Subgroups — C₁₃₇×#22694 #46872080

Order 35072.

812 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2957×C₁₃₇	C₂	1	2	yes	maximal
2	C₂×#443×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#2916×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	#1953×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#1861×C₁₃₇	C₂	1	4	yes	maximal, complemented
6	#1848×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#1814×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	#1772×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₁₃₇	C₂²	1	1	yes	characteristic
10	#568×C₁₃₇	C₂²	1	4	yes	—
11	C₂×#131×C₁₃₇	C₂²	1	2	yes	—
12	#444×C₁₃₇	—	2	4	no	—
13	#443×C₁₃₇	—	2	4	no	—
14	#443×C₁₃₇	C₄	1	2	yes	complemented
15	#443×C₁₃₇	C₄	1	2	yes	complemented
16	#443×C₁₃₇	C₄	1	2	yes	complemented
17	#443×C₁₃₇	C₄	1	2	yes	complemented
18	#430×C₁₃₇	—	2	4	no	—
19	#430×C₁₃₇	C₂²	1	4	yes	—
20	C₂×#108×C₁₃₇	—	2	4	no	—
21	C₂×#108×C₁₃₇	C₂²	1	2	yes	—
22	C₂×#108×C₁₃₇	C₂²	1	2	yes	—
23	C₂×#108×C₁₃₇	C₂²	1	1	yes	characteristic
24	C₂×#108×C₁₃₇	C₂²	1	1	yes	characteristic
25	#427×C₁₃₇	C₂²	1	1	yes	characteristic
26	#423×C₁₃₇	C₂²	1	2	yes	—
27	#421×C₁₃₇	C₂²	1	4	yes	—
28	C₂×#104×C₁₃₇	C₂²	1	2	yes	—
29	#405×C₁₃₇	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
31	#344×C₁₃₇	C₂²	1	4	yes	—
32	#339×C₁₃₇	C₂²	1	2	yes	—
33	C₂×#41×C₁₃₇	C₂³	1	2	yes	—
34	#135×C₁₃₇	C₂×C₄	1	2	yes	—
35	#135×C₁₃₇	C₂×C₄	1	2	yes	—
36	#135×C₁₃₇	D₄	1	2	yes	—
37	#135×C₁₃₇	D₄	1	2	yes	—
38	C₂×Q₁₆×C₁₃₇	Q₈	1	1	yes	characteristic
39	C₂×SD₁₆×C₁₃₇	Q₈	1	2	yes	—
40	C₂×D₈×C₁₃₇	Q₈	1	1	yes	characteristic
41	#131×C₁₃₇	—	2	2	no	—
42	#131×C₁₃₇	—	2	2	no	—
43	#131×C₁₃₇	—	2	8	no	—
44	#131×C₁₃₇	—	4	4	no	—
45	#131×C₁₃₇	C₂×C₄	1	4	yes	—
46	#131×C₁₃₇	C₂×C₄	1	4	yes	—
47	C₂×M₁₆×C₁₃₇	—	2	2	no	—
48	C₂×M₁₆×C₁₃₇	—	2	8	no	—
49	C₂×M₁₆×C₁₃₇	C₂³	1	4	yes	—
50	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
51	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
52	C₂²×C₈×C₁₃₇	—	2	2	no	—
53	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
55	#117×C₁₃₇	C₂³	1	2	yes	—
56	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
57	#108×C₁₃₇	—	2	8	no	—
58	#108×C₁₃₇	—	2	2	no	—
59	#108×C₁₃₇	—	2	2	no	—
60	#108×C₁₃₇	—	2	2	no	—
61	#108×C₁₃₇	—	2	2	no	—
62	#108×C₁₃₇	C₂×C₄	1	2	yes	—
63	#108×C₁₃₇	C₂×C₄	1	2	yes	—
64	#108×C₁₃₇	D₄	1	2	yes	—
65	#108×C₁₃₇	D₄	1	2	yes	—
66	#107×C₁₃₇	—	2	1	no	—
67	#106×C₁₃₇	—	2	1	no	—
68	C₄⋊C₈×C₁₃₇	—	4	2	no	—
69	#104×C₁₃₇	—	2	8	no	—
70	#104×C₁₃₇	C₂×C₄	1	4	yes	—
71	#104×C₁₃₇	C₂×C₄	1	4	yes	—
72	#97×C₁₃₇	—	2	2	no	—
73	C₄×C₈×C₁₃₇	—	2	2	no	—
74	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
75	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
