Subgroups — #31797567

Order 17536.

452 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578483	C₂	1	1	yes	maximal, characteristic, complemented
2	#22578481	C₂	1	1	yes	maximal, characteristic, complemented
3	#22578448	C₂	1	4	yes	maximal, complemented
4	#22578446	C₂	1	4	yes	maximal, complemented
5	D₁₃₇⋊#108	C₂	1	1	yes	maximal, characteristic, complemented
6	#22578402	C₂	1	1	yes	maximal, characteristic, complemented
7	#22578401	C₂	1	1	yes	maximal, characteristic, complemented
8	#22578366	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#17984109	C₂	1	1	yes	maximal, characteristic
10	#17984176	—	2	2	no	—
11	#17984176	C₂²	1	4	yes	complemented
12	#17984175	—	2	2	no	—
13	#17984175	C₂²	1	4	yes	complemented
14	D₁₃₇⋊M₁₆	—	2	2	no	—
15	D₁₃₇⋊M₁₆	C₂²	1	1	yes	characteristic
16	D₁₃₇⋊M₁₆	C₂²	1	1	yes	characteristic
17	D₁₃₇⋊M₁₆	C₂²	1	1	yes	characteristic
18	C₂×#14555519	—	2	2	no	—
19	#17984167	C₂²	1	1	yes	characteristic
20	#17984159	C₂²	1	2	yes	complemented
21	#17984158	C₂²	1	2	yes	complemented
22	#17984153	C₂²	1	4	yes	complemented
23	#17984152	C₂²	1	4	yes	complemented
24	#17984142	C₂²	1	2	yes	complemented
25	C₈×#10587739	C₂²	1	2	yes	complemented
26	C₂×#14555511	C₂²	1	1	yes	characteristic
27	C₂×#14555509	C₂²	1	1	yes	characteristic
28	#17984109	C₄	1	2	yes	complemented
29	#17984109	C₄	1	2	yes	complemented
30	#17984109	C₂²	1	2	yes	—
31	#17984109	C₂²	1	2	yes	—
32	C₂×#14555500	C₄	1	1	yes	characteristic, complemented
33	C₂×#14555499	C₄	1	1	yes	characteristic, complemented
34	C₂×#14555495	C₄	1	1	yes	characteristic, complemented
35	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
36	C₂×SD₁₆×C₁₃₇	C₄	1	1	yes	characteristic, complemented
37	#14555520	—	2	4	no	—
38	#14555520	D₄	1	2	yes	—
39	#14555520	C₂³	1	2	yes	—
40	#14555520	C₂³	1	2	yes	—
41	#14555519	—	2	4	no	—
42	#14555519	—	2	1	no	—
43	#14555519	—	2	1	no	—
44	#14555519	D₄	1	2	yes	—
45	#14555518	—	2	4	no	—
46	#14555518	—	2	1	no	—
47	#14555518	—	2	1	no	—
48	#14555518	C₂³	1	2	yes	—
49	C₂×#11764047	—	2	4	no	—
50	C₄⋊#10587739	—	2	1	no	—
51	C₂²⋊#10587739	—	2	1	no	—
52	C₄×#10587739	C₂³	1	2	yes	—
53	#14555511	—	2	1	no	—
54	#14555511	—	2	1	no	—
55	#14555511	C₂×C₄	1	2	yes	—
56	#14555511	C₂³	1	2	yes	—
57	#14555509	—	2	1	no	—
58	#14555509	—	2	1	no	—
59	#14555509	C₂×C₄	1	2	yes	—
60	#14555509	C₂³	1	2	yes	—
61	C₂×Dic₂₇₄	C₂×C₄	1	1	yes	characteristic
62	C₂×D₅₄₈	—	2	1	no	—
63	C₂×D₅₄₈	C₂×C₄	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₃₇	—	2	1	no	—
