Subgroups — #31797472

Order 17536.

400 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578428	C₂	1	2	yes	maximal, complemented
2	#22578427	C₂	1	2	yes	maximal, complemented
3	#22578422	C₂	1	1	yes	maximal, characteristic, complemented
4	#22578421	C₂	1	1	yes	maximal, characteristic, complemented
5	#22578420	C₂	1	1	yes	maximal, characteristic, complemented
6	#22578419	C₂	1	1	yes	maximal, characteristic, complemented
7	#22578409	C₂	1	2	yes	maximal, complemented
8	#22578376	C₂	1	2	yes	maximal
9	C₄×#14555519	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄×#11764047	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#14555509	C₂	1	1	yes	maximal, characteristic
12	C₂×#14555519	C₂²	1	2	yes	—
13	C₂²×#11764047	C₂²	1	2	yes	—
14	#17984155	C₂²	1	2	yes	—
15	#17984155	C₂²	1	2	yes	—
16	C₄⋊#11764047	C₂²	1	2	yes	—
17	C₄⋊#11764047	C₂²	1	2	yes	—
18	#17984145	C₂²	1	2	yes	—
19	#17984145	C₂²	1	2	yes	—
20	C₂²⋊#11764047	C₂²	1	2	yes	—
21	C₂²⋊#11764047	C₂²	1	2	yes	—
22	#17984143	C₂²	1	2	yes	—
23	C₄×#11764047	C₂²	1	1	yes	characteristic
24	C₄×#11764047	C₂²	1	1	yes	characteristic
25	C₄×#11764047	C₂²	1	2	yes	—
26	C₄×#11764047	C₂²	1	1	yes	characteristic
27	C₄×#11764047	C₂²	1	1	yes	characteristic
28	C₂×#14555509	C₂²	1	1	yes	characteristic
29	C₄×Dic₂₇₄	C₄	1	1	yes	characteristic
30	#17984053	C₄	1	1	yes	characteristic
31	D₄×Dic₁₃₇	C₄	1	1	yes	characteristic
32	C₄×C₂²⋊D₁₃₇	C₄	1	1	yes	characteristic
33	C₄×C₂²⋊D₁₃₇	C₄	1	1	yes	characteristic
34	#17984048	C₄	1	1	yes	characteristic
35	#17984048	C₄	1	1	yes	characteristic
36	#17984046	C₂²	1	1	yes	characteristic
37	#17984044	C₂²	1	1	yes	characteristic
38	#17984044	C₂²	1	1	yes	characteristic
39	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
40	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
41	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
42	C₄×D₄×C₁₃₇	C₄	1	1	yes	characteristic
43	#14555519	—	2	4	no	—
44	C₂×#11764047	—	2	4	no	—
45	C₂×#11764047	C₂³	1	2	yes	—
46	C₂×#11764047	C₂³	1	2	yes	—
47	C₂×#11764047	C₂³	1	2	yes	—
48	C₂×#11764047	C₂³	1	2	yes	—
49	#14555509	—	2	4	no	—
50	C₂×Dic₂₇₄	C₂×C₄	1	1	yes	characteristic
51	C₂×C₂²⋊D₁₃₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₂²⋊D₁₃₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
55	C₂²×Dic₁₃₇	C₂³	1	1	yes	characteristic
56	C₂²×Dic₁₃₇	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
60	C₄⋊Dic₁₃₇	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	#14555489	C₂×C₄	1	1	yes	characteristic
65	C₂²⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
66	C₂²⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
67	#14555487	C₂×C₄	1	1	yes	characteristic
68	#14555487	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₁₃₇	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	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
77	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
78	C₂×D₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
81	C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
82	C₂²⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
