Subgroups — C₁₃₇×#22695 #46872081

Order 35072.

808 conjugacy classes of subgroups.

The subgroup lattice has 308 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×#444×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	#1849×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#1814×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	#1773×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	#444×C₁₃₇	C₄	1	8	yes	complemented
14	#443×C₁₃₇	—	2	4	no	—
15	#430×C₁₃₇	—	2	4	no	—
16	#430×C₁₃₇	C₂²	1	4	yes	—
17	C₂×#108×C₁₃₇	—	2	4	no	—
18	C₂×#108×C₁₃₇	C₂²	1	2	yes	—
19	C₂×#108×C₁₃₇	C₂²	1	2	yes	—
20	C₂×#108×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₂×#108×C₁₃₇	C₂²	1	1	yes	characteristic
22	#427×C₁₃₇	C₂²	1	1	yes	characteristic
23	#423×C₁₃₇	C₂²	1	2	yes	—
24	#421×C₁₃₇	C₂²	1	4	yes	—
25	C₂×#104×C₁₃₇	C₂²	1	2	yes	—
26	#405×C₁₃₇	C₂²	1	2	yes	—
27	C₂×#97×C₁₃₇	C₂²	1	1	yes	characteristic
28	#344×C₁₃₇	C₂²	1	4	yes	—
29	#339×C₁₃₇	C₂²	1	2	yes	—
30	C₂×#41×C₁₃₇	C₂³	1	2	yes	—
31	#135×C₁₃₇	C₂×C₄	1	4	yes	—
32	#135×C₁₃₇	D₄	1	4	yes	—
33	C₂×Q₁₆×C₁₃₇	Q₈	1	1	yes	characteristic
34	C₂×SD₁₆×C₁₃₇	Q₈	1	2	yes	—
35	C₂×D₈×C₁₃₇	Q₈	1	1	yes	characteristic
36	#131×C₁₃₇	—	2	2	no	—
37	#131×C₁₃₇	—	2	2	no	—
38	#131×C₁₃₇	—	2	8	no	—
39	#131×C₁₃₇	—	4	4	no	—
40	#131×C₁₃₇	C₂×C₄	1	4	yes	—
41	#131×C₁₃₇	C₂×C₄	1	4	yes	—
42	C₂×M₁₆×C₁₃₇	—	2	2	no	—
43	C₂×M₁₆×C₁₃₇	—	2	8	no	—
44	C₂×M₁₆×C₁₃₇	C₂³	1	4	yes	—
45	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
46	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
47	C₂²×C₈×C₁₃₇	—	2	2	no	—
48	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
50	#117×C₁₃₇	C₂³	1	2	yes	—
51	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
52	#108×C₁₃₇	—	2	8	no	—
53	#108×C₁₃₇	—	2	2	no	—
54	#108×C₁₃₇	—	2	2	no	—
55	#108×C₁₃₇	—	2	2	no	—
56	#108×C₁₃₇	—	2	2	no	—
57	#108×C₁₃₇	C₂×C₄	1	4	yes	—
58	#108×C₁₃₇	D₄	1	4	yes	—
59	#107×C₁₃₇	—	2	1	no	—
60	#106×C₁₃₇	—	2	1	no	—
61	C₄⋊C₈×C₁₃₇	—	4	2	no	—
62	#104×C₁₃₇	—	2	8	no	—
63	#104×C₁₃₇	C₂×C₄	1	8	yes	—
64	#97×C₁₃₇	—	2	2	no	—
65	#97×C₁₃₇	C₂×C₄	1	4	yes	—
66	C₄×C₈×C₁₃₇	—	2	2	no	—
67	#41×C₁₃₇	—	2	2	no	—
68	#41×C₁₃₇	—	2	2	no	—
69	#41×C₁₃₇	C₂²×C₄	1	4	yes	—
70	#41×C₁₃₇	C₂×D₄	1	4	yes	—
71	C₂×Q₈×C₁₃₇	C₂×Q₈	1	2	yes	—
72	C₂×D₄×C₁₃₇	—	2	2	no	—
73	C₂×D₄×C₁₃₇	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₁₃₇	—	2	2	no	—
75	C₂²×C₄×C₁₃₇	—	2	1	no	—
76	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
77	Q₁₆×C₁₃₇	C₄⋊C₄	1	4	yes	—
78	SD₁₆×C₁₃₇	C₄⋊C₄	1	8	yes	—
79	D₈×C₁₃₇	C₄⋊C₄	1	4	yes	—
80	M₁₆×C₁₃₇	—	2	2	no	—
81	M₁₆×C₁₃₇	—	2	2	no	—
