Subgroups — C₁₃₇×#21163 #46870549

Order 35072.

882 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#108×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#427×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#2905×C₁₃₇	C₂	1	2	yes	maximal
4	C₂×#405×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#1946×C₁₃₇	C₂	1	4	yes	maximal, complemented
6	#1945×C₁₃₇	C₂	1	4	yes	maximal, complemented
7	#1733×C₁₃₇	C₂	1	2	yes	maximal
8	C₂²×M₁₆×C₁₃₇	C₂²	1	2	yes	—
9	C₂²×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×#117×C₁₃₇	C₂²	1	2	yes	—
12	C₂×#117×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×#108×C₁₃₇	—	2	4	no	—
14	C₂×#108×C₁₃₇	C₂²	1	2	yes	—
15	C₂×#108×C₁₃₇	C₂²	1	2	yes	—
16	#427×C₁₃₇	C₄	1	2	yes	complemented
17	#427×C₁₃₇	C₄	1	2	yes	complemented
18	#427×C₁₃₇	C₂²	1	2	yes	—
19	#427×C₁₃₇	C₂²	1	2	yes	—
20	C₂×#107×C₁₃₇	C₄	1	2	yes	complemented
21	C₂×#106×C₁₃₇	C₄	1	2	yes	complemented
22	#424×C₁₃₇	—	2	4	no	—
23	#424×C₁₃₇	C₂²	1	8	yes	—
24	#423×C₁₃₇	—	2	4	no	—
25	#405×C₁₃₇	—	2	4	no	—
26	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
28	#339×C₁₃₇	C₂²	1	8	yes	—
29	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
30	C₂×M₁₆×C₁₃₇	—	2	2	no	—
31	C₂×M₁₆×C₁₃₇	—	2	2	no	—
32	C₂×M₁₆×C₁₃₇	—	2	2	no	—
33	C₂×M₁₆×C₁₃₇	—	2	4	no	—
34	C₂×M₁₆×C₁₃₇	—	2	4	no	—
35	C₂×M₁₆×C₁₃₇	D₄	1	2	yes	—
36	C₂×M₁₆×C₁₃₇	D₄	1	2	yes	—
37	C₂×M₁₆×C₁₃₇	C₂³	1	4	yes	—
38	C₂²×C₈×C₁₃₇	—	2	2	no	—
39	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₃₇	D₄	1	4	yes	—
44	C₂²×C₈×C₁₃₇	Q₈	1	4	yes	—
45	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
46	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
48	#117×C₁₃₇	—	2	2	no	—
49	#117×C₁₃₇	—	2	2	no	—
50	#117×C₁₃₇	—	2	2	no	—
51	#117×C₁₃₇	C₂×C₄	1	4	yes	—
52	#117×C₁₃₇	Q₈	1	4	yes	—
53	#117×C₁₃₇	C₂³	1	4	yes	—
54	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
55	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
56	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
57	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
58	C₂×C₄²×C₁₃₇	—	2	2	no	—
59	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
60	#108×C₁₃₇	—	2	4	no	—
61	#108×C₁₃₇	—	2	4	no	—
62	#107×C₁₃₇	C₂×C₄	1	4	yes	—
63	#107×C₁₃₇	C₂×C₄	1	4	yes	—
64	#106×C₁₃₇	C₂×C₄	1	4	yes	—
65	#106×C₁₃₇	C₂×C₄	1	4	yes	—
66	C₄⋊C₈×C₁₃₇	—	2	16	no	—
67	#97×C₁₃₇	—	2	4	no	—
68	C₄×C₈×C₁₃₇	—	2	4	no	—
69	C₂⁴×C₁₃₇	#41	1	1	yes	characteristic
70	C₂²×C₄×C₁₃₇	—	2	2	no	—
71	C₂²×C₄×C₁₃₇	—	2	4	no	—
72	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
79	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
80	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
81	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
82	M₁₆×C₁₃₇	—	2	4	no	—
83	M₁₆×C₁₃₇	—	2	4	no	—
84	M₁₆×C₁₃₇	—	2	4	no	—
85	M₁₆×C₁₃₇	—	2	4	no	—
86	M₁₆×C₁₃₇	—	4	4	no	—
87	C₂×C₈×C₁₃₇	—	2	2	no	—
88	C₂×C₈×C₁₃₇	—	2	2	no	—
89	C₂×C₈×C₁₃₇	—	2	4	no	—
90	C₂×C₈×C₁₃₇	—	4	4	no	—
91	C₂×C₈×C₁₃₇	C₄⋊C₄	1	4	yes	—
92	C₂×C₈×C₁₃₇	C₄⋊C₄	1	4	yes	—
93	C₂×C₈×C₁₃₇	C₄⋊C₄	1	4	yes	—
94	C₂×C₈×C₁₃₇	C₄⋊C₄	1	4	yes	—
95	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₁₃₇	C₂×D₄	1	4	yes	—
100	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
103	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
