Subgroups — C₁₃₇×#22650 #46872036

Order 35072.

1096 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#436×C₁₃₇	C₂	1	2	yes	maximal
2	C₂×#423×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	C₄²×D₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1939×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#1851×C₁₃₇	C₂	1	4	yes	maximal, complemented
6	#1842×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#1797×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	C₄×C₄⋊C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×M₁₆×C₁₃₇	C₂²	1	4	yes	—
10	C₂×C₄×D₄×C₁₃₇	C₄	1	1	yes	characteristic
11	C₂×C₄×D₄×C₁₃₇	C₄	1	1	yes	characteristic
12	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₁₃₇	C₂²	1	2	yes	—
14	#436×C₁₃₇	—	2	8	no	—
15	#423×C₁₃₇	—	2	8	no	—
16	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	4	yes	—
20	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
21	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	4	yes	—
23	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₁₃₇	C₄	1	1	yes	characteristic
25	C₄×C₄⋊C₄×C₁₃₇	C₄	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
27	C₄×C₂²⋊C₄×C₁₃₇	C₄	1	2	yes	—
28	C₄×C₂²⋊C₄×C₁₃₇	C₄	1	2	yes	—
29	C₄×C₂²⋊C₄×C₁₃₇	C₂²	1	2	yes	—
30	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
31	#336×C₁₃₇	C₂²	1	2	yes	—
32	#336×C₁₃₇	C₂²	1	4	yes	—
33	C₂²×D₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
35	C₂³×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
36	C₂³×C₄×C₁₃₇	C₂³	1	2	yes	—
37	C₂×M₁₆×C₁₃₇	—	2	8	no	—
38	C₂×M₁₆×C₁₃₇	—	2	8	no	—
39	C₂×M₁₆×C₁₃₇	—	4	4	no	—
40	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
42	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
43	C₄×D₄×C₁₃₇	—	2	4	no	—
44	C₄×D₄×C₁₃₇	—	2	4	no	—
45	C₄×D₄×C₁₃₇	D₄	1	4	yes	—
46	C₄×D₄×C₁₃₇	Q₈	1	4	yes	—
47	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₂²⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
51	C₂×C₂²⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
52	C₂×C₂²⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₁₃₇	—	2	4	no	—
54	C₂×C₄²×C₁₃₇	—	2	4	no	—
55	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
58	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
59	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
63	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
65	C₄⋊C₈×C₁₃₇	—	2	8	no	—
66	C₄⋊C₈×C₁₃₇	—	2	4	no	—
67	C₄⋊C₈×C₁₃₇	—	2	4	no	—
68	C₄⋊C₈×C₁₃₇	—	2	2	no	—
69	C₄⋊C₈×C₁₃₇	—	4	2	no	—
70	C₄⋊C₈×C₁₃₇	D₄	1	4	yes	complemented
71	C₂²⋊C₈×C₁₃₇	—	2	8	no	—
72	C₄×C₈×C₁₃₇	—	2	4	no	—
73	#95×C₁₃₇	C₂×C₄	1	2	yes	—
74	#95×C₁₃₇	C₂×C₄	1	2	yes	—
75	#95×C₁₃₇	C₂×C₄	1	2	yes	—
76	C₂⁴×C₁₃₇	C₂²×C₄	1	2	yes	—
77	C₂×D₄×C₁₃₇	—	2	2	no	—
78	C₂×D₄×C₁₃₇	—	2	2	no	—
79	C₂×D₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
80	C₂²×C₄×C₁₃₇	—	2	2	no	—
81	C₂²×C₄×C₁₃₇	—	2	2	no	—
82	C₂²×C₄×C₁₃₇	—	2	2	no	—
83	C₂²×C₄×C₁₃₇	—	2	2	no	—
84	C₂²×C₄×C₁₃₇	—	2	4	no	—
85	C₂²×C₄×C₁₃₇	—	2	4	no	—
86	C₂²×C₄×C₁₃₇	—	2	4	no	—
87	C₂²×C₄×C₁₃₇	—	2	4	no	—
88	C₂²×C₄×C₁₃₇	—	2	4	no	—
89	C₂²×C₄×C₁₃₇	—	4	2	no	—
90	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₁₃₇	C₂×D₄	1	4	yes	—
100	C₂²×C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
