Subgroups — C₁₃₇×#28823 #46878209

Order 35072.

618 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3164×C₁₃₇	C₂	1	2	yes	maximal
2	#3163×C₁₃₇	C₂	1	2	yes	maximal
3	#2061×C₁₃₇	C₂	1	4	yes	maximal
4	#1952×C₁₃₇	C₂	1	2	yes	maximal
5	#1951×C₁₃₇	C₂	1	2	yes	maximal
6	#1846×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	#1772×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	#1772×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	#544×C₁₃₇	C₂²	1	4	yes	—
10	#501×C₁₃₇	—	2	4	no	—
11	#501×C₁₃₇	—	2	4	no	—
12	#500×C₁₃₇	—	2	2	no	—
13	#499×C₁₃₇	—	2	2	no	—
14	#499×C₁₃₇	—	2	2	no	—
15	#498×C₁₃₇	—	2	2	no	—
16	C₂×#108×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×#108×C₁₃₇	C₂²	1	1	yes	characteristic
18	#428×C₁₃₇	C₂²	1	4	yes	—
19	#428×C₁₃₇	C₂²	1	4	yes	—
20	#427×C₁₃₇	C₂²	1	1	yes	characteristic
21	#427×C₁₃₇	C₂²	1	1	yes	characteristic
22	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
23	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
24	#405×C₁₃₇	C₂²	1	2	yes	—
25	#405×C₁₃₇	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
27	#382×C₁₃₇	C₂²	1	2	yes	—
28	#343×C₁₃₇	C₂²	1	4	yes	—
29	#339×C₁₃₇	C₂²	1	2	yes	—
30	#339×C₁₃₇	C₂²	1	2	yes	—
31	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
32	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
35	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
36	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
37	#125×C₁₃₇	—	2	4	no	—
38	#125×C₁₃₇	—	2	4	no	—
39	C₂²⋊Q₈×C₁₃₇	—	4	4	no	—
40	#117×C₁₃₇	C₂³	1	2	yes	—
41	#117×C₁₃₇	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
43	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
44	#108×C₁₃₇	Q₈	1	4	yes	complemented
45	#108×C₁₃₇	Q₈	1	4	yes	complemented
46	#107×C₁₃₇	—	2	2	no	—
47	#107×C₁₃₇	—	2	2	no	—
48	#107×C₁₃₇	—	2	4	no	—
49	#107×C₁₃₇	—	2	4	no	—
50	#107×C₁₃₇	D₄	1	2	yes	—
51	#107×C₁₃₇	D₄	1	2	yes	—
52	#106×C₁₃₇	—	2	2	no	—
53	#106×C₁₃₇	—	2	2	no	—
54	#106×C₁₃₇	—	2	4	no	—
55	#106×C₁₃₇	—	2	4	no	—
56	#106×C₁₃₇	D₄	1	1	yes	characteristic
57	#106×C₁₃₇	D₄	1	1	yes	characteristic
58	#106×C₁₃₇	D₄	1	2	yes	—
59	#97×C₁₃₇	—	2	2	no	—
60	#97×C₁₃₇	—	2	2	no	—
61	C₄×C₈×C₁₃₇	—	2	2	no	—
62	C₄×C₈×C₁₃₇	—	2	2	no	—
63	C₄×C₈×C₁₃₇	Q₈	1	4	yes	complemented
64	#95×C₁₃₇	—	4	2	no	—
65	C₂×Q₈×C₁₃₇	—	4	4	no	—
66	C₂²×C₄×C₁₃₇	—	2	1	no	—
67	C₂²×C₄×C₁₃₇	—	2	4	no	—
68	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
69	M₁₆×C₁₃₇	—	2	2	no	—
70	M₁₆×C₁₃₇	—	2	2	no	—
71	M₁₆×C₁₃₇	C₂×Q₈	1	4	yes	—
72	M₁₆×C₁₃₇	C₂×Q₈	1	4	yes	—
73	C₂×C₈×C₁₃₇	—	2	2	no	—
74	C₂×C₈×C₁₃₇	—	2	2	no	—
75	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
80	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
81	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
82	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
83	C₄⋊C₄×C₁₃₇	—	2	4	no	—
84	C₄⋊C₄×C₁₃₇	—	2	4	no	—
85	C₄⋊C₄×C₁₃₇	—	4	4	no	—
86	C₄⋊C₄×C₁₃₇	—	4	4	no	—
87	C₄⋊C₄×C₁₃₇	—	4	4	no	—
88	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
89	C₄⋊C₄×C₁₃₇	C₂×D₄	1	4	yes	—
90	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
91	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
92	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
