Subgroups — C₁₃₇×#25878 #46875264

Order 35072.

1346 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3498×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#3142×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#3080×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#2641×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#2037×C₁₃₇	C₂	1	4	yes	maximal, complemented
6	#1965×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#1963×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#1792×C₁₃₇	C₂	1	4	yes	maximal, complemented
9	#1759×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×H(5,𝔽₂)×C₁₃₇	C₂²	1	2	yes	—
11	#550×C₁₃₇	—	2	2	no	—
12	D₄²×C₁₃₇	—	2	4	no	—
13	D₄²×C₁₃₇	—	2	2	no	—
14	#535×C₁₃₇	—	2	2	no	—
15	#535×C₁₃₇	C₂²	1	2	yes	—
16	#532×C₁₃₇	—	2	1	no	—
17	#532×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×#127×C₁₃₇	—	2	1	no	—
19	C₂×#127×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₂×#125×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
21	C₄×#41×C₁₃₇	C₂²	1	1	yes	characteristic
22	#491×C₁₃₇	—	2	1	no	—
23	#490×C₁₃₇	—	2	1	no	—
24	#488×C₁₃₇	—	2	2	no	—
25	#479×C₁₃₇	—	2	2	no	—
26	#478×C₁₃₇	—	2	2	no	—
27	#428×C₁₃₇	C₂²	1	2	yes	complemented
28	#423×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
29	C₂×#104×C₁₃₇	C₂²	1	2	yes	—
30	C₂×#104×C₁₃₇	C₂²	1	4	yes	complemented
31	#417×C₁₃₇	C₂²	1	2	yes	—
32	#417×C₁₃₇	C₂²	1	4	yes	complemented
33	C₂×#100×C₁₃₇	C₂²	1	4	yes	complemented
34	C₄×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
35	#394×C₁₃₇	—	2	4	no	—
36	#390×C₁₃₇	C₂²	1	4	yes	complemented
37	#376×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
38	#339×C₁₃₇	C₂²	1	2	yes	complemented
39	H(5,𝔽₂)×C₁₃₇	—	2	8	no	—
40	H(5,𝔽₂)×C₁₃₇	—	4	4	no	—
41	C₂×#41×C₁₃₇	—	2	2	no	—
42	C₂×#41×C₁₃₇	—	2	2	no	—
43	C₂×#41×C₁₃₇	C₂³	1	2	yes	—
44	C₂×#41×C₁₃₇	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₁₃₇	—	2	4	no	—
46	C₂²×D₄×C₁₃₇	—	2	4	no	—
47	C₂²×D₄×C₁₃₇	—	2	2	no	—
48	C₂²×D₄×C₁₃₇	—	2	2	no	—
49	C₂²×D₄×C₁₃₇	D₄	1	4	yes	—
50	C₂²×D₄×C₁₃₇	C₂³	1	2	yes	—
51	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
52	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
53	C₄⋊Q₈×C₁₃₇	D₄	1	1	yes	characteristic
54	#127×C₁₃₇	—	2	4	no	—
55	#127×C₁₃₇	—	2	2	no	—
56	#127×C₁₃₇	—	2	1	no	—
57	#127×C₁₃₇	—	2	1	no	—
58	#127×C₁₃₇	—	2	2	no	—
59	#127×C₁₃₇	D₄	1	1	yes	characteristic
60	#125×C₁₃₇	—	2	2	no	—
61	#125×C₁₃₇	—	2	1	no	—
62	#125×C₁₃₇	—	2	1	no	—
63	#124×C₁₃₇	—	2	2	no	—
64	#124×C₁₃₇	D₄	1	2	yes	—
65	#121×C₁₃₇	—	2	4	no	—
66	#121×C₁₃₇	—	2	8	no	—
67	#121×C₁₃₇	—	2	2	no	—
68	#121×C₁₃₇	—	2	4	no	—
69	#120×C₁₃₇	—	2	8	no	—
