Subgroups — #31789442

Order 17536.

402 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984045	C₂	1	1	yes	maximal, characteristic
2	#22577463	C₂	1	2	yes	maximal, complemented
3	#22577462	C₂	1	2	yes	maximal, complemented
4	#22577460	C₂	1	1	yes	maximal, characteristic, complemented
5	#22577457	C₂	1	1	yes	maximal, characteristic, complemented
6	#22577453	C₂	1	1	yes	maximal, characteristic, complemented
7	#22577428	C₂	1	2	yes	maximal, complemented
8	#22577365	C₂	1	1	yes	maximal, characteristic
9	C₂²⋊C₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic
10	#22577357	C₂	1	2	yes	maximal, complemented
11	#431×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×#14555493	C₂²	1	2	yes	—
14	C₂×C₈×D₁₃₇	C₂²	1	2	yes	—
15	#17984045	C₄	1	1	yes	characteristic
16	#17984045	C₄	1	1	yes	characteristic
17	#17984045	C₄	1	1	yes	characteristic
18	#17984045	C₄	1	1	yes	characteristic
19	#17984045	C₂²	1	1	yes	characteristic
20	#17984045	C₂²	1	1	yes	characteristic
21	#17984045	C₂²	1	1	yes	characteristic
22	#17984045	C₂²	1	1	yes	characteristic
23	C₂×#14555489	C₄	1	1	yes	characteristic
24	C₂×#14555489	C₄	1	1	yes	characteristic
25	C₂×#14555487	C₄	1	1	yes	characteristic
26	C₂×#14555487	C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
28	C₄⋊#11764040	C₂²	1	2	yes	complemented
29	#17984017	C₂²	1	2	yes	complemented
30	#17984017	C₂²	1	2	yes	complemented
31	C₂²⋊#11764040	C₂²	1	1	yes	characteristic
32	C₂²⋊#11764040	C₂²	1	1	yes	characteristic
33	#17984000	C₂²	1	2	yes	—
34	#17984000	C₂²	1	2	yes	—
35	#17983999	C₂²	1	2	yes	complemented
36	#17983998	C₂²	1	2	yes	complemented
37	C₈×Dic₁₃₇	C₂²	1	2	yes	complemented
38	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	complemented
40	C₂²⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
41	C₂²⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
42	C₄×C₈×C₁₃₇	C₂²	1	2	yes	complemented
43	C₂³×D₁₃₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
46	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₁₃₇	—	2	4	no	—
48	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
51	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
52	#14555493	—	2	4	no	—
53	C₂×#11764040	C₂³	1	2	yes	—
54	C₂×#11764040	C₂³	1	2	yes	—
55	C₈×D₁₃₇	—	2	4	no	—
56	#14555489	C₂×C₄	1	1	yes	characteristic
57	#14555489	C₂×C₄	1	1	yes	characteristic
58	#14555489	C₂×C₄	1	1	yes	characteristic
59	#14555489	C₂×C₄	1	1	yes	characteristic
60	#14555489	C₂×C₄	1	1	yes	characteristic
61	#14555489	C₂×C₄	1	1	yes	characteristic
62	#14555489	C₂×C₄	1	1	yes	characteristic
63	#14555489	C₂×C₄	1	1	yes	characteristic
64	#14555487	C₂×C₄	1	1	yes	characteristic
65	#14555487	C₂×C₄	1	1	yes	characteristic
66	#14555487	C₂×C₄	1	1	yes	characteristic
67	#14555487	C₂×C₄	1	1	yes	characteristic
68	#14555487	C₂×C₄	1	1	yes	characteristic
69	#14555487	C₂×C₄	1	1	yes	characteristic
70	#14555487	C₂×C₄	1	1	yes	characteristic
71	#14555487	C₂×C₄	1	1	yes	characteristic
72	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
73	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
74	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
75	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
79	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
80	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
