Subgroups — C₁₃₇×#33859 #46883245

Order 35072.

646 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3417×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#3415×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#3290×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	#2249×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#2222×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#2221×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#2212×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	#2194×C₁₃₇	C₂	1	2	yes	maximal, complemented
9	#2193×C₁₃₇	C₂	1	2	yes	maximal, complemented
10	#2178×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#2170×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#578×C₁₃₇	C₂²	1	2	yes	complemented
13	#577×C₁₃₇	—	2	2	no	—
14	#576×C₁₃₇	C₂²	1	2	yes	—
15	#510×C₁₃₇	—	2	2	no	—
16	#510×C₁₃₇	C₂²	1	2	yes	complemented
17	#509×C₁₃₇	—	2	2	no	—
18	#509×C₁₃₇	C₂²	1	2	yes	—
19	#508×C₁₃₇	—	2	1	no	—
20	#508×C₁₃₇	—	2	1	no	—
21	#508×C₁₃₇	C₂²	1	1	yes	characteristic
22	C₂×Q₃₂×C₁₃₇	—	2	1	no	—
23	C₂×Q₃₂×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
24	C₂×SD₃₂×C₁₃₇	C₂²	1	2	yes	complemented
25	C₂×D₁₆×C₁₃₇	—	2	1	no	—
26	C₂×D₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
27	#504×C₁₃₇	C₂²	1	1	yes	characteristic
28	#504×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
29	#495×C₁₃₇	C₂²	1	2	yes	complemented
30	#473×C₁₃₇	C₂²	1	2	yes	—
31	#472×C₁₃₇	C₂²	1	2	yes	complemented
32	#471×C₁₃₇	—	2	2	no	—
33	#455×C₁₃₇	—	2	2	no	—
34	#454×C₁₃₇	—	2	2	no	—
35	#443×C₁₃₇	C₂²	1	1	yes	characteristic
36	#443×C₁₃₇	C₂²	1	2	yes	complemented
37	#368×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
38	#364×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
39	#362×C₁₃₇	C₂²	1	2	yes	complemented
40	#361×C₁₃₇	C₂²	1	2	yes	complemented
41	#359×C₁₃₇	C₂²	1	2	yes	complemented
42	#350×C₁₃₇	C₂²	1	1	yes	characteristic
43	#350×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
44	C₄×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
45	2₋¹⁺⁴×C₁₃₇	—	2	2	no	—
46	H(5,𝔽₂)×C₁₃₇	—	2	2	no	—
47	#137×C₁₃₇	—	2	2	no	—
48	#137×C₁₃₇	—	2	2	no	—
49	#137×C₁₃₇	—	4	2	no	—
50	#137×C₁₃₇	—	4	2	no	—
