#14555494	—	2	2	no	—
52	C₂×#11764040	—	2	2	no	—
53	C₂×#11764040	C₂³	1	1	yes	characteristic
54	C₂×#11764040	C₂³	1	1	yes	characteristic
55	C₂×#11764040	C₂³	1	2	yes	—
56	C₄⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
58	#14555489	C₂×C₄	1	2	yes	—
59	#14555489	C₂×C₄	1	2	yes	—
60	C₂²⋊Dic₁₃₇	C₂×C₄	1	2	yes	—
61	#14555487	C₂×C₄	1	2	yes	—
62	#14555487	C₂×C₄	1	2	yes	—
63	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
64	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
65	C₄×Dic₁₃₇	C₂³	1	2	yes	—
66	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
67	M₁₆×C₁₃₇	—	2	2	no	—
68	C₂×C₈×C₁₃₇	—	2	2	no	—
69	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
72	C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
73	C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
74	C₂²⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
75	C₄²×C₁₃₇	C₂³	1	2	yes	—
76	C₂²×D₁₃₇	—	2	2	no	—
77	C₂²×D₁₃₇	—	2	2	no	—
78	C₂²×D₁₃₇	—	4	1	no	—
79	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₁₃₇	—	2	2	no	—
82	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
84	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
85	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
86	C₄×D₁₃₇	—	2	4	no	—
87	C₄×D₁₃₇	—	2	2	no	—
88	C₄×D₁₃₇	—	2	2	no	—
89	C₄×D₁₃₇	—	2	1	no	—
90	C₄×D₁₃₇	—	2	1	no	—
91	C₄×D₁₃₇	—	2	1	no	—
92	C₄×D₁₃₇	—	2	1	no	—
93	C₄×D₁₃₇	—	4	2	no	—
94	#11764040	—	2	1	no	—
95	#11764040	—	2	1	no	—
96	#11764040	—	2	2	no	—
97	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
100	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃₇	#41	1	2	yes	—
102	C₂×C₄×C₁₃₇	#41	1	2	yes	—
103	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
104	C₈×C₁₃₇	—	2	1	no	—
105	C₈×C₁₃₇	—	2	1	no	—
106	C₈×C₁₃₇	—	2	2	no	—
107	C₂²×C₁₃₇	—	2	1	no	—
108	C₂²×C₁₃₇	—	2	1	no	—
109	C₂²×C₁₃₇	#117	1	2	yes	—
110	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
111	C₂×D₁₃₇	—	2	1	no	—
112	C₂×D₁₃₇	—	2	1	no	—
113	C₂×D₁₃₇	—	2	1	no	—
114	C₂×D₁₃₇	—	2	1	no	—
115	C₂×D₁₃₇	—	2	1	no	—
116	C₂×D₁₃₇	—	4	4	no	—
117	C₂×D₁₃₇	#131	1	2	yes	—
118	C₄×C₁₃₇	—	2	2	no	—
119	C₄×C₁₃₇	—	2	2	no	—
120	C₄×C₁₃₇	#117	1	2	yes	—
121	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
122	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
123	Dic₁₃₇	—	2	1	no	—
124	Dic₁₃₇	—	2	1	no	—
125	Dic₁₃₇	—	2	2	no	—
126	Dic₁₃₇	—	2	2	no	—
127	C₂×C₁₃₇	—	2	2	no	—
128	C₂×C₁₃₇	C₂×#117	1	1	yes	characteristic
129	C₂×C₁₃₇	C₂×#131	1	1	yes	characteristic
130	C₂×C₁₃₇	#568	1	1	yes	characteristic
131	D₁₃₇	—	2	2	no	—
132	D₁₃₇	—	4	1	no	—
133	C₁₃₇	#2924	1	1	yes	characteristic, complemented, Sylow
134	#2924	—	137	1	no	maximal, Sylow
135	C₂×#117	—	137	1	no	—
136	#433	—	137	2	no	—
137	#433	—	137	2	no	—
138	#432	—	137	2	no	—
139	#431	—	137	2	no	—
140	#424	—	137	1	no	—
141	#424	D₁₃₇	1	1	yes	characteristic, complemented
142	#407	—	137	1	no	—
143	C₂×C₂²⋊C₈	—	137	1	no	—
144	#405	—	137	1	no	—
145	#405	—	137	1	no	—
146	C₂³×C₄	—	137	1	no	—
147	C₂×M₁₆	—	137	1	no	—
148	C₂×M₁₆	C₂×D₁₃₇	1	1	yes	characteristic
149	C₂²×C₈	—	137	1	no	—
150	C₂²×C₈	C₂×D₁₃₇	1	1	yes	characteristic
151	#117	—	137	1	no	—
152	#117	—	137	1	no	—
153	#117	—	137	1	no	—
154	#117	—	137	2	no	—
155	#117	—	137	2	no	—
156	#117	C₂×D₁₃₇	1	1	yes	characteristic
157	C₂×C₄⋊C₄	—	137	1	no	—
158	C₂×C₄⋊C₄	—	137	1	no	—
159	C₂×C₂²⋊C₄	—	137	2	no	—
160	C₂×C₄²	—	137	2	no	—
161	C₄⋊C₈	—	137	2	no	—
162	C₄⋊C₈	—	137	2	no	—
163	C₄⋊C₈	C₂×D₁₃₇	1	2	yes	complemented
164	C₄⋊C₈	C₂×D₁₃₇	1	2	yes	complemented
165	C₂²⋊C₈	—	137	1	no	—
166	C₂²⋊C₈	—	137	1	no	—
167	C₂²⋊C₈	—	137	1	no	—
168	C₂²⋊C₈	—	137	1	no	—
169	C₂²⋊C₈	—	137	2	no	—
