Subgroups — C₁₃₇×#21362 #46870748

Order 35072.

914 conjugacy classes of subgroups.

The subgroup lattice has 362 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	#2285×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₄×#117×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1948×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#1947×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#1854×C₁₃₇	C₂	1	4	yes	maximal
7	C₄×#107×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	C₄×#106×C₁₃₇	C₂	1	2	yes	maximal, complemented
9	C₂×#117×C₁₃₇	—	2	1	no	—
10	C₂×#117×C₁₃₇	C₂²	1	2	yes	—
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	1	yes	characteristic
16	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
19	#384×C₁₃₇	C₂²	1	2	yes	—
20	#382×C₁₃₇	—	2	1	no	—
21	#380×C₁₃₇	—	2	1	no	—
22	#380×C₁₃₇	C₂²	1	2	yes	—
23	C₄×C₄⋊C₄×C₁₃₇	—	2	1	no	—
24	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
25	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
28	C₄×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
29	C₄×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
30	#376×C₁₃₇	—	2	1	no	—
31	C₄³×C₁₃₇	—	2	1	no	—
32	#340×C₁₃₇	C₂²	1	4	yes	—
33	#340×C₁₃₇	C₂²	1	4	yes	—
34	C₂³×C₄×C₁₃₇	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₁₃₇	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
39	#117×C₁₃₇	—	2	4	no	—
40	#117×C₁₃₇	—	2	4	no	—
41	#117×C₁₃₇	—	4	4	no	—
42	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
45	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
47	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
48	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
49	C₂×C₄²×C₁₃₇	—	2	2	no	—
50	C₂×C₄²×C₁₃₇	—	2	2	no	—
51	C₂×C₄²×C₁₃₇	—	2	1	no	—
52	C₂×C₄²×C₁₃₇	—	2	1	no	—
53	C₂×C₄²×C₁₃₇	—	2	2	no	—
54	C₂×C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
55	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₁₃₇	Q₈	1	4	yes	—
58	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
60	#107×C₁₃₇	—	2	8	no	—
61	#106×C₁₃₇	—	2	8	no	—
62	C₄⋊C₈×C₁₃₇	C₂×C₄	1	4	yes	—
63	C₄⋊C₈×C₁₃₇	C₂×C₄	1	4	yes	—
64	C₂²⋊C₈×C₁₃₇	C₂×C₄	1	8	yes	—
65	C₄×C₈×C₁₃₇	C₂×C₄	1	8	yes	—
66	#95×C₁₃₇	—	2	1	no	—
67	#95×C₁₃₇	—	2	1	no	—
68	#95×C₁₃₇	—	2	2	no	—
69	#95×C₁₃₇	—	2	2	no	—
70	#95×C₁₃₇	—	2	2	no	—
71	C₂⁴×C₁₃₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃₇	—	2	4	no	—
73	C₂²×C₄×C₁₃₇	—	2	2	no	—
74	C₂²×C₄×C₁₃₇	—	2	2	no	—
75	C₂²×C₄×C₁₃₇	—	2	2	no	—
76	C₂²×C₄×C₁₃₇	—	2	2	no	—
77	C₂²×C₄×C₁₃₇	—	2	2	no	—
78	C₂²×C₄×C₁₃₇	—	2	2	no	—
79	C₂²×C₄×C₁₃₇	—	2	4	no	—
80	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
81	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
82	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
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×Q₈	1	2	yes	—
89	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
90	C₂×C₈×C₁₃₇	—	2	4	no	—
91	C₂×C₈×C₁₃₇	—	2	4	no	—
92	C₂×C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
93	C₂×C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
94	C₄⋊C₄×C₁₃₇	—	2	2	no	—
95	C₄⋊C₄×C₁₃₇	—	2	2	no	—
96	C₄⋊C₄×C₁₃₇	—	2	4	no	—
97	C₄⋊C₄×C₁₃₇	—	2	4	no	—
98	C₄⋊C₄×C₁₃₇	—	4	4	no	—
