Subgroups — C₁₃₇×#10806 #46860192

Order 35072.

914 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄⋊C₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#340×C₁₃₇	C₂	1	2	yes	maximal
3	C₂×#340×C₁₃₇	C₂	1	2	yes	maximal
4	C₂×#336×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#95×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₁₃₇	C₂²	1	2	yes	—
7	C₂²×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
8	C₂²×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₁₃₇	—	2	1	no	—
11	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₁₃₇	—	2	2	no	—
13	C₄×C₄⋊C₄×C₁₃₇	C₄	1	4	yes	complemented
14	C₂×#95×C₁₃₇	—	2	1	no	—
15	C₂×#95×C₁₃₇	—	2	1	no	—
16	C₂×#95×C₁₃₇	—	2	1	no	—
17	#340×C₁₃₇	—	2	4	no	—
18	#340×C₁₃₇	C₄	1	8	yes	complemented
19	#336×C₁₃₇	—	2	2	no	—
20	C₂³×C₄×C₁₃₇	—	2	1	no	—
21	C₂³×C₄×C₁₃₇	—	2	1	no	—
22	C₂³×C₄×C₁₃₇	—	2	2	no	—
23	C₂³×C₄×C₁₃₇	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₃₇	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₁₃₇	—	2	2	no	—
27	C₂²×C₈×C₁₃₇	—	2	2	no	—
28	C₂²×C₈×C₁₃₇	—	2	2	no	—
29	C₂²×C₈×C₁₃₇	—	2	2	no	—
30	C₂²×C₈×C₁₃₇	—	2	2	no	—
31	C₂²×C₈×C₁₃₇	—	2	2	no	—
32	C₂²×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
34	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
35	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
37	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
38	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
44	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
46	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
48	C₂×C₄⋊C₄×C₁₃₇	Q₈	1	2	yes	—
49	C₂×C₄²×C₁₃₇	—	2	2	no	—
50	C₂×C₄²×C₁₃₇	—	2	2	no	—
51	C₂×C₄²×C₁₃₇	—	2	4	no	—
52	C₂×C₄²×C₁₃₇	—	4	2	no	—
53	C₂×C₄²×C₁₃₇	—	4	2	no	—
54	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
56	#95×C₁₃₇	—	2	4	no	—
57	#95×C₁₃₇	—	4	2	no	—
58	#95×C₁₃₇	—	4	2	no	—
59	C₂⁴×C₁₃₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃₇	—	2	2	no	—
61	C₂²×C₄×C₁₃₇	—	2	2	no	—
62	C₂²×C₄×C₁₃₇	—	2	2	no	—
63	C₂²×C₄×C₁₃₇	—	2	2	no	—
64	C₂²×C₄×C₁₃₇	—	2	4	no	—
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	—	2	1	no	—
70	C₂²×C₄×C₁₃₇	—	2	1	no	—
71	C₂²×C₄×C₁₃₇	—	2	4	no	—
72	C₂²×C₄×C₁₃₇	—	4	1	no	—
73	C₂²×C₄×C₁₃₇	—	4	1	no	—
74	C₂²×C₄×C₁₃₇	—	4	1	no	—
75	C₂²×C₄×C₁₃₇	—	4	1	no	—
76	C₂²×C₄×C₁₃₇	—	4	2	no	—
77	C₂²×C₄×C₁₃₇	—	4	1	no	—
78	C₂²×C₄×C₁₃₇	—	4	1	no	—
79	C₂²×C₄×C₁₃₇	—	4	2	no	—
80	C₂²×C₄×C₁₃₇	—	4	2	no	—
81	C₂²×C₄×C₁₃₇	—	4	2	no	—
82	C₂²×C₄×C₁₃₇	—	4	2	no	—
83	C₂²×C₄×C₁₃₇	—	4	2	no	—
84	C₂²×C₄×C₁₃₇	—	4	2	no	—
85	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
