Subgroups — C₁₃₇×#28829 #46878215

Order 35072.

678 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2552×C₁₃₇	C₂	1	2	yes	maximal
2	#2084×C₁₃₇	C₂	1	4	yes	maximal
3	#1948×C₁₃₇	C₂	1	2	yes	maximal
4	#1941×C₁₃₇	C₂	1	2	yes	maximal
5	#1844×C₁₃₇	C₂	1	2	yes	maximal
6	#1844×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	C₄×#106×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₈×C₄⋊C₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#125×C₁₃₇	—	2	2	no	—
10	C₂×#106×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×#106×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
13	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
14	C₂×#106×C₁₃₇	C₂²	1	4	yes	—
15	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	#398×C₁₃₇	—	2	4	no	—
19	#389×C₁₃₇	—	2	2	no	—
20	#389×C₁₃₇	C₂²	1	4	yes	—
21	#384×C₁₃₇	C₂²	1	4	yes	—
22	#384×C₁₃₇	C₂²	1	2	yes	—
23	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
25	#340×C₁₃₇	C₂²	1	2	yes	—
26	#340×C₁₃₇	C₂²	1	4	yes	—
27	#336×C₁₃₇	C₂²	1	2	yes	—
28	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
29	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
31	#125×C₁₃₇	—	4	8	no	—
32	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
33	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
34	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
35	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
36	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
38	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
39	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
41	C₂×C₄²×C₁₃₇	—	2	2	no	—
42	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
43	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
44	#106×C₁₃₇	—	2	2	no	—
45	#106×C₁₃₇	—	2	2	no	—
46	#106×C₁₃₇	—	2	4	no	—
47	#106×C₁₃₇	—	4	2	no	—
48	#106×C₁₃₇	—	4	4	no	—
49	C₄⋊C₈×C₁₃₇	Q₈	1	4	yes	—
50	C₄×C₈×C₁₃₇	—	2	4	no	—
51	C₄×C₈×C₁₃₇	Q₈	1	4	yes	—
52	#95×C₁₃₇	—	2	2	no	—
53	#95×C₁₃₇	—	2	4	no	—
54	#95×C₁₃₇	D₄	1	2	yes	—
55	C₂²×C₄×C₁₃₇	—	2	2	no	—
56	C₂²×C₄×C₁₃₇	—	2	2	no	—
57	C₂²×C₄×C₁₃₇	—	2	4	no	—
58	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃₇	C₂×D₄	1	4	yes	—
60	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
62	C₂×C₈×C₁₃₇	—	2	2	no	—
63	C₂×C₈×C₁₃₇	—	2	1	no	—
64	C₂×C₈×C₁₃₇	—	2	2	no	—
65	C₂×C₈×C₁₃₇	—	2	1	no	—
66	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
67	C₂×C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
68	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
69	C₂×C₈×C₁₃₇	#41	1	2	yes	—
70	C₂×C₈×C₁₃₇	#41	1	2	yes	—
71	C₄⋊C₄×C₁₃₇	—	2	2	no	—
72	C₄⋊C₄×C₁₃₇	—	2	2	no	—
73	C₄⋊C₄×C₁₃₇	—	2	4	no	—
74	C₄⋊C₄×C₁₃₇	—	2	2	no	—
75	C₄⋊C₄×C₁₃₇	—	2	4	no	—
76	C₄⋊C₄×C₁₃₇	—	2	2	no	—
77	C₄⋊C₄×C₁₃₇	—	4	8	no	—
78	C₄⋊C₄×C₁₃₇	—	4	8	no	—
79	C₄⋊C₄×C₁₃₇	—	4	4	no	—
80	C₄⋊C₄×C₁₃₇	—	8	4	no	—
81	C₄⋊C₄×C₁₃₇	—	8	2	no	—
82	C₄⋊C₄×C₁₃₇	SD₁₆	1	8	yes	—
83	C₄²×C₁₃₇	—	2	2	no	—
84	C₄²×C₁₃₇	—	2	2	no	—
85	C₄²×C₁₃₇	—	4	4	no	—
86	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
87	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
