Subgroups — C₁₃₇×#20384 #46869770

Order 35072.

1566 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#131×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#408×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#405×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#97×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#2297×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	#1753×C₁₃₇	C₂	1	4	yes	maximal, complemented
7	#1752×C₁₃₇	C₂	1	4	yes	maximal, complemented
8	#1729×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#1728×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×#131×C₁₃₇	—	2	4	no	—
12	C₂×#131×C₁₃₇	—	2	4	no	—
13	C₂×#131×C₁₃₇	C₄	1	8	yes	complemented
14	C₂²×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₁₃₇	C₂²	1	2	yes	—
17	C₂³×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₁₃₇	C₂²	1	2	yes	—
19	C₂³×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₂×#117×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
22	#408×C₁₃₇	C₄	1	8	yes	complemented
23	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	4	yes	—
24	#405×C₁₃₇	—	2	4	no	—
25	C₂×#97×C₁₃₇	—	2	4	no	—
26	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
27	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
28	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
30	#391×C₁₃₇	C₄	1	2	yes	—
31	#390×C₁₃₇	C₄	1	2	yes	—
32	#386×C₁₃₇	C₄	1	4	yes	—
33	C₄×C₂²⋊C₄×C₁₃₇	C₂²	1	4	yes	—
34	#336×C₁₃₇	C₂²	1	4	yes	—
35	#336×C₁₃₇	C₂²	1	4	yes	—
36	C₂×#41×C₁₃₇	—	2	2	no	—
37	C₂×#41×C₁₃₇	—	2	2	no	—
38	C₂×#41×C₁₃₇	—	2	2	no	—
39	C₂×#41×C₁₃₇	—	2	2	no	—
40	C₂×#41×C₁₃₇	C₂×C₄	1	4	yes	—
41	C₂×#41×C₁₃₇	C₂×C₄	1	4	yes	—
42	C₂²×Q₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×D₄×C₁₃₇	C₂×C₄	1	2	yes	—
44	C₂²×D₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
46	C₂³×C₄×C₁₃₇	C₂³	1	2	yes	—
47	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
48	#131×C₁₃₇	—	2	32	no	—
49	C₂×M₁₆×C₁₃₇	—	2	2	no	—
50	C₂×M₁₆×C₁₃₇	—	2	2	no	—
51	C₂×M₁₆×C₁₃₇	—	2	2	no	—
52	C₂×M₁₆×C₁₃₇	—	2	2	no	—
53	C₂×M₁₆×C₁₃₇	—	2	4	no	—
54	C₂×M₁₆×C₁₃₇	—	2	4	no	—
55	C₂×M₁₆×C₁₃₇	C₂×C₄	1	4	yes	—
56	C₂×M₁₆×C₁₃₇	C₂×C₄	1	4	yes	—
57	C₂×M₁₆×C₁₃₇	C₂×C₄	1	8	yes	complemented
58	C₂²×C₈×C₁₃₇	—	2	2	no	—
59	C₂²×C₈×C₁₃₇	—	2	2	no	—
60	C₂²×C₈×C₁₃₇	—	2	4	no	—
61	C₂²×C₈×C₁₃₇	—	2	4	no	—
62	C₂²×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
63	C₂²×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
64	C₂²×C₈×C₁₃₇	C₂×C₄	1	8	yes	complemented
65	C₂²×C₈×C₁₃₇	D₄	1	4	yes	—
66	C₂²×C₈×C₁₃₇	D₄	1	4	yes	—
67	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
68	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
69	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
70	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
72	#117×C₁₃₇	—	2	2	no	—
73	#117×C₁₃₇	—	2	2	no	—
74	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
75	C₂×C₂²⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