70	#120×C₁₃₇	—	2	4	no	—
71	C₄×Q₈×C₁₃₇	D₄	1	2	yes	—
72	C₄×D₄×C₁₃₇	—	2	4	no	—
73	C₄×D₄×C₁₃₇	—	2	2	no	—
74	C₄×D₄×C₁₃₇	—	2	2	no	—
75	C₄×D₄×C₁₃₇	D₄	1	2	yes	—
76	#117×C₁₃₇	—	2	1	no	—
77	#117×C₁₃₇	C₂³	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
79	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
80	C₂×C₂²⋊C₄×C₁₃₇	—	2	8	no	—
81	C₂×C₄²×C₁₃₇	—	2	1	no	—
82	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
83	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
84	#107×C₁₃₇	—	2	2	no	—
85	#106×C₁₃₇	—	2	2	no	—
86	C₄⋊C₈×C₁₃₇	—	2	2	no	—
87	#104×C₁₃₇	—	2	4	no	—
88	#104×C₁₃₇	—	4	4	no	—
89	#103×C₁₃₇	—	2	2	no	—
90	#103×C₁₃₇	—	2	4	no	—
91	#102×C₁₃₇	—	2	2	no	—
92	#102×C₁₃₇	—	2	4	no	—
93	#100×C₁₃₇	—	4	4	no	—
94	#97×C₁₃₇	—	2	1	no	—
95	C₄×C₈×C₁₃₇	—	2	1	no	—
96	#95×C₁₃₇	—	2	2	no	—
97	C₂⁴×C₁₃₇	—	2	4	no	—
98	C₂⁴×C₁₃₇	C₂×D₄	1	4	yes	—
99	#41×C₁₃₇	—	2	2	no	—
100	#41×C₁₃₇	—	2	4	no	—
101	#41×C₁₃₇	—	2	4	no	—
102	#41×C₁₃₇	—	4	8	no	—
103	#41×C₁₃₇	—	4	4	no	—
104	#41×C₁₃₇	—	4	4	no	—
105	#41×C₁₃₇	#41	1	4	yes	—
106	C₂×Q₈×C₁₃₇	C₂×D₄	1	2	yes	—
107	C₂×Q₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
108	C₂×D₄×C₁₃₇	—	2	1	no	—
109	C₂×D₄×C₁₃₇	—	2	4	no	—
110	C₂×D₄×C₁₃₇	—	2	4	no	—
111	C₂×D₄×C₁₃₇	—	2	2	no	—
112	C₂×D₄×C₁₃₇	—	2	8	no	—
113	C₂×D₄×C₁₃₇	—	2	4	no	—
114	C₂×D₄×C₁₃₇	—	2	2	no	—
115	C₂×D₄×C₁₃₇	—	2	4	no	—
116	C₂×D₄×C₁₃₇	—	2	2	no	—
117	C₂×D₄×C₁₃₇	—	2	2	no	—
118	C₂×D₄×C₁₃₇	—	2	4	no	—
119	C₂×D₄×C₁₃₇	—	2	4	no	—
120	C₂×D₄×C₁₃₇	—	4	8	no	—
121	C₂×D₄×C₁₃₇	—	4	16	no	—
122	C₂×D₄×C₁₃₇	—	4	4	no	—
123	C₂×D₄×C₁₃₇	—	4	4	no	—
124	C₂×D₄×C₁₃₇	—	8	8	no	—
125	C₂×D₄×C₁₃₇	—	8	4	no	—
126	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
127	C₂×D₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₁₃₇	—	2	1	no	—
129	C₂²×C₄×C₁₃₇	—	2	4	no	—
130	C₂²×C₄×C₁₃₇	—	2	2	no	—
131	C₂²×C₄×C₁₃₇	—	2	2	no	—
132	C₂²×C₄×C₁₃₇	—	2	2	no	—
133	C₂²×C₄×C₁₃₇	—	2	2	no	—
134	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
136	M₁₆×C₁₃₇	—	2	2	no	—
137	M₁₆×C₁₃₇	—	2	2	no	—
138	M₁₆×C₁₃₇	—	2	4	no	—
139	C₂×C₈×C₁₃₇	—	2	2	no	—
140	C₂×C₈×C₁₃₇	—	2	2	no	—
141	C₄⋊C₄×C₁₃₇	—	2	2	no	—
142	C₄⋊C₄×C₁₃₇	—	2	2	no	—
143	C₄⋊C₄×C₁₃₇	—	2	2	no	—
144	C₄⋊C₄×C₁₃₇	—	2	2	no	—
145	C₄⋊C₄×C₁₃₇	—	4	2	no	—
146	C₄⋊C₄×C₁₃₇	—	4	2	no	—
147	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
148	C₂²⋊C₄×C₁₃₇	—	2	8	no	—
149	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
150	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
151	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
152	C₂²⋊C₄×C₁₃₇	—	8	8	no	—
153	C₄²×C₁₃₇	—	2	2	no	—
154	C₄²×C₁₃₇	—	2	1	no	—
155	C₄²×C₁₃₇	—	2	2	no	—
156	C₄²×C₁₃₇	—	2	2	no	—
