Subgroups — C₁₃₇×#28814 #46878200

Order 35072.

718 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#499×C₁₃₇	C₂	1	2	yes	maximal
2	#2552×C₁₃₇	C₂	1	2	yes	maximal
3	#2059×C₁₃₇	C₂	1	4	yes	maximal
4	#1948×C₁₃₇	C₂	1	2	yes	maximal
5	#1942×C₁₃₇	C₂	1	2	yes	maximal
6	#1845×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	C₄×#107×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₈×C₄⋊C₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#125×C₁₃₇	C₂²	1	4	yes	—
10	#499×C₁₃₇	—	2	8	no	—
11	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
12	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
13	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×#106×C₁₃₇	C₂²	1	4	yes	—
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	1	yes	characteristic
19	#398×C₁₃₇	—	2	4	no	—
20	#389×C₁₃₇	—	2	2	no	—
21	#389×C₁₃₇	C₂²	1	4	yes	—
22	#384×C₁₃₇	—	2	2	no	—
23	#384×C₁₃₇	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
26	#340×C₁₃₇	C₂²	1	2	yes	—
27	#340×C₁₃₇	C₂²	1	4	yes	—
28	#336×C₁₃₇	C₂²	1	2	yes	—
29	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
30	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
32	#125×C₁₃₇	—	2	8	no	—
33	#125×C₁₃₇	—	4	4	no	—
34	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
36	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
37	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
38	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
42	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
44	C₂×C₄²×C₁₃₇	—	2	2	no	—
45	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
46	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
47	#107×C₁₃₇	—	2	4	no	—
48	#107×C₁₃₇	—	4	2	no	—
49	#107×C₁₃₇	D₄	1	4	yes	—
50	#107×C₁₃₇	Q₈	1	4	yes	—
51	#106×C₁₃₇	—	2	8	no	—
52	C₄⋊C₈×C₁₃₇	Q₈	1	4	yes	—
53	C₄×C₈×C₁₃₇	—	2	4	no	—
54	C₄×C₈×C₁₃₇	Q₈	1	4	yes	—
55	#95×C₁₃₇	—	2	2	no	—
56	#95×C₁₃₇	—	2	4	no	—
57	#95×C₁₃₇	D₄	1	2	yes	—
58	C₂²×C₄×C₁₃₇	—	2	2	no	—
59	C₂²×C₄×C₁₃₇	—	2	2	no	—
60	C₂²×C₄×C₁₃₇	—	2	4	no	—
61	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
66	C₂×C₈×C₁₃₇	—	2	2	no	—
67	C₂×C₈×C₁₃₇	—	2	1	no	—
68	C₂×C₈×C₁₃₇	—	2	1	no	—
69	C₂×C₈×C₁₃₇	—	2	2	no	—
70	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
72	C₂×C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
73	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
74	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
75	C₄⋊C₄×C₁₃₇	—	2	2	no	—
76	C₄⋊C₄×C₁₃₇	—	2	2	no	—
77	C₄⋊C₄×C₁₃₇	—	2	2	no	—
78	C₄⋊C₄×C₁₃₇	—	2	2	no	—
79	C₄⋊C₄×C₁₃₇	—	2	4	no	—
80	C₄⋊C₄×C₁₃₇	—	2	4	no	—
81	C₄⋊C₄×C₁₃₇	—	4	8	no	—
82	C₄⋊C₄×C₁₃₇	—	4	8	no	—
83	C₄⋊C₄×C₁₃₇	—	4	4	no	—
84	C₄⋊C₄×C₁₃₇	—	8	2	no	—
85	C₄⋊C₄×C₁₃₇	—	8	4	no	—
86	C₄⋊C₄×C₁₃₇	C₂×D₄	1	4	yes	—
87	C₄⋊C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
88	C₄²×C₁₃₇	—	2	2	no	—
89	C₄²×C₁₃₇	—	2	2	no	—
90	C₄²×C₁₃₇	—	4	4	no	—
91	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
92	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
93	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃₇	—	2	1	no	—
95	C₂×C₄×C₁₃₇	—	2	1	no	—
96	C₂×C₄×C₁₃₇	—	2	2	no	—
97	C₂×C₄×C₁₃₇	—	2	2	no	—
98	C₂×C₄×C₁₃₇	—	2	4	no	—
99	C₂×C₄×C₁₃₇	—	2	4	no	—
100	C₂×C₄×C₁₃₇	—	2	2	no	—
101	C₂×C₄×C₁₃₇	—	2	2	no	—
102	C₂×C₄×C₁₃₇	—	2	2	no	—
