Subgroups — #31797466

Order 17536.

484 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578425	C₂	1	2	yes	maximal, complemented
2	#22578424	C₂	1	2	yes	maximal, complemented
3	#22578422	C₂	1	2	yes	maximal, complemented
4	#22578419	C₂	1	2	yes	maximal, complemented
5	#22578407	C₂	1	2	yes	maximal, complemented
6	#22578374	C₂	1	2	yes	maximal
7	C₄×#14555520	C₂	1	2	yes	maximal, complemented
8	C₂×C₄²×D₁₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#14555520	C₂²	1	4	yes	—
10	C₄⋊#11764047	C₂²	1	4	yes	complemented
11	C₄⋊#11764047	C₂²	1	4	yes	complemented
12	#17984145	C₂²	1	4	yes	—
13	C₂²⋊#11764047	C₂²	1	4	yes	—
14	#17984143	C₂²	1	4	yes	complemented
15	C₄×#11764047	C₂²	1	4	yes	complemented
16	C₂²×C₄×D₁₃₇	C₄	1	1	yes	characteristic
17	C₂²×C₄×D₁₃₇	C₄	1	1	yes	characteristic
18	C₂²×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×Dic₁₃₇	C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
22	C₄²×D₁₃₇	C₄	1	1	yes	characteristic
23	C₄²×D₁₃₇	C₄	1	1	yes	characteristic
24	C₄²×D₁₃₇	C₄	1	1	yes	characteristic
25	C₄²×D₁₃₇	C₄	1	1	yes	characteristic
26	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
27	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
28	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
29	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₁₃₇	C₄	1	1	yes	characteristic
31	#14555520	—	2	8	no	—
32	C₂×#11764047	C₂³	1	8	yes	—
33	C₂³×D₁₃₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
36	C₂²×Dic₁₃₇	C₂³	1	1	yes	characteristic
37	C₂×C₄×D₁₃₇	—	2	4	no	—
38	C₂×C₄×D₁₃₇	—	2	4	no	—
39	C₂×C₄×D₁₃₇	—	2	4	no	—
40	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
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	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	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
53	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
54	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
55	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
56	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
57	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
58	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
59	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
60	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
61	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
62	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
63	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
67	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
68	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
69	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
70	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
71	#11764047	—	2	8	no	—
72	C₂²×D₁₃₇	—	2	2	no	—
73	C₂²×D₁₃₇	—	2	2	no	—
74	C₂²×D₁₃₇	—	2	2	no	—
75	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×Dic₁₃₇	—	2	2	no	—
78	C₂×Dic₁₃₇	—	2	2	no	—
79	C₂×Dic₁₃₇	—	2	2	no	—
80	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₁₃₇	C₂⁴	1	1	yes	characteristic