157	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
158	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
159	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
160	C₂³×C₁₃₇	—	2	1	no	—
161	C₂³×C₁₃₇	—	2	4	no	—
162	C₂³×C₁₃₇	—	2	2	no	—
163	C₂³×C₁₃₇	—	2	8	no	—
164	C₂³×C₁₃₇	—	4	8	no	—
165	C₂³×C₁₃₇	—	4	8	no	—
166	C₂³×C₁₃₇	—	8	8	no	—
167	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
168	Q₈×C₁₃₇	—	2	1	no	—
169	Q₈×C₁₃₇	—	2	4	no	—
170	Q₈×C₁₃₇	#124	1	2	yes	—
171	D₄×C₁₃₇	—	2	4	no	—
172	D₄×C₁₃₇	—	2	1	no	—
173	D₄×C₁₃₇	—	2	4	no	—
174	D₄×C₁₃₇	—	4	8	no	—
175	D₄×C₁₃₇	—	4	8	no	—
176	D₄×C₁₃₇	—	4	4	no	—
177	D₄×C₁₃₇	—	4	8	no	—
178	D₄×C₁₃₇	—	4	4	no	—
179	D₄×C₁₃₇	—	4	2	no	—
180	D₄×C₁₃₇	—	4	2	no	—
181	D₄×C₁₃₇	—	4	4	no	—
182	D₄×C₁₃₇	—	8	4	no	—
183	D₄×C₁₃₇	—	8	8	no	—
184	D₄×C₁₃₇	#124	1	2	yes	—
185	C₂×C₄×C₁₃₇	—	2	1	no	—
186	C₂×C₄×C₁₃₇	—	2	4	no	—
187	C₂×C₄×C₁₃₇	—	2	2	no	—
188	C₂×C₄×C₁₃₇	—	2	2	no	—
189	C₂×C₄×C₁₃₇	—	4	2	no	—
190	C₂×C₄×C₁₃₇	—	4	8	no	—
191	C₂×C₄×C₁₃₇	—	4	4	no	—
192	C₂×C₄×C₁₃₇	—	4	2	no	—
193	C₂×C₄×C₁₃₇	—	4	2	no	—
194	C₂×C₄×C₁₃₇	—	4	4	no	—
195	C₂×C₄×C₁₃₇	—	4	4	no	—
196	C₂×C₄×C₁₃₇	—	4	2	no	—
197	C₂×C₄×C₁₃₇	—	4	2	no	—
198	C₂×C₄×C₁₃₇	#120	1	2	yes	—
199	C₂×C₄×C₁₃₇	#120	1	2	yes	—
200	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
201	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
202	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
203	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	2	yes	—
204	C₈×C₁₃₇	—	4	2	no	—
205	C₈×C₁₃₇	—	4	2	no	—
206	C₂²×C₁₃₇	—	2	2	no	—
207	C₂²×C₁₃₇	—	2	2	no	—
208	C₂²×C₁₃₇	—	4	2	no	—
209	C₂²×C₁₃₇	—	4	2	no	—
210	C₂²×C₁₃₇	—	4	4	no	—
211	C₂²×C₁₃₇	—	4	8	no	—
212	C₂²×C₁₃₇	—	4	4	no	—
213	C₂²×C₁₃₇	—	8	8	no	—
214	C₂²×C₁₃₇	—	16	8	no	—
215	C₂²×C₁₃₇	C₂×#120	1	1	yes	characteristic
216	C₂²×C₁₃₇	#546	1	2	yes	—
217	C₄×C₁₃₇	—	2	2	no	—
218	C₄×C₁₃₇	—	2	2	no	—
219	C₄×C₁₃₇	—	2	2	no	—
220	C₄×C₁₃₇	—	4	2	no	—
221	C₄×C₁₃₇	—	4	2	no	—
222	C₄×C₁₃₇	—	8	2	no	—
223	C₄×C₁₃₇	—	8	2	no	—
224	C₄×C₁₃₇	C₂×#124	1	1	yes	characteristic
225	C₄×C₁₃₇	#535	1	1	yes	characteristic
226	C₄×C₁₃₇	#546	1	2	yes	—
227	C₂×C₁₃₇	—	2	2	no	—
228	C₂×C₁₃₇	—	4	2	no	—
229	C₂×C₁₃₇	—	8	4	no	—
230	C₂×C₁₃₇	—	8	2	no	—
231	C₂×C₁₃₇	#2435	1	1	yes	characteristic
232	C₂×C₁₃₇	#3019	1	2	yes	—
233	#25878	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
234	C₁₃₇	#25878	1	1	yes	characteristic, direct product factor, Sylow
235	#3498	C₂×C₁₃₇	1	1	yes	characteristic
236	#3142	C₂×C₁₃₇	1	1	yes	characteristic, complemented
237	#3080	C₂×C₁₃₇	1	1	yes	characteristic, complemented
238	#2641	C₂×C₁₃₇	1	1	yes	characteristic, complemented