103	C₂×C₄×C₁₃₇	—	2	2	no	—
104	C₂×C₄×C₁₃₇	—	4	2	no	—
105	C₂×C₄×C₁₃₇	—	4	2	no	—
106	C₂×C₄×C₁₃₇	—	4	2	no	—
107	C₂×C₄×C₁₃₇	—	4	2	no	—
108	C₂×C₄×C₁₃₇	—	4	4	no	—
109	C₂×C₄×C₁₃₇	—	4	4	no	—
110	C₂×C₄×C₁₃₇	—	4	4	no	—
111	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
112	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
113	C₂×C₄×C₁₃₇	#135	1	2	yes	—
114	C₂×C₄×C₁₃₇	#135	1	2	yes	—
115	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
120	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
121	C₈×C₁₃₇	—	2	2	no	—
122	C₈×C₁₃₇	—	2	4	no	—
123	C₈×C₁₃₇	C₄⋊Q₈	1	4	yes	—
124	C₂²×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
125	C₂²×C₁₃₇	D₄²	1	1	yes	characteristic
126	C₂²×C₁₃₇	D₄×Q₈	1	1	yes	characteristic
127	C₂²×C₁₃₇	C₂×#135	1	1	yes	characteristic
128	C₂²×C₁₃₇	C₂×#135	1	1	yes	characteristic
129	C₂²×C₁₃₇	#576	1	1	yes	characteristic
130	C₂²×C₁₃₇	#578	1	1	yes	characteristic
131	C₄×C₁₃₇	—	2	2	no	—
132	C₄×C₁₃₇	—	4	4	no	—
133	C₄×C₁₃₇	—	4	2	no	—
134	C₄×C₁₃₇	—	4	4	no	—
135	C₄×C₁₃₇	—	4	2	no	—
136	C₄×C₁₃₇	—	8	4	no	—
137	C₄×C₁₃₇	#499	1	4	yes	complemented
138	C₄×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
139	C₄×C₁₃₇	#544	1	1	yes	characteristic
140	C₄×C₁₃₇	D₄×Q₈	1	1	yes	characteristic
141	C₄×C₁₃₇	Q₈²	1	1	yes	characteristic
142	C₂×C₁₃₇	#2551	1	1	yes	characteristic
143	C₂×C₁₃₇	C₂×#499	1	1	yes	characteristic
144	C₂×C₁₃₇	#3164	1	1	yes	characteristic
145	C₂×C₁₃₇	#3283	1	1	yes	characteristic
146	C₂×C₁₃₇	#3287	1	1	yes	characteristic
147	C₂×C₁₃₇	#3380	1	1	yes	characteristic
148	C₂×C₁₃₇	#3383	1	1	yes	characteristic
149	#28814	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₁₃₇	#28814	1	1	yes	characteristic, direct product factor, Sylow
151	C₂×#499	C₂×C₁₃₇	1	2	yes	—
152	#2552	C₂×C₁₃₇	1	2	yes	—
153	#2059	C₂×C₁₃₇	1	4	yes	—
154	#1948	C₂×C₁₃₇	1	2	yes	—
155	#1942	C₂×C₁₃₇	1	2	yes	—
156	#1845	C₂×C₁₃₇	1	1	yes	characteristic
157	C₄×#107	C₂×C₁₃₇	1	1	yes	characteristic
158	C₈×C₄⋊C₄	C₂×C₁₃₇	1	1	yes	characteristic
159	C₂×#125	C₂²×C₁₃₇	1	4	yes	—
160	#499	—	2	8	no	—
161	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
162	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
163	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
164	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
165	C₂×#106	C₂²×C₁₃₇	1	4	yes	—
166	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
167	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
168	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
169	#398	—	2	4	no	—
170	#389	—	2	2	no	—
171	#389	C₂²×C₁₃₇	1	4	yes	—
172	#384	—	2	2	no	—
173	#384	C₂²×C₁₃₇	1	2	yes	—
174	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
175	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
176	#340	C₂²×C₁₃₇	1	2	yes	—
177	#340	C₂²×C₁₃₇	1	4	yes	—
178	#336	C₂²×C₁₃₇	1	2	yes	—
179	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
180	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
181	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
182	#125	—	2	8	no	—
183	#125	—	4	4	no	—
184	C₂×C₄⋊C₄	—	2	4	no	—
185	C₂×C₄⋊C₄	—	2	4	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	2	4	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
190	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