51	AGL(1,ℤ/8)×C₁₃₇	—	2	2	no	—
52	AGL(1,ℤ/8)×C₁₃₇	—	2	2	no	—
53	AGL(1,ℤ/8)×C₁₃₇	—	4	2	no	—
54	AGL(1,ℤ/8)×C₁₃₇	—	4	2	no	—
55	#135×C₁₃₇	—	2	2	no	—
56	#135×C₁₃₇	—	2	2	no	—
57	#135×C₁₃₇	—	4	2	no	—
58	#135×C₁₃₇	D₄	1	1	yes	characteristic
59	#135×C₁₃₇	C₂³	1	2	yes	—
60	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
61	C₂×Q₁₆×C₁₃₇	C₂³	1	2	yes	—
62	C₂×SD₁₆×C₁₃₇	—	2	2	no	—
63	C₂×SD₁₆×C₁₃₇	—	4	2	no	—
64	C₂×D₈×C₁₃₇	—	2	2	no	—
65	C₂×D₈×C₁₃₇	C₂³	1	2	yes	—
66	#131×C₁₃₇	—	2	2	no	—
67	#131×C₁₃₇	C₂³	1	1	yes	characteristic
68	#131×C₁₃₇	C₂³	1	1	yes	characteristic
69	#124×C₁₃₇	—	2	2	no	—
70	Q₃₂×C₁₃₇	—	2	2	no	—
71	Q₃₂×C₁₃₇	—	2	1	no	—
72	Q₃₂×C₁₃₇	—	2	1	no	—
73	Q₃₂×C₁₃₇	D₄	1	1	yes	characteristic
74	SD₃₂×C₁₃₇	—	2	2	no	—
75	SD₃₂×C₁₃₇	—	2	2	no	—
76	SD₃₂×C₁₃₇	—	4	2	no	—
77	SD₃₂×C₁₃₇	D₄	1	2	yes	—
78	D₁₆×C₁₃₇	—	2	2	no	—
79	D₁₆×C₁₃₇	—	2	1	no	—
80	D₁₆×C₁₃₇	—	2	1	no	—
81	D₁₆×C₁₃₇	D₄	1	1	yes	characteristic
82	M₃₂×C₁₃₇	—	2	1	no	—
83	M₃₂×C₁₃₇	—	2	1	no	—
84	M₃₂×C₁₃₇	C₂³	1	1	yes	characteristic
85	M₃₂×C₁₃₇	C₂³	1	1	yes	characteristic
86	C₂×C₁₆×C₁₃₇	—	2	1	no	—
87	C₂×C₁₆×C₁₃₇	—	2	1	no	—
88	C₂×C₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
89	C₂×C₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
90	#108×C₁₃₇	D₄	1	1	yes	characteristic
91	#108×C₁₃₇	C₂³	1	2	yes	—
92	#104×C₁₃₇	—	2	2	no	—
93	#104×C₁₃₇	D₄	1	1	yes	characteristic
94	#104×C₁₃₇	D₄	1	1	yes	characteristic
95	#101×C₁₃₇	—	2	2	no	—
96	#100×C₁₃₇	—	2	2	no	—
97	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
98	#41×C₁₃₇	—	2	2	no	—
99	#41×C₁₃₇	—	4	2	no	—
100	#41×C₁₃₇	—	4	2	no	—
101	#41×C₁₃₇	—	4	2	no	—
102	#41×C₁₃₇	—	4	2	no	—
103	#41×C₁₃₇	—	8	2	no	—
104	#41×C₁₃₇	C₂×D₄	1	1	yes	characteristic
105	#41×C₁₃₇	C₂×D₄	1	1	yes	characteristic
106	C₂×Q₈×C₁₃₇	—	2	2	no	—
107	C₂×Q₈×C₁₃₇	—	4	2	no	—
108	C₂×Q₈×C₁₃₇	—	4	2	no	—
109	C₂×D₄×C₁₃₇	—	2	2	no	—
110	C₂×D₄×C₁₃₇	—	4	2	no	—
111	C₂×D₄×C₁₃₇	—	4	2	no	—
112	C₂×D₄×C₁₃₇	—	8	2	no	—
113	Q₁₆×C₁₃₇	—	2	2	no	—
114	Q₁₆×C₁₃₇	—	2	2	no	—
115	Q₁₆×C₁₃₇	—	4	2	no	—
116	Q₁₆×C₁₃₇	—	4	2	no	—
117	Q₁₆×C₁₃₇	D₈	1	1	yes	characteristic
118	Q₁₆×C₁₃₇	C₂×D₄	1	2	yes	—
119	SD₁₆×C₁₃₇	—	2	2	no	—
120	SD₁₆×C₁₃₇	—	4	2	no	—
121	SD₁₆×C₁₃₇	—	4	2	no	—
122	SD₁₆×C₁₃₇	—	4	2	no	—