239	#2037	C₂×C₁₃₇	1	4	yes	complemented
240	#1965	C₂×C₁₃₇	1	1	yes	characteristic, complemented
241	#1963	C₂×C₁₃₇	1	1	yes	characteristic, complemented
242	#1792	C₂×C₁₃₇	1	4	yes	complemented
243	#1759	C₂×C₁₃₇	1	1	yes	characteristic, complemented
244	C₂×H(5,𝔽₂)	C₂²×C₁₃₇	1	2	yes	—
245	#550	—	2	2	no	—
246	D₄²	—	2	4	no	—
247	D₄²	—	2	2	no	—
248	#535	—	2	2	no	—
249	#535	C₂²×C₁₃₇	1	2	yes	—
250	#532	—	2	1	no	—
251	#532	C₂²×C₁₃₇	1	1	yes	characteristic
252	C₂×#127	—	2	1	no	—
253	C₂×#127	C₂²×C₁₃₇	1	1	yes	characteristic
254	C₂×#125	C₂²×C₁₃₇	1	1	yes	characteristic, complemented
255	C₄×#41	C₂²×C₁₃₇	1	1	yes	characteristic
256	#491	—	2	1	no	—
257	#490	—	2	1	no	—
258	#488	—	2	2	no	—
259	#479	—	2	2	no	—
260	#478	—	2	2	no	—
261	#428	C₂²×C₁₃₇	1	2	yes	complemented
262	#423	C₂²×C₁₃₇	1	1	yes	characteristic, complemented
263	C₂×#104	C₂²×C₁₃₇	1	2	yes	—
264	C₂×#104	C₂²×C₁₃₇	1	4	yes	complemented
265	#417	C₂²×C₁₃₇	1	2	yes	—
266	#417	C₂²×C₁₃₇	1	4	yes	complemented
267	C₂×#100	C₂²×C₁₃₇	1	4	yes	complemented
268	C₄×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic, complemented
269	#394	—	2	4	no	—
270	#390	C₂²×C₁₃₇	1	4	yes	complemented
271	#376	C₂²×C₁₃₇	1	1	yes	characteristic, complemented
272	#339	C₂²×C₁₃₇	1	2	yes	complemented
273	H(5,𝔽₂)	—	2	8	no	—
274	H(5,𝔽₂)	—	4	4	no	—
275	C₂×#41	—	2	2	no	—
276	C₂×#41	—	2	2	no	—
277	C₂×#41	C₂³×C₁₃₇	1	2	yes	—
278	C₂×#41	C₂³×C₁₃₇	1	1	yes	characteristic
279	C₂²×D₄	—	2	4	no	—
280	C₂²×D₄	—	2	4	no	—
281	C₂²×D₄	—	2	2	no	—
282	C₂²×D₄	—	2	2	no	—
283	C₂²×D₄	D₄×C₁₃₇	1	4	yes	—
284	C₂²×D₄	C₂³×C₁₃₇	1	2	yes	—
285	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
286	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
287	C₄⋊Q₈	D₄×C₁₃₇	1	1	yes	characteristic
288	#127	—	2	4	no	—
289	#127	—	2	2	no	—
290	#127	—	2	1	no	—
291	#127	—	2	1	no	—
292	#127	—	2	2	no	—
293	#127	D₄×C₁₃₇	1	1	yes	characteristic
294	#125	—	2	2	no	—
295	#125	—	2	1	no	—
296	#125	—	2	1	no	—
297	#124	—	2	2	no	—
298	#124	D₄×C₁₃₇	1	2	yes	—
299	#121	—	2	4	no	—
300	#121	—	2	8	no	—
301	#121	—	2	2	no	—
302	#121	—	2	4	no	—
303	#120	—	2	8	no	—
304	#120	—	2	4	no	—
305	C₄×Q₈	D₄×C₁₃₇	1	2	yes	—
306	C₄×D₄	—	2	4	no	—
307	C₄×D₄	—	2	2	no	—
308	C₄×D₄	—	2	2	no	—
309	C₄×D₄	D₄×C₁₃₇	1	2	yes	—
310	#117	—	2	1	no	—
311	#117	C₂³×C₁₃₇	1	1	yes	characteristic
312	C₂×C₄⋊C₄	—	2	2	no	—
313	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
314	C₂×C₂²⋊C₄	—	2	8	no	—
315	C₂×C₄²	—	2	1	no	—
316	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
317	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
318	#107	—	2	2	no	—
319	#106	—	2	2	no	—
320	C₄⋊C₈	—	2	2	no	—
321	#104	—	2	4	no	—
322	#104	—	4	4	no	—