82	M₁₆×C₁₃₇	—	2	4	no	—
83	M₁₆×C₁₃₇	—	2	4	no	—
84	M₁₆×C₁₃₇	—	2	4	no	—
85	M₁₆×C₁₃₇	—	2	4	no	—
86	M₁₆×C₁₃₇	—	4	8	no	—
87	M₁₆×C₁₃₇	C₂²×C₄	1	4	yes	—
88	M₁₆×C₁₃₇	C₂×D₄	1	4	yes	—
89	C₂×C₈×C₁₃₇	—	2	2	no	—
90	C₂×C₈×C₁₃₇	—	2	2	no	—
91	C₂×C₈×C₁₃₇	—	2	4	no	—
92	C₂×C₈×C₁₃₇	—	2	4	no	—
93	C₂×C₈×C₁₃₇	—	2	4	no	—
94	C₂×C₈×C₁₃₇	—	4	8	no	—
95	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
100	C₂×C₈×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
101	C₂×C₈×C₁₃₇	#41	1	1	yes	characteristic
102	C₂×C₈×C₁₃₇	#41	1	1	yes	characteristic
103	C₄⋊C₄×C₁₃₇	—	2	2	no	—
104	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
105	C₄²×C₁₃₇	—	2	1	no	—
106	C₄²×C₁₃₇	—	2	2	no	—
107	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
108	C₂³×C₁₃₇	—	2	2	no	—
109	C₂³×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
110	Q₈×C₁₃₇	—	2	2	no	—
111	Q₈×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
112	D₄×C₁₃₇	—	2	4	no	—
113	D₄×C₁₃₇	—	2	4	no	—
114	D₄×C₁₃₇	—	2	2	no	—
115	D₄×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
116	C₂×C₄×C₁₃₇	—	2	2	no	—
117	C₂×C₄×C₁₃₇	—	2	4	no	—
118	C₂×C₄×C₁₃₇	—	2	4	no	—
119	C₂×C₄×C₁₃₇	—	2	2	no	—
120	C₂×C₄×C₁₃₇	—	4	2	no	—
121	C₂×C₄×C₁₃₇	—	8	1	no	—
122	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
128	C₈×C₁₃₇	—	2	4	no	—
129	C₈×C₁₃₇	—	2	4	no	—
130	C₈×C₁₃₇	—	4	4	no	—
131	C₈×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
132	C₈×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
133	C₈×C₁₃₇	C₄×D₄	1	2	yes	—
134	C₈×C₁₃₇	C₄×D₄	1	2	yes	—
135	C₂²×C₁₃₇	—	2	4	no	—
136	C₂²×C₁₃₇	—	4	1	no	—
137	C₂²×C₁₃₇	—	8	2	no	—
138	C₂²×C₁₃₇	#520	1	1	yes	characteristic
139	C₂²×C₁₃₇	#547	1	1	yes	characteristic
140	C₂²×C₁₃₇	D₄×Q₈	1	1	yes	characteristic
141	C₄×C₁₃₇	—	2	4	no	—
142	C₄×C₁₃₇	—	4	2	no	—
143	C₄×C₁₃₇	—	8	2	no	—
144	C₄×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
145	C₄×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
146	C₄×C₁₃₇	D₄²	1	1	yes	characteristic
147	C₄×C₁₃₇	#554	1	1	yes	characteristic
148	C₂×C₁₃₇	—	2	1	no	—
149	C₂×C₁₃₇	—	2	1	no	—
150	C₂×C₁₃₇	—	2	1	no	—
151	C₂×C₁₃₇	—	4	4	no	—
152	C₂×C₁₃₇	D₄×C₄⋊C₄	1	1	yes	characteristic
153	#22695	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₁₃₇	#22695	1	1	yes	characteristic, direct product factor, Sylow
155	#2957	C₂×C₁₃₇	1	2	yes	—
156	C₂×#444	C₂×C₁₃₇	1	1	yes	characteristic
157	#2916	C₂×C₁₃₇	1	2	yes	complemented
158	#1953	C₂×C₁₃₇	1	2	yes	complemented
159	#1861	C₂×C₁₃₇	1	4	yes	complemented
160	#1849	C₂×C₁₃₇	1	1	yes	characteristic, complemented
161	#1814	C₂×C₁₃₇	1	2	yes	complemented