76	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
77	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
78	#41×C₁₃₇	—	2	2	no	—
79	#41×C₁₃₇	—	2	2	no	—
80	#41×C₁₃₇	C₂²×C₄	1	4	yes	—
81	#41×C₁₃₇	C₂×D₄	1	4	yes	—
82	C₂×Q₈×C₁₃₇	C₂×Q₈	1	2	yes	—
83	C₂×D₄×C₁₃₇	—	2	2	no	—
84	C₂×D₄×C₁₃₇	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₁₃₇	—	2	2	no	—
86	C₂²×C₄×C₁₃₇	—	2	1	no	—
87	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
88	Q₁₆×C₁₃₇	C₄⋊C₄	1	2	yes	—
89	Q₁₆×C₁₃₇	C₄⋊C₄	1	2	yes	—
90	SD₁₆×C₁₃₇	C₄⋊C₄	1	4	yes	—
91	SD₁₆×C₁₃₇	C₄⋊C₄	1	4	yes	—
92	D₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
93	D₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
94	M₁₆×C₁₃₇	—	2	2	no	—
95	M₁₆×C₁₃₇	—	2	2	no	—
96	M₁₆×C₁₃₇	—	2	4	no	—
97	M₁₆×C₁₃₇	—	2	4	no	—
98	M₁₆×C₁₃₇	—	2	4	no	—
99	M₁₆×C₁₃₇	—	2	4	no	—
100	M₁₆×C₁₃₇	—	4	8	no	—
101	M₁₆×C₁₃₇	C₂²×C₄	1	4	yes	—
102	M₁₆×C₁₃₇	C₂×D₄	1	4	yes	—
103	C₂×C₈×C₁₃₇	—	2	2	no	—
104	C₂×C₈×C₁₃₇	—	2	2	no	—
105	C₂×C₈×C₁₃₇	—	2	4	no	—
106	C₂×C₈×C₁₃₇	—	2	4	no	—
107	C₂×C₈×C₁₃₇	—	2	4	no	—
108	C₂×C₈×C₁₃₇	—	4	8	no	—
109	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
118	C₂×C₈×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
119	C₂×C₈×C₁₃₇	#41	1	1	yes	characteristic
120	C₂×C₈×C₁₃₇	#41	1	1	yes	characteristic
121	C₄⋊C₄×C₁₃₇	—	2	2	no	—
122	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
123	C₄²×C₁₃₇	—	2	1	no	—
124	C₄²×C₁₃₇	—	2	2	no	—
125	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
126	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
127	C₂³×C₁₃₇	—	2	2	no	—
128	C₂³×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
129	Q₈×C₁₃₇	—	2	2	no	—
130	Q₈×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
131	D₄×C₁₃₇	—	2	4	no	—
132	D₄×C₁₃₇	—	2	4	no	—
133	D₄×C₁₃₇	—	2	2	no	—
134	D₄×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
135	C₂×C₄×C₁₃₇	—	2	1	no	—
136	C₂×C₄×C₁₃₇	—	2	2	no	—
137	C₂×C₄×C₁₃₇	—	2	4	no	—
138	C₂×C₄×C₁₃₇	—	2	4	no	—
139	C₂×C₄×C₁₃₇	—	2	1	no	—
140	C₂×C₄×C₁₃₇	—	4	2	no	—
141	C₂×C₄×C₁₃₇	—	4	2	no	—
142	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
146	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
147	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
148	C₈×C₁₃₇	—	2	4	no	—
149	C₈×C₁₃₇	—	2	4	no	—
150	C₈×C₁₃₇	—	4	4	no	—
151	C₈×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₈×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₈×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₈×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₈×C₁₃₇	C₄×D₄	1	1	yes	characteristic
156	C₈×C₁₃₇	C₄×D₄	1	1	yes	characteristic
157	C₈×C₁₃₇	C₄×D₄	1	1	yes	characteristic
158	C₈×C₁₃₇	C₄×D₄	1	1	yes	characteristic