123	SD₁₆×C₁₃₇	—	4	2	no	—
124	SD₁₆×C₁₃₇	—	4	2	no	—
125	SD₁₆×C₁₃₇	—	4	2	no	—
126	SD₁₆×C₁₃₇	D₈	1	1	yes	characteristic
127	SD₁₆×C₁₃₇	D₈	1	1	yes	characteristic
128	D₈×C₁₃₇	—	2	2	no	—
129	D₈×C₁₃₇	—	2	2	no	—
130	D₈×C₁₃₇	—	4	2	no	—
131	D₈×C₁₃₇	—	4	2	no	—
132	D₈×C₁₃₇	D₈	1	1	yes	characteristic
133	D₈×C₁₃₇	C₂×D₄	1	2	yes	—
134	M₁₆×C₁₃₇	—	2	1	no	—
135	M₁₆×C₁₃₇	—	2	1	no	—
136	M₁₆×C₁₃₇	—	2	2	no	—
137	M₁₆×C₁₃₇	—	4	2	no	—
138	M₁₆×C₁₃₇	C₂×D₄	1	1	yes	characteristic
139	M₁₆×C₁₃₇	C₂×D₄	1	1	yes	characteristic
140	C₂×C₈×C₁₃₇	—	2	1	no	—
141	C₂×C₈×C₁₃₇	—	2	1	no	—
142	C₂×C₈×C₁₃₇	—	4	2	no	—
143	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
144	C₂×C₈×C₁₃₇	C₂⁴	1	1	yes	characteristic
145	C₂²⋊C₄×C₁₃₇	—	8	2	no	—
146	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
147	C₁₆×C₁₃₇	—	2	1	no	—
148	C₁₆×C₁₃₇	—	2	1	no	—
149	C₁₆×C₁₃₇	—	2	1	no	—
150	C₁₆×C₁₃₇	C₂×D₄	1	1	yes	characteristic
151	C₁₆×C₁₃₇	C₂×D₄	1	1	yes	characteristic
152	C₂³×C₁₃₇	—	4	2	no	—
153	C₂³×C₁₃₇	—	8	2	no	—
154	Q₈×C₁₃₇	—	2	2	no	—
155	Q₈×C₁₃₇	—	4	2	no	—
156	Q₈×C₁₃₇	—	4	2	no	—
157	Q₈×C₁₃₇	—	8	2	no	—
158	Q₈×C₁₃₇	C₂×D₈	1	1	yes	characteristic
159	Q₈×C₁₃₇	C₂×D₈	1	1	yes	characteristic
160	D₄×C₁₃₇	—	2	1	no	—
161	D₄×C₁₃₇	—	2	1	no	—
162	D₄×C₁₃₇	—	2	2	no	—
163	D₄×C₁₃₇	—	4	2	no	—
164	D₄×C₁₃₇	—	4	2	no	—
165	D₄×C₁₃₇	—	4	2	no	—
166	D₄×C₁₃₇	—	8	2	no	—
167	D₄×C₁₃₇	—	8	2	no	—
168	D₄×C₁₃₇	—	8	2	no	—
169	D₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
170	D₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
171	C₂×C₄×C₁₃₇	—	2	1	no	—
172	C₂×C₄×C₁₃₇	—	2	1	no	—
173	C₂×C₄×C₁₃₇	—	2	1	no	—
174	C₂×C₄×C₁₃₇	—	4	2	no	—
175	C₂×C₄×C₁₃₇	—	4	2	no	—
176	C₂×C₄×C₁₃₇	—	8	2	no	—
177	C₂×C₄×C₁₃₇	—	8	2	no	—
178	C₂×C₄×C₁₃₇	—	8	2	no	—
179	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
180	C₈×C₁₃₇	—	2	1	no	—
181	C₈×C₁₃₇	—	2	1	no	—
182	C₈×C₁₃₇	—	4	2	no	—
183	C₈×C₁₃₇	C₂×D₈	1	1	yes	characteristic
184	C₈×C₁₃₇	C₂×D₈	1	1	yes	characteristic
185	C₈×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
186	C₈×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
187	C₂²×C₁₃₇	—	2	1	no	—
188	C₂²×C₁₃₇	—	2	1	no	—
189	C₂²×C₁₃₇	—	4	2	no	—
190	C₂²×C₁₃₇	—	8	2	no	—
191	C₂²×C₁₃₇	—	16	2	no	—