101	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
102	M₁₆×C₁₃₇	—	4	16	no	—
103	C₂×C₈×C₁₃₇	—	2	2	no	—
104	C₂×C₈×C₁₃₇	—	2	2	no	—
105	C₂×C₈×C₁₃₇	—	2	2	no	—
106	C₂×C₈×C₁₃₇	—	2	4	no	—
107	C₂×C₈×C₁₃₇	—	2	4	no	—
108	C₂×C₈×C₁₃₇	—	2	2	no	—
109	C₂×C₈×C₁₃₇	—	2	4	no	—
110	C₂×C₈×C₁₃₇	—	2	2	no	—
111	C₂×C₈×C₁₃₇	C₂×D₄	1	4	yes	—
112	C₄⋊C₄×C₁₃₇	—	2	2	no	—
113	C₄⋊C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
114	C₄⋊C₄×C₁₃₇	M₁₆	1	4	yes	complemented
115	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
116	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
117	C₂²⋊C₄×C₁₃₇	C₄⋊C₄	1	8	yes	—
118	C₄²×C₁₃₇	—	2	4	no	—
119	C₄²×C₁₃₇	—	2	4	no	—
120	C₄²×C₁₃₇	—	2	2	no	—
121	C₄²×C₁₃₇	—	2	2	no	—
122	C₄²×C₁₃₇	—	2	2	no	—
123	C₄²×C₁₃₇	—	2	2	no	—
124	C₄²×C₁₃₇	—	2	1	no	—
125	C₄²×C₁₃₇	—	2	1	no	—
126	C₄²×C₁₃₇	—	4	1	no	—
127	C₄²×C₁₃₇	M₁₆	1	4	yes	complemented
128	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
129	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
130	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
131	C₄²×C₁₃₇	#41	1	2	yes	—
132	C₂³×C₁₃₇	—	2	2	no	—
133	C₂³×C₁₃₇	—	2	2	no	—
134	C₂³×C₁₃₇	—	2	2	no	—
135	C₂³×C₁₃₇	—	2	2	no	—
136	C₂³×C₁₃₇	—	2	2	no	—
137	C₂³×C₁₃₇	—	2	2	no	—
138	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
139	C₂³×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
140	D₄×C₁₃₇	—	2	8	no	—
141	C₂×C₄×C₁₃₇	—	2	2	no	—
142	C₂×C₄×C₁₃₇	—	2	2	no	—
143	C₂×C₄×C₁₃₇	—	2	1	no	—
144	C₂×C₄×C₁₃₇	—	2	2	no	—
145	C₂×C₄×C₁₃₇	—	2	1	no	—
146	C₂×C₄×C₁₃₇	—	2	1	no	—
147	C₂×C₄×C₁₃₇	—	2	1	no	—
148	C₂×C₄×C₁₃₇	—	2	1	no	—
149	C₂×C₄×C₁₃₇	—	2	1	no	—
150	C₂×C₄×C₁₃₇	—	2	2	no	—
151	C₂×C₄×C₁₃₇	—	2	4	no	—
152	C₂×C₄×C₁₃₇	—	2	4	no	—
153	C₂×C₄×C₁₃₇	—	2	4	no	—
154	C₂×C₄×C₁₃₇	—	2	4	no	—
155	C₂×C₄×C₁₃₇	—	2	2	no	—
156	C₂×C₄×C₁₃₇	—	2	1	no	—
157	C₂×C₄×C₁₃₇	—	2	1	no	—
158	C₂×C₄×C₁₃₇	—	2	2	no	—
159	C₂×C₄×C₁₃₇	—	2	1	no	—
160	C₂×C₄×C₁₃₇	—	2	2	no	—
161	C₂×C₄×C₁₃₇	—	4	8	no	—
162	C₂×C₄×C₁₃₇	—	4	8	no	—
163	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
164	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
165	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
166	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
167	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
168	C₂×C₄×C₁₃₇	C₂×M₁₆	1	2	yes	—
169	C₂×C₄×C₁₃₇	C₂×M₁₆	1	4	yes	—
170	C₂×C₄×C₁₃₇	C₂×M₁₆	1	2	yes	—
171	C₂×C₄×C₁₃₇	C₂×M₁₆	1	2	yes	—
172	C₂×C₄×C₁₃₇	C₂×M₁₆	1	2	yes	—
173	C₂×C₄×C₁₃₇	#131	1	2	yes	—
174	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
177	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
178	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
179	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
180	C₈×C₁₃₇	—	2	4	no	—
181	C₈×C₁₃₇	—	4	4	no	—
182	C₈×C₁₃₇	—	4	2	no	—
183	C₂²×C₁₃₇	—	2	4	no	—
184	C₂²×C₁₃₇	—	2	4	no	—
185	C₂²×C₁₃₇	—	2	4	no	—
186	C₂²×C₁₃₇	—	4	2	no	—
187	C₂²×C₁₃₇	—	4	2	no	—
188	C₂²×C₁₃₇	—	4	2	no	—
189	C₂²×C₁₃₇	—	4	2	no	—
190	C₂²×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
191	C₂²×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
192	C₂²×C₁₃₇	#520	1	1	yes	characteristic
193	C₂²×C₁₃₇	C₂²×M₁₆	1	1	yes	characteristic
194	C₂²×C₁₃₇	C₂²×M₁₆	1	1	yes	characteristic
195	C₂²×C₁₃₇	C₂×#131	1	1	yes	characteristic