86	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
87	C₂²×C₄×C₁₃₇	D₈	1	2	yes	—
88	C₂²×C₄×C₁₃₇	SD₁₆	1	2	yes	—
89	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
94	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
95	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
96	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
97	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
98	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
99	C₂×C₈×C₁₃₇	—	2	4	no	—
100	C₂×C₈×C₁₃₇	—	2	4	no	—
101	C₂×C₈×C₁₃₇	—	2	4	no	—
102	C₂×C₈×C₁₃₇	—	4	2	no	—
103	C₂×C₈×C₁₃₇	—	4	2	no	—
104	C₂×C₈×C₁₃₇	—	4	2	no	—
105	C₂×C₈×C₁₃₇	—	4	2	no	—
106	C₄⋊C₄×C₁₃₇	—	2	2	no	—
107	C₄⋊C₄×C₁₃₇	—	2	2	no	—
108	C₄⋊C₄×C₁₃₇	—	2	4	no	—
109	C₄⋊C₄×C₁₃₇	—	2	2	no	—
110	C₄⋊C₄×C₁₃₇	—	2	2	no	—
111	C₄⋊C₄×C₁₃₇	—	2	4	no	—
112	C₄⋊C₄×C₁₃₇	—	2	4	no	—
113	C₄⋊C₄×C₁₃₇	—	4	2	no	—
114	C₄⋊C₄×C₁₃₇	—	4	2	no	—
115	C₄⋊C₄×C₁₃₇	C₄²	1	4	yes	—
116	C₄⋊C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
117	C₄²×C₁₃₇	—	2	2	no	—
118	C₄²×C₁₃₇	—	2	2	no	—
119	C₄²×C₁₃₇	—	2	4	no	—
120	C₄²×C₁₃₇	—	4	8	no	—
121	C₂³×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂³×C₁₃₇	C₄×D₄	1	2	yes	—
123	C₂³×C₁₃₇	#121	1	1	yes	characteristic
124	C₂³×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂³×C₁₃₇	#123	1	2	yes	—
126	C₂³×C₁₃₇	C₂×D₈	1	1	yes	characteristic
127	C₂³×C₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
128	C₂³×C₁₃₇	#135	1	2	yes	—
129	C₂³×C₁₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₂³×C₁₃₇	#137	1	1	yes	characteristic
131	C₂³×C₁₃₇	#137	1	2	yes	—
132	C₂×C₄×C₁₃₇	—	2	2	no	—
133	C₂×C₄×C₁₃₇	—	2	2	no	—
134	C₂×C₄×C₁₃₇	—	2	4	no	—
135	C₂×C₄×C₁₃₇	—	2	4	no	—
136	C₂×C₄×C₁₃₇	—	2	4	no	—
137	C₂×C₄×C₁₃₇	—	2	4	no	—
138	C₂×C₄×C₁₃₇	—	2	2	no	—
139	C₂×C₄×C₁₃₇	—	2	2	no	—
140	C₂×C₄×C₁₃₇	—	2	2	no	—
141	C₂×C₄×C₁₃₇	—	4	2	no	—
142	C₂×C₄×C₁₃₇	—	4	2	no	—
143	C₂×C₄×C₁₃₇	—	4	2	no	—
144	C₂×C₄×C₁₃₇	—	4	2	no	—
145	C₂×C₄×C₁₃₇	—	4	4	no	—
146	C₂×C₄×C₁₃₇	—	4	4	no	—
147	C₂×C₄×C₁₃₇	—	4	2	no	—
148	C₂×C₄×C₁₃₇	—	4	2	no	—
149	C₂×C₄×C₁₃₇	—	4	2	no	—
150	C₂×C₄×C₁₃₇	—	4	2	no	—
151	C₂×C₄×C₁₃₇	—	4	2	no	—
152	C₂×C₄×C₁₃₇	—	4	2	no	—
153	C₂×C₄×C₁₃₇	—	4	4	no	—
154	C₂×C₄×C₁₃₇	—	4	4	no	—
155	C₂×C₄×C₁₃₇	—	4	4	no	—
156	C₂×C₄×C₁₃₇	—	8	2	no	—
157	C₂×C₄×C₁₃₇	—	8	2	no	—
158	C₂×C₄×C₁₃₇	—	8	2	no	—
159	C₂×C₄×C₁₃₇	—	8	2	no	—
160	C₂×C₄×C₁₃₇	#102	1	4	yes	—
161	C₂×C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
162	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₁₃₇	#117	1	1	yes	characteristic
164	C₂×C₄×C₁₃₇	#117	1	1	yes	characteristic