76	C₂×C₂²⋊C₄×C₁₃₇	C₂×C₄	1	8	yes	—
77	C₂×C₄²×C₁₃₇	—	2	2	no	—
78	C₂×C₄²×C₁₃₇	—	2	2	no	—
79	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
80	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
81	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
82	C₂²⋊C₈×C₁₃₇	C₂×C₄	1	16	yes	complemented
83	#97×C₁₃₇	—	2	4	no	—
84	#97×C₁₃₇	—	2	4	no	—
85	#97×C₁₃₇	—	2	4	no	—
86	C₄×C₈×C₁₃₇	—	2	4	no	—
87	#95×C₁₃₇	C₂×C₄	1	8	yes	—
88	C₂⁴×C₁₃₇	C₂²×C₄	1	2	yes	—
89	C₂⁴×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
90	#41×C₁₃₇	—	2	8	no	—
91	#41×C₁₃₇	—	2	8	no	—
92	#41×C₁₃₇	—	2	8	no	—
93	#41×C₁₃₇	—	2	8	no	—
94	C₂×Q₈×C₁₃₇	—	2	2	no	—
95	C₂×Q₈×C₁₃₇	—	2	2	no	—
96	C₂×Q₈×C₁₃₇	C₄²	1	4	yes	—
97	C₂×D₄×C₁₃₇	—	2	4	no	—
98	C₂×D₄×C₁₃₇	—	2	4	no	—
99	C₂×D₄×C₁₃₇	—	2	2	no	—
100	C₂×D₄×C₁₃₇	—	2	2	no	—
101	C₂×D₄×C₁₃₇	C₄²	1	8	yes	—
102	C₂×D₄×C₁₃₇	C₄²	1	4	yes	—
103	C₂²×C₄×C₁₃₇	—	2	2	no	—
104	C₂²×C₄×C₁₃₇	—	2	2	no	—
105	C₂²×C₄×C₁₃₇	—	2	2	no	—
106	C₂²×C₄×C₁₃₇	—	2	2	no	—
107	C₂²×C₄×C₁₃₇	—	2	4	no	—
108	C₂²×C₄×C₁₃₇	—	4	4	no	—
109	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
110	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
111	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
112	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
113	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
114	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
116	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
117	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
118	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
120	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
121	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
122	M₁₆×C₁₃₇	—	2	8	no	—
123	M₁₆×C₁₃₇	—	2	8	no	—
124	M₁₆×C₁₃₇	—	2	16	no	—
125	C₂×C₈×C₁₃₇	—	2	8	no	—
126	C₂×C₈×C₁₃₇	—	2	2	no	—
127	C₂×C₈×C₁₃₇	—	2	2	no	—
128	C₂×C₈×C₁₃₇	—	2	2	no	—
129	C₂×C₈×C₁₃₇	—	2	2	no	—
130	C₂×C₈×C₁₃₇	—	2	2	no	—
131	C₂×C₈×C₁₃₇	—	2	2	no	—
132	C₂×C₈×C₁₃₇	—	2	16	no	—
133	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	16	yes	complemented
134	C₂×C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
135	C₂×C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
136	C₂×C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
137	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
138	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
139	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
140	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
141	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
142	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
143	C₄⋊C₄×C₁₃₇	—	2	4	no	—
144	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
145	C₂²⋊C₄×C₁₃₇	—	4	8	no	—
146	C₄²×C₁₃₇	—	2	4	no	—
147	C₄²×C₁₃₇	—	2	2	no	—
148	C₄²×C₁₃₇	—	2	2	no	—
149	C₄²×C₁₃₇	—	2	2	no	—
150	C₄²×C₁₃₇	—	2	2	no	—
151	C₂³×C₁₃₇	—	2	4	no	—
152	C₂³×C₁₃₇	—	2	2	no	—
153	C₂³×C₁₃₇	—	2	2	no	—
154	C₂³×C₁₃₇	—	2	8	no	—