93	C₄²×C₁₃₇	—	2	2	no	—
94	C₄²×C₁₃₇	—	2	2	no	—
95	C₄²×C₁₃₇	—	2	1	no	—
96	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
97	C₂³×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
98	Q₈×C₁₃₇	—	16	4	no	—
99	C₂×C₄×C₁₃₇	—	2	2	no	—
100	C₂×C₄×C₁₃₇	—	2	2	no	—
101	C₂×C₄×C₁₃₇	—	2	4	no	—
102	C₂×C₄×C₁₃₇	—	2	2	no	—
103	C₂×C₄×C₁₃₇	—	4	2	no	—
104	C₂×C₄×C₁₃₇	—	4	4	no	—
105	C₂×C₄×C₁₃₇	—	8	4	no	—
106	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
112	C₈×C₁₃₇	—	2	4	no	—
113	C₈×C₁₃₇	—	2	4	no	—
114	C₈×C₁₃₇	C₄⋊Q₈	1	2	yes	—
115	C₈×C₁₃₇	C₄⋊Q₈	1	2	yes	—
116	C₈×C₁₃₇	C₄⋊Q₈	1	2	yes	—
117	C₈×C₁₃₇	C₄⋊Q₈	1	2	yes	—
118	C₂²×C₁₃₇	—	2	2	no	—
119	C₂²×C₁₃₇	#544	1	1	yes	characteristic
120	C₂²×C₁₃₇	D₄²	1	1	yes	characteristic
121	C₂²×C₁₃₇	Q₈²	1	1	yes	characteristic
122	C₄×C₁₃₇	—	4	2	no	—
123	C₄×C₁₃₇	—	4	2	no	—
124	C₄×C₁₃₇	—	4	4	no	—
125	C₄×C₁₃₇	—	4	2	no	—
126	C₄×C₁₃₇	—	8	4	no	—
127	C₄×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
128	C₄×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
129	C₄×C₁₃₇	D₄×Q₈	1	1	yes	characteristic
130	C₄×C₁₃₇	D₄×Q₈	1	1	yes	characteristic
131	C₂×C₁₃₇	—	2	1	no	—
132	C₂×C₁₃₇	—	2	1	no	—
133	C₂×C₁₃₇	#2551	1	1	yes	characteristic
134	C₂×C₁₃₇	#3290	1	2	yes	—
135	#28823	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₁₃₇	#28823	1	1	yes	characteristic, direct product factor, Sylow
137	#3164	C₂×C₁₃₇	1	2	yes	—
138	#3163	C₂×C₁₃₇	1	2	yes	—
139	#2061	C₂×C₁₃₇	1	4	yes	—
140	#1952	C₂×C₁₃₇	1	2	yes	—
141	#1951	C₂×C₁₃₇	1	2	yes	—
142	#1846	C₂×C₁₃₇	1	1	yes	characteristic
143	#1772	C₂×C₁₃₇	1	1	yes	characteristic
144	#1772	C₂×C₁₃₇	1	1	yes	characteristic
145	#544	C₂²×C₁₃₇	1	4	yes	—
146	#501	—	2	4	no	—
147	#501	—	2	4	no	—
148	#500	—	2	2	no	—
149	#499	—	2	2	no	—
150	#499	—	2	2	no	—
151	#498	—	2	2	no	—
152	C₂×#108	C₂²×C₁₃₇	1	1	yes	characteristic
153	C₂×#108	C₂²×C₁₃₇	1	1	yes	characteristic
154	#428	C₂²×C₁₃₇	1	4	yes	—
155	#428	C₂²×C₁₃₇	1	4	yes	—
156	#427	C₂²×C₁₃₇	1	1	yes	characteristic
157	#427	C₂²×C₁₃₇	1	1	yes	characteristic
158	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
159	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
160	#405	C₂²×C₁₃₇	1	2	yes	—
161	#405	C₂²×C₁₃₇	1	2	yes	—
162	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
163	#382	C₂²×C₁₃₇	1	2	yes	—
164	#343	C₂²×C₁₃₇	1	4	yes	—
165	#339	C₂²×C₁₃₇	1	2	yes	—
166	#339	C₂²×C₁₃₇	1	2	yes	—
167	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
168	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
169	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
170	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
171	C₄⋊Q₈	—	2	4	no	—
172	C₄⋊Q₈	—	2	4	no	—
173	#125	—	2	4	no	—
174	#125	—	2	4	no	—
175	C₂²⋊Q₈	—	4	4	no	—
176	#117	C₂³×C₁₃₇	1	2	yes	—
177	#117	C₂³×C₁₃₇	1	2	yes	—
178	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
179	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
180	#108	Q₈×C₁₃₇	1	4	yes	complemented
181	#108	Q₈×C₁₃₇	1	4	yes	complemented
182	#107	—	2	2	no	—
183	#107	—	2	2	no	—
184	#107	—	2	4	no	—
185	#107	—	2	4	no	—
186	#107	D₄×C₁₃₇	1	2	yes	—
187	#107	D₄×C₁₃₇	1	2	yes	—
188	#106	—	2	2	no	—
189	#106	—	2	2	no	—
190	#106	—	2	4	no	—
191	#106	—	2	4	no	—