196	C₂²×C₁₃₇	#568	1	1	yes	characteristic
197	C₄×C₁₃₇	—	2	4	no	—
198	C₄×C₁₃₇	—	2	2	no	—
199	C₄×C₁₃₇	—	2	2	no	—
200	C₄×C₁₃₇	—	2	4	no	—
201	C₄×C₁₃₇	—	2	2	no	—
202	C₄×C₁₃₇	—	4	2	no	—
203	C₄×C₁₃₇	—	4	1	no	—
204	C₄×C₁₃₇	#423	1	4	yes	complemented
205	C₄×C₁₃₇	#436	1	4	yes	complemented
206	C₄×C₁₃₇	D₄²	1	1	yes	characteristic
207	C₄×C₁₃₇	#547	1	1	yes	characteristic
208	C₄×C₁₃₇	D₄×Q₈	1	1	yes	characteristic
209	C₄×C₁₃₇	#554	1	1	yes	characteristic
210	C₂×C₁₃₇	—	4	4	no	—
211	C₂×C₁₃₇	D₄×C₄⋊C₄	1	1	yes	characteristic
212	C₂×C₁₃₇	C₂×#423	1	1	yes	characteristic
213	C₂×C₁₃₇	#2906	1	1	yes	characteristic
214	C₂×C₁₃₇	C₂×#436	1	1	yes	characteristic
215	C₂×C₁₃₇	D₄×M₁₆	1	1	yes	characteristic
216	C₂×C₁₃₇	#2989	1	1	yes	characteristic
217	C₂×C₁₃₇	#2991	1	1	yes	characteristic
218	#22650	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
219	C₁₃₇	#22650	1	1	yes	characteristic, direct product factor, Sylow
220	C₂×#436	C₂×C₁₃₇	1	2	yes	—
221	C₂×#423	C₂×C₁₃₇	1	2	yes	complemented
222	C₄²×D₄	C₂×C₁₃₇	1	1	yes	characteristic
223	#1939	C₂×C₁₃₇	1	2	yes	complemented
224	#1851	C₂×C₁₃₇	1	4	yes	complemented
225	#1842	C₂×C₁₃₇	1	1	yes	characteristic, complemented
226	#1797	C₂×C₁₃₇	1	2	yes	complemented
227	C₄×C₄⋊C₈	C₂×C₁₃₇	1	1	yes	characteristic, complemented
228	C₂²×M₁₆	C₂²×C₁₃₇	1	4	yes	—
229	C₂×C₄×D₄	C₄×C₁₃₇	1	1	yes	characteristic
230	C₂×C₄×D₄	C₄×C₁₃₇	1	1	yes	characteristic
231	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
232	C₂²×C₄²	C₂²×C₁₃₇	1	2	yes	—
233	#436	—	2	8	no	—
234	#423	—	2	8	no	—
235	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
236	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
237	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
238	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	4	yes	—
239	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
240	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
241	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	4	yes	—
242	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
243	C₄×C₄⋊C₄	C₄×C₁₃₇	1	1	yes	characteristic
244	C₄×C₄⋊C₄	C₄×C₁₃₇	1	1	yes	characteristic
245	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
246	C₄×C₂²⋊C₄	C₄×C₁₃₇	1	2	yes	—
247	C₄×C₂²⋊C₄	C₄×C₁₃₇	1	2	yes	—
248	C₄×C₂²⋊C₄	C₂²×C₁₃₇	1	2	yes	—
249	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
250	#336	C₂²×C₁₃₇	1	4	yes	—
251	#336	C₂²×C₁₃₇	1	2	yes	—
252	C₂²×D₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
253	C₂³×C₄	C₂×C₄×C₁₃₇	1	2	yes	—
254	C₂³×C₄	C₂×C₄×C₁₃₇	1	2	yes	—
255	C₂³×C₄	C₂³×C₁₃₇	1	2	yes	—
256	C₂×M₁₆	—	2	8	no	—
257	C₂×M₁₆	—	2	8	no	—
258	C₂×M₁₆	—	4	4	no	—
259	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
260	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
261	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
262	C₄×D₄	—	2	4	no	—
263	C₄×D₄	—	2	4	no	—
264	C₄×D₄	D₄×C₁₃₇	1	4	yes	—
265	C₄×D₄	Q₈×C₁₃₇	1	4	yes	—
266	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
267	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
268	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
269	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₇	1	2	yes	—
270	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₇	1	2	yes	—
