Subgroups — #31797568

Order 17536.

452 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578482	C₂	1	2	yes	maximal, complemented
2	#22578450	C₂	1	2	yes	maximal, complemented
3	#22578449	C₂	1	2	yes	maximal, complemented
4	#22578446	C₂	1	4	yes	maximal, complemented
5	C₂×#17984158	C₂	1	1	yes	maximal, characteristic, complemented
6	#22578400	C₂	1	2	yes	maximal, complemented
7	#22578366	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₁₃₇	C₂	1	1	yes	maximal, characteristic
9	#17984176	C₂²	1	4	yes	complemented
10	#17984175	—	2	4	no	—
11	#17984175	C₂²	1	4	yes	complemented
12	C₂×#14555520	—	2	2	no	—
13	C₂×#14555520	C₂²	1	2	yes	—
14	D₁₃₇⋊M₁₆	—	2	2	no	—
15	D₁₃₇⋊M₁₆	C₂²	1	1	yes	characteristic
16	#17984167	C₂²	1	1	yes	characteristic
17	#17984158	C₂²	1	2	yes	complemented
18	#17984158	C₂²	1	2	yes	complemented
19	#17984153	C₂²	1	4	yes	complemented
20	#17984152	C₂²	1	4	yes	complemented
21	#17984142	C₂²	1	2	yes	complemented
22	C₈×#10587739	C₂²	1	2	yes	complemented
23	#41×D₁₃₇	C₂²	1	2	yes	—
24	#17984111	C₂²	1	1	yes	characteristic
25	#17984110	C₄	1	1	yes	characteristic, complemented
26	#17984109	C₂²	1	2	yes	—
27	#17984108	C₄	1	2	yes	complemented
28	#17984107	C₂²	1	1	yes	characteristic
29	Q₁₆×D₁₃₇	C₄	1	1	yes	characteristic, complemented
30	SD₁₆×D₁₃₇	C₄	1	2	yes	complemented
31	D₈×D₁₃₇	C₄	1	1	yes	characteristic, complemented
32	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
33	#135×C₁₃₇	C₄	1	1	yes	characteristic, complemented
34	#14555520	—	2	4	no	—
35	#14555520	—	2	2	no	—
36	#14555520	D₄	1	2	yes	—
37	#14555520	C₂³	1	4	yes	—
38	#14555519	—	2	4	no	—
39	#14555519	D₄	1	2	yes	—
40	#14555518	—	2	4	no	—
41	#14555518	C₂³	1	2	yes	—
42	C₂×#11764047	—	2	4	no	—
43	C₂×#11764047	—	2	2	no	—
44	C₄⋊#10587739	—	2	1	no	—
45	C₂²⋊#10587739	—	2	1	no	—
46	C₄×#10587739	C₂³	1	2	yes	—
47	#14555511	C₂³	1	2	yes	—
48	#14555510	—	2	2	no	—
49	#14555510	C₂×C₄	1	2	yes	—
50	#14555509	—	2	2	no	—
51	#14555509	C₂³	1	2	yes	—
52	Q₈×D₁₃₇	C₂×C₄	1	2	yes	—
53	D₄×D₁₃₇	—	2	2	no	—
54	D₄×D₁₃₇	C₂×C₄	1	2	yes	—
55	C₂×C₄×D₁₃₇	—	2	2	no	—
56	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
57	Dic₅₄₈	C₂×C₄	1	1	yes	characteristic
58	#14555500	C₂×C₄	1	2	yes	—
59	#14555499	C₂×C₄	1	2	yes	—
60	#14555498	C₂×C₄	1	2	yes	—
61	#14555497	C₂×C₄	1	2	yes	—
62	D₁₀₉₆	C₂×C₄	1	1	yes	characteristic
63	#14555495	C₂×C₄	1	2	yes	—
64	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
65	C₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic
66	C₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic
67	C₈×D₁₃₇	C₂³	1	1	yes	characteristic
68	C₈×D₁₃₇	C₂³	1	1	yes	characteristic
69	#41×C₁₃₇	C₂×C₄	1	2	yes	—
70	Q₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
71	SD₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
72	D₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
74	C₂×#10587739	—	2	2	no	—
75	#11764047	—	2	4	no	—
76	#11764047	C₂×D₄	1	4	yes	—
