Subgroups — C₁₃₇×#29129 #46878515

Order 35072.

674 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	#2864×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#2100×C₁₃₇	C₂	1	2	yes	maximal
4	#2043×C₁₃₇	C₂	1	2	yes	maximal
5	#1935×C₁₃₇	C₂	1	4	yes	maximal
6	#1805×C₁₃₇	C₂	1	2	yes	maximal
7	#1805×C₁₃₇	C₂	1	2	yes	maximal
8	#1767×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₁₃₇	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
12	C₂×#103×C₁₃₇	C₂²	1	4	yes	—
13	C₂×#103×C₁₃₇	C₂²	1	4	yes	—
14	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
15	#391×C₁₃₇	C₂²	1	2	yes	—
16	#389×C₁₃₇	—	2	2	no	—
17	#389×C₁₃₇	C₂²	1	2	yes	—
18	#389×C₁₃₇	C₂²	1	2	yes	—
19	#387×C₁₃₇	—	2	2	no	—
20	#387×C₁₃₇	—	2	2	no	—
21	#384×C₁₃₇	C₂²	1	1	yes	characteristic
22	#382×C₁₃₇	C₂²	1	1	yes	characteristic
23	#382×C₁₃₇	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₁₃₇	—	2	2	no	—
25	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
26	#340×C₁₃₇	C₂²	1	4	yes	—
27	#340×C₁₃₇	C₂²	1	4	yes	—
28	#336×C₁₃₇	C₂²	1	2	yes	—
29	C₂²×Q₈×C₁₃₇	C₂³	1	2	yes	—
30	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
31	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
32	C₄×Q₈×C₁₃₇	—	2	4	no	—
33	C₄×Q₈×C₁₃₇	—	2	4	no	—
34	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
41	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
42	C₂×C₄²×C₁₃₇	—	2	2	no	—
43	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
44	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
45	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₁₃₇	—	2	4	no	—
49	#103×C₁₃₇	—	2	8	no	—
50	#103×C₁₃₇	—	4	4	no	—
51	#97×C₁₃₇	—	4	2	no	—
52	#95×C₁₃₇	—	2	4	no	—
53	#95×C₁₃₇	—	2	1	no	—
54	#95×C₁₃₇	—	2	1	no	—
55	#95×C₁₃₇	—	2	1	no	—
56	#95×C₁₃₇	—	2	1	no	—
57	#95×C₁₃₇	—	2	2	no	—
58	#95×C₁₃₇	—	2	2	no	—
59	C₂×Q₈×C₁₃₇	—	2	2	no	—
60	C₂×Q₈×C₁₃₇	—	2	2	no	—
61	C₂×Q₈×C₁₃₇	—	2	2	no	—
62	C₂×Q₈×C₁₃₇	—	2	2	no	—
63	C₂×Q₈×C₁₃₇	—	2	2	no	—
64	C₂×Q₈×C₁₃₇	#41	1	4	yes	—
65	C₂²×C₄×C₁₃₇	—	2	2	no	—
66	C₂²×C₄×C₁₃₇	—	2	2	no	—
67	C₂²×C₄×C₁₃₇	—	2	2	no	—
68	C₂²×C₄×C₁₃₇	—	2	2	no	—
69	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
74	C₂×C₈×C₁₃₇	—	2	4	no	—
75	C₂×C₈×C₁₃₇	—	2	4	no	—
76	C₂×C₈×C₁₃₇	—	2	4	no	—
77	C₄⋊C₄×C₁₃₇	—	2	8	no	—
78	C₄⋊C₄×C₁₃₇	—	2	4	no	—
79	C₄⋊C₄×C₁₃₇	—	2	2	no	—
80	C₄⋊C₄×C₁₃₇	—	2	2	no	—
81	C₄⋊C₄×C₁₃₇	—	2	2	no	—
82	C₄⋊C₄×C₁₃₇	—	2	2	no	—
83	C₄⋊C₄×C₁₃₇	—	4	4	no	—
84	C₄⋊C₄×C₁₃₇	—	4	4	no	—
85	C₄⋊C₄×C₁₃₇	—	4	4	no	—
86	C₄⋊C₄×C₁₃₇	—	4	4	no	—
87	C₄²×C₁₃₇	—	2	4	no	—
88	C₄²×C₁₃₇	—	2	2	no	—
89	C₄²×C₁₃₇	—	2	1	no	—
90	C₄²×C₁₃₇	—	2	1	no	—
91	C₄²×C₁₃₇	—	4	2	no	—
92	C₄²×C₁₃₇	—	4	4	no	—
93	C₄²×C₁₃₇	#41	1	4	yes	—
94	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
95	Q₈×C₁₃₇	—	2	4	no	—
96	Q₈×C₁₃₇	—	2	4	no	—