65	C₂²×Dic₁₃₇	—	2	1	no	—
66	C₂×C₄×D₁₃₇	—	2	1	no	—
67	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
68	#14555500	C₂×C₄	1	2	yes	—
69	#14555500	C₂×C₄	1	2	yes	—
70	#14555499	C₂×C₄	1	2	yes	—
71	#14555499	C₂×C₄	1	2	yes	—
72	#14555495	C₂×C₄	1	2	yes	—
73	#14555495	C₂×C₄	1	2	yes	—
74	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
75	C₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic
76	C₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic
77	C₈×D₁₃₇	C₂³	1	1	yes	characteristic
78	C₈×D₁₃₇	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
80	C₂×D₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
81	SD₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
82	SD₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
83	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
84	C₂×#10587739	—	2	2	no	—
85	#11764047	—	2	2	no	—
86	#11764047	—	2	2	no	—
87	#11764047	C₂×D₄	1	4	yes	—
88	C₂²×D₁₃₇	—	2	1	no	—
89	C₂²×D₁₃₇	#41	1	1	yes	characteristic
90	Dic₂₇₄	—	2	1	no	—
91	Dic₂₇₄	C₂²×C₄	1	2	yes	—
92	D₅₄₈	—	2	2	no	—
93	D₅₄₈	—	2	2	no	—
94	D₅₄₈	—	2	1	no	—
95	D₅₄₈	C₂²×C₄	1	2	yes	—
96	C₂²⋊D₁₃₇	—	2	2	no	—
97	C₂²⋊D₁₃₇	—	2	2	no	—
98	C₂×Dic₁₃₇	—	2	1	no	—
99	C₂×Dic₁₃₇	—	2	2	no	—
100	C₂×Dic₁₃₇	—	2	2	no	—
101	C₂×Dic₁₃₇	C₂×D₄	1	1	yes	characteristic
102	C₄×D₁₃₇	—	2	2	no	—
103	C₄×D₁₃₇	—	2	2	no	—
104	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
105	C₄×D₁₃₇	C₂×D₄	1	1	yes	characteristic
106	C₄×D₁₃₇	#41	1	1	yes	characteristic
107	C₄×D₁₃₇	C₂⁴	1	1	yes	characteristic
108	#11764040	C₂²×C₄	1	1	yes	characteristic
109	#11764040	C₂²×C₄	1	1	yes	characteristic
110	C₂³×C₁₃₇	—	2	1	no	—
111	Q₈×C₁₃₇	—	2	1	no	—
112	Q₈×C₁₃₇	C₂²×C₄	1	2	yes	—
113	D₄×C₁₃₇	—	2	1	no	—
114	D₄×C₁₃₇	C₂²×C₄	1	2	yes	—
115	C₂×C₄×C₁₃₇	—	2	1	no	—
116	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
117	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
118	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₁₃₇	—	2	2	no	—
120	C₂²×C₁₃₇	—	8	1	no	—
121	C₂²×C₁₃₇	C₄×D₄	1	1	yes	characteristic
122	C₂×D₁₃₇	—	2	2	no	—
123	C₂×D₁₃₇	—	4	1	no	—
124	C₂×D₁₃₇	—	8	1	no	—
125	C₂×D₁₃₇	C₄×D₄	1	1	yes	characteristic
126	C₂×D₁₃₇	C₂×#41	1	1	yes	characteristic
127	#10587739	—	4	2	no	—
128	C₄×C₁₃₇	—	2	2	no	—
129	C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
130	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
131	Dic₁₃₇	—	2	2	no	—
132	Dic₁₃₇	C₄×D₄	1	1	yes	characteristic
