Subgroups — C₁₃₇×#20535 #46869921

Order 35072.

1038 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#432×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#431×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊C₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1771×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#1770×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	C₈×C₄⋊C₄×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#1749×C₁₃₇	C₂	1	2	yes	maximal
8	C₄×C₂²⋊C₈×C₁₃₇	C₂	1	2	yes	maximal
9	C₂²×C₄⋊C₄×C₁₃₇	C₄	1	2	yes	—
10	C₂²×C₄²×C₁₃₇	C₂²	1	2	yes	—
11	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
12	#432×C₁₃₇	C₄	1	8	yes	complemented
13	#431×C₁₃₇	C₄	1	8	yes	complemented
14	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	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×C₂²⋊C₈×C₁₃₇	C₂²	1	2	yes	—
19	C₂×#97×C₁₃₇	C₂²	1	4	yes	—
20	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₃₇	C₂²	1	4	yes	—
22	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
23	C₄×C₄⋊C₄×C₁₃₇	C₄	1	2	yes	—
24	C₄×C₄⋊C₄×C₁₃₇	C₄	1	2	yes	—
25	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
28	C₂×#95×C₁₃₇	C₄	1	2	yes	—
29	#336×C₁₃₇	C₂²	1	4	yes	—
30	#336×C₁₃₇	C₂²	1	4	yes	—
31	C₂³×C₄×C₁₃₇	C₂×C₄	1	4	yes	—
32	C₂³×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
37	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
38	C₂×C₄⋊C₄×C₁₃₇	—	2	8	no	—
39	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
41	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₁₃₇	—	4	4	no	—
43	C₂×C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
49	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
52	C₄⋊C₈×C₁₃₇	C₂×C₄	1	8	yes	complemented
53	C₄⋊C₈×C₁₃₇	C₂×C₄	1	8	yes	complemented
54	C₂²⋊C₈×C₁₃₇	C₂×C₄	1	4	yes	—
55	C₂²⋊C₈×C₁₃₇	C₂×C₄	1	4	yes	—
56	C₂²⋊C₈×C₁₃₇	C₂×C₄	1	8	yes	—
57	#97×C₁₃₇	—	4	4	no	—
58	#97×C₁₃₇	C₂×C₄	1	8	yes	complemented
59	C₄×C₈×C₁₃₇	—	4	4	no	—
60	C₄×C₈×C₁₃₇	C₂×C₄	1	8	yes	complemented
61	#95×C₁₃₇	C₂×C₄	1	2	yes	—
62	#95×C₁₃₇	C₂×C₄	1	4	yes	—
63	#95×C₁₃₇	C₂×C₄	1	2	yes	—
64	C₂⁴×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃₇	—	2	4	no	—
66	C₂²×C₄×C₁₃₇	—	2	4	no	—
67	C₂²×C₄×C₁₃₇	—	2	2	no	—
68	C₂²×C₄×C₁₃₇	—	2	2	no	—
69	C₂²×C₄×C₁₃₇	—	2	8	no	—
70	C₂²×C₄×C₁₃₇	—	2	8	no	—
71	C₂²×C₄×C₁₃₇	—	4	4	no	—
72	C₂²×C₄×C₁₃₇	C₄²	1	4	yes	—
73	C₂²×C₄×C₁₃₇	C₄²	1	4	yes	—
74	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
82	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
84	C₂×C₈×C₁₃₇	—	2	2	no	—
85	C₂×C₈×C₁₃₇	—	2	4	no	—
86	C₂×C₈×C₁₃₇	—	2	2	no	—
87	C₂×C₈×C₁₃₇	—	2	2	no	—
88	C₂×C₈×C₁₃₇	—	2	4	no	—
89	C₂×C₈×C₁₃₇	—	2	2	no	—
90	C₂×C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
92	C₂×C₈×C₁₃₇	C₂²×C₄	1	8	yes	—
93	C₄⋊C₄×C₁₃₇	—	2	8	no	—
94	C₄⋊C₄×C₁₃₇	—	2	4	no	—
95	C₄⋊C₄×C₁₃₇	M₁₆	1	8	yes	—
96	C₄²×C₁₃₇	—	2	4	no	—
97	C₄²×C₁₃₇	—	2	4	no	—
98	C₄²×C₁₃₇	—	4	4	no	—
99	C₄²×C₁₃₇	C₄²	1	4	yes	—
100	C₄²×C₁₃₇	C₄²	1	4	yes	—