271	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₇	1	2	yes	—
272	C₂×C₄²	—	2	4	no	—
273	C₂×C₄²	—	2	4	no	—
274	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
275	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
276	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
277	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
278	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
279	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
280	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
281	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
282	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
283	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
284	C₄⋊C₈	—	2	8	no	—
285	C₄⋊C₈	—	2	4	no	—
286	C₄⋊C₈	—	2	4	no	—
287	C₄⋊C₈	—	2	2	no	—
288	C₄⋊C₈	—	4	2	no	—
289	C₄⋊C₈	D₄×C₁₃₇	1	4	yes	complemented
290	C₂²⋊C₈	—	2	8	no	—
291	C₄×C₈	—	2	4	no	—
292	#95	C₂×C₄×C₁₃₇	1	2	yes	—
293	#95	C₂×C₄×C₁₃₇	1	2	yes	—
294	#95	C₂×C₄×C₁₃₇	1	2	yes	—
295	C₂⁴	C₂²×C₄×C₁₃₇	1	2	yes	—
296	C₂×D₄	—	2	2	no	—
297	C₂×D₄	—	2	2	no	—
298	C₂×D₄	C₄⋊C₄×C₁₃₇	1	4	yes	—
299	C₂²×C₄	—	2	2	no	—
300	C₂²×C₄	—	2	2	no	—
301	C₂²×C₄	—	2	2	no	—
302	C₂²×C₄	—	2	2	no	—
303	C₂²×C₄	—	2	4	no	—
304	C₂²×C₄	—	2	4	no	—
305	C₂²×C₄	—	2	4	no	—
306	C₂²×C₄	—	2	4	no	—
307	C₂²×C₄	—	2	4	no	—
308	C₂²×C₄	—	4	2	no	—
309	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
310	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
311	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
312	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
313	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
314	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
315	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
316	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
317	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
318	C₂²×C₄	C₂×D₄×C₁₃₇	1	4	yes	—
319	C₂²×C₄	C₂×Q₈×C₁₃₇	1	4	yes	—
320	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
321	M₁₆	—	4	16	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	2	4	no	—
326	C₂×C₈	—	2	4	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	4	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	C₂×D₄×C₁₃₇	1	4	yes	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	C₄⋊C₄×C₁₃₇	1	4	yes	—
333	C₄⋊C₄	M₁₆×C₁₃₇	1	4	yes	complemented
334	C₂²⋊C₄	—	2	4	no	—
335	C₂²⋊C₄	—	2	4	no	—
336	C₂²⋊C₄	C₄⋊C₄×C₁₃₇	1	8	yes	—
337	C₄²	—	2	4	no	—
338	C₄²	—	2	4	no	—
339	C₄²	—	2	2	no	—
340	C₄²	—	2	2	no	—
341	C₄²	—	2	2	no	—
342	C₄²	—	2	2	no	—
343	C₄²	—	2	1	no	—
344	C₄²	—	2	1	no	—
345	C₄²	—	4	1	no	—
346	C₄²	M₁₆×C₁₃₇	1	4	yes	complemented
347	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
348	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
349	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
350	C₄²	#41×C₁₃₇	1	2	yes	—
351	C₂³	—	2	2	no	—
352	C₂³	—	2	2	no	—
353	C₂³	—	2	2	no	—
354	C₂³	—	2	2	no	—
355	C₂³	—	2	2	no	—
356	C₂³	—	2	2	no	—
357	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
358	C₂³	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
359	D₄	—	2	8	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	4	no	—
371	C₂×C₄	—	2	4	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	2	4	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	4	8	no	—