192	C₂²×C₁₃₇	—	32	2	no	—
193	C₂²×C₁₃₇	#576	1	1	yes	characteristic
194	C₄×C₁₃₇	—	2	1	no	—
195	C₄×C₁₃₇	—	2	1	no	—
196	C₄×C₁₃₇	—	4	1	no	—
197	C₄×C₁₃₇	—	4	2	no	—
198	C₄×C₁₃₇	—	8	2	no	—
199	C₄×C₁₃₇	D₄²	1	1	yes	characteristic
200	C₄×C₁₃₇	C₂²×D₈	1	1	yes	characteristic
201	C₂×C₁₃₇	—	2	1	no	—
202	C₂×C₁₃₇	—	4	1	no	—
203	C₂×C₁₃₇	—	4	1	no	—
204	C₂×C₁₃₇	—	8	2	no	—
205	C₂×C₁₃₇	—	16	2	no	—
206	C₂×C₁₃₇	D₄×D₈	1	1	yes	characteristic
207	#33859	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
208	C₁₃₇	#33859	1	1	yes	characteristic, direct product factor, Sylow
209	#3417	C₂×C₁₃₇	1	1	yes	characteristic, complemented
210	#3415	C₂×C₁₃₇	1	1	yes	characteristic, complemented
211	#3290	C₂×C₁₃₇	1	2	yes	complemented
212	#2249	C₂×C₁₃₇	1	1	yes	characteristic, complemented
213	#2222	C₂×C₁₃₇	1	1	yes	characteristic, complemented
214	#2221	C₂×C₁₃₇	1	1	yes	characteristic, complemented
215	#2212	C₂×C₁₃₇	1	2	yes	complemented
216	#2194	C₂×C₁₃₇	1	2	yes	complemented
217	#2193	C₂×C₁₃₇	1	2	yes	complemented
218	#2178	C₂×C₁₃₇	1	1	yes	characteristic, complemented
219	#2170	C₂×C₁₃₇	1	1	yes	characteristic, complemented
220	#578	C₂²×C₁₃₇	1	2	yes	complemented
221	#577	—	2	2	no	—
222	#576	C₂²×C₁₃₇	1	2	yes	—
223	#510	—	2	2	no	—
224	#510	C₂²×C₁₃₇	1	2	yes	complemented
225	#509	—	2	2	no	—
226	#509	C₂²×C₁₃₇	1	2	yes	—
227	#508	—	2	1	no	—
228	#508	—	2	1	no	—
229	#508	C₂²×C₁₃₇	1	1	yes	characteristic
230	C₂×Q₃₂	—	2	1	no	—
231	C₂×Q₃₂	C₂²×C₁₃₇	1	1	yes	characteristic, complemented
232	C₂×SD₃₂	C₂²×C₁₃₇	1	2	yes	complemented
233	C₂×D₁₆	—	2	1	no	—
234	C₂×D₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
235	#504	C₂²×C₁₃₇	1	1	yes	characteristic
236	#504	C₂²×C₁₃₇	1	1	yes	characteristic, complemented
237	#495	C₂²×C₁₃₇	1	2	yes	complemented
238	#473	C₂²×C₁₃₇	1	2	yes	—
239	#472	C₂²×C₁₃₇	1	2	yes	complemented
240	#471	—	2	2	no	—
241	#455	—	2	2	no	—
242	#454	—	2	2	no	—
243	#443	C₂²×C₁₃₇	1	1	yes	characteristic
244	#443	C₂²×C₁₃₇	1	2	yes	complemented
245	#368	C₂²×C₁₃₇	1	1	yes	characteristic, complemented
246	#364	C₂²×C₁₃₇	1	1	yes	characteristic, complemented
247	#362	C₂²×C₁₃₇	1	2	yes	complemented
248	#361	C₂²×C₁₃₇	1	2	yes	complemented
249	#359	C₂²×C₁₃₇	1	2	yes	complemented