323	#103	—	2	2	no	—
324	#103	—	2	4	no	—
325	#102	—	2	2	no	—
326	#102	—	2	4	no	—
327	#100	—	4	4	no	—
328	#97	—	2	1	no	—
329	C₄×C₈	—	2	1	no	—
330	#95	—	2	2	no	—
331	C₂⁴	—	2	4	no	—
332	C₂⁴	C₂×D₄×C₁₃₇	1	4	yes	—
333	#41	—	2	2	no	—
334	#41	—	2	4	no	—
335	#41	—	2	4	no	—
336	#41	—	4	8	no	—
337	#41	—	4	4	no	—
338	#41	—	4	4	no	—
339	#41	#41×C₁₃₇	1	4	yes	—
340	C₂×Q₈	C₂×D₄×C₁₃₇	1	2	yes	—
341	C₂×Q₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
342	C₂×D₄	—	2	1	no	—
343	C₂×D₄	—	2	4	no	—
344	C₂×D₄	—	2	4	no	—
345	C₂×D₄	—	2	2	no	—
346	C₂×D₄	—	2	8	no	—
347	C₂×D₄	—	2	4	no	—
348	C₂×D₄	—	2	2	no	—
349	C₂×D₄	—	2	4	no	—
350	C₂×D₄	—	2	2	no	—
351	C₂×D₄	—	2	2	no	—
352	C₂×D₄	—	2	4	no	—
353	C₂×D₄	—	2	4	no	—
354	C₂×D₄	—	4	8	no	—
355	C₂×D₄	—	4	16	no	—
356	C₂×D₄	—	4	4	no	—
357	C₂×D₄	—	4	4	no	—
358	C₂×D₄	—	8	8	no	—
359	C₂×D₄	—	8	4	no	—
360	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
361	C₂×D₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	4	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	2	2	no	—
367	C₂²×C₄	—	2	2	no	—
368	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
369	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
370	M₁₆	—	2	2	no	—
371	M₁₆	—	2	2	no	—
372	M₁₆	—	2	4	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	2	2	no	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	2	2	no	—
377	C₄⋊C₄	—	2	2	no	—
378	C₄⋊C₄	—	2	2	no	—
379	C₄⋊C₄	—	4	2	no	—
380	C₄⋊C₄	—	4	2	no	—
381	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
382	C₂²⋊C₄	—	2	8	no	—
383	C₂²⋊C₄	—	2	1	no	—
384	C₂²⋊C₄	—	2	2	no	—
385	C₂²⋊C₄	—	2	4	no	—
386	C₂²⋊C₄	—	8	8	no	—
387	C₄²	—	2	2	no	—
388	C₄²	—	2	1	no	—
389	C₄²	—	2	2	no	—
390	C₄²	—	2	2	no	—
391	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
392	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
393	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
394	C₂³	—	2	1	no	—
395	C₂³	—	2	4	no	—
396	C₂³	—	2	2	no	—
397	C₂³	—	2	8	no	—
398	C₂³	—	4	8	no	—
399	C₂³	—	4	8	no	—
400	C₂³	—	8	8	no	—
401	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
402	Q₈	—	2	1	no	—
403	Q₈	—	2	4	no	—
404	Q₈	#124×C₁₃₇	1	2	yes	—
405	D₄	—	2	4	no	—
406	D₄	—	2	1	no	—
407	D₄	—	2	4	no	—
408	D₄	—	4	8	no	—
409	D₄	—	4	8	no	—
410	D₄	—	4	4	no	—
411	D₄	—	4	8	no	—
412	D₄	—	4	4	no	—
413	D₄	—	4	2	no	—
414	D₄	—	4	2	no	—
415	D₄	—	4	4	no	—
416	D₄	—	8	4	no	—
417	D₄	—	8	8	no	—
418	D₄	#124×C₁₃₇	1	2	yes	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	4	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	8	no	—
425	C₂×C₄	—	4	4	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	4	no	—
