Subgroups — C₁₃₇×#21361 #46870747

Order 35072.

958 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#431×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#2283×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊C₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1949×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#1853×C₁₃₇	C₂	1	2	yes	maximal
6	#1852×C₁₃₇	C₂	1	2	yes	maximal
7	C₄×#107×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	C₄×#106×C₁₃₇	C₂	1	2	yes	maximal, complemented
9	C₂²×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₃₇	—	2	1	no	—
11	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
12	#431×C₁₃₇	C₄	1	8	yes	complemented
13	C₂×#107×C₁₃₇	C₂²	1	4	yes	—
14	C₂×#106×C₁₃₇	C₂²	1	4	yes	—
15	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
19	#383×C₁₃₇	—	2	1	no	—
20	#382×C₁₃₇	C₂²	1	2	yes	—
21	#381×C₁₃₇	—	2	1	no	—
22	#380×C₁₃₇	C₂²	1	2	yes	—
23	C₄×C₄⋊C₄×C₁₃₇	—	2	1	no	—
24	C₄×C₄⋊C₄×C₁₃₇	—	2	1	no	—
25	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
29	C₄×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
30	#376×C₁₃₇	—	2	1	no	—
31	C₂×#95×C₁₃₇	—	2	1	no	—
32	#340×C₁₃₇	C₂²	1	4	yes	—
33	#340×C₁₃₇	C₂²	1	4	yes	—
34	C₂³×C₄×C₁₃₇	—	2	2	no	—
35	C₂³×C₄×C₁₃₇	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₁₃₇	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
39	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₃₇	—	2	8	no	—
42	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
44	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
45	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
46	C₂×C₄²×C₁₃₇	—	2	2	no	—
47	C₂×C₄²×C₁₃₇	—	2	1	no	—
48	C₂×C₄²×C₁₃₇	—	2	1	no	—
49	C₂×C₄²×C₁₃₇	—	2	2	no	—
50	C₂×C₄²×C₁₃₇	—	4	4	no	—
51	C₂×C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₁₃₇	Q₈	1	4	yes	—
55	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
57	#107×C₁₃₇	—	2	8	no	—
58	#106×C₁₃₇	—	2	8	no	—
59	C₄⋊C₈×C₁₃₇	C₂×C₄	1	8	yes	complemented
60	C₂²⋊C₈×C₁₃₇	C₂×C₄	1	4	yes	—
61	C₂²⋊C₈×C₁₃₇	C₂×C₄	1	4	yes	—
62	C₄×C₈×C₁₃₇	C₂×C₄	1	8	yes	complemented
63	#95×C₁₃₇	—	2	2	no	—
64	#95×C₁₃₇	—	2	2	no	—
65	#95×C₁₃₇	—	2	2	no	—
66	#95×C₁₃₇	—	2	2	no	—
67	#95×C₁₃₇	—	2	2	no	—
68	#95×C₁₃₇	—	2	1	no	—
69	#95×C₁₃₇	—	2	1	no	—
70	C₂⁴×C₁₃₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃₇	—	2	4	no	—
72	C₂²×C₄×C₁₃₇	—	2	2	no	—
73	C₂²×C₄×C₁₃₇	—	2	2	no	—
74	C₂²×C₄×C₁₃₇	—	2	4	no	—
75	C₂²×C₄×C₁₃₇	—	2	8	no	—
76	C₂²×C₄×C₁₃₇	—	2	4	no	—
77	C₂²×C₄×C₁₃₇	—	2	2	no	—
78	C₂²×C₄×C₁₃₇	—	2	2	no	—
79	C₂²×C₄×C₁₃₇	—	4	4	no	—
80	C₂²×C₄×C₁₃₇	—	4	4	no	—
81	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
82	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
83	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
90	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
91	C₂×C₈×C₁₃₇	—	2	4	no	—
92	C₂×C₈×C₁₃₇	—	2	4	no	—
93	C₂×C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
94	C₂×C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
95	C₄⋊C₄×C₁₃₇	—	2	8	no	—
96	C₄⋊C₄×C₁₃₇	—	2	4	no	—