97	Q₈×C₁₃₇	—	4	2	no	—
98	Q₈×C₁₃₇	—	4	2	no	—
99	C₂×C₄×C₁₃₇	—	2	2	no	—
100	C₂×C₄×C₁₃₇	—	2	2	no	—
101	C₂×C₄×C₁₃₇	—	2	2	no	—
102	C₂×C₄×C₁₃₇	—	2	2	no	—
103	C₂×C₄×C₁₃₇	—	2	4	no	—
104	C₂×C₄×C₁₃₇	—	2	2	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	—	4	2	no	—
109	C₂×C₄×C₁₃₇	—	4	2	no	—
110	C₂×C₄×C₁₃₇	—	4	2	no	—
111	C₂×C₄×C₁₃₇	—	4	2	no	—
112	C₂×C₄×C₁₃₇	—	4	2	no	—
113	C₂×C₄×C₁₃₇	—	4	2	no	—
114	C₂×C₄×C₁₃₇	—	4	2	no	—
115	C₂×C₄×C₁₃₇	—	4	4	no	—
116	C₂×C₄×C₁₃₇	—	4	2	no	—
117	C₂×C₄×C₁₃₇	#124	1	2	yes	—
118	C₂×C₄×C₁₃₇	#124	1	2	yes	—
119	C₂×C₄×C₁₃₇	#137	1	4	yes	—
120	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
121	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
122	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
124	C₈×C₁₃₇	—	4	4	no	—
125	C₂²×C₁₃₇	#525	1	1	yes	characteristic
126	C₂²×C₁₃₇	C₂×#124	1	1	yes	characteristic
127	C₂²×C₁₃₇	#537	1	1	yes	characteristic
128	C₂²×C₁₃₇	C₂×#137	1	2	yes	—
129	C₂²×C₁₃₇	#575	1	2	yes	—
130	C₄×C₁₃₇	—	2	4	no	—
131	C₄×C₁₃₇	—	2	2	no	—
132	C₄×C₁₃₇	—	2	2	no	—
133	C₄×C₁₃₇	—	4	4	no	—
134	C₄×C₁₃₇	—	4	2	no	—
135	C₄×C₁₃₇	—	8	2	no	—
136	C₄×C₁₃₇	—	8	2	no	—
137	C₄×C₁₃₇	—	8	2	no	—
138	C₄×C₁₃₇	#552	1	2	yes	—
139	C₄×C₁₃₇	#553	1	2	yes	—
140	C₂×C₁₃₇	#2562	1	1	yes	characteristic
141	C₂×C₁₃₇	#3117	1	2	yes	—
142	C₂×C₁₃₇	#3135	1	1	yes	characteristic
143	C₂×C₁₃₇	#3139	1	1	yes	characteristic
144	C₂×C₁₃₇	#3252	1	2	yes	—
145	#29129	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₁₃₇	#29129	1	1	yes	characteristic, direct product factor, Sylow
147	#2864	C₂×C₁₃₇	1	1	yes	characteristic
148	C₄×C₄⋊Q₈	C₂×C₁₃₇	1	1	yes	characteristic
149	#2100	C₂×C₁₃₇	1	2	yes	—
150	#2043	C₂×C₁₃₇	1	2	yes	—
151	#1935	C₂×C₁₃₇	1	4	yes	—
152	#1805	C₂×C₁₃₇	1	2	yes	—
153	#1805	C₂×C₁₃₇	1	2	yes	—
154	#1767	C₂×C₁₃₇	1	1	yes	characteristic
155	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
156	C₂×C₄×Q₈	C₂²×C₁₃₇	1	2	yes	—
157	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
158	C₂×#103	C₂²×C₁₃₇	1	4	yes	—
159	C₂×#103	C₂²×C₁₃₇	1	4	yes	—
160	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
161	#391	C₂²×C₁₃₇	1	2	yes	—
162	#389	—	2	2	no	—
163	#389	C₂²×C₁₃₇	1	2	yes	—
164	#389	C₂²×C₁₃₇	1	2	yes	—
165	#387	—	2	2	no	—
166	#387	—	2	2	no	—
167	#384	C₂²×C₁₃₇	1	1	yes	characteristic
168	#382	C₂²×C₁₃₇	1	1	yes	characteristic
169	#382	C₂²×C₁₃₇	1	1	yes	characteristic
170	C₄×C₄⋊C₄	—	2	2	no	—
171	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
172	#340	C₂²×C₁₃₇	1	4	yes	—
173	#340	C₂²×C₁₃₇	1	4	yes	—
174	#336	C₂²×C₁₃₇	1	2	yes	—
175	C₂²×Q₈	C₂³×C₁₃₇	1	2	yes	—
176	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
177	C₄⋊Q₈	—	2	4	no	—
178	C₄×Q₈	—	2	4	no	—
179	C₄×Q₈	—	2	4	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