133	Dic₁₃₇	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₁₃₇	—	2	1	no	—
135	C₂×C₁₃₇	—	4	2	no	—
136	C₂×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
137	D₁₃₇	—	2	1	no	—
138	D₁₃₇	—	2	1	no	—
139	D₁₃₇	—	4	2	no	—
140	C₁₃₇	#2957	1	1	yes	characteristic, complemented, Sylow
141	#2957	—	137	1	no	maximal, Sylow
142	C₂×#135	—	137	1	no	—
143	#568	—	137	1	no	—
144	#568	—	137	1	no	—
145	#444	—	137	4	no	—
146	#443	—	137	4	no	—
147	C₂×#108	—	137	1	no	—
148	#421	—	137	1	no	—
149	#421	—	137	1	no	—
150	#405	—	137	1	no	—
151	C₂×#41	—	137	1	no	—
152	C₂×#41	—	137	1	no	—
153	#135	—	137	2	no	—
154	#135	—	137	2	no	—
155	#135	—	137	2	no	—
156	#135	—	137	2	no	—
157	C₂×Q₁₆	—	137	1	no	—
158	C₂×SD₁₆	—	137	1	no	—
159	C₂×SD₁₆	#10587739	1	1	yes	characteristic, complemented
160	C₂×D₈	—	137	1	no	—
161	#131	—	274	2	no	—
162	#131	—	274	2	no	—
163	#131	—	137	4	no	—
164	#131	—	137	4	no	—
165	C₂×M₁₆	—	274	2	no	—
166	C₂×M₁₆	—	274	2	no	—
167	C₂×M₁₆	—	137	1	no	—
168	C₂×M₁₆	—	137	1	no	—
169	C₂×M₁₆	—	137	1	no	—
170	C₂²×C₈	—	137	1	no	—
171	#117	—	137	1	no	—
172	#108	—	137	2	no	—
173	#108	—	137	2	no	—
174	#104	—	137	4	no	—
175	#104	—	137	4	no	—
176	#97	—	137	2	no	—
177	C₄×C₈	—	137	2	no	—
178	#41	—	274	1	no	—
179	#41	—	274	1	no	—
180	#41	—	274	1	no	—
181	#41	—	274	1	no	—
182	#41	—	137	2	no	—
183	#41	—	137	2	no	—
184	#41	—	137	2	no	—
185	#41	—	137	2	no	—
186	C₂×Q₈	—	137	1	no	—
187	C₂×Q₈	C₂×#10587739	1	1	yes	characteristic
188	C₂×D₄	—	274	1	no	—
189	C₂×D₄	—	274	1	no	—
190	C₂×D₄	—	137	1	no	—
191	C₂×D₄	C₂×#10587739	1	1	yes	characteristic
192	C₂²×C₄	—	274	1	no	—
193	C₂²×C₄	—	274	1	no	—
194	C₂²×C₄	—	137	1	no	—
195	Q₁₆	—	137	2	no	—
196	Q₁₆	—	137	2	no	—
197	SD₁₆	—	137	2	no	—
198	SD₁₆	—	137	2	no	—
199	SD₁₆	C₂×#10587739	1	2	yes	—
200	SD₁₆	C₂×#10587739	1	2	yes	—
201	D₈	—	137	2	no	—
202	D₈	—	137	2	no	—
203	M₁₆	—	274	1	no	—
204	M₁₆	—	274	1	no	—
205	M₁₆	—	274	4	no	—
206	M₁₆	—	274	4	no	—
207	M₁₆	—	137	2	no	—
208	M₁₆	—	137	2	no	—
209	M₁₆	—	137	2	no	—
210	M₁₆	—	137	2	no	—
211	C₂×C₈	—	274	1	no	—
212	C₂×C₈	—	274	1	no	—
213	C₂×C₈	—	274	4	no	—
214	C₂×C₈	—	274	4	no	—
215	C₂×C₈	—	137	2	no	—
216	C₂×C₈	—	137	1	no	—
217	C₂×C₈	—	137	1	no	—
218	C₂×C₈	—	137	1	no	—
219	C₂×C₈	—	137	1	no	—
220	C₂×C₈	—	137	1	no	—
221	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
222	C₄⋊C₄	—	274	1	no	—
223	C₂²⋊C₄	—	274	1	no	—
224	C₄²	—	137	2	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	274	1	no	—
227	C₂³	—	137	1	no	—
228	Q₈	—	2	1	no	—
