Subgroups — #31797566

Order 17536.

452 conjugacy classes of subgroups.

The subgroup lattice has 254 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	#22578448	C₂	1	4	yes	maximal, complemented
3	#22578447	C₂	1	2	yes	maximal, complemented
4	#22578445	C₂	1	2	yes	maximal, complemented
5	C₂×#17984159	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	#17984159	C₂²	1	2	yes	complemented
18	#17984159	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	1	yes	characteristic
37	#14555520	D₄	1	1	yes	characteristic
38	#14555520	C₂³	1	4	yes	—
39	#14555519	—	2	4	no	—
40	#14555519	D₄	1	2	yes	—
41	#14555518	—	2	4	no	—
42	#14555518	C₂³	1	2	yes	—
43	C₂×#11764047	—	2	4	no	—
44	C₂×#11764047	—	2	2	no	—
45	C₄⋊#10587739	—	2	1	no	—
46	C₂²⋊#10587739	—	2	1	no	—
47	C₄×#10587739	C₂³	1	2	yes	—
48	#14555511	C₂³	1	2	yes	—
49	#14555510	—	2	2	no	—
50	#14555510	C₂×C₄	1	2	yes	—
51	#14555509	—	2	2	no	—
52	#14555509	C₂³	1	2	yes	—
53	Q₈×D₁₃₇	C₂×C₄	1	2	yes	—
54	D₄×D₁₃₇	—	2	2	no	—
55	D₄×D₁₃₇	C₂×C₄	1	2	yes	—
56	C₂×C₄×D₁₃₇	—	2	2	no	—
57	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
58	Dic₅₄₈	C₂×C₄	1	1	yes	characteristic
59	#14555500	C₂×C₄	1	2	yes	—
60	#14555499	C₂×C₄	1	2	yes	—
61	#14555498	C₂×C₄	1	2	yes	—
62	#14555497	C₂×C₄	1	2	yes	—
63	D₁₀₉₆	C₂×C₄	1	1	yes	characteristic
64	#14555495	C₂×C₄	1	2	yes	—
65	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
66	C₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic
67	C₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic
68	C₈×D₁₃₇	C₂³	1	1	yes	characteristic
69	C₈×D₁₃₇	C₂³	1	1	yes	characteristic
70	#41×C₁₃₇	C₂×C₄	1	2	yes	—
71	Q₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
72	SD₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
73	D₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
75	C₂×#10587739	—	2	2	no	—
76	#11764047	—	2	4	no	—
77	#11764047	C₂×D₄	1	2	yes	—
78	#11764047	C₂×D₄	1	2	yes	—
79	C₂²×D₁₃₇	—	2	2	no	—
80	C₂²×D₁₃₇	#41	1	1	yes	characteristic
81	Dic₂₇₄	—	2	2	no	—
82	Dic₂₇₄	C₂²×C₄	1	2	yes	—
83	D₅₄₈	—	2	2	no	—
84	D₅₄₈	C₂²×C₄	1	2	yes	—
85	C₂²⋊D₁₃₇	—	2	2	no	—
86	C₂²⋊D₁₃₇	—	2	2	no	—
87	C₂²⋊D₁₃₇	—	2	2	no	—
88	C₂×Dic₁₃₇	—	2	2	no	—
89	C₂×Dic₁₃₇	C₂×D₄	1	1	yes	characteristic
90	C₄×D₁₃₇	—	2	2	no	—
91	C₄×D₁₃₇	—	2	2	no	—
92	C₄×D₁₃₇	—	2	2	no	—
93	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
94	C₄×D₁₃₇	C₂×D₄	1	1	yes	characteristic
95	C₄×D₁₃₇	#41	1	1	yes	characteristic
96	C₄×D₁₃₇	C₂⁴	1	1	yes	characteristic
97	#11764040	C₂²×C₄	1	1	yes	characteristic
98	#11764040	C₂²×C₄	1	1	yes	characteristic
99	Q₈×C₁₃₇	C₂²×C₄	1	2	yes	—
100	D₄×C₁₃₇	—	2	2	no	—
101	D₄×C₁₃₇	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₁₃₇	—	2	2	no	—
103	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
104	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
105	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₁₃₇	—	2	2	no	—
107	C₂²×C₁₃₇	C₄×D₄	1	1	yes	characteristic
108	C₂×D₁₃₇	—	2	2	no	—
109	C₂×D₁₃₇	—	4	1	no	—
110	C₂×D₁₃₇	—	8	2	no	—
111	C₂×D₁₃₇	C₄×D₄	1	1	yes	characteristic
112	C₂×D₁₃₇	C₂×#41	1	1	yes	characteristic
113	#10587739	—	4	2	no	—
114	C₄×C₁₃₇	—	2	2	no	—
115	C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
116	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
117	Dic₁₃₇	—	2	2	no	—
118	Dic₁₃₇	C₄×D₄	1	1	yes	characteristic
119	Dic₁₃₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₁₃₇	—	2	1	no	—
121	C₂×C₁₃₇	—	4	2	no	—
122	C₂×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
123	D₁₃₇	—	2	1	no	—
124	D₁₃₇	—	2	1	no	—
125	D₁₃₇	—	4	2	no	—
126	C₁₃₇	#2957	1	1	yes	characteristic, complemented, Sylow
127	#2957	—	137	1	no	maximal, Sylow
128	C₂×#135	—	137	1	no	—