81	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
82	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
83	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
84	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
85	C₂²×D₁₃₇	—	2	2	no	—
86	C₂²×D₁₃₇	—	2	2	no	—
87	C₂²×D₁₃₇	—	4	1	no	—
88	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
89	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×Dic₁₃₇	—	2	2	no	—
91	C₂×Dic₁₃₇	—	2	2	no	—
92	C₂×Dic₁₃₇	—	4	1	no	—
93	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
99	C₄×D₁₃₇	—	2	2	no	—
100	C₄×D₁₃₇	—	2	2	no	—
101	C₄×D₁₃₇	—	2	2	no	—
102	C₄×D₁₃₇	#41	1	4	yes	—
103	#11764040	—	2	2	no	—
104	#11764040	—	2	2	no	—
105	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃₇	—	2	2	no	—
107	C₂×C₄×C₁₃₇	—	2	2	no	—
108	C₂×C₄×C₁₃₇	—	4	1	no	—
109	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
115	C₈×C₁₃₇	—	2	2	no	—
116	C₈×C₁₃₇	—	2	2	no	—
117	C₂²×C₁₃₇	—	2	1	no	—
118	C₂²×C₁₃₇	—	2	1	no	—
119	C₂²×C₁₃₇	#131	1	2	yes	—
120	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
121	C₂×D₁₃₇	—	2	1	no	—
122	C₂×D₁₃₇	—	2	1	no	—
123	C₂×D₁₃₇	—	2	1	no	—
124	C₂×D₁₃₇	—	2	1	no	—
125	C₂×D₁₃₇	—	2	1	no	—
126	C₂×D₁₃₇	—	4	4	no	—
127	C₂×D₁₃₇	#117	1	2	yes	—
128	C₄×C₁₃₇	—	2	1	no	—
129	C₄×C₁₃₇	—	2	1	no	—
130	C₄×C₁₃₇	—	2	1	no	—
131	C₄×C₁₃₇	—	2	1	no	—
132	C₄×C₁₃₇	—	2	1	no	—
133	C₄×C₁₃₇	C₂×#41	1	2	yes	—
134	Dic₁₃₇	—	2	1	no	—
135	Dic₁₃₇	—	2	1	no	—
136	Dic₁₃₇	—	2	1	no	—
137	Dic₁₃₇	—	2	1	no	—
138	Dic₁₃₇	—	2	1	no	—
139	Dic₁₃₇	#117	1	2	yes	—
140	C₂×C₁₃₇	—	2	2	no	—
141	C₂×C₁₃₇	C₂×#117	1	1	yes	characteristic
142	C₂×C₁₃₇	C₂×#131	1	1	yes	characteristic
143	C₂×C₁₃₇	#568	1	1	yes	characteristic
144	D₁₃₇	—	2	2	no	—
145	D₁₃₇	—	4	1	no	—
146	C₁₃₇	#2924	1	1	yes	characteristic, complemented, Sylow
147	#2924	—	137	1	no	maximal, Sylow
148	C₂×#117	—	137	1	no	—
149	#433	—	137	2	no	—
150	#433	—	137	2	no	—
151	#432	—	137	1	no	—
152	#432	—	137	1	no	—
153	#431	—	137	1	no	—
154	#431	D₁₃₇	1	1	yes	characteristic, complemented
155	#424	—	137	2	no	—
156	#407	—	137	1	no	—
157	C₂×C₂²⋊C₈	—	137	1	no	—
158	#405	—	137	2	no	—
159	C₂³×C₄	—	137	1	no	—
160	C₂×M₁₆	—	137	2	no	—
161	C₂²×C₈	—	137	2	no	—
162	#117	—	137	1	no	—
163	#117	—	137	1	no	—
164	#117	—	137	1	no	—
165	#117	—	137	1	no	—
166	#117	—	137	1	no	—
167	#117	—	137	1	no	—
168	#117	—	137	1	no	—
169	#117	—	137	1	no	—
170	C₂×C₄⋊C₄	—	137	1	no	—
171	C₂×C₄⋊C₄	—	137	1	no	—
172	C₂×C₂²⋊C₄	—	137	1	no	—
173	C₂×C₂²⋊C₄	—	137	1	no	—
174	C₂×C₄²	—	137	1	no	—
175	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
176	C₄⋊C₈	—	137	2	no	—
177	C₄⋊C₈	—	137	2	no	—
178	C₄⋊C₈	—	137	2	no	—
179	C₄⋊C₈	C₂×D₁₃₇	1	2	yes	complemented
180	C₂²⋊C₈	—	137	1	no	—
181	C₂²⋊C₈	—	137	1	no	—
182	C₂²⋊C₈	—	137	2	no	—
183	C₂²⋊C₈	—	137	2	no	—
184	C₂²⋊C₈	C₂×D₁₃₇	1	1	yes	characteristic
185	C₂²⋊C₈	C₂×D₁₃₇	1	1	yes	characteristic
186	#97	—	137	2	no	—
187	#97	—	137	2	no	—
188	C₄×C₈	—	137	2	no	—
189	C₄×C₈	C₂×D₁₃₇	1	2	yes	complemented
190	C₂⁴	—	137	1	no	—
191	C₂²×C₄	—	274	4	no	—
192	C₂²×C₄	—	137	1	no	—
193	C₂²×C₄	—	137	1	no	—
194	C₂²×C₄	—	137	1	no	—
195	C₂²×C₄	—	137	1	no	—
196	C₂²×C₄	—	137	1	no	—
197	C₂²×C₄	—	137	1	no	—
198	C₂²×C₄	—	137	1	no	—
199	C₂²×C₄	C₄×D₁₃₇	1	1	yes	characteristic
200	C₂²×C₄	C₄×D₁₃₇	1	1	yes	characteristic
201	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
202	M₁₆	—	274	4	no	—