99	C₄⋊C₄×C₁₃₇	—	4	4	no	—
100	C₄⋊C₄×C₁₃₇	#41	1	4	yes	—
101	C₄⋊C₄×C₁₃₇	#41	1	4	yes	—
102	C₂²⋊C₄×C₁₃₇	—	2	8	no	—
103	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
104	C₄²×C₁₃₇	—	2	8	no	—
105	C₄²×C₁₃₇	—	4	4	no	—
106	C₄²×C₁₃₇	—	4	2	no	—
107	C₄²×C₁₃₇	—	4	4	no	—
108	C₄²×C₁₃₇	—	4	2	no	—
109	C₄²×C₁₃₇	C₄⋊C₄	1	4	yes	—
110	C₄²×C₁₃₇	C₄⋊C₄	1	4	yes	—
111	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
112	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
113	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
114	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
115	C₂³×C₁₃₇	—	2	2	no	—
116	C₂³×C₁₃₇	—	2	2	no	—
117	C₂³×C₁₃₇	—	2	1	no	—
118	C₂³×C₁₃₇	—	2	1	no	—
119	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
120	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃₇	—	2	2	no	—
122	C₂×C₄×C₁₃₇	—	2	2	no	—
123	C₂×C₄×C₁₃₇	—	2	1	no	—
124	C₂×C₄×C₁₃₇	—	2	1	no	—
125	C₂×C₄×C₁₃₇	—	2	2	no	—
126	C₂×C₄×C₁₃₇	—	2	2	no	—
127	C₂×C₄×C₁₃₇	—	2	4	no	—
128	C₂×C₄×C₁₃₇	—	2	8	no	—
129	C₂×C₄×C₁₃₇	—	2	8	no	—
130	C₂×C₄×C₁₃₇	—	2	4	no	—
131	C₂×C₄×C₁₃₇	—	2	4	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	—	4	4	no	—
137	C₂×C₄×C₁₃₇	—	4	4	no	—
138	C₂×C₄×C₁₃₇	—	4	4	no	—
139	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
140	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
141	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
142	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
143	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
144	C₂×C₄×C₁₃₇	#135	1	4	yes	—
145	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
146	C₂×C₄×C₁₃₇	#137	1	2	yes	—
147	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
149	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
150	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
151	C₈×C₁₃₇	—	2	4	no	—
152	C₈×C₁₃₇	—	2	4	no	—
153	C₂²×C₁₃₇	—	2	2	no	—
154	C₂²×C₁₃₇	—	2	2	no	—
155	C₂²×C₁₃₇	—	2	2	no	—
156	C₂²×C₁₃₇	—	4	2	no	—
157	C₂²×C₁₃₇	—	4	2	no	—
158	C₂²×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
159	C₂²×C₁₃₇	#525	1	2	yes	—
160	C₂²×C₁₃₇	C₂×#135	1	2	yes	—
161	C₂²×C₁₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₂²×C₁₃₇	C₂×#137	1	1	yes	characteristic
163	C₄×C₁₃₇	—	2	2	no	—
164	C₄×C₁₃₇	—	2	2	no	—
165	C₄×C₁₃₇	—	2	2	no	—
166	C₄×C₁₃₇	—	4	4	no	—
167	C₄×C₁₃₇	—	4	4	no	—
168	C₄×C₁₃₇	—	8	4	no	—
169	C₄×C₁₃₇	#427	1	4	yes	—
170	C₄×C₁₃₇	#428	1	4	yes	—
171	C₄×C₁₃₇	C₄×#41	1	1	yes	characteristic
172	C₄×C₁₃₇	C₄×#41	1	1	yes	characteristic
173	C₄×C₁₃₇	#533	1	2	yes	—
174	C₂×C₁₃₇	—	4	2	no	—
175	C₂×C₁₃₇	#2296	1	1	yes	characteristic
176	C₂×C₁₃₇	C₂×#427	1	1	yes	characteristic
177	C₂×C₁₃₇	C₂×#428	1	1	yes	characteristic
178	C₂×C₁₃₇	#3106	1	2	yes	—
179	C₂×C₁₃₇	#3107	1	2	yes	—
180	#21362	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
181	C₁₃₇	#21362	1	1	yes	characteristic, direct product factor, Sylow
182	C₂×#431	C₂×C₁₃₇	1	1	yes	characteristic
183	#2285	C₂×C₁₃₇	1	1	yes	characteristic
184	C₄×#117	C₂×C₁₃₇	1	1	yes	characteristic
185	#1948	C₂×C₁₃₇	1	2	yes	complemented
186	#1947	C₂×C₁₃₇	1	2	yes	complemented
187	#1854	C₂×C₁₃₇	1	4	yes	—
188	C₄×#107	C₂×C₁₃₇	1	2	yes	complemented