250	#350	C₂²×C₁₃₇	1	1	yes	characteristic
251	#350	C₂²×C₁₃₇	1	1	yes	characteristic, complemented
252	C₄×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic, complemented
253	2₋¹⁺⁴	—	2	2	no	—
254	H(5,𝔽₂)	—	2	2	no	—
255	#137	—	2	2	no	—
256	#137	—	2	2	no	—
257	#137	—	4	2	no	—
258	#137	—	4	2	no	—
259	AGL(1,ℤ/8)	—	2	2	no	—
260	AGL(1,ℤ/8)	—	2	2	no	—
261	AGL(1,ℤ/8)	—	4	2	no	—
262	AGL(1,ℤ/8)	—	4	2	no	—
263	#135	—	2	2	no	—
264	#135	—	2	2	no	—
265	#135	—	4	2	no	—
266	#135	D₄×C₁₃₇	1	1	yes	characteristic
267	#135	C₂³×C₁₃₇	1	2	yes	—
268	C₂×Q₁₆	—	2	2	no	—
269	C₂×Q₁₆	C₂³×C₁₃₇	1	2	yes	—
270	C₂×SD₁₆	—	2	2	no	—
271	C₂×SD₁₆	—	4	2	no	—
272	C₂×D₈	—	2	2	no	—
273	C₂×D₈	C₂³×C₁₃₇	1	2	yes	—
274	#131	—	2	2	no	—
275	#131	C₂³×C₁₃₇	1	1	yes	characteristic
276	#131	C₂³×C₁₃₇	1	1	yes	characteristic
277	#124	—	2	2	no	—
278	Q₃₂	—	2	2	no	—
279	Q₃₂	—	2	1	no	—
280	Q₃₂	—	2	1	no	—
281	Q₃₂	D₄×C₁₃₇	1	1	yes	characteristic
282	SD₃₂	—	2	2	no	—
283	SD₃₂	—	2	2	no	—
284	SD₃₂	—	4	2	no	—
285	SD₃₂	D₄×C₁₃₇	1	2	yes	—
286	D₁₆	—	2	2	no	—
287	D₁₆	—	2	1	no	—
288	D₁₆	—	2	1	no	—
289	D₁₆	D₄×C₁₃₇	1	1	yes	characteristic
290	M₃₂	—	2	1	no	—
291	M₃₂	—	2	1	no	—
292	M₃₂	C₂³×C₁₃₇	1	1	yes	characteristic
293	M₃₂	C₂³×C₁₃₇	1	1	yes	characteristic
294	C₂×C₁₆	—	2	1	no	—
295	C₂×C₁₆	—	2	1	no	—
296	C₂×C₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
297	C₂×C₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
298	#108	D₄×C₁₃₇	1	1	yes	characteristic
299	#108	C₂³×C₁₃₇	1	2	yes	—
300	#104	—	2	2	no	—
301	#104	D₄×C₁₃₇	1	1	yes	characteristic
302	#104	D₄×C₁₃₇	1	1	yes	characteristic
303	#101	—	2	2	no	—
304	#100	—	2	2	no	—
305	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
306	#41	—	2	2	no	—
307	#41	—	4	2	no	—
308	#41	—	4	2	no	—
309	#41	—	4	2	no	—
310	#41	—	4	2	no	—
311	#41	—	8	2	no	—
312	#41	C₂×D₄×C₁₃₇	1	1	yes	characteristic
313	#41	C₂×D₄×C₁₃₇	1	1	yes	characteristic
314	C₂×Q₈	—	2	2	no	—
315	C₂×Q₈	—	4	2	no	—
316	C₂×Q₈	—	4	2	no	—
317	C₂×D₄	—	2	2	no	—
318	C₂×D₄	—	4	2	no	—
319	C₂×D₄	—	4	2	no	—
320	C₂×D₄	—	8	2	no	—
321	Q₁₆	—	2	2	no	—
322	Q₁₆	—	2	2	no	—
323	Q₁₆	—	4	2	no	—
324	Q₁₆	—	4	2	no	—
325	Q₁₆	D₈×C₁₃₇	1	1	yes	characteristic
326	Q₁₆	C₂×D₄×C₁₃₇	1	2	yes	—
327	SD₁₆	—	2	2	no	—