191	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
192	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
193	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
196	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
197	#107	—	2	4	no	—
198	#107	—	4	2	no	—
199	#107	D₄×C₁₃₇	1	4	yes	—
200	#107	Q₈×C₁₃₇	1	4	yes	—
201	#106	—	2	8	no	—
202	C₄⋊C₈	Q₈×C₁₃₇	1	4	yes	—
203	C₄×C₈	—	2	4	no	—
204	C₄×C₈	Q₈×C₁₃₇	1	4	yes	—
205	#95	—	2	2	no	—
206	#95	—	2	4	no	—
207	#95	D₄×C₁₃₇	1	2	yes	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
212	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
213	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
215	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
221	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
222	C₂×C₈	C₂×Q₈×C₁₃₇	1	4	yes	—
223	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
224	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	4	8	no	—
232	C₄⋊C₄	—	4	8	no	—
233	C₄⋊C₄	—	4	4	no	—
234	C₄⋊C₄	—	8	2	no	—
235	C₄⋊C₄	—	8	4	no	—
236	C₄⋊C₄	C₂×D₄×C₁₃₇	1	4	yes	—
237	C₄⋊C₄	C₂×Q₈×C₁₃₇	1	4	yes	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	4	4	no	—
241	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
242	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
243	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
262	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
263	C₂×C₄	#135×C₁₃₇	1	2	yes	—
264	C₂×C₄	#135×C₁₃₇	1	2	yes	—
265	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
266	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
267	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
268	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
269	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
270	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	—	2	4	no	—
273	C₈	C₄⋊Q₈×C₁₃₇	1	4	yes	—
274	C₂²	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
275	C₂²	D₄²×C₁₃₇	1	1	yes	characteristic
276	C₂²	D₄×Q₈×C₁₃₇	1	1	yes	characteristic
277	C₂²	C₂×#135×C₁₃₇	1	1	yes	characteristic
278	C₂²	C₂×#135×C₁₃₇	1	1	yes	characteristic
279	C₂²	#576×C₁₃₇	1	1	yes	characteristic
280	C₂²	#578×C₁₃₇	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	4	4	no	—
283	C₄	—	4	2	no	—
284	C₄	—	4	4	no	—
285	C₄	—	4	2	no	—
286	C₄	—	8	4	no	—
287	C₄	#499×C₁₃₇	1	4	yes	complemented
288	C₄	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
289	C₄	#544×C₁₃₇	1	1	yes	characteristic
290	C₄	D₄×Q₈×C₁₃₇	1	1	yes	characteristic
291	C₄	Q₈²×C₁₃₇	1	1	yes	characteristic
292	C₂	#2551×C₁₃₇	1	1	yes	characteristic
293	C₂	C₂×#499×C₁₃₇	1	1	yes	characteristic
294	C₂	#3164×C₁₃₇	1	1	yes	characteristic
295	C₂	#3283×C₁₃₇	1	1	yes	characteristic
296	C₂	#3287×C₁₃₇	1	1	yes	characteristic
297	C₂	#3380×C₁₃₇	1	1	yes	characteristic
298	C₂	#3383×C₁₃₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
137. C₄×C₁₃₇	#499	semidirect
149. #28814	C₁₃₇	direct (N × H)
150. C₁₃₇	#28814	direct (N × H)
287. C₄	#499×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#499×C₁₃₇	2
#2552×C₁₃₇	2
#2059×C₁₃₇	2
#1948×C₁₃₇	2
#1942×C₁₃₇	2
#1845×C₁₃₇	2
C₄×#107×C₁₃₇	2
C₈×C₄⋊C₄×C₁₃₇	2
#28814	137

Subgroups — C137×#28814 #46878200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28814 #46878200