83	C₂²⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
84	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
85	#11764047	—	2	2	no	—
86	#11764047	—	2	2	no	—
87	#11764047	—	2	2	no	—
88	#11764047	—	2	2	no	—
89	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
90	Dic₂₇₄	—	2	2	no	—
91	C₂²⋊D₁₃₇	—	2	2	no	—
92	C₂²⋊D₁₃₇	—	2	2	no	—
93	C₂×Dic₁₃₇	—	2	2	no	—
94	C₂×Dic₁₃₇	—	2	2	no	—
95	C₂×Dic₁₃₇	—	2	2	no	—
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	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
100	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
101	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
102	C₂×Dic₁₃₇	#41	1	4	yes	—
103	C₂×Dic₁₃₇	C₂⁴	1	1	yes	characteristic
104	C₄×D₁₃₇	—	2	2	no	—
105	C₄×D₁₃₇	—	2	2	no	—
106	C₄×D₁₃₇	—	2	2	no	—
107	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
108	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
109	D₄×C₁₃₇	—	2	2	no	—
110	C₂×C₄×C₁₃₇	—	2	2	no	—
111	C₂×C₄×C₁₃₇	—	2	2	no	—
112	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
117	C₂²×C₁₃₇	—	2	1	no	—
118	C₂²×C₁₃₇	—	2	1	no	—
119	C₂²×C₁₃₇	—	2	1	no	—
120	C₂²×C₁₃₇	—	2	1	no	—
121	C₂²×C₁₃₇	—	2	1	no	—
122	C₂²×C₁₃₇	#117	1	2	yes	—
123	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
124	C₂×D₁₃₇	—	2	1	no	—
125	C₂×D₁₃₇	—	2	1	no	—
126	C₂×D₁₃₇	—	2	1	no	—
127	C₄×C₁₃₇	—	2	1	no	—
128	C₄×C₁₃₇	—	2	1	no	—
129	C₄×C₁₃₇	—	2	1	no	—
130	C₄×C₁₃₇	—	2	1	no	—
131	C₄×C₁₃₇	#131	1	2	yes	—
132	Dic₁₃₇	—	2	1	no	—
133	Dic₁₃₇	—	2	1	no	—
134	Dic₁₃₇	—	2	1	no	—
135	Dic₁₃₇	—	2	1	no	—
136	Dic₁₃₇	#117	1	2	yes	—
137	Dic₁₃₇	#131	1	2	yes	—
138	Dic₁₃₇	C₂×#41	1	2	yes	—
139	C₂×C₁₃₇	—	2	2	no	—
140	C₂×C₁₃₇	—	4	1	no	—
141	C₂×C₁₃₇	C₂×#117	1	1	yes	characteristic
142	C₂×C₁₃₇	C₂×#131	1	1	yes	characteristic
143	C₂×C₁₃₇	C₂×#131	1	1	yes	characteristic
144	D₁₃₇	—	4	1	no	—
145	C₁₃₇	#2922	1	1	yes	characteristic, complemented, Sylow
146	#2922	—	137	1	no	maximal, Sylow
147	C₄×#41	—	137	1	no	—
148	#433	—	137	2	no	—
149	#433	—	137	2	no	—
150	#431	—	137	1	no	—
151	#431	—	137	1	no	—
152	#431	—	137	1	no	—
153	#431	—	137	1	no	—
154	#424	—	137	2	no	—
155	#408	—	137	2	no	—
156	C₄×M₁₆	—	137	1	no	—
157	C₂×C₄×C₈	—	137	1	no	—
158	C₂×#41	—	137	1	no	—
159	C₂×M₁₆	—	137	2	no	—
160	C₂²×C₈	—	137	2	no	—
161	C₄×Q₈	—	137	1	no	—
162	C₄×Q₈	—	137	1	no	—
163	C₄×D₄	—	137	1	no	—
164	C₄×D₄	—	137	1	no	—
165	C₄×D₄	—	137	1	no	—
166	C₄×D₄	—	137	1	no	—
167	C₄×D₄	—	137	1	no	—
168	C₄×D₄	#10587739	1	1	yes	characteristic
169	#117	—	137	1	no	—
170	#117	—	137	1	no	—
171	#117	—	137	1	no	—
172	C₂×C₄²	—	137	1	no	—
173	C₂×C₄²	—	137	1	no	—
174	C₂×C₄²	—	137	1	no	—
175	C₄⋊C₈	—	137	2	no	—
176	C₄⋊C₈	—	137	2	no	—
177	C₄⋊C₈	—	137	2	no	—
178	C₄⋊C₈	—	137	2	no	—
179	C₂²⋊C₈	—	137	2	no	—
180	C₂²⋊C₈	—	137	2	no	—
181	C₂²⋊C₈	—	137	2	no	—
182	C₂²⋊C₈	—	137	2	no	—
183	#97	—	137	2	no	—
184	C₄×C₈	—	137	1	no	—
185	C₄×C₈	—	137	1	no	—
186	C₄×C₈	—	137	2	no	—
187	C₄×C₈	—	137	1	no	—
188	C₄×C₈	—	137	1	no	—
189	#41	—	274	4	no	—
190	C₂×Q₈	—	137	1	no	—
191	C₂×D₄	—	137	1	no	—
192	C₂×D₄	—	137	1	no	—