203	C₂×C₈	—	274	4	no	—
204	C₂×C₈	—	137	2	no	—
205	C₂×C₈	—	137	2	no	—
206	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
207	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
208	C₄⋊C₄	—	137	1	no	—
209	C₄⋊C₄	—	137	1	no	—
210	C₄⋊C₄	—	137	1	no	—
211	C₄⋊C₄	—	137	1	no	—
212	C₄⋊C₄	—	137	1	no	—
213	C₄⋊C₄	—	137	1	no	—
214	C₄⋊C₄	—	137	1	no	—
215	C₄⋊C₄	—	137	1	no	—
216	C₂²⋊C₄	—	137	1	no	—
217	C₂²⋊C₄	—	137	1	no	—
218	C₂²⋊C₄	—	137	1	no	—
219	C₂²⋊C₄	—	137	1	no	—
220	C₂²⋊C₄	—	137	1	no	—
221	C₂²⋊C₄	—	137	1	no	—
222	C₂²⋊C₄	—	137	1	no	—
223	C₂²⋊C₄	—	137	1	no	—
224	C₄²	—	137	1	no	—
225	C₄²	—	137	1	no	—
226	C₄²	—	137	1	no	—
227	C₄²	—	137	1	no	—
228	C₄²	C₄×D₁₃₇	1	1	yes	characteristic
229	C₄²	C₄×D₁₃₇	1	1	yes	characteristic
230	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
231	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
232	C₂³	—	274	2	no	—
233	C₂³	—	274	2	no	—
234	C₂³	—	548	1	no	—
235	C₂³	—	137	1	no	—
236	C₂³	—	137	1	no	—
237	C₂³	C₂×C₄×D₁₃₇	1	1	yes	characteristic
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	274	2	no	—
242	C₂×C₄	—	274	2	no	—
243	C₂×C₄	—	274	2	no	—
244	C₂×C₄	—	274	2	no	—
245	C₂×C₄	—	274	2	no	—
246	C₂×C₄	—	548	1	no	—
247	C₂×C₄	—	137	1	no	—
248	C₂×C₄	—	137	1	no	—
249	C₂×C₄	—	137	1	no	—
250	C₂×C₄	—	137	1	no	—
251	C₂×C₄	—	137	1	no	—
252	C₂×C₄	—	137	1	no	—
253	C₂×C₄	—	137	4	no	—
254	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
255	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
256	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
258	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
259	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	—	2	2	no	—
262	C₈	—	274	2	no	—
263	C₈	—	274	2	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	274	1	no	—
267	C₂²	—	274	1	no	—
268	C₂²	—	274	1	no	—
269	C₂²	—	274	1	no	—
270	C₂²	—	274	1	no	—
271	C₂²	—	548	4	no	—
272	C₂²	—	137	2	no	—
273	C₂²	#17984094	1	2	yes	—
274	C₂²	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	274	1	no	—
281	C₄	—	274	1	no	—
282	C₄	—	274	1	no	—
283	C₄	—	274	1	no	—
284	C₄	—	274	1	no	—
285	C₄	—	137	2	no	—
286	C₄	#41×D₁₃₇	1	2	yes	—
287	C₂	—	2	2	no	—
288	C₂	—	274	2	no	—
289	C₂	—	548	1	no	—
290	C₂	#117×D₁₃₇	1	1	yes	characteristic
291	C₂	C₂×#17984094	1	1	yes	characteristic
292	C₂	#22578169	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂³×C₄
Fitting	#431×C₁₃₇	C₂
prime radical	C₂³×D₁₃₇	C₂×C₄
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	#431	D₁₃₇
hypocenter	C₁₃₇	#2924
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
2. #22577463	C₂	semidirect
3. #22577462	C₂	semidirect
4. #22577460	C₂	semidirect
5. #22577457	C₂	semidirect
6. #22577453	C₂	semidirect
7. #22577428	C₂	semidirect
10. #22577357	C₂	semidirect
11. #431×C₁₃₇	C₂	semidirect
28. C₄⋊#11764040	C₂²	semidirect
29. #17984017	C₂²	semidirect
30. #17984017	C₂²	semidirect
35. #17983999	C₂²	semidirect
36. #17983998	C₂²	semidirect
37. C₈×Dic₁₃₇	C₂²	semidirect
39. C₄⋊C₈×C₁₃₇	C₂²	semidirect
42. C₄×C₈×C₁₃₇	C₂²	semidirect
146. C₁₃₇	#2924	semidirect
154. #431	D₁₃₇	semidirect
179. C₄⋊C₈	C₂×D₁₃₇	semidirect
189. 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
C₂×#17984045	2
#22577463	2
#22577462	2
#22577460	2
#22577457	2
#22577453	2
#22577428	2
#22577365	2
C₂²⋊C₈×D₁₃₇	2
#22577357	2
#431×C₁₃₇	2
#2924	137