429	C₂×C₄	—	4	4	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	#120×C₁₃₇	1	2	yes	—
433	C₂×C₄	#120×C₁₃₇	1	2	yes	—
434	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
435	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
436	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
437	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	2	yes	—
438	C₈	—	4	2	no	—
439	C₈	—	4	2	no	—
440	C₂²	—	2	2	no	—
441	C₂²	—	2	2	no	—
442	C₂²	—	4	2	no	—
443	C₂²	—	4	2	no	—
444	C₂²	—	4	4	no	—
445	C₂²	—	4	8	no	—
446	C₂²	—	4	4	no	—
447	C₂²	—	8	8	no	—
448	C₂²	—	16	8	no	—
449	C₂²	C₂×#120×C₁₃₇	1	1	yes	characteristic
450	C₂²	#546×C₁₃₇	1	2	yes	—
451	C₄	—	2	2	no	—
452	C₄	—	2	2	no	—
453	C₄	—	2	2	no	—
454	C₄	—	4	2	no	—
455	C₄	—	4	2	no	—
456	C₄	—	8	2	no	—
457	C₄	—	8	2	no	—
458	C₄	C₂×#124×C₁₃₇	1	1	yes	characteristic
459	C₄	#535×C₁₃₇	1	1	yes	characteristic
460	C₄	#546×C₁₃₇	1	2	yes	—
461	C₂	—	2	2	no	—
462	C₂	—	4	2	no	—
463	C₂	—	8	4	no	—
464	C₂	—	8	2	no	—
465	C₂	#2435×C₁₃₇	1	1	yes	characteristic
466	C₂	#3019×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	#3498×C₁₃₇	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ 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. #3142×C₁₃₇	C₂	semidirect
3. #3080×C₁₃₇	C₂	semidirect
4. #2641×C₁₃₇	C₂	semidirect
5. #2037×C₁₃₇	C₂	semidirect
6. #1965×C₁₃₇	C₂	semidirect
7. #1963×C₁₃₇	C₂	semidirect
8. #1792×C₁₃₇	C₂	semidirect
9. #1759×C₁₃₇	C₂	semidirect
20. C₂×#125×C₁₃₇	C₂²	semidirect
27. #428×C₁₃₇	C₂²	semidirect
28. #423×C₁₃₇	C₂²	semidirect
30. C₂×#104×C₁₃₇	C₂²	semidirect
32. #417×C₁₃₇	C₂²	semidirect
33. C₂×#100×C₁₃₇	C₂²	semidirect
34. C₄×M₁₆×C₁₃₇	C₂²	semidirect
36. #390×C₁₃₇	C₂²	semidirect
37. #376×C₁₃₇	C₂²	semidirect
38. #339×C₁₃₇	C₂²	semidirect
233. #25878	C₁₃₇	direct (N × H)
234. C₁₃₇	#25878	direct (N × H)
236. #3142	C₂×C₁₃₇	semidirect
237. #3080	C₂×C₁₃₇	semidirect
238. #2641	C₂×C₁₃₇	semidirect
239. #2037	C₂×C₁₃₇	semidirect
240. #1965	C₂×C₁₃₇	semidirect
241. #1963	C₂×C₁₃₇	semidirect
242. #1792	C₂×C₁₃₇	semidirect
243. #1759	C₂×C₁₃₇	semidirect
254. C₂×#125	C₂²×C₁₃₇	semidirect
261. #428	C₂²×C₁₃₇	semidirect
262. #423	C₂²×C₁₃₇	semidirect
264. C₂×#104	C₂²×C₁₃₇	semidirect
266. #417	C₂²×C₁₃₇	semidirect
267. C₂×#100	C₂²×C₁₃₇	semidirect
268. C₄×M₁₆	C₂²×C₁₃₇	semidirect
270. #390	C₂²×C₁₃₇	semidirect
271. #376	C₂²×C₁₃₇	semidirect
272. #339	C₂²×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3498×C₁₃₇	2
#3142×C₁₃₇	2
#3080×C₁₃₇	2
#2641×C₁₃₇	2
#2037×C₁₃₇	2
#1965×C₁₃₇	2
#1963×C₁₃₇	2
#1792×C₁₃₇	2
#1759×C₁₃₇	2
#25878	137

Subgroups — C137×#25878 #46875264

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#25878 #46875264