381	C₂×C₄	—	4	8	no	—
382	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
383	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
384	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
385	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
386	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
387	C₂×C₄	C₂×M₁₆×C₁₃₇	1	2	yes	—
388	C₂×C₄	C₂×M₁₆×C₁₃₇	1	4	yes	—
389	C₂×C₄	C₂×M₁₆×C₁₃₇	1	2	yes	—
390	C₂×C₄	C₂×M₁₆×C₁₃₇	1	2	yes	—
391	C₂×C₄	C₂×M₁₆×C₁₃₇	1	2	yes	—
392	C₂×C₄	#131×C₁₃₇	1	2	yes	—
393	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
394	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
395	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
396	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
397	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
398	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
399	C₈	—	2	4	no	—
400	C₈	—	4	4	no	—
401	C₈	—	4	2	no	—
402	C₂²	—	2	4	no	—
403	C₂²	—	2	4	no	—
404	C₂²	—	2	4	no	—
405	C₂²	—	4	2	no	—
406	C₂²	—	4	2	no	—
407	C₂²	—	4	2	no	—
408	C₂²	—	4	2	no	—
409	C₂²	C₂²×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
410	C₂²	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
411	C₂²	#520×C₁₃₇	1	1	yes	characteristic
412	C₂²	C₂²×M₁₆×C₁₃₇	1	1	yes	characteristic
413	C₂²	C₂²×M₁₆×C₁₃₇	1	1	yes	characteristic
414	C₂²	C₂×#131×C₁₃₇	1	1	yes	characteristic
415	C₂²	#568×C₁₃₇	1	1	yes	characteristic
416	C₄	—	2	4	no	—
417	C₄	—	2	2	no	—
418	C₄	—	2	2	no	—
419	C₄	—	2	4	no	—
420	C₄	—	2	2	no	—
421	C₄	—	4	2	no	—
422	C₄	—	4	1	no	—
423	C₄	#423×C₁₃₇	1	4	yes	complemented
424	C₄	#436×C₁₃₇	1	4	yes	complemented
425	C₄	D₄²×C₁₃₇	1	1	yes	characteristic
426	C₄	#547×C₁₃₇	1	1	yes	characteristic
427	C₄	D₄×Q₈×C₁₃₇	1	1	yes	characteristic
428	C₄	#554×C₁₃₇	1	1	yes	characteristic
429	C₂	—	4	4	no	—
430	C₂	D₄×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
431	C₂	C₂×#423×C₁₃₇	1	1	yes	characteristic
432	C₂	#2906×C₁₃₇	1	1	yes	characteristic
433	C₂	C₂×#436×C₁₃₇	1	1	yes	characteristic
434	C₂	D₄×M₁₆×C₁₃₇	1	1	yes	characteristic
435	C₂	#2989×C₁₃₇	1	1	yes	characteristic
436	C₂	#2991×C₁₃₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×#423×C₁₃₇	C₂	semidirect
4. #1939×C₁₃₇	C₂	semidirect
5. #1851×C₁₃₇	C₂	semidirect
6. #1842×C₁₃₇	C₂	semidirect
7. #1797×C₁₃₇	C₂	semidirect
8. C₄×C₄⋊C₈×C₁₃₇	C₂	semidirect
70. C₄⋊C₈×C₁₃₇	D₄	semidirect
114. C₄⋊C₄×C₁₃₇	M₁₆	semidirect
127. C₄²×C₁₃₇	M₁₆	semidirect
204. C₄×C₁₃₇	#423	semidirect
205. C₄×C₁₃₇	#436	semidirect
218. #22650	C₁₃₇	direct (N × H)
219. C₁₃₇	#22650	direct (N × H)
221. C₂×#423	C₂×C₁₃₇	semidirect
223. #1939	C₂×C₁₃₇	semidirect
224. #1851	C₂×C₁₃₇	semidirect
225. #1842	C₂×C₁₃₇	semidirect
226. #1797	C₂×C₁₃₇	semidirect
227. C₄×C₄⋊C₈	C₂×C₁₃₇	semidirect
289. C₄⋊C₈	D₄×C₁₃₇	semidirect
333. C₄⋊C₄	M₁₆×C₁₃₇	semidirect
346. C₄²	M₁₆×C₁₃₇	semidirect
423. C₄	#423×C₁₃₇	semidirect
424. C₄	#436×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#436×C₁₃₇	2
C₂×#423×C₁₃₇	2
C₄²×D₄×C₁₃₇	2
#1939×C₁₃₇	2
#1851×C₁₃₇	2
#1842×C₁₃₇	2
#1797×C₁₃₇	2
C₄×C₄⋊C₈×C₁₃₇	2
#22650	137

Subgroups — C137×#22650 #46872036

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#22650 #46872036