165	C₂×C₄×C₁₃₇	#117	1	2	yes	—
166	C₂×C₄×C₁₃₇	#117	1	2	yes	—
167	C₂×C₄×C₁₃₇	#117	1	1	yes	characteristic
168	C₂×C₄×C₁₃₇	#117	1	1	yes	characteristic
169	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
170	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
171	C₂×C₄×C₁₃₇	C₄×Q₈	1	2	yes	—
172	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
173	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
174	C₂×C₄×C₁₃₇	#123	1	2	yes	—
175	C₂×C₄×C₁₃₇	#124	1	2	yes	—
176	C₂×C₄×C₁₃₇	#125	1	2	yes	—
177	C₂×C₄×C₁₃₇	#126	1	2	yes	—
178	C₂×C₄×C₁₃₇	#126	1	2	yes	—
179	C₈×C₁₃₇	—	4	4	no	—
180	C₂²×C₁₃₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
181	C₂²×C₁₃₇	#380	1	1	yes	characteristic
182	C₂²×C₁₃₇	#381	1	1	yes	characteristic
183	C₂²×C₁₃₇	#387	1	2	yes	—
184	C₂²×C₁₃₇	#388	1	2	yes	—
185	C₂²×C₁₃₇	C₂×#102	1	1	yes	characteristic
186	C₂²×C₁₃₇	#416	1	1	yes	characteristic
187	C₂²×C₁₃₇	#417	1	1	yes	characteristic
188	C₂²×C₁₃₇	#419	1	1	yes	characteristic
189	C₂²×C₁₃₇	C₄×D₈	1	2	yes	—
190	C₂²×C₁₃₇	C₄×SD₁₆	1	2	yes	—
191	C₂²×C₁₃₇	#440	1	2	yes	—
192	C₂²×C₁₃₇	#441	1	2	yes	—
193	C₂²×C₁₃₇	#459	1	1	yes	characteristic
194	C₂²×C₁₃₇	#461	1	1	yes	characteristic
195	C₂²×C₁₃₇	#464	1	2	yes	—
196	C₂²×C₁₃₇	#474	1	1	yes	characteristic
197	C₂²×C₁₃₇	#476	1	1	yes	characteristic
198	C₂²×C₁₃₇	#479	1	2	yes	—
199	C₂²×C₁₃₇	#480	1	2	yes	—
200	C₂²×C₁₃₇	#482	1	2	yes	—
201	C₂²×C₁₃₇	#483	1	2	yes	—
202	C₂²×C₁₃₇	#485	1	2	yes	—
203	C₄×C₁₃₇	—	2	4	no	—
204	C₄×C₁₃₇	—	4	4	no	—
205	C₄×C₁₃₇	—	4	2	no	—
206	C₄×C₁₃₇	—	4	4	no	—
207	C₄×C₁₃₇	—	8	4	no	—
208	C₄×C₁₃₇	#376	1	1	yes	characteristic
209	C₄×C₁₃₇	C₄×C₄⋊C₄	1	1	yes	characteristic
210	C₄×C₁₃₇	#382	1	1	yes	characteristic
211	C₄×C₁₃₇	#383	1	1	yes	characteristic
212	C₄×C₁₃₇	#387	1	2	yes	—
213	C₄×C₁₃₇	#388	1	2	yes	—
214	C₂×C₁₃₇	#1434	1	1	yes	characteristic
215	C₂×C₁₃₇	C₄×#102	1	1	yes	characteristic
216	C₂×C₁₃₇	#1763	1	1	yes	characteristic
217	C₂×C₁₃₇	#1802	1	1	yes	characteristic
218	C₂×C₁₃₇	#1805	1	1	yes	characteristic
219	C₂×C₁₃₇	#1823	1	1	yes	characteristic
220	C₂×C₁₃₇	#1825	1	1	yes	characteristic
221	C₂×C₁₃₇	#1918	1	2	yes	—
222	C₂×C₁₃₇	#1920	1	2	yes	—
223	C₂×C₁₃₇	#1925	1	2	yes	—
224	C₂×C₁₃₇	#1927	1	2	yes	—
225	#10806	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
226	C₁₃₇	#10806	1	1	yes	characteristic, direct product factor, Sylow
227	C₂×C₄×C₄⋊C₄	C₂×C₁₃₇	1	1	yes	characteristic
228	C₂×#340	C₂×C₁₃₇	1	2	yes	—
229	C₂×#340	C₂×C₁₃₇	1	2	yes	—
230	C₂×#336	C₂×C₁₃₇	1	1	yes	characteristic
231	C₄⋊#95	C₂×C₁₃₇	1	1	yes	characteristic
232	C₂³×C₈	C₂²×C₁₃₇	1	2	yes	—