97	C₄⋊C₄×C₁₃₇	—	2	4	no	—
98	C₄⋊C₄×C₁₃₇	—	2	4	no	—
99	C₄⋊C₄×C₁₃₇	#41	1	8	yes	—
100	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
101	C₄²×C₁₃₇	—	4	4	no	—
102	C₄²×C₁₃₇	—	4	2	no	—
103	C₄²×C₁₃₇	—	4	4	no	—
104	C₄²×C₁₃₇	—	4	2	no	—
105	C₄²×C₁₃₇	C₄⋊C₄	1	4	yes	—
106	C₄²×C₁₃₇	C₄⋊C₄	1	4	yes	—
107	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
108	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
109	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
110	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
111	C₂³×C₁₃₇	—	2	2	no	—
112	C₂³×C₁₃₇	—	2	1	no	—
113	C₂³×C₁₃₇	—	2	1	no	—
114	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
115	C₂³×C₁₃₇	#135	1	4	yes	—
116	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃₇	—	2	2	no	—
118	C₂×C₄×C₁₃₇	—	2	2	no	—
119	C₂×C₄×C₁₃₇	—	2	2	no	—
120	C₂×C₄×C₁₃₇	—	2	1	no	—
121	C₂×C₄×C₁₃₇	—	2	1	no	—
122	C₂×C₄×C₁₃₇	—	2	8	no	—
123	C₂×C₄×C₁₃₇	—	2	4	no	—
124	C₂×C₄×C₁₃₇	—	2	4	no	—
125	C₂×C₄×C₁₃₇	—	2	8	no	—
126	C₂×C₄×C₁₃₇	—	2	8	no	—
127	C₂×C₄×C₁₃₇	—	4	16	no	—
128	C₂×C₄×C₁₃₇	—	4	4	no	—
129	C₂×C₄×C₁₃₇	—	4	4	no	—
130	C₂×C₄×C₁₃₇	—	4	2	no	—
131	C₂×C₄×C₁₃₇	—	4	2	no	—
132	C₂×C₄×C₁₃₇	—	4	2	no	—
133	C₂×C₄×C₁₃₇	—	4	2	no	—
134	C₂×C₄×C₁₃₇	—	4	2	no	—
135	C₂×C₄×C₁₃₇	—	4	2	no	—
136	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
138	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
139	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
140	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
141	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
142	C₂×C₄×C₁₃₇	#137	1	2	yes	—
143	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
145	C₂×C₄×C₁₃₇	C₂×#41	1	4	yes	—
146	C₈×C₁₃₇	—	2	4	no	—
147	C₈×C₁₃₇	—	2	4	no	—
148	C₂²×C₁₃₇	—	2	4	no	—
149	C₂²×C₁₃₇	—	2	4	no	—
150	C₂²×C₁₃₇	—	2	2	no	—
151	C₂²×C₁₃₇	—	2	2	no	—
152	C₂²×C₁₃₇	#427	1	4	yes	—
153	C₂²×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
154	C₂²×C₁₃₇	#525	1	2	yes	—
155	C₂²×C₁₃₇	C₂×#135	1	2	yes	—
156	C₂²×C₁₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₂²×C₁₃₇	C₂×#137	1	1	yes	characteristic
158	C₄×C₁₃₇	—	2	2	no	—
159	C₄×C₁₃₇	—	2	2	no	—
160	C₄×C₁₃₇	—	2	2	no	—
161	C₄×C₁₃₇	—	2	2	no	—
162	C₄×C₁₃₇	—	4	8	no	—
163	C₄×C₁₃₇	—	8	2	no	—
164	C₄×C₁₃₇	—	8	2	no	—
165	C₄×C₁₃₇	#428	1	4	yes	—
166	C₄×C₁₃₇	C₄×#41	1	2	yes	—
167	C₄×C₁₃₇	#533	1	2	yes	—
168	C₂×C₁₃₇	—	2	4	no	—
169	C₂×C₁₃₇	#2296	1	1	yes	characteristic
170	C₂×C₁₃₇	C₂×#427	1	1	yes	characteristic
171	C₂×C₁₃₇	C₂×#428	1	1	yes	characteristic
172	C₂×C₁₃₇	#3105	1	2	yes	—
173	C₂×C₁₃₇	#3108	1	2	yes	—
174	#21361	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
175	C₁₃₇	#21361	1	1	yes	characteristic, direct product factor, Sylow
176	C₂×#431	C₂×C₁₃₇	1	1	yes	characteristic
177	#2283	C₂×C₁₃₇	1	1	yes	characteristic
178	C₂×C₄×C₄⋊C₄	C₂×C₁₃₇	1	1	yes	characteristic
179	#1949	C₂×C₁₃₇	1	4	yes	complemented
180	#1853	C₂×C₁₃₇	1	2	yes	—
181	#1852	C₂×C₁₃₇	1	2	yes	—
182	C₄×#107	C₂×C₁₃₇	1	2	yes	complemented
183	C₄×#106	C₂×C₁₃₇	1	2	yes	complemented
184	C₂²×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