88	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃₇	—	2	4	no	—
90	C₂×C₄×C₁₃₇	—	2	4	no	—
91	C₂×C₄×C₁₃₇	—	2	2	no	—
92	C₂×C₄×C₁₃₇	—	4	2	no	—
93	C₂×C₄×C₁₃₇	—	4	2	no	—
94	C₂×C₄×C₁₃₇	—	4	4	no	—
95	C₂×C₄×C₁₃₇	—	4	2	no	—
96	C₂×C₄×C₁₃₇	—	4	2	no	—
97	C₂×C₄×C₁₃₇	—	4	4	no	—
98	C₂×C₄×C₁₃₇	—	4	2	no	—
99	C₂×C₄×C₁₃₇	—	4	4	no	—
100	C₂×C₄×C₁₃₇	—	4	2	no	—
101	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
102	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
103	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	8	yes	—
104	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	4	yes	—
105	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
110	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
111	C₈×C₁₃₇	—	2	2	no	—
112	C₈×C₁₃₇	—	2	4	no	—
113	C₈×C₁₃₇	#125	1	4	yes	complemented
114	C₂²×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
115	C₂²×C₁₃₇	#547	1	1	yes	characteristic
116	C₂²×C₁₃₇	#550	1	1	yes	characteristic
117	C₂²×C₁₃₇	C₂²×SD₁₆	1	2	yes	—
118	C₂²×C₁₃₇	#577	1	2	yes	—
119	C₄×C₁₃₇	—	2	8	no	—
120	C₄×C₁₃₇	—	2	2	no	—
121	C₄×C₁₃₇	—	8	2	no	—
122	C₄×C₁₃₇	—	8	2	no	—
123	C₄×C₁₃₇	—	8	4	no	—
124	C₄×C₁₃₇	#498	1	4	yes	—
125	C₄×C₁₃₇	C₂×#125	1	1	yes	characteristic
126	C₄×C₁₃₇	#544	1	1	yes	characteristic
127	C₄×C₁₃₇	#554	1	1	yes	characteristic
128	C₄×C₁₃₇	#557	1	1	yes	characteristic
129	C₂×C₁₃₇	#2552	1	1	yes	characteristic
130	C₂×C₁₃₇	C₂×#498	1	1	yes	characteristic
131	C₂×C₁₃₇	#3163	1	1	yes	characteristic
132	C₂×C₁₃₇	#3335	1	2	yes	—
133	C₂×C₁₃₇	#3392	1	2	yes	—
134	#28829	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₁₃₇	#28829	1	1	yes	characteristic, direct product factor, Sylow
136	#2552	C₂×C₁₃₇	1	2	yes	—
137	#2084	C₂×C₁₃₇	1	4	yes	—
138	#1948	C₂×C₁₃₇	1	2	yes	—
139	#1941	C₂×C₁₃₇	1	2	yes	—
140	#1844	C₂×C₁₃₇	1	2	yes	—
141	#1844	C₂×C₁₃₇	1	1	yes	characteristic
142	C₄×#106	C₂×C₁₃₇	1	1	yes	characteristic
143	C₈×C₄⋊C₄	C₂×C₁₃₇	1	1	yes	characteristic
144	C₂×#125	—	2	2	no	—
145	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
146	C₂×#106	C₂²×C₁₃₇	1	1	yes	characteristic
147	C₂×#106	C₂²×C₁₃₇	1	1	yes	characteristic
148	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
149	C₂×#106	C₂²×C₁₃₇	1	4	yes	—
150	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
151	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
152	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
153	#398	—	2	4	no	—
154	#389	—	2	2	no	—
155	#389	C₂²×C₁₃₇	1	4	yes	—
156	#384	C₂²×C₁₃₇	1	4	yes	—
157	#384	C₂²×C₁₃₇	1	2	yes	—
158	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
159	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
160	#340	C₂²×C₁₃₇	1	2	yes	—
161	#340	C₂²×C₁₃₇	1	4	yes	—
162	#336	C₂²×C₁₃₇	1	2	yes	—
163	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
164	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
165	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
166	#125	—	4	8	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	—	2	4	no	—
169	C₂×C₄⋊C₄	—	2	4	no	—
170	C₂×C₄⋊C₄	—	2	4	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
173	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