233	C₂²×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
234	C₂²×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
235	C₂²×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
236	C₂²×C₄²	—	2	1	no	—
237	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
238	C₄×C₄⋊C₄	—	2	2	no	—
239	C₄×C₄⋊C₄	C₄×C₁₃₇	1	4	yes	complemented
240	C₂×#95	—	2	1	no	—
241	C₂×#95	—	2	1	no	—
242	C₂×#95	—	2	1	no	—
243	#340	—	2	4	no	—
244	#340	C₄×C₁₃₇	1	8	yes	complemented
245	#336	—	2	2	no	—
246	C₂³×C₄	—	2	1	no	—
247	C₂³×C₄	—	2	1	no	—
248	C₂³×C₄	—	2	2	no	—
249	C₂³×C₄	D₄×C₁₃₇	1	1	yes	characteristic
250	C₂³×C₄	D₄×C₁₃₇	1	1	yes	characteristic
251	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
252	C₂²×C₈	—	2	2	no	—
253	C₂²×C₈	—	2	2	no	—
254	C₂²×C₈	—	2	2	no	—
255	C₂²×C₈	—	2	2	no	—
256	C₂²×C₈	—	2	2	no	—
257	C₂²×C₈	—	2	2	no	—
258	C₂²×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
259	C₂×C₄⋊C₄	—	2	1	no	—
260	C₂×C₄⋊C₄	—	2	1	no	—
261	C₂×C₄⋊C₄	—	2	2	no	—
262	C₂×C₄⋊C₄	—	2	1	no	—
263	C₂×C₄⋊C₄	—	2	1	no	—
264	C₂×C₄⋊C₄	—	2	2	no	—
265	C₂×C₄⋊C₄	—	2	2	no	—
266	C₂×C₄⋊C₄	—	2	2	no	—
267	C₂×C₄⋊C₄	—	2	2	no	—
268	C₂×C₄⋊C₄	—	2	2	no	—
269	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	4	yes	—
270	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	4	yes	—
271	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	2	yes	—
272	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	2	yes	—
273	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
274	C₂×C₄⋊C₄	Q₈×C₁₃₇	1	2	yes	—
275	C₂×C₄²	—	2	2	no	—
276	C₂×C₄²	—	2	2	no	—
277	C₂×C₄²	—	2	4	no	—
278	C₂×C₄²	—	4	2	no	—
279	C₂×C₄²	—	4	2	no	—
280	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
281	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
282	#95	—	2	4	no	—
283	#95	—	4	2	no	—
284	#95	—	4	2	no	—
285	C₂⁴	C₂×D₄×C₁₃₇	1	1	yes	characteristic
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	2	2	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	—	2	4	no	—
291	C₂²×C₄	—	2	2	no	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	—	2	2	no	—
295	C₂²×C₄	—	2	1	no	—
296	C₂²×C₄	—	2	1	no	—
297	C₂²×C₄	—	2	4	no	—
298	C₂²×C₄	—	4	1	no	—
299	C₂²×C₄	—	4	1	no	—
300	C₂²×C₄	—	4	1	no	—
301	C₂²×C₄	—	4	1	no	—
302	C₂²×C₄	—	4	2	no	—
303	C₂²×C₄	—	4	1	no	—
304	C₂²×C₄	—	4	1	no	—
305	C₂²×C₄	—	4	2	no	—
306	C₂²×C₄	—	4	2	no	—
307	C₂²×C₄	—	4	2	no	—
308	C₂²×C₄	—	4	2	no	—
309	C₂²×C₄	—	4	2	no	—
310	C₂²×C₄	—	4	2	no	—
311	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	2	yes	—
312	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	2	yes	—
313	C₂²×C₄	D₈×C₁₃₇	1	2	yes	—