193	C₂×D₄	C₂×#10587739	1	1	yes	characteristic
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	—	137	1	no	—
200	C₂²×C₄	—	137	1	no	—
201	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
202	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
203	M₁₆	—	274	4	no	—
204	C₂×C₈	—	274	4	no	—
205	C₂×C₈	—	137	2	no	—
206	C₂×C₈	—	137	2	no	—
207	C₂×C₈	—	137	2	no	—
208	C₂×C₈	—	137	2	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	C₂×#10587739	1	1	yes	characteristic
215	C₂²⋊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	C₂×#10587739	1	1	yes	characteristic
220	C₂²⋊C₄	C₂×#10587739	1	1	yes	characteristic
221	C₄²	—	137	1	no	—
222	C₄²	—	137	1	no	—
223	C₄²	—	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	—	137	1	no	—
229	C₄²	—	137	1	no	—
230	C₄²	C₂×#10587739	1	1	yes	characteristic
231	C₂³	—	137	1	no	—
232	C₂³	C₂²×#10587739	1	1	yes	characteristic
233	C₂³	C₂²×#10587739	1	1	yes	characteristic
234	Q₈	—	274	2	no	—
235	D₄	—	2	2	no	—
236	D₄	—	274	2	no	—
237	D₄	—	274	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	274	2	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	—	137	1	no	—
247	C₂×C₄	—	137	1	no	—
248	C₂×C₄	—	137	1	no	—
249	C₂×C₄	—	137	1	no	—
250	C₂×C₄	—	137	4	no	—
251	C₂×C₄	—	137	1	no	—
252	C₂×C₄	—	137	1	no	—
253	C₂×C₄	—	137	1	no	—
254	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
255	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
256	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
257	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
258	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
259	C₈	—	274	2	no	—
260	C₈	—	274	2	no	—
261	C₈	—	274	2	no	—
262	C₈	—	274	2	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	274	1	no	—
269	C₂²	—	274	1	no	—
270	C₂²	—	274	1	no	—
271	C₂²	#17984167	1	2	yes	—
272	C₂²	C₂³×#10587739	1	1	yes	characteristic
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	274	1	no	—
278	C₄	—	274	1	no	—
279	C₄	—	274	1	no	—
280	C₄	—	274	1	no	—
281	C₄	—	137	2	no	—
282	C₄	—	137	2	no	—
283	C₄	—	137	2	no	—
284	C₄	#17984175	1	2	yes	—
285	C₂	—	2	2	no	—
286	C₂	—	4	1	no	—
287	C₂	—	548	1	no	—
288	C₂	C₂×#17984167	1	1	yes	characteristic
289	C₂	C₂×#17984175	1	1	yes	characteristic
290	C₂	D₁₃₇⋊#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#10587739
commutator, socle	C₂²×C₁₃₇	C₂³×C₄
Frattini	C₂²	C₂³×#10587739
Fitting	C₄×D₄×C₁₃₇	C₄
prime radical	C₂×#14555509	C₂²
cosocle	C₂×Dic₁₃₇	C₂⁴
hypercenter	C₄×D₄	#10587739
hypocenter	C₁₃₇	#2922
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #22578428	C₂	semidirect
2. #22578427	C₂	semidirect
3. #22578422	C₂	semidirect
4. #22578421	C₂	semidirect
5. #22578420	C₂	semidirect
6. #22578419	C₂	semidirect
7. #22578409	C₂	semidirect
9. C₄×#14555519	C₂	semidirect
10. C₂×C₄×#11764047	C₂	semidirect
145. C₁₃₇	#2922	semidirect

Maximal subgroups

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

subgroup	index
#22578428	2
#22578427	2
#22578422	2
#22578421	2
#22578420	2
#22578419	2
#22578409	2
#22578376	2
C₄×#14555519	2
C₂×C₄×#11764047	2
C₄×#14555509	2
#2922	137