101	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
102	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
103	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
104	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
105	C₂³×C₁₃₇	—	2	1	no	—
106	C₂³×C₁₃₇	—	2	1	no	—
107	C₂³×C₁₃₇	—	2	2	no	—
108	C₂³×C₁₃₇	C₂×C₄²	1	2	yes	—
109	C₂³×C₁₃₇	#131	1	4	yes	—
110	C₂³×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃₇	—	2	4	no	—
112	C₂×C₄×C₁₃₇	—	2	1	no	—
113	C₂×C₄×C₁₃₇	—	2	1	no	—
114	C₂×C₄×C₁₃₇	—	2	2	no	—
115	C₂×C₄×C₁₃₇	—	2	2	no	—
116	C₂×C₄×C₁₃₇	—	2	2	no	—
117	C₂×C₄×C₁₃₇	—	2	4	no	—
118	C₂×C₄×C₁₃₇	—	2	8	no	—
119	C₂×C₄×C₁₃₇	—	2	4	no	—
120	C₂×C₄×C₁₃₇	—	2	4	no	—
121	C₂×C₄×C₁₃₇	—	2	8	no	—
122	C₂×C₄×C₁₃₇	—	2	4	no	—
123	C₂×C₄×C₁₃₇	—	2	4	no	—
124	C₂×C₄×C₁₃₇	—	2	4	no	—
125	C₂×C₄×C₁₃₇	—	4	16	no	—
126	C₂×C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
127	C₂×C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
128	C₂×C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
129	C₂×C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
130	C₂×C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
131	C₂×C₄×C₁₃₇	C₂×M₁₆	1	8	yes	—
132	C₂×C₄×C₁₃₇	C₂×M₁₆	1	4	yes	—
133	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	2	yes	—
136	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
137	C₈×C₁₃₇	—	2	4	no	—
138	C₈×C₁₃₇	—	2	8	no	—
139	C₈×C₁₃₇	—	2	4	no	—
140	C₂²×C₁₃₇	—	2	2	no	—
141	C₂²×C₁₃₇	—	2	4	no	—
142	C₂²×C₁₃₇	—	2	4	no	—
143	C₂²×C₁₃₇	—	2	2	no	—
144	C₂²×C₁₃₇	#405	1	4	yes	—
145	C₂²×C₁₃₇	C₂²×C₄²	1	1	yes	characteristic
146	C₂²×C₁₃₇	#518	1	1	yes	characteristic
147	C₂²×C₁₃₇	#519	1	1	yes	characteristic
148	C₂²×C₁₃₇	C₂²×M₁₆	1	2	yes	—
149	C₂²×C₁₃₇	C₂×#131	1	2	yes	—
150	C₄×C₁₃₇	—	2	2	no	—
151	C₄×C₁₃₇	—	2	2	no	—
152	C₄×C₁₃₇	—	2	2	no	—
153	C₄×C₁₃₇	—	2	8	no	—
154	C₄×C₁₃₇	—	2	2	no	—
155	C₄×C₁₃₇	—	4	4	no	—
156	C₄×C₁₃₇	C₄×M₁₆	1	4	yes	—
157	C₄×C₁₃₇	#520	1	2	yes	—
158	C₄×C₁₃₇	#520	1	2	yes	—
159	C₂×C₁₃₇	—	2	4	no	—
160	C₂×C₁₃₇	#2274	1	1	yes	characteristic
161	C₂×C₁₃₇	C₂×C₄×M₁₆	1	1	yes	characteristic
162	C₂×C₁₃₇	C₂×#405	1	1	yes	characteristic
163	C₂×C₁₃₇	#2974	1	2	yes	—
164	C₂×C₁₃₇	#2984	1	2	yes	—
165	#20535	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₁₃₇	#20535	1	1	yes	characteristic, direct product factor, Sylow
167	C₂×#432	C₂×C₁₃₇	1	1	yes	characteristic
168	C₂×#431	C₂×C₁₃₇	1	1	yes	characteristic
169	C₂×C₄×C₄⋊C₄	C₂×C₁₃₇	1	1	yes	characteristic
170	#1771	C₂×C₁₃₇	1	4	yes	complemented
171	#1770	C₂×C₁₃₇	1	2	yes	complemented
172	C₈×C₄⋊C₄	C₂×C₁₃₇	1	2	yes	complemented
173	#1749	C₂×C₁₃₇	1	2	yes	—
174	C₄×C₂²⋊C₈	C₂×C₁₃₇	1	2	yes	—
175	C₂²×C₄⋊C₄	C₄×C₁₃₇	1	2	yes	—
176	C₂²×C₄²	C₂²×C₁₃₇	1	2	yes	—
177	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
178	#432	C₄×C₁₃₇	1	8	yes	complemented
179	#431	C₄×C₁₃₇	1	8	yes	complemented
180	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
181	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
182	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