159	C₂²×C₁₃₇	—	2	4	no	—
160	C₂²×C₁₃₇	—	4	1	no	—
161	C₂²×C₁₃₇	—	8	2	no	—
162	C₂²×C₁₃₇	#520	1	1	yes	characteristic
163	C₂²×C₁₃₇	#547	1	1	yes	characteristic
164	C₂²×C₁₃₇	D₄×Q₈	1	1	yes	characteristic
165	C₄×C₁₃₇	—	2	2	no	—
166	C₄×C₁₃₇	—	2	4	no	—
167	C₄×C₁₃₇	—	4	1	no	—
168	C₄×C₁₃₇	—	8	2	no	—
169	C₄×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
170	C₄×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
171	C₄×C₁₃₇	D₄²	1	1	yes	characteristic
172	C₄×C₁₃₇	#554	1	1	yes	characteristic
173	C₂×C₁₃₇	—	2	1	no	—
174	C₂×C₁₃₇	—	2	1	no	—
175	C₂×C₁₃₇	—	2	1	no	—
176	C₂×C₁₃₇	—	4	4	no	—
177	C₂×C₁₃₇	D₄×C₄⋊C₄	1	1	yes	characteristic
178	#22694	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
179	C₁₃₇	#22694	1	1	yes	characteristic, direct product factor, Sylow
180	#2957	C₂×C₁₃₇	1	2	yes	—
181	C₂×#443	C₂×C₁₃₇	1	1	yes	characteristic
182	#2916	C₂×C₁₃₇	1	2	yes	complemented
183	#1953	C₂×C₁₃₇	1	2	yes	complemented
184	#1861	C₂×C₁₃₇	1	4	yes	complemented
185	#1848	C₂×C₁₃₇	1	1	yes	characteristic, complemented
186	#1814	C₂×C₁₃₇	1	2	yes	complemented
187	#1772	C₂×C₁₃₇	1	1	yes	characteristic, complemented
188	C₂×#135	C₂²×C₁₃₇	1	1	yes	characteristic
189	#568	C₂²×C₁₃₇	1	4	yes	—
190	C₂×#131	C₂²×C₁₃₇	1	2	yes	—
191	#444	—	2	4	no	—
192	#443	—	2	4	no	—
193	#443	C₄×C₁₃₇	1	2	yes	complemented
194	#443	C₄×C₁₃₇	1	2	yes	complemented
195	#443	C₄×C₁₃₇	1	2	yes	complemented
196	#443	C₄×C₁₃₇	1	2	yes	complemented
197	#430	—	2	4	no	—
198	#430	C₂²×C₁₃₇	1	4	yes	—
199	C₂×#108	—	2	4	no	—
200	C₂×#108	C₂²×C₁₃₇	1	2	yes	—
201	C₂×#108	C₂²×C₁₃₇	1	2	yes	—
202	C₂×#108	C₂²×C₁₃₇	1	1	yes	characteristic
203	C₂×#108	C₂²×C₁₃₇	1	1	yes	characteristic
204	#427	C₂²×C₁₃₇	1	1	yes	characteristic
205	#423	C₂²×C₁₃₇	1	2	yes	—
206	#421	C₂²×C₁₃₇	1	4	yes	—
207	C₂×#104	C₂²×C₁₃₇	1	2	yes	—
208	#405	C₂²×C₁₃₇	1	2	yes	—
209	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
210	#344	C₂²×C₁₃₇	1	4	yes	—
211	#339	C₂²×C₁₃₇	1	2	yes	—
212	C₂×#41	C₂³×C₁₃₇	1	2	yes	—
213	#135	C₂×C₄×C₁₃₇	1	2	yes	—
214	#135	C₂×C₄×C₁₃₇	1	2	yes	—
215	#135	D₄×C₁₃₇	1	2	yes	—
216	#135	D₄×C₁₃₇	1	2	yes	—
217	C₂×Q₁₆	Q₈×C₁₃₇	1	1	yes	characteristic
218	C₂×SD₁₆	Q₈×C₁₃₇	1	2	yes	—
219	C₂×D₈	Q₈×C₁₃₇	1	1	yes	characteristic
220	#131	—	2	2	no	—
221	#131	—	2	2	no	—
222	#131	—	2	8	no	—
223	#131	—	4	4	no	—
224	#131	C₂×C₄×C₁₃₇	1	4	yes	—
225	#131	C₂×C₄×C₁₃₇	1	4	yes	—
226	C₂×M₁₆	—	2	2	no	—
227	C₂×M₁₆	—	2	8	no	—
228	C₂×M₁₆	C₂³×C₁₃₇	1	4	yes	—
229	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
230	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
231	C₂²×C₈	—	2	2	no	—
232	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
233	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