162	#1773	C₂×C₁₃₇	1	1	yes	characteristic, complemented
163	C₂×#135	C₂²×C₁₃₇	1	1	yes	characteristic
164	#568	C₂²×C₁₃₇	1	4	yes	—
165	C₂×#131	C₂²×C₁₃₇	1	2	yes	—
166	#444	—	2	4	no	—
167	#444	C₄×C₁₃₇	1	8	yes	complemented
168	#443	—	2	4	no	—
169	#430	—	2	4	no	—
170	#430	C₂²×C₁₃₇	1	4	yes	—
171	C₂×#108	—	2	4	no	—
172	C₂×#108	C₂²×C₁₃₇	1	2	yes	—
173	C₂×#108	C₂²×C₁₃₇	1	2	yes	—
174	C₂×#108	C₂²×C₁₃₇	1	1	yes	characteristic
175	C₂×#108	C₂²×C₁₃₇	1	1	yes	characteristic
176	#427	C₂²×C₁₃₇	1	1	yes	characteristic
177	#423	C₂²×C₁₃₇	1	2	yes	—
178	#421	C₂²×C₁₃₇	1	4	yes	—
179	C₂×#104	C₂²×C₁₃₇	1	2	yes	—
180	#405	C₂²×C₁₃₇	1	2	yes	—
181	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
182	#344	C₂²×C₁₃₇	1	4	yes	—
183	#339	C₂²×C₁₃₇	1	2	yes	—
184	C₂×#41	C₂³×C₁₃₇	1	2	yes	—
185	#135	C₂×C₄×C₁₃₇	1	4	yes	—
186	#135	D₄×C₁₃₇	1	4	yes	—
187	C₂×Q₁₆	Q₈×C₁₃₇	1	1	yes	characteristic
188	C₂×SD₁₆	Q₈×C₁₃₇	1	2	yes	—
189	C₂×D₈	Q₈×C₁₃₇	1	1	yes	characteristic
190	#131	—	2	2	no	—
191	#131	—	2	2	no	—
192	#131	—	2	8	no	—
193	#131	—	4	4	no	—
194	#131	C₂×C₄×C₁₃₇	1	4	yes	—
195	#131	C₂×C₄×C₁₃₇	1	4	yes	—
196	C₂×M₁₆	—	2	2	no	—
197	C₂×M₁₆	—	2	8	no	—
198	C₂×M₁₆	C₂³×C₁₃₇	1	4	yes	—
199	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
200	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
201	C₂²×C₈	—	2	2	no	—
202	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
203	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
204	#117	C₂³×C₁₃₇	1	2	yes	—
205	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
206	#108	—	2	8	no	—
207	#108	—	2	2	no	—
208	#108	—	2	2	no	—
209	#108	—	2	2	no	—
210	#108	—	2	2	no	—
211	#108	C₂×C₄×C₁₃₇	1	4	yes	—
212	#108	D₄×C₁₃₇	1	4	yes	—
213	#107	—	2	1	no	—
214	#106	—	2	1	no	—
215	C₄⋊C₈	—	4	2	no	—
216	#104	—	2	8	no	—
217	#104	C₂×C₄×C₁₃₇	1	8	yes	—
218	#97	—	2	2	no	—
219	#97	C₂×C₄×C₁₃₇	1	4	yes	—
220	C₄×C₈	—	2	2	no	—
221	#41	—	2	2	no	—
222	#41	—	2	2	no	—
223	#41	C₂²×C₄×C₁₃₇	1	4	yes	—
224	#41	C₂×D₄×C₁₃₇	1	4	yes	—
225	C₂×Q₈	C₂×Q₈×C₁₃₇	1	2	yes	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	C₂×Q₈×C₁₃₇	1	2	yes	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
231	Q₁₆	C₄⋊C₄×C₁₃₇	1	4	yes	—
232	SD₁₆	C₄⋊C₄×C₁₃₇	1	8	yes	—
233	D₈	C₄⋊C₄×C₁₃₇	1	4	yes	—
234	M₁₆	—	2	2	no	—
235	M₁₆	—	2	2	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	2	4	no	—
240	M₁₆	—	4	8	no	—
241	M₁₆	C₂²×C₄×C₁₃₇	1	4	yes	—
242	M₁₆	C₂×D₄×C₁₃₇	1	4	yes	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	4	8	no	—
249	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
250	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