155	C₂³×C₁₃₇	C₂×C₄²	1	4	yes	—
156	C₂³×C₁₃₇	C₂×C₄²	1	2	yes	—
157	C₂³×C₁₃₇	C₄×D₄	1	2	yes	—
158	C₂³×C₁₃₇	C₄×D₄	1	2	yes	—
159	C₂³×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
160	Q₈×C₁₃₇	—	2	8	no	—
161	D₄×C₁₃₇	—	2	16	no	—
162	D₄×C₁₃₇	—	2	8	no	—
163	C₂×C₄×C₁₃₇	—	2	2	no	—
164	C₂×C₄×C₁₃₇	—	2	2	no	—
165	C₂×C₄×C₁₃₇	—	2	4	no	—
166	C₂×C₄×C₁₃₇	—	2	4	no	—
167	C₂×C₄×C₁₃₇	—	2	4	no	—
168	C₂×C₄×C₁₃₇	—	2	4	no	—
169	C₂×C₄×C₁₃₇	—	2	4	no	—
170	C₂×C₄×C₁₃₇	—	2	4	no	—
171	C₂×C₄×C₁₃₇	—	2	4	no	—
172	C₂×C₄×C₁₃₇	—	2	8	no	—
173	C₂×C₄×C₁₃₇	—	2	4	no	—
174	C₂×C₄×C₁₃₇	—	4	8	no	—
175	C₂×C₄×C₁₃₇	C₂×C₄²	1	4	yes	—
176	C₂×C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
177	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	4	yes	—
178	C₂×C₄×C₁₃₇	#117	1	4	yes	—
179	C₂×C₄×C₁₃₇	C₄×D₄	1	4	yes	—
180	C₂×C₄×C₁₃₇	C₄×D₄	1	4	yes	—
181	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
182	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
183	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
184	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
185	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
188	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
189	C₈×C₁₃₇	—	2	4	no	—
190	C₈×C₁₃₇	—	2	4	no	—
191	C₈×C₁₃₇	—	2	4	no	—
192	C₈×C₁₃₇	C₂×C₂²⋊C₄	1	4	yes	—
193	C₈×C₁₃₇	C₂×C₂²⋊C₄	1	4	yes	—
194	C₂²×C₁₃₇	—	2	4	no	—
195	C₂²×C₁₃₇	—	2	4	no	—
196	C₂²×C₁₃₇	—	2	4	no	—
197	C₂²×C₁₃₇	—	2	8	no	—
198	C₂²×C₁₃₇	—	2	4	no	—
199	C₂²×C₁₃₇	—	4	4	no	—
200	C₂²×C₁₃₇	—	4	4	no	—
201	C₂²×C₁₃₇	C₄×C₂²⋊C₄	1	4	yes	—
202	C₂²×C₁₃₇	C₂²×C₄²	1	1	yes	characteristic
203	C₂²×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
204	C₂²×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
205	C₂²×C₁₃₇	#568	1	4	yes	—
206	C₄×C₁₃₇	—	2	4	no	—
207	C₄×C₁₃₇	—	4	4	no	—
208	C₄×C₁₃₇	—	4	4	no	—
209	C₄×C₁₃₇	C₄×C₂²⋊C₄	1	4	yes	—
210	C₄×C₁₃₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
211	C₄×C₁₃₇	C₂×#117	1	1	yes	characteristic
212	C₄×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
213	C₄×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
214	C₂×C₁₃₇	—	2	4	no	—
215	C₂×C₁₃₇	—	4	4	no	—
216	C₂×C₁₃₇	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
217	C₂×C₁₃₇	#2875	1	2	yes	—
218	C₂×C₁₃₇	#2930	1	2	yes	—
219	C₂×C₁₃₇	#2932	1	2	yes	—
220	#20384	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
221	C₁₃₇	#20384	1	1	yes	characteristic, direct product factor, Sylow
222	C₂²×#131	C₂×C₁₃₇	1	1	yes	characteristic
223	C₂×#408	C₂×C₁₃₇	1	1	yes	characteristic
224	C₂×#405	C₂×C₁₃₇	1	1	yes	characteristic, complemented
225	C₂²×#97	C₂×C₁₃₇	1	1	yes	characteristic, complemented
226	#2297	C₂×C₁₃₇	1	1	yes	characteristic
227	#1753	C₂×C₁₃₇	1	4	yes	complemented
228	#1752	C₂×C₁₃₇	1	4	yes	complemented
229	#1729	C₂×C₁₃₇	1	1	yes	characteristic, complemented
230	#1728	C₂×C₁₃₇	1	1	yes	characteristic, complemented
231	C₂²×#41	C₂²×C₁₃₇	1	1	yes	characteristic
232	C₂×#131	—	2	4	no	—
233	C₂×#131	—	2	4	no	—