104	C₄⋊C₄×C₁₃₇	—	2	4	no	—
105	C₄⋊C₄×C₁₃₇	—	2	4	no	—
106	C₄⋊C₄×C₁₃₇	C₂²×C₄	1	8	yes	—
107	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
108	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
109	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
110	C₄²×C₁₃₇	—	2	4	no	—
111	C₄²×C₁₃₇	—	2	2	no	—
112	C₄²×C₁₃₇	—	2	4	no	—
113	C₄²×C₁₃₇	C₂×Q₈	1	4	yes	—
114	C₂³×C₁₃₇	—	4	2	no	—
115	C₂³×C₁₃₇	C₄×D₄	1	1	yes	characteristic
116	C₂³×C₁₃₇	C₄×D₄	1	1	yes	characteristic
117	C₂³×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
118	C₂³×C₁₃₇	#125	1	2	yes	—
119	C₂³×C₁₃₇	C₂×#41	1	1	yes	characteristic
120	C₂×C₄×C₁₃₇	—	2	8	no	—
121	C₂×C₄×C₁₃₇	—	2	4	no	—
122	C₂×C₄×C₁₃₇	—	4	4	no	—
123	C₂×C₄×C₁₃₇	—	8	4	no	—
124	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃₇	C₄×Q₈	1	2	yes	—
129	C₂×C₄×C₁₃₇	C₄×Q₈	1	2	yes	—
130	C₂×C₄×C₁₃₇	#121	1	2	yes	—
131	C₂×C₄×C₁₃₇	#121	1	2	yes	—
132	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
133	C₂×C₄×C₁₃₇	#125	1	2	yes	—
134	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
135	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
136	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
140	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
141	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
142	C₈×C₁₃₇	—	2	4	no	—
143	C₈×C₁₃₇	—	4	4	no	—
144	C₈×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
145	C₈×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
146	C₈×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
147	C₈×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
148	C₂²×C₁₃₇	—	2	2	no	—
149	C₂²×C₁₃₇	—	2	2	no	—
150	C₂²×C₁₃₇	—	2	2	no	—
151	C₂²×C₁₃₇	—	4	2	no	—
152	C₂²×C₁₃₇	—	4	2	no	—
153	C₂²×C₁₃₇	#389	1	2	yes	—
154	C₂²×C₁₃₇	#389	1	2	yes	—
155	C₂²×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
156	C₂²×C₁₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
157	C₂²×C₁₃₇	C₂×#125	1	1	yes	characteristic
158	C₄×C₁₃₇	—	4	4	no	—
159	C₄×C₁₃₇	—	4	8	no	—
160	C₄×C₁₃₇	#389	1	2	yes	—
161	C₄×C₁₃₇	#389	1	2	yes	—
162	C₄×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
163	C₄×C₁₃₇	C₂×C₄×Q₈	1	1	yes	characteristic
164	C₄×C₁₃₇	C₂×#121	1	1	yes	characteristic
165	C₄×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
166	C₂×C₁₃₇	—	2	1	no	—
167	C₂×C₁₃₇	—	2	1	no	—
168	C₂×C₁₃₇	—	2	2	no	—
169	C₂×C₁₃₇	—	2	2	no	—
170	C₂×C₁₃₇	C₂×#389	1	1	yes	characteristic
171	C₂×C₁₃₇	#2957	1	2	yes	—
172	#21163	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₁₃₇	#21163	1	1	yes	characteristic, direct product factor, Sylow
174	C₂²×#108	C₂×C₁₃₇	1	1	yes	characteristic
175	C₂×#427	C₂×C₁₃₇	1	1	yes	characteristic
176	#2905	C₂×C₁₃₇	1	2	yes	—
177	C₂×#405	C₂×C₁₃₇	1	1	yes	characteristic
178	#1946	C₂×C₁₃₇	1	4	yes	complemented
179	#1945	C₂×C₁₃₇	1	4	yes	complemented
180	#1733	C₂×C₁₃₇	1	2	yes	—
181	C₂²×M₁₆	C₂²×C₁₃₇	1	2	yes	—
182	C₂²×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
183	C₂³×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
184	C₂×#117	C₂²×C₁₃₇	1	2	yes	—
185	C₂×#117	C₂²×C₁₃₇	1	1	yes	characteristic
186	C₂×#108	—	2	4	no	—
187	C₂×#108	C₂²×C₁₃₇	1	2	yes	—
188	C₂×#108	C₂²×C₁₃₇	1	2	yes	—
189	#427	C₄×C₁₃₇	1	2	yes	complemented
190	#427	C₄×C₁₃₇	1	2	yes	complemented