189	C₄×#106	C₂×C₁₃₇	1	2	yes	complemented
190	C₂×#117	—	2	1	no	—
191	C₂×#117	C₂²×C₁₃₇	1	2	yes	—
192	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
193	#431	C₄×C₁₃₇	1	8	yes	complemented
194	C₂×#107	C₂²×C₁₃₇	1	4	yes	—
195	C₂×#106	C₂²×C₁₃₇	1	4	yes	—
196	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
197	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
198	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	2	yes	—
199	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
200	#384	C₂²×C₁₃₇	1	2	yes	—
201	#382	—	2	1	no	—
202	#380	—	2	1	no	—
203	#380	C₂²×C₁₃₇	1	2	yes	—
204	C₄×C₄⋊C₄	—	2	1	no	—
205	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
206	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
207	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
208	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
209	C₄×C₂²⋊C₄	—	2	1	no	—
210	C₄×C₂²⋊C₄	—	2	1	no	—
211	#376	—	2	1	no	—
212	C₄³	—	2	1	no	—
213	#340	C₂²×C₁₃₇	1	4	yes	—
214	#340	C₂²×C₁₃₇	1	4	yes	—
215	C₂³×C₄	D₄×C₁₃₇	1	1	yes	characteristic
216	C₂³×C₄	D₄×C₁₃₇	1	1	yes	characteristic
217	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
218	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
219	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
220	#117	—	2	4	no	—
221	#117	—	2	4	no	—
222	#117	—	4	4	no	—
223	C₂×C₄⋊C₄	—	2	2	no	—
224	C₂×C₄⋊C₄	—	2	2	no	—
225	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
226	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
227	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
228	C₂×C₂²⋊C₄	—	2	2	no	—
229	C₂×C₂²⋊C₄	—	2	2	no	—
230	C₂×C₄²	—	2	2	no	—
231	C₂×C₄²	—	2	2	no	—
232	C₂×C₄²	—	2	2	no	—
233	C₂×C₄²	—	2	1	no	—
234	C₂×C₄²	—	2	1	no	—
235	C₂×C₄²	C₂×C₄×C₁₃₇	1	4	yes	—
236	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
237	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
238	C₂×C₄²	Q₈×C₁₃₇	1	4	yes	—
239	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
240	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
241	#107	—	2	8	no	—
242	#106	—	2	8	no	—
243	C₄⋊C₈	C₂×C₄×C₁₃₇	1	4	yes	—
244	C₄⋊C₈	C₂×C₄×C₁₃₇	1	4	yes	—
245	C₂²⋊C₈	C₂×C₄×C₁₃₇	1	8	yes	—
246	C₄×C₈	C₂×C₄×C₁₃₇	1	8	yes	—
247	#95	—	2	1	no	—
248	#95	—	2	1	no	—
249	#95	—	2	2	no	—
250	#95	—	2	2	no	—
251	#95	—	2	2	no	—
252	C₂⁴	C₂×D₄×C₁₃₇	1	1	yes	characteristic
253	C₂²×C₄	—	2	4	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	2	4	no	—
261	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	4	yes	—
262	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	4	yes	—
263	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
264	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
265	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
266	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
269	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
270	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
271	C₂×C₈	—	2	4	no	—
272	C₂×C₈	—	2	4	no	—
273	C₂×C₈	C₂²×C₄×C₁₃₇	1	4	yes	—
274	C₂×C₈	C₂²×C₄×C₁₃₇	1	4	yes	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	2	4	no	—
278	C₄⋊C₄	—	2	4	no	—
279	C₄⋊C₄	—	4	4	no	—