86	C₄×D₁₃₇	—	2	2	no	—
87	C₄×D₁₃₇	—	2	2	no	—
88	C₄×D₁₃₇	—	2	2	no	—
89	C₄×D₁₃₇	—	2	2	no	—
90	C₄×D₁₃₇	—	2	2	no	—
91	C₄×D₁₃₇	—	2	2	no	—
92	C₄×D₁₃₇	—	2	2	no	—
93	C₄×D₁₃₇	—	2	2	no	—
94	C₄×D₁₃₇	—	2	2	no	—
95	C₄×D₁₃₇	—	2	2	no	—
96	C₄×D₁₃₇	M₁₆	1	4	yes	complemented
97	C₄×D₁₃₇	#41	1	4	yes	—
98	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃₇	—	2	2	no	—
100	C₂×C₄×C₁₃₇	—	2	2	no	—
101	C₂×C₄×C₁₃₇	M₁₆	1	4	yes	complemented
102	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₁₃₇	—	2	1	no	—
109	C₂²×C₁₃₇	—	2	1	no	—
110	C₂²×C₁₃₇	C₂×M₁₆	1	2	yes	—
111	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×D₁₃₇	—	2	8	no	—
113	C₂×D₁₃₇	—	2	1	no	—
114	C₂×D₁₃₇	—	2	1	no	—
115	C₂×D₁₃₇	—	2	1	no	—
116	C₂×D₁₃₇	—	2	1	no	—
117	C₂×D₁₃₇	#117	1	2	yes	—
118	C₂×D₁₃₇	C₂×M₁₆	1	2	yes	—
119	C₄×C₁₃₇	—	2	1	no	—
120	C₄×C₁₃₇	—	2	1	no	—
121	C₄×C₁₃₇	#117	1	2	yes	—
122	C₄×C₁₃₇	C₂×M₁₆	1	2	yes	—
123	C₄×C₁₃₇	C₂×M₁₆	1	2	yes	—
124	C₄×C₁₃₇	C₂×M₁₆	1	2	yes	—
125	Dic₁₃₇	—	2	1	no	—
126	Dic₁₃₇	—	2	1	no	—
127	Dic₁₃₇	—	2	1	no	—
128	Dic₁₃₇	—	2	1	no	—
129	Dic₁₃₇	C₂×M₁₆	1	2	yes	—
130	Dic₁₃₇	C₂×#41	1	2	yes	—
131	C₂×C₁₃₇	—	2	2	no	—
132	C₂×C₁₃₇	C₂×#117	1	1	yes	characteristic
133	C₂×C₁₃₇	C₂²×M₁₆	1	1	yes	characteristic
134	C₂×C₁₃₇	C₂²×M₁₆	1	1	yes	characteristic
135	D₁₃₇	—	2	2	no	—
136	D₁₃₇	—	2	2	no	—
137	C₁₃₇	#2920	1	1	yes	characteristic, complemented, Sylow
138	#2920	—	137	1	no	maximal, Sylow
139	C₂²×C₄²	—	137	1	no	—
140	#432	—	137	2	no	—
141	#432	—	137	2	no	—
142	#431	—	137	2	no	—
143	#431	—	137	2	no	—
144	#423	—	137	2	no	—
145	#407	—	137	2	no	—
146	C₄×M₁₆	—	137	2	no	—
147	C₂³×C₄	—	137	1	no	—
148	C₂³×C₄	—	137	1	no	—
149	C₂³×C₄	—	137	1	no	—
150	C₂×M₁₆	—	137	4	no	—
151	C₂×C₄²	—	137	1	no	—
152	C₂×C₄²	—	137	1	no	—
153	C₂×C₄²	—	137	1	no	—
154	C₂×C₄²	—	137	1	no	—
155	C₂×C₄²	—	137	1	no	—
156	C₂×C₄²	—	137	1	no	—
157	C₂×C₄²	—	137	1	no	—
158	C₂×C₄²	—	137	1	no	—
159	C₂×C₄²	—	137	1	no	—
160	C₂×C₄²	—	137	1	no	—
161	C₂×C₄²	—	137	1	no	—
162	C₂×C₄²	#10587739	1	1	yes	characteristic
163	C₄⋊C₈	—	137	4	no	—
164	C₄⋊C₈	—	137	4	no	—
165	C₂²⋊C₈	—	137	4	no	—
166	C₂²⋊C₈	—	137	4	no	—
167	#97	—	137	4	no	—
168	C₄×C₈	—	137	4	no	—
169	C₂⁴	—	137	1	no	—
170	C₂²×C₄	—	274	4	no	—
171	C₂²×C₄	—	274	4	no	—
172	C₂²×C₄	—	274	4	no	—
173	C₂²×C₄	—	137	1	no	—
174	C₂²×C₄	—	137	1	no	—
175	C₂²×C₄	—	137	1	no	—
176	C₂²×C₄	—	137	1	no	—
177	C₂²×C₄	—	137	1	no	—
178	C₂²×C₄	—	137	1	no	—
179	C₂²×C₄	—	137	1	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	1	no	—
185	C₂²×C₄	—	137	1	no	—
186	C₂²×C₄	—	137	1	no	—
187	C₂²×C₄	—	137	1	no	—
188	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
189	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
190	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
191	M₁₆	—	274	8	no	—
192	C₂×C₈	—	137	8	no	—
193	C₄²	—	137	1	no	—
194	C₄²	—	137	1	no	—
195	C₄²	—	137	1	no	—
196	C₄²	—	137	1	no	—
197	C₄²	—	137	1	no	—
198	C₄²	—	137	1	no	—
199	C₄²	—	137	1	no	—
200	C₄²	—	137	1	no	—
201	C₄²	—	137	1	no	—
202	C₄²	—	137	1	no	—