192	#106	D₄×C₁₃₇	1	1	yes	characteristic
193	#106	D₄×C₁₃₇	1	1	yes	characteristic
194	#106	D₄×C₁₃₇	1	2	yes	—
195	#97	—	2	2	no	—
196	#97	—	2	2	no	—
197	C₄×C₈	—	2	2	no	—
198	C₄×C₈	—	2	2	no	—
199	C₄×C₈	Q₈×C₁₃₇	1	4	yes	complemented
200	#95	—	4	2	no	—
201	C₂×Q₈	—	4	4	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
205	M₁₆	—	2	2	no	—
206	M₁₆	—	2	2	no	—
207	M₁₆	C₂×Q₈×C₁₃₇	1	4	yes	—
208	M₁₆	C₂×Q₈×C₁₃₇	1	4	yes	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
212	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
213	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
214	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
215	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
216	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
217	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
218	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	4	4	no	—
222	C₄⋊C₄	—	4	4	no	—
223	C₄⋊C₄	—	4	4	no	—
224	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
225	C₄⋊C₄	C₂×D₄×C₁₃₇	1	4	yes	—
226	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
227	C₂²⋊C₄	—	2	2	no	—
228	C₂²⋊C₄	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	1	no	—
232	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
233	C₂³	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
234	Q₈	—	16	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	8	4	no	—
242	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
243	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
244	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
245	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
246	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
247	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
248	C₈	—	2	4	no	—
249	C₈	—	2	4	no	—
250	C₈	C₄⋊Q₈×C₁₃₇	1	2	yes	—
251	C₈	C₄⋊Q₈×C₁₃₇	1	2	yes	—
252	C₈	C₄⋊Q₈×C₁₃₇	1	2	yes	—
253	C₈	C₄⋊Q₈×C₁₃₇	1	2	yes	—
254	C₂²	—	2	2	no	—
255	C₂²	#544×C₁₃₇	1	1	yes	characteristic
256	C₂²	D₄²×C₁₃₇	1	1	yes	characteristic
257	C₂²	Q₈²×C₁₃₇	1	1	yes	characteristic
258	C₄	—	4	2	no	—
259	C₄	—	4	2	no	—
260	C₄	—	4	4	no	—
261	C₄	—	4	2	no	—
262	C₄	—	8	4	no	—
263	C₄	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
264	C₄	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
265	C₄	D₄×Q₈×C₁₃₇	1	1	yes	characteristic
266	C₄	D₄×Q₈×C₁₃₇	1	1	yes	characteristic
267	C₂	—	2	1	no	—
268	C₂	—	2	1	no	—
269	C₂	#2551×C₁₃₇	1	1	yes	characteristic
270	C₂	#3290×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₇	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂³×C₁₃₇	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂²×C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
44. #108×C₁₃₇	Q₈	semidirect
45. #108×C₁₃₇	Q₈	semidirect
63. C₄×C₈×C₁₃₇	Q₈	semidirect
135. #28823	C₁₃₇	direct (N × H)
136. C₁₃₇	#28823	direct (N × H)
180. #108	Q₈×C₁₃₇	semidirect
181. #108	Q₈×C₁₃₇	semidirect
199. C₄×C₈	Q₈×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3164×C₁₃₇	2
#3163×C₁₃₇	2
#2061×C₁₃₇	2
#1952×C₁₃₇	2
#1951×C₁₃₇	2
#1846×C₁₃₇	2
#1772×C₁₃₇	2
#28823	137

Subgroups — C137×#28823 #46878209

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28823 #46878209