129	#568	—	137	2	no	—
130	#444	—	137	4	no	—
131	#443	—	137	2	no	—
132	#443	—	137	2	no	—
133	C₂×#108	—	137	1	no	—
134	#421	—	137	2	no	—
135	#405	—	137	1	no	—
136	C₂×#41	—	137	2	no	—
137	#135	—	137	2	no	—
138	#135	—	137	1	no	—
139	#135	—	137	2	no	—
140	#135	—	137	1	no	—
141	#135	—	137	1	no	—
142	#135	#10587739	1	1	yes	characteristic, complemented
143	C₂×Q₁₆	—	137	1	no	—
144	C₂×SD₁₆	—	137	2	no	—
145	C₂×D₈	—	137	1	no	—
146	#131	—	274	4	no	—
147	#131	—	137	4	no	—
148	#131	—	137	4	no	—
149	C₂×M₁₆	—	274	2	no	—
150	C₂×M₁₆	—	274	2	no	—
151	C₂×M₁₆	—	137	2	no	—
152	C₂×M₁₆	—	137	1	no	—
153	C₂²×C₈	—	137	1	no	—
154	#117	—	137	1	no	—
155	#108	—	137	2	no	—
156	#108	—	137	2	no	—
157	#104	—	137	4	no	—
158	#104	—	137	4	no	—
159	#97	—	137	2	no	—
160	C₄×C₈	—	137	2	no	—
161	#41	—	274	2	no	—
162	#41	—	274	2	no	—
163	#41	—	137	2	no	—
164	#41	—	137	2	no	—
165	#41	—	137	2	no	—
166	#41	C₂×#10587739	1	2	yes	—
167	C₂×Q₈	—	137	2	no	—
168	C₂×D₄	—	274	2	no	—
169	C₂×D₄	—	137	2	no	—
170	C₂²×C₄	—	274	2	no	—
171	C₂²×C₄	—	137	1	no	—
172	Q₁₆	—	137	2	no	—
173	Q₁₆	—	137	1	no	—
174	Q₁₆	C₂×#10587739	1	1	yes	characteristic
175	SD₁₆	—	137	2	no	—
176	SD₁₆	—	137	2	no	—
177	SD₁₆	—	137	2	no	—
178	SD₁₆	C₂×#10587739	1	2	yes	—
179	D₈	—	137	2	no	—
180	D₈	—	137	1	no	—
181	D₈	C₂×#10587739	1	1	yes	characteristic
182	M₁₆	—	274	2	no	—
183	M₁₆	—	274	4	no	—
184	M₁₆	—	274	4	no	—
185	M₁₆	—	137	4	no	—
186	M₁₆	—	137	2	no	—
187	M₁₆	—	137	1	no	—
188	M₁₆	—	137	1	no	—
189	C₂×C₈	—	274	2	no	—
190	C₂×C₈	—	274	4	no	—
191	C₂×C₈	—	274	4	no	—
192	C₂×C₈	—	137	2	no	—
193	C₂×C₈	—	137	1	no	—
194	C₂×C₈	—	137	1	no	—
195	C₂×C₈	—	137	1	no	—
196	C₂×C₈	—	137	1	no	—
197	C₂×C₈	—	137	1	no	—
198	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
199	C₄⋊C₄	—	274	1	no	—
200	C₂²⋊C₄	—	274	1	no	—
201	C₄²	—	137	2	no	—
202	C₂³	—	274	2	no	—
203	C₂³	—	137	1	no	—
204	Q₈	—	274	2	no	—
205	Q₈	—	137	2	no	—
206	Q₈	C₂²×#10587739	1	2	yes	—
207	D₄	—	2	2	no	—
208	D₄	—	274	2	no	—
209	D₄	—	274	2	no	—
210	D₄	—	274	2	no	—
211	D₄	—	274	2	no	—
212	D₄	—	137	2	no	—
213	D₄	C₂²×#10587739	1	2	yes	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	274	2	no	—
216	C₂×C₄	—	274	2	no	—
217	C₂×C₄	—	274	2	no	—
218	C₂×C₄	—	274	2	no	—
219	C₂×C₄	—	274	2	no	—
220	C₂×C₄	—	137	1	no	—
221	C₂×C₄	—	137	1	no	—
222	C₂×C₄	—	137	1	no	—
223	C₂×C₄	—	137	1	no	—
224	C₂×C₄	—	137	1	no	—
225	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
226	C₈	—	274	4	no	—
227	C₈	—	137	2	no	—
228	C₈	—	137	2	no	—
229	C₈	—	137	1	no	—
230	C₈	—	137	1	no	—
231	C₈	C₂²×#10587739	1	1	yes	characteristic
232	C₈	C₂²×#10587739	1	1	yes	characteristic
233	C₂²	—	2	2	no	—
234	C₂²	—	274	2	no	—
235	C₂²	—	548	1	no	—
236	C₂²	—	1096	2	no	—
237	C₂²	—	137	1	no	—
238	C₂²	—	137	1	no	—
239	C₂²	D₄×#10587739	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	274	2	no	—
242	C₄	—	548	2	no	—
243	C₄	—	137	1	no	—
244	C₄	—	137	1	no	—
245	C₄	D₄×#10587739	1	1	yes	characteristic
246	C₄	C₂³×#10587739	1	1	yes	characteristic
247	C₂	—	2	1	no	—
248	C₂	—	4	2	no	—
249	C₂	—	274	1	no	—
250	C₂	—	274	1	no	—
251	C₂	—	548	2	no	—
252	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. #22578448	C₂	semidirect
3. #22578447	C₂	semidirect
4. #22578445	C₂	semidirect
5. C₂×#17984159	C₂	semidirect
6. #22578400	C₂	semidirect
7. #22578366	C₂	semidirect
9. #17984176	C₂²	semidirect
11. #17984175	C₂²	semidirect
17. #17984159	C₂²	semidirect
18. #17984159	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
126. C₁₃₇	#2957	semidirect
142. #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
#22578448	2
#22578447	2
#22578445	2
C₂×#17984159	2
#22578400	2
#22578366	2
#135×D₁₃₇	2
#2957	137