234	C₂×#131	C₄×C₁₃₇	1	8	yes	complemented
235	C₂²×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
236	C₂²×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
237	C₂²×M₁₆	C₂²×C₁₃₇	1	2	yes	—
238	C₂³×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
239	C₂³×C₈	C₂²×C₁₃₇	1	2	yes	—
240	C₂³×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
241	C₂×#117	C₂²×C₁₃₇	1	1	yes	characteristic
242	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
243	#408	C₄×C₁₃₇	1	8	yes	complemented
244	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	4	yes	—
245	#405	—	2	4	no	—
246	C₂×#97	—	2	4	no	—
247	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
248	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
249	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
250	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
251	#391	C₄×C₁₃₇	1	2	yes	—
252	#390	C₄×C₁₃₇	1	2	yes	—
253	#386	C₄×C₁₃₇	1	4	yes	—
254	C₄×C₂²⋊C₄	C₂²×C₁₃₇	1	4	yes	—
255	#336	C₂²×C₁₃₇	1	4	yes	—
256	#336	C₂²×C₁₃₇	1	4	yes	—
257	C₂×#41	—	2	2	no	—
258	C₂×#41	—	2	2	no	—
259	C₂×#41	—	2	2	no	—
260	C₂×#41	—	2	2	no	—
261	C₂×#41	C₂×C₄×C₁₃₇	1	4	yes	—
262	C₂×#41	C₂×C₄×C₁₃₇	1	4	yes	—
263	C₂²×Q₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
264	C₂²×D₄	C₂×C₄×C₁₃₇	1	2	yes	—
265	C₂²×D₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
266	C₂³×C₄	C₂×C₄×C₁₃₇	1	2	yes	—
267	C₂³×C₄	C₂³×C₁₃₇	1	2	yes	—
268	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
269	#131	—	2	32	no	—
270	C₂×M₁₆	—	2	2	no	—
271	C₂×M₁₆	—	2	2	no	—
272	C₂×M₁₆	—	2	2	no	—
273	C₂×M₁₆	—	2	2	no	—
274	C₂×M₁₆	—	2	4	no	—
275	C₂×M₁₆	—	2	4	no	—
276	C₂×M₁₆	C₂×C₄×C₁₃₇	1	4	yes	—
277	C₂×M₁₆	C₂×C₄×C₁₃₇	1	4	yes	—
278	C₂×M₁₆	C₂×C₄×C₁₃₇	1	8	yes	complemented
279	C₂²×C₈	—	2	2	no	—
280	C₂²×C₈	—	2	2	no	—
281	C₂²×C₈	—	2	4	no	—
282	C₂²×C₈	—	2	4	no	—
283	C₂²×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
284	C₂²×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
285	C₂²×C₈	C₂×C₄×C₁₃₇	1	8	yes	complemented
286	C₂²×C₈	D₄×C₁₃₇	1	4	yes	—
287	C₂²×C₈	D₄×C₁₃₇	1	4	yes	—
288	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
289	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
290	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
291	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
292	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
293	#117	—	2	2	no	—
294	#117	—	2	2	no	—
295	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	2	yes	—
296	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₇	1	2	yes	—
297	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₇	1	8	yes	—
298	C₂×C₄²	—	2	2	no	—
299	C₂×C₄²	—	2	2	no	—
300	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
301	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
302	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
303	C₂²⋊C₈	C₂×C₄×C₁₃₇	1	16	yes	complemented
304	#97	—	2	4	no	—
305	#97	—	2	4	no	—
306	#97	—	2	4	no	—
307	C₄×C₈	—	2	4	no	—
