Subgroups — C₁₃₇×#9774 #46859160

Order 35072.

458 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#486×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1848×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#1549×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#1548×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#1545×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	#1545×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	#1452×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂×#127×C₁₃₇	C₂²	1	1	yes	characteristic
10	#486×C₁₃₇	—	2	2	no	—
11	#486×C₁₃₇	C₄	1	4	yes	—
12	#431×C₁₃₇	C₂²	1	1	yes	characteristic
13	#431×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×#108×C₁₃₇	—	2	2	no	—
15	#423×C₁₃₇	C₂²	1	1	yes	characteristic
16	#423×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×#102×C₁₃₇	—	2	2	no	—
18	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
19	#333×C₁₃₇	—	2	2	no	—
20	#332×C₁₃₇	—	2	2	no	—
21	#331×C₁₃₇	—	2	2	no	—
22	#331×C₁₃₇	—	2	2	no	—
23	#322×C₁₃₇	—	2	2	no	—
24	C₂²×Q₈×C₁₃₇	—	2	1	no	—
25	C₂²×D₄×C₁₃₇	—	2	1	no	—
26	C₂²×D₄×C₁₃₇	—	2	1	no	—
27	C₂²×D₄×C₁₃₇	—	2	1	no	—
28	C₂×M₁₆×C₁₃₇	—	2	1	no	—
29	C₂×M₁₆×C₁₃₇	—	2	1	no	—
30	C₂²×C₈×C₁₃₇	D₄	1	2	yes	—
31	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
32	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
33	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
34	C₄⋊Q₈×C₁₃₇	C₂×C₄	1	2	yes	—
35	#127×C₁₃₇	—	2	1	no	—
36	#127×C₁₃₇	—	2	1	no	—
37	#127×C₁₃₇	—	2	1	no	—
38	#127×C₁₃₇	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
40	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
41	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
42	#108×C₁₃₇	—	4	4	no	—
43	C₄⋊C₈×C₁₃₇	—	2	1	no	—
44	C₄⋊C₈×C₁₃₇	—	2	1	no	—
45	C₄⋊C₈×C₁₃₇	—	2	1	no	—
46	C₄⋊C₈×C₁₃₇	—	2	1	no	—
47	C₄⋊C₈×C₁₃₇	D₄	1	2	yes	—
48	C₄⋊C₈×C₁₃₇	D₄	1	2	yes	—
49	#102×C₁₃₇	—	2	8	no	—
50	C₂²⋊C₈×C₁₃₇	—	2	1	no	—
51	C₂²⋊C₈×C₁₃₇	—	2	1	no	—
52	C₄×C₈×C₁₃₇	—	2	1	no	—
53	C₄×C₈×C₁₃₇	—	2	1	no	—
54	C₄×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
56	C₂⁴×C₁₃₇	—	4	1	no	—
57	C₂×Q₈×C₁₃₇	—	2	2	no	—
58	C₂×Q₈×C₁₃₇	—	4	1	no	—
59	C₂×Q₈×C₁₃₇	—	8	2	no	—
60	C₂×D₄×C₁₃₇	—	2	2	no	—
61	C₂×D₄×C₁₃₇	—	2	2	no	—
62	C₂×D₄×C₁₃₇	—	2	2	no	—
63	C₂×D₄×C₁₃₇	—	4	1	no	—
64	C₂×D₄×C₁₃₇	—	4	1	no	—
65	C₂×D₄×C₁₃₇	—	4	1	no	—
66	C₂×D₄×C₁₃₇	—	4	2	no	—
67	C₂×D₄×C₁₃₇	—	4	2	no	—
68	C₂×D₄×C₁₃₇	—	8	2	no	—
69	C₂×D₄×C₁₃₇	—	8	2	no	—
70	C₂²×C₄×C₁₃₇	—	4	1	no	—
71	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
73	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
74	M₁₆×C₁₃₇	—	4	2	no	—
75	M₁₆×C₁₃₇	—	4	2	no	—
76	C₂×C₈×C₁₃₇	—	2	2	no	—
77	C₂×C₈×C₁₃₇	—	2	2	no	—
78	C₂×C₈×C₁₃₇	—	2	2	no	—
79	C₂×C₈×C₁₃₇	—	2	2	no	—
80	C₂×C₈×C₁₃₇	—	4	1	no	—
81	C₂×C₈×C₁₃₇	—	4	1	no	—
82	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	4	yes	—
83	C₄⋊C₄×C₁₃₇	—	2	2	no	—
84	C₄⋊C₄×C₁₃₇	—	2	2	no	—
85	C₄⋊C₄×C₁₃₇	—	4	1	no	—
86	C₄⋊C₄×C₁₃₇	—	4	1	no	—
87	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
91	C₂³×C₁₃₇	—	4	2	no	—
92	C₂³×C₁₃₇	—	8	2	no	—
93	C₂³×C₁₃₇	—	8	2	no	—
94	C₂³×C₁₃₇	—	8	2	no	—
95	C₂³×C₁₃₇	#124	1	1	yes	characteristic