183	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
184	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	2	yes	—
185	C₂×#97	C₂²×C₁₃₇	1	4	yes	—
186	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
187	C₂×C₄×C₈	C₂²×C₁₃₇	1	4	yes	—
188	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
189	C₄×C₄⋊C₄	C₄×C₁₃₇	1	2	yes	—
190	C₄×C₄⋊C₄	C₄×C₁₃₇	1	2	yes	—
191	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
192	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
193	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
194	C₂×#95	C₄×C₁₃₇	1	2	yes	—
195	#336	C₂²×C₁₃₇	1	4	yes	—
196	#336	C₂²×C₁₃₇	1	4	yes	—
197	C₂³×C₄	C₂×C₄×C₁₃₇	1	4	yes	—
198	C₂³×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
199	C₂³×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
200	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
201	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
202	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
203	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
204	C₂×C₄⋊C₄	—	2	8	no	—
205	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	2	yes	—
206	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	4	yes	—
207	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	2	yes	—
208	C₂×C₄²	—	4	4	no	—
209	C₂×C₄²	C₂×C₄×C₁₃₇	1	4	yes	—
210	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
211	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
212	C₂×C₄²	C₂×C₄×C₁₃₇	1	4	yes	—
213	C₂×C₄²	C₂×C₄×C₁₃₇	1	4	yes	—
214	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
215	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
216	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
217	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
218	C₄⋊C₈	C₂×C₄×C₁₃₇	1	8	yes	complemented
219	C₄⋊C₈	C₂×C₄×C₁₃₇	1	8	yes	complemented
220	C₂²⋊C₈	C₂×C₄×C₁₃₇	1	4	yes	—
221	C₂²⋊C₈	C₂×C₄×C₁₃₇	1	4	yes	—
222	C₂²⋊C₈	C₂×C₄×C₁₃₇	1	8	yes	—
223	#97	—	4	4	no	—
224	#97	C₂×C₄×C₁₃₇	1	8	yes	complemented
225	C₄×C₈	—	4	4	no	—
226	C₄×C₈	C₂×C₄×C₁₃₇	1	8	yes	complemented
227	#95	C₂×C₄×C₁₃₇	1	2	yes	—
228	#95	C₂×C₄×C₁₃₇	1	4	yes	—
229	#95	C₂×C₄×C₁₃₇	1	2	yes	—
230	C₂⁴	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	8	no	—
236	C₂²×C₄	—	2	8	no	—
237	C₂²×C₄	—	4	4	no	—
238	C₂²×C₄	C₄²×C₁₃₇	1	4	yes	—
239	C₂²×C₄	C₄²×C₁₃₇	1	4	yes	—
240	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
241	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
242	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
243	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
244	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
245	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
246	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
247	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
248	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
249	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	C₂²×C₄×C₁₃₇	1	4	yes	—