280	C₄⋊C₄	—	4	4	no	—
281	C₄⋊C₄	#41×C₁₃₇	1	4	yes	—
282	C₄⋊C₄	#41×C₁₃₇	1	4	yes	—
283	C₂²⋊C₄	—	2	8	no	—
284	C₂²⋊C₄	—	4	4	no	—
285	C₄²	—	2	8	no	—
286	C₄²	—	4	4	no	—
287	C₄²	—	4	2	no	—
288	C₄²	—	4	4	no	—
289	C₄²	—	4	2	no	—
290	C₄²	C₄⋊C₄×C₁₃₇	1	4	yes	—
291	C₄²	C₄⋊C₄×C₁₃₇	1	4	yes	—
292	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
293	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
294	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
295	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
296	C₂³	—	2	2	no	—
297	C₂³	—	2	2	no	—
298	C₂³	—	2	1	no	—
299	C₂³	—	2	1	no	—
300	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
301	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	8	no	—
307	C₂×C₄	—	2	8	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	4	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	2	4	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	4	no	—
318	C₂×C₄	—	4	4	no	—
319	C₂×C₄	—	4	4	no	—
320	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
321	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
322	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
323	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
324	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
325	C₂×C₄	#135×C₁₃₇	1	4	yes	—
326	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
327	C₂×C₄	#137×C₁₃₇	1	2	yes	—
328	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
329	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
330	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
331	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
332	C₈	—	2	4	no	—
333	C₈	—	2	4	no	—
334	C₂²	—	2	2	no	—
335	C₂²	—	2	2	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	4	2	no	—
338	C₂²	—	4	2	no	—
339	C₂²	C₂²×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
340	C₂²	#525×C₁₃₇	1	2	yes	—
341	C₂²	C₂×#135×C₁₃₇	1	2	yes	—
342	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
343	C₂²	C₂×#137×C₁₃₇	1	1	yes	characteristic
344	C₄	—	2	2	no	—
345	C₄	—	2	2	no	—
346	C₄	—	2	2	no	—
347	C₄	—	4	4	no	—
348	C₄	—	4	4	no	—
349	C₄	—	8	4	no	—
350	C₄	#427×C₁₃₇	1	4	yes	—
351	C₄	#428×C₁₃₇	1	4	yes	—
352	C₄	C₄×#41×C₁₃₇	1	1	yes	characteristic
353	C₄	C₄×#41×C₁₃₇	1	1	yes	characteristic
354	C₄	#533×C₁₃₇	1	2	yes	—
355	C₂	—	4	2	no	—
356	C₂	#2296×C₁₃₇	1	1	yes	characteristic
357	C₂	C₂×#427×C₁₃₇	1	1	yes	characteristic
358	C₂	C₂×#428×C₁₃₇	1	1	yes	characteristic
359	C₂	#3106×C₁₃₇	1	2	yes	—
360	C₂	#3107×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. #1948×C₁₃₇	C₂	semidirect
5. #1947×C₁₃₇	C₂	semidirect
7. C₄×#107×C₁₃₇	C₂	semidirect
8. C₄×#106×C₁₃₇	C₂	semidirect
12. #431×C₁₃₇	C₄	semidirect
180. #21362	C₁₃₇	direct (N × H)
181. C₁₃₇	#21362	direct (N × H)
185. #1948	C₂×C₁₃₇	semidirect
186. #1947	C₂×C₁₃₇	semidirect
188. C₄×#107	C₂×C₁₃₇	semidirect
189. C₄×#106	C₂×C₁₃₇	semidirect
193. #431	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
#2285×C₁₃₇	2
C₄×#117×C₁₃₇	2
#1948×C₁₃₇	2
#1947×C₁₃₇	2
#1854×C₁₃₇	2
C₄×#107×C₁₃₇	2
C₄×#106×C₁₃₇	2
#21362	137

Subgroups — C137×#21362 #46870748

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#21362 #46870748