186	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
187	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
190	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
191	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
192	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
193	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
194	C₄⋊C₈	—	2	4	no	—
195	#103	—	2	8	no	—
196	#103	—	4	4	no	—
197	#97	—	4	2	no	—
198	#95	—	2	4	no	—
199	#95	—	2	1	no	—
200	#95	—	2	1	no	—
201	#95	—	2	1	no	—
202	#95	—	2	1	no	—
203	#95	—	2	2	no	—
204	#95	—	2	2	no	—
205	C₂×Q₈	—	2	2	no	—
206	C₂×Q₈	—	2	2	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	#41×C₁₃₇	1	4	yes	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
216	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
219	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₄⋊C₄	—	2	8	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	4	4	no	—
230	C₄⋊C₄	—	4	4	no	—
231	C₄⋊C₄	—	4	4	no	—
232	C₄⋊C₄	—	4	4	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	1	no	—
236	C₄²	—	2	1	no	—
237	C₄²	—	4	2	no	—
238	C₄²	—	4	4	no	—
239	C₄²	#41×C₁₃₇	1	4	yes	—
240	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
241	Q₈	—	2	4	no	—
242	Q₈	—	2	4	no	—
243	Q₈	—	4	2	no	—
244	Q₈	—	4	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	#124×C₁₃₇	1	2	yes	—
264	C₂×C₄	#124×C₁₃₇	1	2	yes	—
265	C₂×C₄	#137×C₁₃₇	1	4	yes	—
266	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
267	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
268	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
269	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
270	C₈	—	4	4	no	—
271	C₂²	#525×C₁₃₇	1	1	yes	characteristic
272	C₂²	C₂×#124×C₁₃₇	1	1	yes	characteristic
273	C₂²	#537×C₁₃₇	1	1	yes	characteristic
274	C₂²	C₂×#137×C₁₃₇	1	2	yes	—
275	C₂²	#575×C₁₃₇	1	2	yes	—
276	C₄	—	2	4	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	4	4	no	—
280	C₄	—	4	2	no	—
281	C₄	—	8	2	no	—
282	C₄	—	8	2	no	—
283	C₄	—	8	2	no	—
284	C₄	#552×C₁₃₇	1	2	yes	—
285	C₄	#553×C₁₃₇	1	2	yes	—
286	C₂	#2562×C₁₃₇	1	1	yes	characteristic
287	C₂	#3117×C₁₃₇	1	2	yes	—
288	C₂	#3135×C₁₃₇	1	1	yes	characteristic
289	C₂	#3139×C₁₃₇	1	1	yes	characteristic
290	C₂	#3252×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 ◁ C₄³×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
145. #29129	C₁₃₇	direct (N × H)
146. C₁₃₇	#29129	direct (N × H)

Maximal subgroups

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

subgroup	index
#2864×C₁₃₇	2
C₄×C₄⋊Q₈×C₁₃₇	2
#2100×C₁₃₇	2
#2043×C₁₃₇	2
#1935×C₁₃₇	2
#1805×C₁₃₇	2
#1767×C₁₃₇	2
#29129	137

Subgroups — C137×#29129 #46878515

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#29129 #46878515