314	C₂²×C₄	SD₁₆×C₁₃₇	1	2	yes	—
315	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
316	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
317	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
318	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
319	C₂²×C₄	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
320	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
321	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
322	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
323	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
324	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
325	C₂×C₈	—	2	4	no	—
326	C₂×C₈	—	2	4	no	—
327	C₂×C₈	—	2	4	no	—
328	C₂×C₈	—	4	2	no	—
329	C₂×C₈	—	4	2	no	—
330	C₂×C₈	—	4	2	no	—
331	C₂×C₈	—	4	2	no	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	2	2	no	—
334	C₄⋊C₄	—	2	4	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	2	4	no	—
338	C₄⋊C₄	—	2	4	no	—
339	C₄⋊C₄	—	4	2	no	—
340	C₄⋊C₄	—	4	2	no	—
341	C₄⋊C₄	C₄²×C₁₃₇	1	4	yes	—
342	C₄⋊C₄	C₄⋊C₄×C₁₃₇	1	4	yes	—
343	C₄²	—	2	2	no	—
344	C₄²	—	2	2	no	—
345	C₄²	—	2	4	no	—
346	C₄²	—	4	8	no	—
347	C₂³	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
348	C₂³	C₄×D₄×C₁₃₇	1	2	yes	—
349	C₂³	#121×C₁₃₇	1	1	yes	characteristic
350	C₂³	C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
351	C₂³	#123×C₁₃₇	1	2	yes	—
352	C₂³	C₂×D₈×C₁₃₇	1	1	yes	characteristic
353	C₂³	C₂×SD₁₆×C₁₃₇	1	1	yes	characteristic
354	C₂³	#135×C₁₃₇	1	2	yes	—
355	C₂³	AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
356	C₂³	#137×C₁₃₇	1	1	yes	characteristic
357	C₂³	#137×C₁₃₇	1	2	yes	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	4	no	—
361	C₂×C₄	—	2	4	no	—
362	C₂×C₄	—	2	4	no	—
363	C₂×C₄	—	2	4	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	—	4	4	no	—
372	C₂×C₄	—	4	4	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	4	2	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	4	2	no	—
379	C₂×C₄	—	4	4	no	—
380	C₂×C₄	—	4	4	no	—
381	C₂×C₄	—	4	4	no	—
382	C₂×C₄	—	8	2	no	—
383	C₂×C₄	—	8	2	no	—
384	C₂×C₄	—	8	2	no	—
385	C₂×C₄	—	8	2	no	—
386	C₂×C₄	#102×C₁₃₇	1	4	yes	—
387	C₂×C₄	C₂×C₄²×C₁₃₇	1	1	yes	characteristic
388	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
389	C₂×C₄	#117×C₁₃₇	1	1	yes	characteristic
390	C₂×C₄	#117×C₁₃₇	1	1	yes	characteristic
391	C₂×C₄	#117×C₁₃₇	1	2	yes	—
392	C₂×C₄	#117×C₁₃₇	1	2	yes	—
393	C₂×C₄	#117×C₁₃₇	1	1	yes	characteristic
394	C₂×C₄	#117×C₁₃₇	1	1	yes	characteristic
395	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