185	C₂²×C₄²	—	2	1	no	—
186	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
187	#431	C₄×C₁₃₇	1	8	yes	complemented
188	C₂×#107	C₂²×C₁₃₇	1	4	yes	—
189	C₂×#106	C₂²×C₁₃₇	1	4	yes	—
190	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
191	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
192	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
193	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
194	#383	—	2	1	no	—
195	#382	C₂²×C₁₃₇	1	2	yes	—
196	#381	—	2	1	no	—
197	#380	C₂²×C₁₃₇	1	2	yes	—
198	C₄×C₄⋊C₄	—	2	1	no	—
199	C₄×C₄⋊C₄	—	2	1	no	—
200	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
201	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
202	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
203	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
204	C₄×C₂²⋊C₄	—	2	1	no	—
205	#376	—	2	1	no	—
206	C₂×#95	—	2	1	no	—
207	#340	C₂²×C₁₃₇	1	4	yes	—
208	#340	C₂²×C₁₃₇	1	4	yes	—
209	C₂³×C₄	—	2	2	no	—
210	C₂³×C₄	D₄×C₁₃₇	1	1	yes	characteristic
211	C₂³×C₄	D₄×C₁₃₇	1	1	yes	characteristic
212	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
213	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
214	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	8	no	—
217	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
218	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
219	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
220	C₂×C₂²⋊C₄	—	2	2	no	—
221	C₂×C₄²	—	2	1	no	—
222	C₂×C₄²	—	2	1	no	—
223	C₂×C₄²	—	2	2	no	—
224	C₂×C₄²	—	2	2	no	—
225	C₂×C₄²	—	4	4	no	—
226	C₂×C₄²	C₂×C₄×C₁₃₇	1	4	yes	—
227	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
228	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
229	C₂×C₄²	Q₈×C₁₃₇	1	4	yes	—
230	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
231	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
232	#107	—	2	8	no	—
233	#106	—	2	8	no	—
234	C₄⋊C₈	C₂×C₄×C₁₃₇	1	8	yes	complemented
235	C₂²⋊C₈	C₂×C₄×C₁₃₇	1	4	yes	—
236	C₂²⋊C₈	C₂×C₄×C₁₃₇	1	4	yes	—
237	C₄×C₈	C₂×C₄×C₁₃₇	1	8	yes	complemented
238	#95	—	2	2	no	—
239	#95	—	2	2	no	—
240	#95	—	2	2	no	—
241	#95	—	2	2	no	—
242	#95	—	2	2	no	—
243	#95	—	2	1	no	—
244	#95	—	2	1	no	—
245	C₂⁴	C₂×D₄×C₁₃₇	1	1	yes	characteristic
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	2	8	no	—
251	C₂²×C₄	—	2	4	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	4	4	no	—
255	C₂²×C₄	—	4	4	no	—
256	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	4	yes	—
257	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	4	yes	—
258	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
259	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
260	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
261	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
263	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
264	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
265	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
266	C₂×C₈	—	2	4	no	—
267	C₂×C₈	—	2	4	no	—
268	C₂×C₈	C₂²×C₄×C₁₃₇	1	4	yes	—