308	#95	C₂×C₄×C₁₃₇	1	8	yes	—
309	C₂⁴	C₂²×C₄×C₁₃₇	1	2	yes	—
310	C₂⁴	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
311	#41	—	2	8	no	—
312	#41	—	2	8	no	—
313	#41	—	2	8	no	—
314	#41	—	2	8	no	—
315	C₂×Q₈	—	2	2	no	—
316	C₂×Q₈	—	2	2	no	—
317	C₂×Q₈	C₄²×C₁₃₇	1	4	yes	—
318	C₂×D₄	—	2	4	no	—
319	C₂×D₄	—	2	4	no	—
320	C₂×D₄	—	2	2	no	—
321	C₂×D₄	—	2	2	no	—
322	C₂×D₄	C₄²×C₁₃₇	1	8	yes	—
323	C₂×D₄	C₄²×C₁₃₇	1	4	yes	—
324	C₂²×C₄	—	2	2	no	—
325	C₂²×C₄	—	2	2	no	—
326	C₂²×C₄	—	2	2	no	—
327	C₂²×C₄	—	2	2	no	—
328	C₂²×C₄	—	2	4	no	—
329	C₂²×C₄	—	4	4	no	—
330	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
331	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
332	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
333	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
334	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
335	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
336	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
337	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
338	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
339	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
340	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
341	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
342	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
343	M₁₆	—	2	8	no	—
344	M₁₆	—	2	8	no	—
345	M₁₆	—	2	16	no	—
346	C₂×C₈	—	2	8	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	2	2	no	—
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	—	2	2	no	—
352	C₂×C₈	—	2	2	no	—
353	C₂×C₈	—	2	16	no	—
354	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	16	yes	complemented
355	C₂×C₈	C₂²×C₄×C₁₃₇	1	4	yes	—
356	C₂×C₈	C₂²×C₄×C₁₃₇	1	4	yes	—
357	C₂×C₈	C₂²×C₄×C₁₃₇	1	4	yes	—
358	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
359	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
360	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
361	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
362	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
363	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
364	C₄⋊C₄	—	2	4	no	—
365	C₂²⋊C₄	—	2	4	no	—
366	C₂²⋊C₄	—	4	8	no	—
367	C₄²	—	2	4	no	—
368	C₄²	—	2	2	no	—
369	C₄²	—	2	2	no	—
370	C₄²	—	2	2	no	—
371	C₄²	—	2	2	no	—
372	C₂³	—	2	4	no	—
373	C₂³	—	2	2	no	—
374	C₂³	—	2	2	no	—
375	C₂³	—	2	8	no	—
376	C₂³	C₂×C₄²×C₁₃₇	1	4	yes	—
377	C₂³	C₂×C₄²×C₁₃₇	1	2	yes	—
378	C₂³	C₄×D₄×C₁₃₇	1	2	yes	—
379	C₂³	C₄×D₄×C₁₃₇	1	2	yes	—
380	C₂³	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
381	Q₈	—	2	8	no	—
382	D₄	—	2	16	no	—
383	D₄	—	2	8	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	4	no	—
387	C₂×C₄	—	2	4	no	—
388	C₂×C₄	—	2	4	no	—
389	C₂×C₄	—	2	4	no	—
390	C₂×C₄	—	2	4	no	—
391	C₂×C₄	—	2	4	no	—
392	C₂×C₄	—	2	4	no	—
393	C₂×C₄	—	2	8	no	—
394	C₂×C₄	—	2	4	no	—
395	C₂×C₄	—	4	8	no	—
396	C₂×C₄	C₂×C₄²×C₁₃₇	1	4	yes	—
397	C₂×C₄	C₂×C₄²×C₁₃₇	1	2	yes	—
398	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	4	yes	—