96	Q₈×C₁₃₇	—	8	2	no	—
97	Q₈×C₁₃₇	—	8	2	no	—
98	D₄×C₁₃₇	—	4	4	no	—
99	D₄×C₁₃₇	—	8	2	no	—
100	D₄×C₁₃₇	—	8	2	no	—
101	D₄×C₁₃₇	—	8	2	no	—
102	D₄×C₁₃₇	—	8	2	no	—
103	D₄×C₁₃₇	—	8	1	no	—
104	D₄×C₁₃₇	—	8	1	no	—
105	C₂×C₄×C₁₃₇	—	2	2	no	—
106	C₂×C₄×C₁₃₇	—	2	2	no	—
107	C₂×C₄×C₁₃₇	—	4	2	no	—
108	C₂×C₄×C₁₃₇	—	8	2	no	—
109	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
113	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
114	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
115	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
116	C₂×C₄×C₁₃₇	#137	1	2	yes	—
117	C₈×C₁₃₇	—	2	4	no	—
118	C₈×C₁₃₇	—	4	2	no	—
119	C₈×C₁₃₇	—	4	2	no	—
120	C₈×C₁₃₇	—	8	1	no	—
121	C₈×C₁₃₇	—	8	1	no	—
122	C₂²×C₁₃₇	—	2	1	no	—
123	C₂²×C₁₃₇	—	2	1	no	—
124	C₂²×C₁₃₇	—	8	2	no	—
125	C₂²×C₁₃₇	—	8	2	no	—
126	C₂²×C₁₃₇	—	8	2	no	—
127	C₂²×C₁₃₇	—	16	4	no	—
128	C₂²×C₁₃₇	#390	1	1	yes	characteristic
129	C₂²×C₁₃₇	#489	1	2	yes	—
130	C₄×C₁₃₇	—	2	1	no	—
131	C₄×C₁₃₇	—	2	1	no	—
132	C₄×C₁₃₇	—	8	2	no	—
133	C₄×C₁₃₇	#417	1	2	yes	—
134	C₄×C₁₃₇	#471	1	2	yes	—
135	C₄×C₁₃₇	#472	1	2	yes	—
136	C₄×C₁₃₇	#496	1	1	yes	characteristic
137	C₄×C₁₃₇	#497	1	1	yes	characteristic
138	C₂×C₁₃₇	—	2	2	no	—
139	C₂×C₁₃₇	—	16	2	no	—
140	C₂×C₁₃₇	#1959	1	1	yes	characteristic
141	C₂×C₁₃₇	#1977	1	1	yes	characteristic
142	C₂×C₁₃₇	#1978	1	1	yes	characteristic
143	#9774	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₁₃₇	#9774	1	1	yes	characteristic, direct product factor, Sylow
145	C₂×#486	C₂×C₁₃₇	1	1	yes	characteristic
146	#1848	C₂×C₁₃₇	1	1	yes	characteristic, complemented
147	#1549	C₂×C₁₃₇	1	1	yes	characteristic, complemented
148	#1548	C₂×C₁₃₇	1	1	yes	characteristic, complemented
149	#1545	C₂×C₁₃₇	1	1	yes	characteristic
150	#1545	C₂×C₁₃₇	1	1	yes	characteristic
151	#1452	C₂×C₁₃₇	1	1	yes	characteristic, complemented
152	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
153	C₂×#127	C₂²×C₁₃₇	1	1	yes	characteristic
154	#486	—	2	2	no	—
155	#486	C₄×C₁₃₇	1	4	yes	—
156	#431	C₂²×C₁₃₇	1	1	yes	characteristic
157	#431	C₂²×C₁₃₇	1	1	yes	characteristic
158	C₂×#108	—	2	2	no	—
159	#423	C₂²×C₁₃₇	1	1	yes	characteristic
160	#423	C₂²×C₁₃₇	1	1	yes	characteristic
161	C₂×#102	—	2	2	no	—
162	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
163	#333	—	2	2	no	—
164	#332	—	2	2	no	—
165	#331	—	2	2	no	—
166	#331	—	2	2	no	—
167	#322	—	2	2	no	—
168	C₂²×Q₈	—	2	1	no	—
169	C₂²×D₄	—	2	1	no	—
170	C₂²×D₄	—	2	1	no	—
171	C₂²×D₄	—	2	1	no	—
172	C₂×M₁₆	—	2	1	no	—
173	C₂×M₁₆	—	2	1	no	—
174	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
175	C₄⋊Q₈	—	2	1	no	—
176	C₄⋊Q₈	—	2	1	no	—
177	C₄⋊Q₈	—	2	1	no	—
178	C₄⋊Q₈	C₂×C₄×C₁₃₇	1	2	yes	—
179	#127	—	2	1	no	—
180	#127	—	2	1	no	—
181	#127	—	2	1	no	—
182	#127	C₂×C₄×C₁₃₇	1	2	yes	—
183	C₂×C₄⋊C₄	—	2	1	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
186	#108	—	4	4	no	—
187	C₄⋊C₈	—	2	1	no	—
188	C₄⋊C₈	—	2	1	no	—
189	C₄⋊C₈	—	2	1	no	—
190	C₄⋊C₈	—	2	1	no	—
191	C₄⋊C₈	D₄×C₁₃₇	1	2	yes	—
192	C₄⋊C₈	D₄×C₁₃₇	1	2	yes	—
193	#102	—	2	8	no	—
194	C₂²⋊C₈	—	2	1	no	—
195	C₂²⋊C₈	—	2	1	no	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	—	2	1	no	—
198	C₄×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