77	C₂²×D₁₃₇	—	2	2	no	—
78	C₂²×D₁₃₇	#41	1	1	yes	characteristic
79	Dic₂₇₄	—	2	2	no	—
80	Dic₂₇₄	C₂²×C₄	1	2	yes	—
81	D₅₄₈	—	2	2	no	—
82	D₅₄₈	C₂²×C₄	1	2	yes	—
83	C₂²⋊D₁₃₇	—	2	2	no	—
84	C₂²⋊D₁₃₇	—	2	2	no	—
85	C₂²⋊D₁₃₇	—	2	2	no	—
86	C₂×Dic₁₃₇	—	2	2	no	—
87	C₂×Dic₁₃₇	C₂×D₄	1	1	yes	characteristic
88	C₄×D₁₃₇	—	2	2	no	—
89	C₄×D₁₃₇	—	2	2	no	—
90	C₄×D₁₃₇	—	2	2	no	—
91	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
92	C₄×D₁₃₇	C₂×D₄	1	1	yes	characteristic
93	C₄×D₁₃₇	#41	1	1	yes	characteristic
94	C₄×D₁₃₇	C₂⁴	1	1	yes	characteristic
95	#11764040	C₂²×C₄	1	1	yes	characteristic
96	#11764040	C₂²×C₄	1	1	yes	characteristic
97	Q₈×C₁₃₇	C₂²×C₄	1	2	yes	—
98	D₄×C₁₃₇	—	2	2	no	—
99	D₄×C₁₃₇	C₂²×C₄	1	2	yes	—
100	C₂×C₄×C₁₃₇	—	2	2	no	—
101	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
102	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
103	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₁₃₇	—	2	2	no	—
105	C₂²×C₁₃₇	C₄×D₄	1	1	yes	characteristic
106	C₂×D₁₃₇	—	2	2	no	—
107	C₂×D₁₃₇	—	4	1	no	—
108	C₂×D₁₃₇	—	8	2	no	—
109	C₂×D₁₃₇	C₄×D₄	1	1	yes	characteristic
110	C₂×D₁₃₇	C₂×#41	1	1	yes	characteristic
111	#10587739	—	4	2	no	—
112	C₄×C₁₃₇	—	2	2	no	—
113	C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
114	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
115	Dic₁₃₇	—	2	2	no	—
116	Dic₁₃₇	C₄×D₄	1	1	yes	characteristic
117	Dic₁₃₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₁₃₇	—	2	1	no	—
119	C₂×C₁₃₇	—	4	2	no	—
120	C₂×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
121	D₁₃₇	—	2	1	no	—
122	D₁₃₇	—	2	1	no	—
123	D₁₃₇	—	4	2	no	—
124	C₁₃₇	#2957	1	1	yes	characteristic, complemented, Sylow
125	#2957	—	137	1	no	maximal, Sylow
126	C₂×#135	—	137	1	no	—
127	#568	—	137	2	no	—
128	#444	—	137	2	no	—
129	#444	—	137	2	no	—
130	#443	—	137	4	no	—
131	C₂×#108	—	137	1	no	—
132	#421	—	137	2	no	—
133	#405	—	137	1	no	—
134	C₂×#41	—	137	2	no	—
135	#135	—	137	2	no	—
136	#135	—	137	1	no	—
137	#135	—	137	2	no	—
138	#135	—	137	1	no	—
139	#135	—	137	1	no	—
140	#135	#10587739	1	1	yes	characteristic, complemented
141	C₂×Q₁₆	—	137	1	no	—
142	C₂×SD₁₆	—	137	2	no	—
143	C₂×D₈	—	137	1	no	—
144	#131	—	274	4	no	—
145	#131	—	137	4	no	—
146	#131	—	137	4	no	—
147	C₂×M₁₆	—	274	2	no	—
148	C₂×M₁₆	—	274	2	no	—
149	C₂×M₁₆	—	137	2	no	—
150	C₂×M₁₆	—	137	1	no	—
151	C₂²×C₈	—	137	1	no	—
152	#117	—	137	1	no	—
153	#108	—	137	2	no	—
154	#108	—	137	2	no	—
155	#104	—	137	4	no	—
156	#104	—	137	4	no	—
157	#97	—	137	2	no	—
158	C₄×C₈	—	137	2	no	—
159	#41	—	274	2	no	—
160	#41	—	274	2	no	—
161	#41	—	137	2	no	—
162	#41	—	137	2	no	—
163	#41	—	137	2	no	—
164	#41	C₂×#10587739	1	2	yes	—
165	C₂×Q₈	—	137	2	no	—
166	C₂×D₄	—	274	2	no	—
167	C₂×D₄	—	137	2	no	—
168	C₂²×C₄	—	274	2	no	—
169	C₂²×C₄	—	137	1	no	—
170	Q₁₆	—	137	2	no	—
171	Q₁₆	—	137	1	no	—
172	Q₁₆	C₂×#10587739	1	1	yes	characteristic
173	SD₁₆	—	137	2	no	—
174	SD₁₆	—	137	2	no	—
175	SD₁₆	—	137	2	no	—
176	SD₁₆	C₂×#10587739	1	2	yes	—