251	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
252	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
253	C₂×C₈	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
254	C₂×C₈	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
255	C₂×C₈	#41×C₁₃₇	1	1	yes	characteristic
256	C₂×C₈	#41×C₁₃₇	1	1	yes	characteristic
257	C₄⋊C₄	—	2	2	no	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	2	2	no	—
261	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
262	C₂³	—	2	2	no	—
263	C₂³	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
264	Q₈	—	2	2	no	—
265	Q₈	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
266	D₄	—	2	4	no	—
267	D₄	—	2	4	no	—
268	D₄	—	2	2	no	—
269	D₄	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	8	1	no	—
276	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
277	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
278	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
279	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
280	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
281	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
282	C₈	—	2	4	no	—
283	C₈	—	2	4	no	—
284	C₈	—	4	4	no	—
285	C₈	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
286	C₈	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
287	C₈	C₄×D₄×C₁₃₇	1	2	yes	—
288	C₈	C₄×D₄×C₁₃₇	1	2	yes	—
289	C₂²	—	2	4	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	8	2	no	—
292	C₂²	#520×C₁₃₇	1	1	yes	characteristic
293	C₂²	#547×C₁₃₇	1	1	yes	characteristic
294	C₂²	D₄×Q₈×C₁₃₇	1	1	yes	characteristic
295	C₄	—	2	4	no	—
296	C₄	—	4	2	no	—
297	C₄	—	8	2	no	—
298	C₄	C₂²×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
299	C₄	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
300	C₄	D₄²×C₁₃₇	1	1	yes	characteristic
301	C₄	#554×C₁₃₇	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	4	4	no	—
306	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. #1849×C₁₃₇	C₂	semidirect
7. #1814×C₁₃₇	C₂	semidirect
8. #1773×C₁₃₇	C₂	semidirect
13. #444×C₁₃₇	C₄	semidirect
153. #22695	C₁₃₇	direct (N × H)
154. C₁₃₇	#22695	direct (N × H)
157. #2916	C₂×C₁₃₇	semidirect
158. #1953	C₂×C₁₃₇	semidirect
159. #1861	C₂×C₁₃₇	semidirect
160. #1849	C₂×C₁₃₇	semidirect
161. #1814	C₂×C₁₃₇	semidirect
162. #1773	C₂×C₁₃₇	semidirect
167. #444	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×#444×C₁₃₇	2
#2916×C₁₃₇	2
#1953×C₁₃₇	2
#1861×C₁₃₇	2
#1849×C₁₃₇	2
#1814×C₁₃₇	2
#1773×C₁₃₇	2
#22695	137

Subgroups — C137×#22695 #46872081

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#22695 #46872081