269	C₂×C₈	C₂²×C₄×C₁₃₇	1	4	yes	—
270	C₄⋊C₄	—	2	8	no	—
271	C₄⋊C₄	—	2	4	no	—
272	C₄⋊C₄	—	2	4	no	—
273	C₄⋊C₄	—	2	4	no	—
274	C₄⋊C₄	#41×C₁₃₇	1	8	yes	—
275	C₂²⋊C₄	—	4	4	no	—
276	C₄²	—	4	4	no	—
277	C₄²	—	4	2	no	—
278	C₄²	—	4	4	no	—
279	C₄²	—	4	2	no	—
280	C₄²	C₄⋊C₄×C₁₃₇	1	4	yes	—
281	C₄²	C₄⋊C₄×C₁₃₇	1	4	yes	—
282	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
283	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
284	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
285	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
286	C₂³	—	2	2	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
290	C₂³	#135×C₁₃₇	1	4	yes	—
291	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
292	C₂×C₄	—	2	8	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	8	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	8	no	—
302	C₂×C₄	—	4	16	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	4	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
312	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
313	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
314	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
315	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
316	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
317	C₂×C₄	#137×C₁₃₇	1	2	yes	—
318	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
319	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
320	C₂×C₄	C₂×#41×C₁₃₇	1	4	yes	—
321	C₈	—	2	4	no	—
322	C₈	—	2	4	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	4	no	—
325	C₂²	—	2	4	no	—
326	C₂²	—	2	2	no	—
327	C₂²	#427×C₁₃₇	1	4	yes	—
328	C₂²	C₂²×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
329	C₂²	#525×C₁₃₇	1	2	yes	—
330	C₂²	C₂×#135×C₁₃₇	1	2	yes	—
331	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
332	C₂²	C₂×#137×C₁₃₇	1	1	yes	characteristic
333	C₄	—	2	2	no	—
334	C₄	—	2	2	no	—
335	C₄	—	2	2	no	—
336	C₄	—	2	2	no	—
337	C₄	—	4	8	no	—
338	C₄	—	8	2	no	—
339	C₄	—	8	2	no	—
340	C₄	#428×C₁₃₇	1	4	yes	—
341	C₄	C₄×#41×C₁₃₇	1	2	yes	—
342	C₄	#533×C₁₃₇	1	2	yes	—
343	C₂	—	2	4	no	—
344	C₂	#2296×C₁₃₇	1	1	yes	characteristic
345	C₂	C₂×#427×C₁₃₇	1	1	yes	characteristic
346	C₂	C₂×#428×C₁₃₇	1	1	yes	characteristic
347	C₂	#3105×C₁₃₇	1	2	yes	—
348	C₂	#3108×C₁₃₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. #1949×C₁₃₇	C₂	semidirect
7. C₄×#107×C₁₃₇	C₂	semidirect
8. C₄×#106×C₁₃₇	C₂	semidirect
12. #431×C₁₃₇	C₄	semidirect
59. C₄⋊C₈×C₁₃₇	C₂×C₄	semidirect
62. C₄×C₈×C₁₃₇	C₂×C₄	semidirect
174. #21361	C₁₃₇	direct (N × H)
175. C₁₃₇	#21361	direct (N × H)
179. #1949	C₂×C₁₃₇	semidirect
182. C₄×#107	C₂×C₁₃₇	semidirect
183. C₄×#106	C₂×C₁₃₇	semidirect
187. #431	C₄×C₁₃₇	semidirect
234. C₄⋊C₈	C₂×C₄×C₁₃₇	semidirect
237. C₄×C₈	C₂×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×#431×C₁₃₇	2
#2283×C₁₃₇	2
C₂×C₄×C₄⋊C₄×C₁₃₇	2
#1949×C₁₃₇	2
#1853×C₁₃₇	2
#1852×C₁₃₇	2
C₄×#107×C₁₃₇	2
C₄×#106×C₁₃₇	2
#21361	137

Subgroups — C137×#21361 #46870747

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#21361 #46870747