229	Q₈	—	274	1	no	—
230	Q₈	—	137	2	no	—
231	Q₈	C₂²×#10587739	1	2	yes	—
232	D₄	—	2	1	no	—
233	D₄	—	274	2	no	—
234	D₄	—	274	2	no	—
235	D₄	—	274	2	no	—
236	D₄	—	274	2	no	—
237	D₄	—	274	1	no	—
238	D₄	—	137	2	no	—
239	D₄	C₂²×#10587739	1	2	yes	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	274	2	no	—
242	C₂×C₄	—	274	2	no	—
243	C₂×C₄	—	274	1	no	—
244	C₂×C₄	—	274	2	no	—
245	C₂×C₄	—	274	2	no	—
246	C₂×C₄	—	274	2	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	C₂²×#10587739	1	1	yes	characteristic
253	C₈	—	274	2	no	—
254	C₈	—	274	2	no	—
255	C₈	—	137	4	no	—
256	C₈	—	137	1	no	—
257	C₈	—	137	1	no	—
258	C₈	C₂²×#10587739	1	1	yes	characteristic
259	C₈	C₂²×#10587739	1	1	yes	characteristic
260	C₂²	—	2	2	no	—
261	C₂²	—	8	1	no	—
262	C₂²	—	274	2	no	—
263	C₂²	—	548	1	no	—
264	C₂²	—	1096	1	no	—
265	C₂²	—	137	1	no	—
266	C₂²	—	137	1	no	—
267	C₂²	D₄×#10587739	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	274	2	no	—
270	C₄	—	548	2	no	—
271	C₄	—	137	1	no	—
272	C₄	—	137	1	no	—
273	C₄	D₄×#10587739	1	1	yes	characteristic
274	C₄	C₂³×#10587739	1	1	yes	characteristic
275	C₂	—	2	1	no	—
276	C₂	—	4	2	no	—
277	C₂	—	274	1	no	—
278	C₂	—	274	1	no	—
279	C₂	—	548	2	no	—
280	C₂	C₂×D₄×#10587739	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#10587739
commutator	C₄×C₁₃₇	C₂³×C₄
Frattini	C₄	C₂³×#10587739
Fitting	C₂×SD₁₆×C₁₃₇	C₄
socle	C₂×C₁₃₇	C₂×C₄×D₄
prime radical	C₂×#17984109	C₂
cosocle	C₄×D₁₃₇	C₂⁴
hypercenter	C₂×SD₁₆	#10587739
hypocenter	C₁₃₇	#2957
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #22578483	C₂	semidirect
2. #22578481	C₂	semidirect
3. #22578448	C₂	semidirect
4. #22578446	C₂	semidirect
5. D₁₃₇⋊#108	C₂	semidirect
6. #22578402	C₂	semidirect
7. #22578401	C₂	semidirect
8. #22578366	C₂	semidirect
11. #17984176	C₂²	semidirect
13. #17984175	C₂²	semidirect
20. #17984159	C₂²	semidirect
21. #17984158	C₂²	semidirect
22. #17984153	C₂²	semidirect
23. #17984152	C₂²	semidirect
24. #17984142	C₂²	semidirect
25. C₈×#10587739	C₂²	semidirect
28. #17984109	C₄	semidirect
29. #17984109	C₄	semidirect
32. C₂×#14555500	C₄	semidirect
33. C₂×#14555499	C₄	semidirect
34. C₂×#14555495	C₄	semidirect
36. C₂×SD₁₆×C₁₃₇	C₄	semidirect
140. C₁₃₇	#2957	semidirect
159. C₂×SD₁₆	#10587739	semidirect

Maximal subgroups

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

subgroup	index
#22578483	2
#22578481	2
#22578448	2
#22578446	2
D₁₃₇⋊#108	2
#22578402	2
#22578401	2
#22578366	2
C₂×#17984109	2
#2957	137