257	C₂×C₈	C₂²×C₄×C₁₃₇	1	4	yes	—
258	C₂×C₈	C₂²×C₄×C₁₃₇	1	8	yes	—
259	C₄⋊C₄	—	2	8	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	M₁₆×C₁₃₇	1	8	yes	—
262	C₄²	—	2	4	no	—
263	C₄²	—	2	4	no	—
264	C₄²	—	4	4	no	—
265	C₄²	C₄²×C₁₃₇	1	4	yes	—
266	C₄²	C₄²×C₁₃₇	1	4	yes	—
267	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
268	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
269	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
270	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
271	C₂³	—	2	1	no	—
272	C₂³	—	2	1	no	—
273	C₂³	—	2	2	no	—
274	C₂³	C₂×C₄²×C₁₃₇	1	2	yes	—
275	C₂³	#131×C₁₃₇	1	4	yes	—
276	C₂³	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	8	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	4	16	no	—
292	C₂×C₄	C₂×C₄²×C₁₃₇	1	2	yes	—
293	C₂×C₄	C₂×C₄²×C₁₃₇	1	2	yes	—
294	C₂×C₄	C₂×C₄²×C₁₃₇	1	2	yes	—
295	C₂×C₄	C₂×C₄²×C₁₃₇	1	2	yes	—
296	C₂×C₄	C₂×C₄²×C₁₃₇	1	2	yes	—
297	C₂×C₄	C₂×M₁₆×C₁₃₇	1	8	yes	—
298	C₂×C₄	C₂×M₁₆×C₁₃₇	1	4	yes	—
299	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
300	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
301	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	2	yes	—
302	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
303	C₈	—	2	4	no	—
304	C₈	—	2	8	no	—
305	C₈	—	2	4	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	4	no	—
309	C₂²	—	2	4	no	—
310	C₂²	#405×C₁₃₇	1	4	yes	—
311	C₂²	C₂²×C₄²×C₁₃₇	1	1	yes	characteristic
312	C₂²	#518×C₁₃₇	1	1	yes	characteristic
313	C₂²	#519×C₁₃₇	1	1	yes	characteristic
314	C₂²	C₂²×M₁₆×C₁₃₇	1	2	yes	—
315	C₂²	C₂×#131×C₁₃₇	1	2	yes	—
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	8	no	—
321	C₄	—	4	4	no	—
322	C₄	C₄×M₁₆×C₁₃₇	1	4	yes	—
323	C₄	#520×C₁₃₇	1	2	yes	—
324	C₄	#520×C₁₃₇	1	2	yes	—
325	C₂	—	2	4	no	—
326	C₂	#2274×C₁₃₇	1	1	yes	characteristic
327	C₂	C₂×C₄×M₁₆×C₁₃₇	1	1	yes	characteristic
328	C₂	C₂×#405×C₁₃₇	1	1	yes	characteristic
329	C₂	#2974×C₁₃₇	1	2	yes	—
330	C₂	#2984×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	C₂⁴×C₁₃₇	C₂²×C₄
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
4. #1771×C₁₃₇	C₂	semidirect
5. #1770×C₁₃₇	C₂	semidirect
6. C₈×C₄⋊C₄×C₁₃₇	C₂	semidirect
12. #432×C₁₃₇	C₄	semidirect
13. #431×C₁₃₇	C₄	semidirect
52. C₄⋊C₈×C₁₃₇	C₂×C₄	semidirect
53. C₄⋊C₈×C₁₃₇	C₂×C₄	semidirect
58. #97×C₁₃₇	C₂×C₄	semidirect
60. C₄×C₈×C₁₃₇	C₂×C₄	semidirect
165. #20535	C₁₃₇	direct (N × H)
166. C₁₃₇	#20535	direct (N × H)
170. #1771	C₂×C₁₃₇	semidirect
171. #1770	C₂×C₁₃₇	semidirect
172. C₈×C₄⋊C₄	C₂×C₁₃₇	semidirect
178. #432	C₄×C₁₃₇	semidirect
179. #431	C₄×C₁₃₇	semidirect
218. C₄⋊C₈	C₂×C₄×C₁₃₇	semidirect
219. C₄⋊C₈	C₂×C₄×C₁₃₇	semidirect
224. #97	C₂×C₄×C₁₃₇	semidirect
226. 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×#432×C₁₃₇	2
C₂×#431×C₁₃₇	2
C₂×C₄×C₄⋊C₄×C₁₃₇	2
#1771×C₁₃₇	2
#1770×C₁₃₇	2
C₈×C₄⋊C₄×C₁₃₇	2
#1749×C₁₃₇	2
C₄×C₂²⋊C₈×C₁₃₇	2
#20535	137

Subgroups — C137×#20535 #46869921

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#20535 #46869921