399	C₂×C₄	#117×C₁₃₇	1	4	yes	—
400	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
401	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
402	C₂×C₄	C₄×D₄×C₁₃₇	1	4	yes	—
403	C₂×C₄	C₄×D₄×C₁₃₇	1	4	yes	—
404	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
405	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
406	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
407	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
408	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
409	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
410	C₈	—	2	4	no	—
411	C₈	—	2	4	no	—
412	C₈	—	2	4	no	—
413	C₈	C₂×C₂²⋊C₄×C₁₃₇	1	4	yes	—
414	C₈	C₂×C₂²⋊C₄×C₁₃₇	1	4	yes	—
415	C₂²	—	2	4	no	—
416	C₂²	—	2	4	no	—
417	C₂²	—	2	4	no	—
418	C₂²	—	2	8	no	—
419	C₂²	—	2	4	no	—
420	C₂²	—	4	4	no	—
421	C₂²	—	4	4	no	—
422	C₂²	C₄×C₂²⋊C₄×C₁₃₇	1	4	yes	—
423	C₂²	C₂²×C₄²×C₁₃₇	1	1	yes	characteristic
424	C₂²	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
425	C₂²	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
426	C₂²	#568×C₁₃₇	1	4	yes	—
427	C₄	—	2	4	no	—
428	C₄	—	4	4	no	—
429	C₄	—	4	4	no	—
430	C₄	C₄×C₂²⋊C₄×C₁₃₇	1	4	yes	—
431	C₄	C₂²×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
432	C₄	C₂×#117×C₁₃₇	1	1	yes	characteristic
433	C₄	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
434	C₄	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
435	C₂	—	2	4	no	—
436	C₂	—	4	4	no	—
437	C₂	C₂×C₄×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
438	C₂	#2875×C₁₃₇	1	2	yes	—
439	C₂	#2930×C₁₃₇	1	2	yes	—
440	C₂	#2932×C₁₃₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃₇	C₂⁴
commutator	C₂²	C₂²×C₄²×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
socle	C₂³×C₁₃₇	C₂³×C₄
prime radical	C₂²×#41×C₁₃₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#405×C₁₃₇	C₂	semidirect
4. C₂²×#97×C₁₃₇	C₂	semidirect
6. #1753×C₁₃₇	C₂	semidirect
7. #1752×C₁₃₇	C₂	semidirect
8. #1729×C₁₃₇	C₂	semidirect
9. #1728×C₁₃₇	C₂	semidirect
13. C₂×#131×C₁₃₇	C₄	semidirect
22. #408×C₁₃₇	C₄	semidirect
57. C₂×M₁₆×C₁₃₇	C₂×C₄	semidirect
64. C₂²×C₈×C₁₃₇	C₂×C₄	semidirect
82. C₂²⋊C₈×C₁₃₇	C₂×C₄	semidirect
133. C₂×C₈×C₁₃₇	C₂²⋊C₄	semidirect
220. #20384	C₁₃₇	direct (N × H)
221. C₁₃₇	#20384	direct (N × H)
224. C₂×#405	C₂×C₁₃₇	semidirect
225. C₂²×#97	C₂×C₁₃₇	semidirect
227. #1753	C₂×C₁₃₇	semidirect
228. #1752	C₂×C₁₃₇	semidirect
229. #1729	C₂×C₁₃₇	semidirect
230. #1728	C₂×C₁₃₇	semidirect
234. C₂×#131	C₄×C₁₃₇	semidirect
243. #408	C₄×C₁₃₇	semidirect
278. C₂×M₁₆	C₂×C₄×C₁₃₇	semidirect
285. C₂²×C₈	C₂×C₄×C₁₃₇	semidirect
303. C₂²⋊C₈	C₂×C₄×C₁₃₇	semidirect
354. C₂×C₈	C₂²⋊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×#131×C₁₃₇	2
C₂×#408×C₁₃₇	2
C₂×#405×C₁₃₇	2
C₂²×#97×C₁₃₇	2
#2297×C₁₃₇	2
#1753×C₁₃₇	2
#1752×C₁₃₇	2
#1729×C₁₃₇	2
#1728×C₁₃₇	2
#20384	137

Subgroups — C137×#20384 #46869770

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#20384 #46869770