191	#427	C₂²×C₁₃₇	1	2	yes	—
192	#427	C₂²×C₁₃₇	1	2	yes	—
193	C₂×#107	C₄×C₁₃₇	1	2	yes	complemented
194	C₂×#106	C₄×C₁₃₇	1	2	yes	complemented
195	#424	—	2	4	no	—
196	#424	C₂²×C₁₃₇	1	8	yes	—
197	#423	—	2	4	no	—
198	#405	—	2	4	no	—
199	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
200	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
201	#339	C₂²×C₁₃₇	1	8	yes	—
202	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
203	C₂×M₁₆	—	2	2	no	—
204	C₂×M₁₆	—	2	2	no	—
205	C₂×M₁₆	—	2	2	no	—
206	C₂×M₁₆	—	2	4	no	—
207	C₂×M₁₆	—	2	4	no	—
208	C₂×M₁₆	D₄×C₁₃₇	1	2	yes	—
209	C₂×M₁₆	D₄×C₁₃₇	1	2	yes	—
210	C₂×M₁₆	C₂³×C₁₃₇	1	4	yes	—
211	C₂²×C₈	—	2	2	no	—
212	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
213	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
214	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
215	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
216	C₂²×C₈	D₄×C₁₃₇	1	4	yes	—
217	C₂²×C₈	Q₈×C₁₃₇	1	4	yes	—
218	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
219	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
220	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
221	#117	—	2	2	no	—
222	#117	—	2	2	no	—
223	#117	—	2	2	no	—
224	#117	C₂×C₄×C₁₃₇	1	4	yes	—
225	#117	Q₈×C₁₃₇	1	4	yes	—
226	#117	C₂³×C₁₃₇	1	4	yes	—
227	C₂×C₄⋊C₄	—	2	1	no	—
228	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	4	yes	—
229	C₂×C₂²⋊C₄	—	2	2	no	—
230	C₂×C₂²⋊C₄	—	2	1	no	—
231	C₂×C₄²	—	2	2	no	—
232	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
233	#108	—	2	4	no	—
234	#108	—	2	4	no	—
235	#107	C₂×C₄×C₁₃₇	1	4	yes	—
236	#107	C₂×C₄×C₁₃₇	1	4	yes	—
237	#106	C₂×C₄×C₁₃₇	1	4	yes	—
238	#106	C₂×C₄×C₁₃₇	1	4	yes	—
239	C₄⋊C₈	—	2	16	no	—
240	#97	—	2	4	no	—
241	C₄×C₈	—	2	4	no	—
242	C₂⁴	#41×C₁₃₇	1	1	yes	characteristic
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
246	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
248	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
250	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
251	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
252	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
253	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
254	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
255	M₁₆	—	2	4	no	—
256	M₁₆	—	2	4	no	—
257	M₁₆	—	2	4	no	—
258	M₁₆	—	2	4	no	—
259	M₁₆	—	4	4	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	4	4	no	—
264	C₂×C₈	C₄⋊C₄×C₁₃₇	1	4	yes	—
265	C₂×C₈	C₄⋊C₄×C₁₃₇	1	4	yes	—
266	C₂×C₈	C₄⋊C₄×C₁₃₇	1	4	yes	—
267	C₂×C₈	C₄⋊C₄×C₁₃₇	1	4	yes	—
268	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
269	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
270	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
271	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
272	C₂×C₈	C₂×D₄×C₁₃₇	1	4	yes	—
273	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
274	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
275	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
276	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
277	C₄⋊C₄	—	2	4	no	—
278	C₄⋊C₄	—	2	4	no	—
279	C₄⋊C₄	C₂²×C₄×C₁₃₇	1	8	yes	—
280	C₂²⋊C₄	—	2	4	no	—
281	C₂²⋊C₄	—	2	4	no	—
282	C₂²⋊C₄	—	2	4	no	—