396	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
397	C₂×C₄	C₄×Q₈×C₁₃₇	1	2	yes	—
398	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
399	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
400	C₂×C₄	#123×C₁₃₇	1	2	yes	—
401	C₂×C₄	#124×C₁₃₇	1	2	yes	—
402	C₂×C₄	#125×C₁₃₇	1	2	yes	—
403	C₂×C₄	#126×C₁₃₇	1	2	yes	—
404	C₂×C₄	#126×C₁₃₇	1	2	yes	—
405	C₈	—	4	4	no	—
406	C₂²	C₄×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
407	C₂²	#380×C₁₃₇	1	1	yes	characteristic
408	C₂²	#381×C₁₃₇	1	1	yes	characteristic
409	C₂²	#387×C₁₃₇	1	2	yes	—
410	C₂²	#388×C₁₃₇	1	2	yes	—
411	C₂²	C₂×#102×C₁₃₇	1	1	yes	characteristic
412	C₂²	#416×C₁₃₇	1	1	yes	characteristic
413	C₂²	#417×C₁₃₇	1	1	yes	characteristic
414	C₂²	#419×C₁₃₇	1	1	yes	characteristic
415	C₂²	C₄×D₈×C₁₃₇	1	2	yes	—
416	C₂²	C₄×SD₁₆×C₁₃₇	1	2	yes	—
417	C₂²	#440×C₁₃₇	1	2	yes	—
418	C₂²	#441×C₁₃₇	1	2	yes	—
419	C₂²	#459×C₁₃₇	1	1	yes	characteristic
420	C₂²	#461×C₁₃₇	1	1	yes	characteristic
421	C₂²	#464×C₁₃₇	1	2	yes	—
422	C₂²	#474×C₁₃₇	1	1	yes	characteristic
423	C₂²	#476×C₁₃₇	1	1	yes	characteristic
424	C₂²	#479×C₁₃₇	1	2	yes	—
425	C₂²	#480×C₁₃₇	1	2	yes	—
426	C₂²	#482×C₁₃₇	1	2	yes	—
427	C₂²	#483×C₁₃₇	1	2	yes	—
428	C₂²	#485×C₁₃₇	1	2	yes	—
429	C₄	—	2	4	no	—
430	C₄	—	4	4	no	—
431	C₄	—	4	4	no	—
432	C₄	—	4	2	no	—
433	C₄	—	8	4	no	—
434	C₄	#376×C₁₃₇	1	1	yes	characteristic
435	C₄	C₄×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
436	C₄	#382×C₁₃₇	1	1	yes	characteristic
437	C₄	#383×C₁₃₇	1	1	yes	characteristic
438	C₄	#387×C₁₃₇	1	2	yes	—
439	C₄	#388×C₁₃₇	1	2	yes	—
440	C₂	#1434×C₁₃₇	1	1	yes	characteristic
441	C₂	C₄×#102×C₁₃₇	1	1	yes	characteristic
442	C₂	#1763×C₁₃₇	1	1	yes	characteristic
443	C₂	#1802×C₁₃₇	1	1	yes	characteristic
444	C₂	#1805×C₁₃₇	1	1	yes	characteristic
445	C₂	#1823×C₁₃₇	1	1	yes	characteristic
446	C₂	#1825×C₁₃₇	1	1	yes	characteristic
447	C₂	#1918×C₁₃₇	1	2	yes	—
448	C₂	#1920×C₁₃₇	1	2	yes	—
449	C₂	#1925×C₁₃₇	1	2	yes	—
450	C₂	#1927×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₃₇	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₁₃₇
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃₇ ◁ C₂²×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₄×C₄⋊C₄×C₁₃₇	C₄	semidirect
18. #340×C₁₃₇	C₄	semidirect
225. #10806	C₁₃₇	direct (N × H)
226. C₁₃₇	#10806	direct (N × H)
239. C₄×C₄⋊C₄	C₄×C₁₃₇	semidirect
244. #340	C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₄⋊C₄×C₁₃₇	2
C₂×#340×C₁₃₇	2
C₂×#336×C₁₃₇	2
C₄⋊#95×C₁₃₇	2
#10806	137

Subgroups — C137×#10806 #46860192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#10806 #46860192