177	D₈	—	137	2	no	—
178	D₈	—	137	1	no	—
179	D₈	C₂×#10587739	1	1	yes	characteristic
180	M₁₆	—	274	2	no	—
181	M₁₆	—	274	4	no	—
182	M₁₆	—	274	4	no	—
183	M₁₆	—	137	4	no	—
184	M₁₆	—	137	2	no	—
185	M₁₆	—	137	2	no	—
186	C₂×C₈	—	274	2	no	—
187	C₂×C₈	—	274	4	no	—
188	C₂×C₈	—	274	4	no	—
189	C₂×C₈	—	137	2	no	—
190	C₂×C₈	—	137	1	no	—
191	C₂×C₈	—	137	1	no	—
192	C₂×C₈	—	137	1	no	—
193	C₂×C₈	—	137	1	no	—
194	C₂×C₈	—	137	1	no	—
195	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
196	C₄⋊C₄	—	274	1	no	—
197	C₂²⋊C₄	—	274	1	no	—
198	C₄²	—	137	2	no	—
199	C₂³	—	274	2	no	—
200	C₂³	—	137	1	no	—
201	Q₈	—	274	2	no	—
202	Q₈	—	137	2	no	—
203	Q₈	C₂²×#10587739	1	2	yes	—
204	D₄	—	2	2	no	—
205	D₄	—	274	2	no	—
206	D₄	—	274	2	no	—
207	D₄	—	274	2	no	—
208	D₄	—	274	2	no	—
209	D₄	—	137	2	no	—
210	D₄	C₂²×#10587739	1	2	yes	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	274	2	no	—
213	C₂×C₄	—	274	2	no	—
214	C₂×C₄	—	274	2	no	—
215	C₂×C₄	—	274	2	no	—
216	C₂×C₄	—	274	2	no	—
217	C₂×C₄	—	137	1	no	—
218	C₂×C₄	—	137	1	no	—
219	C₂×C₄	—	137	1	no	—
220	C₂×C₄	—	137	1	no	—
221	C₂×C₄	—	137	1	no	—
222	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
223	C₈	—	274	4	no	—
224	C₈	—	137	4	no	—
225	C₈	—	137	1	no	—
226	C₈	—	137	1	no	—
227	C₈	C₂²×#10587739	1	1	yes	characteristic
228	C₈	C₂²×#10587739	1	1	yes	characteristic
229	C₂²	—	2	2	no	—
230	C₂²	—	274	2	no	—
231	C₂²	—	548	1	no	—
232	C₂²	—	1096	2	no	—
233	C₂²	—	137	1	no	—
234	C₂²	—	137	1	no	—
235	C₂²	D₄×#10587739	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	274	2	no	—
238	C₄	—	548	2	no	—
239	C₄	—	137	1	no	—
240	C₄	—	137	1	no	—
241	C₄	D₄×#10587739	1	1	yes	characteristic
242	C₄	C₂³×#10587739	1	1	yes	characteristic
243	C₂	—	2	1	no	—
244	C₂	—	4	2	no	—
245	C₂	—	274	1	no	—
246	C₂	—	274	1	no	—
247	C₂	—	548	2	no	—
248	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	#135×C₁₃₇	C₄
socle	C₂×C₁₃₇	C₂×C₄×D₄
prime radical	#135×D₁₃₇	C₂
cosocle	C₄×D₁₃₇	C₂⁴
hypercenter	#135	#10587739
hypocenter	C₁₃₇	#2957
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #22578482	C₂	semidirect
2. #22578450	C₂	semidirect
3. #22578449	C₂	semidirect
4. #22578446	C₂	semidirect
5. C₂×#17984158	C₂	semidirect
6. #22578400	C₂	semidirect
7. #22578366	C₂	semidirect
9. #17984176	C₂²	semidirect
11. #17984175	C₂²	semidirect
17. #17984158	C₂²	semidirect
18. #17984158	C₂²	semidirect
19. #17984153	C₂²	semidirect
20. #17984152	C₂²	semidirect
21. #17984142	C₂²	semidirect
22. C₈×#10587739	C₂²	semidirect
25. #17984110	C₄	semidirect
27. #17984108	C₄	semidirect
29. Q₁₆×D₁₃₇	C₄	semidirect
30. SD₁₆×D₁₃₇	C₄	semidirect
31. D₈×D₁₃₇	C₄	semidirect
33. #135×C₁₃₇	C₄	semidirect
124. C₁₃₇	#2957	semidirect
140. #135	#10587739	semidirect

Maximal subgroups

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

subgroup	index
#22578482	2
#22578450	2
#22578449	2
#22578446	2
C₂×#17984158	2
#22578400	2
#22578366	2
#135×D₁₃₇	2
#2957	137