328	SD₁₆	—	4	2	no	—
329	SD₁₆	—	4	2	no	—
330	SD₁₆	—	4	2	no	—
331	SD₁₆	—	4	2	no	—
332	SD₁₆	—	4	2	no	—
333	SD₁₆	—	4	2	no	—
334	SD₁₆	D₈×C₁₃₇	1	1	yes	characteristic
335	SD₁₆	D₈×C₁₃₇	1	1	yes	characteristic
336	D₈	—	2	2	no	—
337	D₈	—	2	2	no	—
338	D₈	—	4	2	no	—
339	D₈	—	4	2	no	—
340	D₈	D₈×C₁₃₇	1	1	yes	characteristic
341	D₈	C₂×D₄×C₁₃₇	1	2	yes	—
342	M₁₆	—	2	1	no	—
343	M₁₆	—	2	1	no	—
344	M₁₆	—	2	2	no	—
345	M₁₆	—	4	2	no	—
346	M₁₆	C₂×D₄×C₁₃₇	1	1	yes	characteristic
347	M₁₆	C₂×D₄×C₁₃₇	1	1	yes	characteristic
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	2	1	no	—
350	C₂×C₈	—	4	2	no	—
351	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
352	C₂×C₈	C₂⁴×C₁₃₇	1	1	yes	characteristic
353	C₂²⋊C₄	—	8	2	no	—
354	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
355	C₁₆	—	2	1	no	—
356	C₁₆	—	2	1	no	—
357	C₁₆	—	2	1	no	—
358	C₁₆	C₂×D₄×C₁₃₇	1	1	yes	characteristic
359	C₁₆	C₂×D₄×C₁₃₇	1	1	yes	characteristic
360	C₂³	—	4	2	no	—
361	C₂³	—	8	2	no	—
362	Q₈	—	2	2	no	—
363	Q₈	—	4	2	no	—
364	Q₈	—	4	2	no	—
365	Q₈	—	8	2	no	—
366	Q₈	C₂×D₈×C₁₃₇	1	1	yes	characteristic
367	Q₈	C₂×D₈×C₁₃₇	1	1	yes	characteristic
368	D₄	—	2	1	no	—
369	D₄	—	2	1	no	—
370	D₄	—	2	2	no	—
371	D₄	—	4	2	no	—
372	D₄	—	4	2	no	—
373	D₄	—	4	2	no	—
374	D₄	—	8	2	no	—
375	D₄	—	8	2	no	—
376	D₄	—	8	2	no	—
377	D₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
378	D₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	8	2	no	—
385	C₂×C₄	—	8	2	no	—
386	C₂×C₄	—	8	2	no	—
387	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
388	C₈	—	2	1	no	—
389	C₈	—	2	1	no	—
390	C₈	—	4	2	no	—
391	C₈	C₂×D₈×C₁₃₇	1	1	yes	characteristic
392	C₈	C₂×D₈×C₁₃₇	1	1	yes	characteristic
393	C₈	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
394	C₈	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
395	C₂²	—	2	1	no	—
396	C₂²	—	2	1	no	—
397	C₂²	—	4	2	no	—
398	C₂²	—	8	2	no	—
399	C₂²	—	16	2	no	—
400	C₂²	—	32	2	no	—
401	C₂²	#576×C₁₃₇	1	1	yes	characteristic
402	C₄	—	2	1	no	—
403	C₄	—	2	1	no	—
404	C₄	—	4	1	no	—
405	C₄	—	4	2	no	—
406	C₄	—	8	2	no	—
407	C₄	D₄²×C₁₃₇	1	1	yes	characteristic
408	C₄	C₂²×D₈×C₁₃₇	1	1	yes	characteristic
409	C₂	—	2	1	no	—
410	C₂	—	4	1	no	—
411	C₂	—	4	1	no	—