170	C₂²⋊C₈	—	137	2	no	—
171	#97	—	137	2	no	—
172	#97	—	137	2	no	—
173	C₄×C₈	—	137	2	no	—
174	C₄×C₈	—	137	2	no	—
175	C₂⁴	—	137	1	no	—
176	C₂²×C₄	—	274	2	no	—
177	C₂²×C₄	—	274	2	no	—
178	C₂²×C₄	—	137	1	no	—
179	C₂²×C₄	—	137	2	no	—
180	C₂²×C₄	—	137	1	no	—
181	C₂²×C₄	—	137	1	no	—
182	C₂²×C₄	—	137	1	no	—
183	C₂²×C₄	—	137	1	no	—
184	C₂²×C₄	—	137	2	no	—
185	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
186	M₁₆	—	2	2	no	—
187	M₁₆	—	274	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	274	2	no	—
190	C₂×C₈	—	137	1	no	—
191	C₂×C₈	—	137	2	no	—
192	C₂×C₈	—	137	1	no	—
193	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
194	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
195	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
196	C₄⋊C₄	—	137	2	no	—
197	C₄⋊C₄	—	137	2	no	—
198	C₄⋊C₄	—	137	1	no	—
199	C₄⋊C₄	—	137	1	no	—
200	C₄⋊C₄	C₄×D₁₃₇	1	1	yes	characteristic
201	C₄⋊C₄	C₄×D₁₃₇	1	1	yes	characteristic
202	C₂²⋊C₄	—	137	2	no	—
203	C₂²⋊C₄	—	137	2	no	—
204	C₂²⋊C₄	—	137	2	no	—
205	C₂²⋊C₄	C₄×D₁₃₇	1	2	yes	—
206	C₄²	—	137	2	no	—
207	C₄²	—	137	2	no	—
208	C₄²	—	137	2	no	—
209	C₄²	C₂²×D₁₃₇	1	2	yes	—
210	C₂³	—	274	2	no	—
211	C₂³	—	274	2	no	—
212	C₂³	—	548	1	no	—
213	C₂³	—	137	1	no	—
214	C₂³	—	137	1	no	—
215	C₂³	C₂×C₄×D₁₃₇	1	1	yes	characteristic
216	C₂×C₄	—	274	4	no	—
217	C₂×C₄	—	274	2	no	—
218	C₂×C₄	—	274	2	no	—
219	C₂×C₄	—	274	2	no	—
220	C₂×C₄	—	274	1	no	—
221	C₂×C₄	—	274	1	no	—
222	C₂×C₄	—	274	1	no	—
223	C₂×C₄	—	274	1	no	—
224	C₂×C₄	—	548	2	no	—
225	C₂×C₄	—	137	1	no	—
226	C₂×C₄	—	137	2	no	—
227	C₂×C₄	—	137	2	no	—
228	C₂×C₄	—	137	1	no	—
229	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
230	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
231	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
232	C₂×C₄	#14555509	1	2	yes	—
233	C₂×C₄	#14555511	1	2	yes	—
234	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	2	2	no	—
238	C₈	—	274	1	no	—
239	C₈	—	274	1	no	—
240	C₈	—	274	2	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	274	1	no	—
244	C₂²	—	274	1	no	—
245	C₂²	—	274	1	no	—
246	C₂²	—	274	1	no	—
247	C₂²	—	274	1	no	—
248	C₂²	—	548	4	no	—
249	C₂²	—	137	2	no	—
250	C₂²	#17984046	1	2	yes	—
251	C₂²	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	274	1	no	—
255	C₄	—	274	1	no	—
256	C₄	—	274	2	no	—
257	C₄	—	274	2	no	—
258	C₄	#17984046	1	2	yes	—
259	C₄	C₂×#14555509	1	1	yes	characteristic
260	C₄	C₂×#14555511	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	274	2	no	—
263	C₂	—	548	1	no	—
264	C₂	C₂×#17984046	1	1	yes	characteristic
265	C₂	#131×D₁₃₇	1	1	yes	characteristic
266	C₂	#22578168	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂³×C₄
Fitting	#424×C₁₃₇	C₂
prime radical	C₂³×D₁₃₇	C₂×C₄
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	#424	D₁₃₇
hypocenter	C₁₃₇	#2924
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
2. #22577463	C₂	semidirect
3. #22577463	C₂	semidirect
4. #22577459	C₂	semidirect
5. #22577455	C₂	semidirect
6. #22577430	C₂	semidirect
9. #22577361	C₂	semidirect
10. #22577358	C₂	semidirect
11. #424×C₁₃₇	C₂	semidirect
24. C₄⋊#11764040	C₂²	semidirect
25. C₄⋊#11764040	C₂²	semidirect
32. #17983998	C₂²	semidirect
33. #17983998	C₂²	semidirect
34. C₈×Dic₁₃₇	C₂²	semidirect
35. C₈×Dic₁₃₇	C₂²	semidirect
39. C₄⋊C₈×C₁₃₇	C₂²	semidirect
40. C₄⋊C₈×C₁₃₇	C₂²	semidirect
133. C₁₃₇	#2924	semidirect
141. #424	D₁₃₇	semidirect
163. C₄⋊C₈	C₂×D₁₃₇	semidirect
164. C₄⋊C₈	C₂×D₁₃₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#117×D₁₃₇	2
#22577463	2
#22577459	2
#22577455	2
#22577430	2
#22577366	2
C₂×#17984000	2
#22577361	2
#22577358	2
#424×C₁₃₇	2
#2924	137