174	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
175	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
176	C₂×C₄²	—	2	2	no	—
177	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
178	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
179	#106	—	2	2	no	—
180	#106	—	2	4	no	—
181	#106	—	2	2	no	—
182	#106	—	4	2	no	—
183	#106	—	4	4	no	—
184	C₄⋊C₈	Q₈×C₁₃₇	1	4	yes	—
185	C₄×C₈	—	2	4	no	—
186	C₄×C₈	Q₈×C₁₃₇	1	4	yes	—
187	#95	—	2	2	no	—
188	#95	—	2	4	no	—
189	#95	D₄×C₁₃₇	1	2	yes	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₄×C₁₃₇	1	4	yes	—
195	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
196	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
202	C₂×C₈	C₂×Q₈×C₁₃₇	1	4	yes	—
203	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
204	C₂×C₈	#41×C₁₃₇	1	2	yes	—
205	C₂×C₈	#41×C₁₃₇	1	2	yes	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	4	8	no	—
213	C₄⋊C₄	—	4	8	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₄⋊C₄	—	8	4	no	—
216	C₄⋊C₄	—	8	2	no	—
217	C₄⋊C₄	SD₁₆×C₁₃₇	1	8	yes	—
218	C₄²	—	2	2	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	4	4	no	—
221	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
222	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
223	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
237	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
238	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	8	yes	—
239	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	4	yes	—
240	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
241	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
242	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
243	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
244	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
245	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	—	2	4	no	—
248	C₈	#125×C₁₃₇	1	4	yes	complemented
249	C₂²	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
250	C₂²	#547×C₁₃₇	1	1	yes	characteristic
251	C₂²	#550×C₁₃₇	1	1	yes	characteristic
252	C₂²	C₂²×SD₁₆×C₁₃₇	1	2	yes	—
253	C₂²	#577×C₁₃₇	1	2	yes	—
254	C₄	—	2	8	no	—
255	C₄	—	2	2	no	—
256	C₄	—	8	2	no	—
257	C₄	—	8	2	no	—
258	C₄	—	8	4	no	—
259	C₄	#498×C₁₃₇	1	4	yes	—
260	C₄	C₂×#125×C₁₃₇	1	1	yes	characteristic
261	C₄	#544×C₁₃₇	1	1	yes	characteristic
262	C₄	#554×C₁₃₇	1	1	yes	characteristic
263	C₄	#557×C₁₃₇	1	1	yes	characteristic
264	C₂	#2552×C₁₃₇	1	1	yes	characteristic
265	C₂	C₂×#498×C₁₃₇	1	1	yes	characteristic
266	C₂	#3163×C₁₃₇	1	1	yes	characteristic
267	C₂	#3335×C₁₃₇	1	2	yes	—
268	C₂	#3392×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₄×C₄⋊C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
113. C₈×C₁₃₇	#125	semidirect
134. #28829	C₁₃₇	direct (N × H)
135. C₁₃₇	#28829	direct (N × H)
248. C₈	#125×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#2552×C₁₃₇	2
#2084×C₁₃₇	2
#1948×C₁₃₇	2
#1941×C₁₃₇	2
#1844×C₁₃₇	2
C₄×#106×C₁₃₇	2
C₈×C₄⋊C₄×C₁₃₇	2
#28829	137

Subgroups — C137×#28829 #46878215

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28829 #46878215