234	#117	C₂³×C₁₃₇	1	2	yes	—
235	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
236	#108	—	2	8	no	—
237	#108	—	2	2	no	—
238	#108	—	2	2	no	—
239	#108	—	2	2	no	—
240	#108	—	2	2	no	—
241	#108	C₂×C₄×C₁₃₇	1	2	yes	—
242	#108	C₂×C₄×C₁₃₇	1	2	yes	—
243	#108	D₄×C₁₃₇	1	2	yes	—
244	#108	D₄×C₁₃₇	1	2	yes	—
245	#107	—	2	1	no	—
246	#106	—	2	1	no	—
247	C₄⋊C₈	—	4	2	no	—
248	#104	—	2	8	no	—
249	#104	C₂×C₄×C₁₃₇	1	4	yes	—
250	#104	C₂×C₄×C₁₃₇	1	4	yes	—
251	#97	—	2	2	no	—
252	C₄×C₈	—	2	2	no	—
253	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
254	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
255	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
256	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
257	#41	—	2	2	no	—
258	#41	—	2	2	no	—
259	#41	C₂²×C₄×C₁₃₇	1	4	yes	—
260	#41	C₂×D₄×C₁₃₇	1	4	yes	—
261	C₂×Q₈	C₂×Q₈×C₁₃₇	1	2	yes	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	C₂×Q₈×C₁₃₇	1	2	yes	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	2	1	no	—
266	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
267	Q₁₆	C₄⋊C₄×C₁₃₇	1	2	yes	—
268	Q₁₆	C₄⋊C₄×C₁₃₇	1	2	yes	—
269	SD₁₆	C₄⋊C₄×C₁₃₇	1	4	yes	—
270	SD₁₆	C₄⋊C₄×C₁₃₇	1	4	yes	—
271	D₈	C₄⋊C₄×C₁₃₇	1	2	yes	—
272	D₈	C₄⋊C₄×C₁₃₇	1	2	yes	—
273	M₁₆	—	2	2	no	—
274	M₁₆	—	2	2	no	—
275	M₁₆	—	2	4	no	—
276	M₁₆	—	2	4	no	—
277	M₁₆	—	2	4	no	—
278	M₁₆	—	2	4	no	—
279	M₁₆	—	4	8	no	—
280	M₁₆	C₂²×C₄×C₁₃₇	1	4	yes	—
281	M₁₆	C₂×D₄×C₁₃₇	1	4	yes	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	4	no	—
285	C₂×C₈	—	2	4	no	—
286	C₂×C₈	—	2	4	no	—
287	C₂×C₈	—	4	8	no	—
288	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
289	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
290	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
291	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
292	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
293	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
294	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
295	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
296	C₂×C₈	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
297	C₂×C₈	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
298	C₂×C₈	#41×C₁₃₇	1	1	yes	characteristic
299	C₂×C₈	#41×C₁₃₇	1	1	yes	characteristic
300	C₄⋊C₄	—	2	2	no	—
301	C₂²⋊C₄	—	2	2	no	—
302	C₄²	—	2	1	no	—
303	C₄²	—	2	2	no	—
304	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
305	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
306	C₂³	—	2	2	no	—
307	C₂³	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
308	Q₈	—	2	2	no	—
309	Q₈	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
310	D₄	—	2	4	no	—
311	D₄	—	2	4	no	—
312	D₄	—	2	2	no	—