199	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
200	C₂⁴	—	4	1	no	—
201	C₂×Q₈	—	2	2	no	—
202	C₂×Q₈	—	4	1	no	—
203	C₂×Q₈	—	8	2	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	4	1	no	—
208	C₂×D₄	—	4	1	no	—
209	C₂×D₄	—	4	1	no	—
210	C₂×D₄	—	4	2	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂×D₄	—	8	2	no	—
213	C₂×D₄	—	8	2	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
216	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
217	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
218	M₁₆	—	4	2	no	—
219	M₁₆	—	4	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	4	yes	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	4	1	no	—
230	C₄⋊C₄	—	4	1	no	—
231	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
232	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
233	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
234	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
235	C₂³	—	4	2	no	—
236	C₂³	—	8	2	no	—
237	C₂³	—	8	2	no	—
238	C₂³	—	8	2	no	—
239	C₂³	#124×C₁₃₇	1	1	yes	characteristic
240	Q₈	—	8	2	no	—
241	Q₈	—	8	2	no	—
242	D₄	—	4	4	no	—
243	D₄	—	8	2	no	—
244	D₄	—	8	2	no	—
245	D₄	—	8	2	no	—
246	D₄	—	8	2	no	—
247	D₄	—	8	1	no	—
248	D₄	—	8	1	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	8	2	no	—
253	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
254	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
255	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
256	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
257	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
258	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
259	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
260	C₂×C₄	#137×C₁₃₇	1	2	yes	—
261	C₈	—	2	4	no	—
262	C₈	—	4	2	no	—
263	C₈	—	4	2	no	—
264	C₈	—	8	1	no	—
265	C₈	—	8	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	8	2	no	—
269	C₂²	—	8	2	no	—
270	C₂²	—	8	2	no	—
271	C₂²	—	16	4	no	—
272	C₂²	#390×C₁₃₇	1	1	yes	characteristic
273	C₂²	#489×C₁₃₇	1	2	yes	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	8	2	no	—
277	C₄	#417×C₁₃₇	1	2	yes	—
278	C₄	#471×C₁₃₇	1	2	yes	—
279	C₄	#472×C₁₃₇	1	2	yes	—
280	C₄	#496×C₁₃₇	1	1	yes	characteristic
281	C₄	#497×C₁₃₇	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	16	2	no	—
284	C₂	#1959×C₁₃₇	1	1	yes	characteristic
285	C₂	#1977×C₁₃₇	1	1	yes	characteristic
286	C₂	#1978×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#390
commutator	C₄²	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
prime radical	C₂×#127×C₁₃₇	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #1848×C₁₃₇	C₂	semidirect
3. #1549×C₁₃₇	C₂	semidirect
4. #1548×C₁₃₇	C₂	semidirect
7. #1452×C₁₃₇	C₂	semidirect
143. #9774	C₁₃₇	direct (N × H)
144. C₁₃₇	#9774	direct (N × H)
146. #1848	C₂×C₁₃₇	semidirect
147. #1549	C₂×C₁₃₇	semidirect
148. #1548	C₂×C₁₃₇	semidirect
151. #1452	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#486×C₁₃₇	2
#1848×C₁₃₇	2
#1549×C₁₃₇	2
#1548×C₁₃₇	2
#1545×C₁₃₇	2
#1452×C₁₃₇	2
#9774	137

Subgroups — C137×#9774 #46859160

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#9774 #46859160