283	C₄²	—	2	4	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	4	no	—
286	C₄²	C₂×Q₈×C₁₃₇	1	4	yes	—
287	C₂³	—	4	2	no	—
288	C₂³	C₄×D₄×C₁₃₇	1	1	yes	characteristic
289	C₂³	C₄×D₄×C₁₃₇	1	1	yes	characteristic
290	C₂³	C₂²⋊Q₈×C₁₃₇	1	2	yes	—
291	C₂³	#125×C₁₃₇	1	2	yes	—
292	C₂³	C₂×#41×C₁₃₇	1	1	yes	characteristic
293	C₂×C₄	—	2	8	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	8	4	no	—
297	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
298	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
299	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
300	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
301	C₂×C₄	C₄×Q₈×C₁₃₇	1	2	yes	—
302	C₂×C₄	C₄×Q₈×C₁₃₇	1	2	yes	—
303	C₂×C₄	#121×C₁₃₇	1	2	yes	—
304	C₂×C₄	#121×C₁₃₇	1	2	yes	—
305	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	2	yes	—
306	C₂×C₄	#125×C₁₃₇	1	2	yes	—
307	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
308	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
309	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
310	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
311	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
312	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
313	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
314	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
315	C₈	—	2	4	no	—
316	C₈	—	4	4	no	—
317	C₈	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
318	C₈	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
319	C₈	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
320	C₈	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	4	2	no	—
325	C₂²	—	4	2	no	—
326	C₂²	#389×C₁₃₇	1	2	yes	—
327	C₂²	#389×C₁₃₇	1	2	yes	—
328	C₂²	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
329	C₂²	C₂×C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
330	C₂²	C₂×#125×C₁₃₇	1	1	yes	characteristic
331	C₄	—	4	4	no	—
332	C₄	—	4	8	no	—
333	C₄	#389×C₁₃₇	1	2	yes	—
334	C₄	#389×C₁₃₇	1	2	yes	—
335	C₄	C₂²×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
336	C₄	C₂×C₄×Q₈×C₁₃₇	1	1	yes	characteristic
337	C₄	C₂×#121×C₁₃₇	1	1	yes	characteristic
338	C₄	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
339	C₂	—	2	1	no	—
340	C₂	—	2	2	no	—
341	C₂	—	2	2	no	—
342	C₂	—	2	1	no	—
343	C₂	C₂×#389×C₁₃₇	1	1	yes	characteristic
344	C₂	#2957×C₁₃₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
socle	C₂²×C₁₃₇	C₂×C₄×D₄
prime radical	C₂⁴×C₁₃₇	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ C₂²×M₁₆×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
5. #1946×C₁₃₇	C₂	semidirect
6. #1945×C₁₃₇	C₂	semidirect
16. #427×C₁₃₇	C₄	semidirect
17. #427×C₁₃₇	C₄	semidirect
20. C₂×#107×C₁₃₇	C₄	semidirect
21. C₂×#106×C₁₃₇	C₄	semidirect
172. #21163	C₁₃₇	direct (N × H)
173. C₁₃₇	#21163	direct (N × H)
178. #1946	C₂×C₁₃₇	semidirect
179. #1945	C₂×C₁₃₇	semidirect
189. #427	C₄×C₁₃₇	semidirect
190. #427	C₄×C₁₃₇	semidirect
193. C₂×#107	C₄×C₁₃₇	semidirect
194. C₂×#106	C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#108×C₁₃₇	2
C₂×#427×C₁₃₇	2
#2905×C₁₃₇	2
C₂×#405×C₁₃₇	2
#1946×C₁₃₇	2
#1945×C₁₃₇	2
#1733×C₁₃₇	2
#21163	137

Subgroups — C137×#21163 #46870549

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#21163 #46870549