412	C₂	—	8	2	no	—
413	C₂	—	16	2	no	—
414	C₂	D₄×D₈×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃₇	D₄×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃₇ ◁ C₂×C₄×C₁₃₇ ◁ #443×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. #3417×C₁₃₇	C₂	semidirect
2. #3415×C₁₃₇	C₂	semidirect
3. #3290×C₁₃₇	C₂	semidirect
4. #2249×C₁₃₇	C₂	semidirect
5. #2222×C₁₃₇	C₂	semidirect
6. #2221×C₁₃₇	C₂	semidirect
7. #2212×C₁₃₇	C₂	semidirect
8. #2194×C₁₃₇	C₂	semidirect
9. #2193×C₁₃₇	C₂	semidirect
10. #2178×C₁₃₇	C₂	semidirect
11. #2170×C₁₃₇	C₂	semidirect
12. #578×C₁₃₇	C₂²	semidirect
16. #510×C₁₃₇	C₂²	semidirect
23. C₂×Q₃₂×C₁₃₇	C₂²	semidirect
24. C₂×SD₃₂×C₁₃₇	C₂²	semidirect
28. #504×C₁₃₇	C₂²	semidirect
29. #495×C₁₃₇	C₂²	semidirect
31. #472×C₁₃₇	C₂²	semidirect
36. #443×C₁₃₇	C₂²	semidirect
37. #368×C₁₃₇	C₂²	semidirect
38. #364×C₁₃₇	C₂²	semidirect
39. #362×C₁₃₇	C₂²	semidirect
40. #361×C₁₃₇	C₂²	semidirect
41. #359×C₁₃₇	C₂²	semidirect
43. #350×C₁₃₇	C₂²	semidirect
44. C₄×C₁₆×C₁₃₇	C₂²	semidirect
207. #33859	C₁₃₇	direct (N × H)
208. C₁₃₇	#33859	direct (N × H)
209. #3417	C₂×C₁₃₇	semidirect
210. #3415	C₂×C₁₃₇	semidirect
211. #3290	C₂×C₁₃₇	semidirect
212. #2249	C₂×C₁₃₇	semidirect
213. #2222	C₂×C₁₃₇	semidirect
214. #2221	C₂×C₁₃₇	semidirect
215. #2212	C₂×C₁₃₇	semidirect
216. #2194	C₂×C₁₃₇	semidirect
217. #2193	C₂×C₁₃₇	semidirect
218. #2178	C₂×C₁₃₇	semidirect
219. #2170	C₂×C₁₃₇	semidirect
220. #578	C₂²×C₁₃₇	semidirect
224. #510	C₂²×C₁₃₇	semidirect
231. C₂×Q₃₂	C₂²×C₁₃₇	semidirect
232. C₂×SD₃₂	C₂²×C₁₃₇	semidirect
236. #504	C₂²×C₁₃₇	semidirect
237. #495	C₂²×C₁₃₇	semidirect
239. #472	C₂²×C₁₃₇	semidirect
244. #443	C₂²×C₁₃₇	semidirect
245. #368	C₂²×C₁₃₇	semidirect
246. #364	C₂²×C₁₃₇	semidirect
247. #362	C₂²×C₁₃₇	semidirect
248. #361	C₂²×C₁₃₇	semidirect
249. #359	C₂²×C₁₃₇	semidirect
251. #350	C₂²×C₁₃₇	semidirect
252. C₄×C₁₆	C₂²×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3417×C₁₃₇	2
#3415×C₁₃₇	2
#3290×C₁₃₇	2
#2249×C₁₃₇	2
#2222×C₁₃₇	2
#2221×C₁₃₇	2
#2212×C₁₃₇	2
#2194×C₁₃₇	2
#2193×C₁₃₇	2
#2178×C₁₃₇	2
#2170×C₁₃₇	2
#33859	137

Subgroups — C137×#33859 #46883245

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#33859 #46883245