313	D₄	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	4	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
322	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
323	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
324	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
325	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
326	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
327	C₈	—	2	4	no	—
328	C₈	—	2	4	no	—
329	C₈	—	4	4	no	—
330	C₈	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
331	C₈	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
332	C₈	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
333	C₈	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
334	C₈	C₄×D₄×C₁₃₇	1	1	yes	characteristic
335	C₈	C₄×D₄×C₁₃₇	1	1	yes	characteristic
336	C₈	C₄×D₄×C₁₃₇	1	1	yes	characteristic
337	C₈	C₄×D₄×C₁₃₇	1	1	yes	characteristic
338	C₂²	—	2	4	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	8	2	no	—
341	C₂²	#520×C₁₃₇	1	1	yes	characteristic
342	C₂²	#547×C₁₃₇	1	1	yes	characteristic
343	C₂²	D₄×Q₈×C₁₃₇	1	1	yes	characteristic
344	C₄	—	2	2	no	—
345	C₄	—	2	4	no	—
346	C₄	—	4	1	no	—
347	C₄	—	8	2	no	—
348	C₄	C₂²×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
349	C₄	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
350	C₄	D₄²×C₁₃₇	1	1	yes	characteristic
351	C₄	#554×C₁₃₇	1	1	yes	characteristic
352	C₂	—	2	1	no	—
353	C₂	—	2	1	no	—
354	C₂	—	2	1	no	—
355	C₂	—	4	4	no	—
356	C₂	D₄×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃₇	D₄²
commutator	C₂×C₄	C₂³×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
socle	C₂×C₁₃₇	D₄×C₄⋊C₄
prime radical	C₂×#135×C₁₃₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃₇ ◁ C₂²×C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #2916×C₁₃₇	C₂	semidirect
4. #1953×C₁₃₇	C₂	semidirect
5. #1861×C₁₃₇	C₂	semidirect
6. #1848×C₁₃₇	C₂	semidirect
7. #1814×C₁₃₇	C₂	semidirect
8. #1772×C₁₃₇	C₂	semidirect
14. #443×C₁₃₇	C₄	semidirect
15. #443×C₁₃₇	C₄	semidirect
16. #443×C₁₃₇	C₄	semidirect
17. #443×C₁₃₇	C₄	semidirect
178. #22694	C₁₃₇	direct (N × H)
179. C₁₃₇	#22694	direct (N × H)
182. #2916	C₂×C₁₃₇	semidirect
183. #1953	C₂×C₁₃₇	semidirect
184. #1861	C₂×C₁₃₇	semidirect
185. #1848	C₂×C₁₃₇	semidirect
186. #1814	C₂×C₁₃₇	semidirect
187. #1772	C₂×C₁₃₇	semidirect
193. #443	C₄×C₁₃₇	semidirect
194. #443	C₄×C₁₃₇	semidirect
195. #443	C₄×C₁₃₇	semidirect
196. #443	C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#2957×C₁₃₇	2
C₂×#443×C₁₃₇	2
#2916×C₁₃₇	2
#1953×C₁₃₇	2
#1861×C₁₃₇	2
#1848×C₁₃₇	2
#1814×C₁₃₇	2
#1772×C₁₃₇	2
#22694	137

Subgroups — C137×#22694 #46872080

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#22694 #46872080