203	C₄²	—	137	1	no	—
204	C₄²	—	137	1	no	—
205	C₄²	C₂×#10587739	1	1	yes	characteristic
206	C₄²	C₂×#10587739	1	1	yes	characteristic
207	C₄²	C₂×#10587739	1	1	yes	characteristic
208	C₄²	C₂×#10587739	1	1	yes	characteristic
209	C₂³	—	274	2	no	—
210	C₂³	—	274	2	no	—
211	C₂³	—	274	2	no	—
212	C₂³	—	137	1	no	—
213	C₂³	—	137	1	no	—
214	C₂³	C₂²×#10587739	1	1	yes	characteristic
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	274	2	no	—
218	C₂×C₄	—	274	2	no	—
219	C₂×C₄	—	274	2	no	—
220	C₂×C₄	—	274	2	no	—
221	C₂×C₄	—	274	2	no	—
222	C₂×C₄	—	274	2	no	—
223	C₂×C₄	—	274	2	no	—
224	C₂×C₄	—	274	2	no	—
225	C₂×C₄	—	274	2	no	—
226	C₂×C₄	—	274	2	no	—
227	C₂×C₄	—	274	2	no	—
228	C₂×C₄	—	274	2	no	—
229	C₂×C₄	—	274	2	no	—
230	C₂×C₄	—	137	1	no	—
231	C₂×C₄	—	137	1	no	—
232	C₂×C₄	—	137	1	no	—
233	C₂×C₄	—	137	1	no	—
234	C₂×C₄	—	137	1	no	—
235	C₂×C₄	—	137	1	no	—
236	C₂×C₄	—	137	4	no	—
237	C₂×C₄	—	137	4	no	—
238	C₂×C₄	#14555520	1	4	yes	complemented
239	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
240	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
241	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
242	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
243	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
244	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
245	C₈	—	274	8	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	274	8	no	—
249	C₂²	—	274	1	no	—
250	C₂²	—	274	1	no	—
251	C₂²	—	274	1	no	—
252	C₂²	—	274	1	no	—
253	C₂²	—	137	2	no	—
254	C₂²	—	137	2	no	—
255	C₂²	C₂×#14555520	1	2	yes	—
256	C₂²	C₂³×#10587739	1	1	yes	characteristic
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	274	1	no	—
260	C₄	—	274	1	no	—
261	C₄	—	274	1	no	—
262	C₄	—	274	1	no	—
263	C₄	—	137	2	no	—
264	C₄	—	137	2	no	—
265	C₄	#17984167	1	2	yes	—
266	C₄	D₁₃₇⋊M₁₆	1	2	yes	—
267	C₄	C₂×#14555520	1	2	yes	—
268	C₄	C₂×#14555520	1	2	yes	—
269	C₂	—	2	2	no	—
270	C₂	—	274	2	no	—
271	C₂	—	274	2	no	—
272	C₂	C₂×#17984167	1	1	yes	characteristic
273	C₂	C₂×D₁₃₇⋊M₁₆	1	1	yes	characteristic
274	C₂	C₂²×#14555520	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#10587739
commutator, socle	C₂²×C₁₃₇	C₂³×C₄
Frattini	C₂²	C₂³×#10587739
Fitting	C₂×C₄²×C₁₃₇	C₄
prime radical	C₂³×D₁₃₇	C₂×C₄
cosocle	C₂×Dic₁₃₇	C₂⁴
hypercenter	C₂×C₄²	#10587739
hypocenter	C₁₃₇	#2920
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. #22578425	C₂	semidirect
2. #22578424	C₂	semidirect
3. #22578422	C₂	semidirect
4. #22578419	C₂	semidirect
5. #22578407	C₂	semidirect
7. C₄×#14555520	C₂	semidirect
10. C₄⋊#11764047	C₂²	semidirect
11. C₄⋊#11764047	C₂²	semidirect
14. #17984143	C₂²	semidirect
15. C₄×#11764047	C₂²	semidirect
96. C₄×D₁₃₇	M₁₆	semidirect
101. C₂×C₄×C₁₃₇	M₁₆	semidirect
137. C₁₃₇	#2920	semidirect
238. C₂×C₄	#14555520	semidirect

Maximal subgroups

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

subgroup	index
#22578425	2
#22578424	2
#22578422	2
#22578419	2
#22578407	2
#22578374	2
C₄×#14555520	2
C₂×C₄²×D₁₃₇	2
#2920	137