Subgroups — C₁₃₇×#22685 #46872071

Order 35072.

910 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₁₆×C₁₃₇	C₂	1	2	yes	maximal
2	C₂×C₄×Q₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#2351×C₁₃₇	C₂	1	2	yes	maximal
4	#1942×C₁₃₇	C₂	1	2	yes	maximal
5	#1865×C₁₃₇	C₂	1	4	yes	maximal
6	#1846×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	#1805×C₁₃₇	C₂	1	2	yes	maximal
8	C₈×C₄⋊C₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₁₃₇	C₂²	1	4	yes	—
11	C₂×C₄×Q₈×C₁₃₇	C₂²	1	2	yes	—
12	C₄×Q₁₆×C₁₃₇	—	2	8	no	—
13	C₄×Q₁₆×C₁₃₇	C₄	1	8	yes	complemented
14	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
15	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×#103×C₁₃₇	C₂²	1	4	yes	—
19	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
22	#389×C₁₃₇	—	2	4	no	—
23	#389×C₁₃₇	C₂²	1	4	yes	—
24	#382×C₁₃₇	—	2	2	no	—
25	#382×C₁₃₇	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₁₃₇	—	2	2	no	—
27	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
28	#340×C₁₃₇	C₂²	1	4	yes	—
29	#336×C₁₃₇	C₂²	1	2	yes	—
30	C₂²×Q₈×C₁₃₇	C₂³	1	2	yes	—
31	C₂×Q₁₆×C₁₃₇	C₂×C₄	1	4	yes	—
32	C₂×Q₁₆×C₁₃₇	D₄	1	4	yes	—
33	C₂×Q₁₆×C₁₃₇	Q₈	1	4	yes	—
34	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
37	C₄×Q₈×C₁₃₇	—	2	4	no	—
38	C₄×Q₈×C₁₃₇	—	2	8	no	—
39	C₄×Q₈×C₁₃₇	—	4	4	no	—
40	C₄×Q₈×C₁₃₇	C₂×C₄	1	8	yes	—
41	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
43	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
44	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
47	C₂×C₄²×C₁₃₇	—	2	2	no	—
48	C₂×C₄²×C₁₃₇	—	2	4	no	—
49	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
51	#107×C₁₃₇	—	2	4	no	—
52	#107×C₁₃₇	C₂×C₄	1	4	yes	—
53	#106×C₁₃₇	—	4	2	no	—
54	C₄⋊C₈×C₁₃₇	—	2	2	no	—
55	#103×C₁₃₇	—	2	8	no	—
56	#103×C₁₃₇	C₂×C₄	1	8	yes	—
57	C₄×C₈×C₁₃₇	—	2	4	no	—
58	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
59	#95×C₁₃₇	—	2	2	no	—
60	#95×C₁₃₇	—	2	4	no	—
61	#95×C₁₃₇	—	2	2	no	—
62	#95×C₁₃₇	D₄	1	1	yes	characteristic
63	#95×C₁₃₇	D₄	1	1	yes	characteristic
64	C₂×Q₈×C₁₃₇	—	2	2	no	—
65	C₂×Q₈×C₁₃₇	—	2	2	no	—
66	C₂×Q₈×C₁₃₇	—	2	2	no	—
67	C₂×Q₈×C₁₃₇	C₂²×C₄	1	4	yes	—
68	C₂×Q₈×C₁₃₇	C₂×D₄	1	4	yes	—
69	C₂×Q₈×C₁₃₇	C₂×Q₈	1	4	yes	—
70	C₂²×C₄×C₁₃₇	—	2	2	no	—
71	C₂²×C₄×C₁₃₇	—	2	2	no	—
72	C₂²×C₄×C₁₃₇	—	2	4	no	—
73	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
78	Q₁₆×C₁₃₇	C₄⋊C₄	1	16	yes	complemented
79	C₂×C₈×C₁₃₇	—	2	1	no	—
80	C₂×C₈×C₁₃₇	—	2	1	no	—
81	C₂×C₈×C₁₃₇	—	2	2	no	—
82	C₂×C₈×C₁₃₇	—	2	2	no	—
83	C₂×C₈×C₁₃₇	—	2	2	no	—
84	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
88	C₄⋊C₄×C₁₃₇	—	2	8	no	—
89	C₄⋊C₄×C₁₃₇	—	2	4	no	—
90	C₄⋊C₄×C₁₃₇	—	2	2	no	—
91	C₄⋊C₄×C₁₃₇	—	2	4	no	—
92	C₄⋊C₄×C₁₃₇	—	2	4	no	—
93	C₄⋊C₄×C₁₃₇	—	4	8	no	—
94	C₄⋊C₄×C₁₃₇	—	4	8	no	—
95	C₄⋊C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
96	C₄²×C₁₃₇	—	2	8	no	—
97	C₄²×C₁₃₇	—	2	2	no	—
98	C₄²×C₁₃₇	—	2	2	no	—
99	C₄²×C₁₃₇	—	4	8	no	—
100	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
101	C₄²×C₁₃₇	#41	1	2	yes	—
102	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
103	Q₈×C₁₃₇	—	2	4	no	—
104	Q₈×C₁₃₇	—	2	4	no	—
105	Q₈×C₁₃₇	—	2	4	no	—
106	Q₈×C₁₃₇	C₂×C₄⋊C₄	1	8	yes	—
107	C₂×C₄×C₁₃₇	—	2	4	no	—
108	C₂×C₄×C₁₃₇	—	2	2	no	—
109	C₂×C₄×C₁₃₇	—	2	4	no	—
110	C₂×C₄×C₁₃₇	—	2	4	no	—
111	C₂×C₄×C₁₃₇	—	2	2	no	—
112	C₂×C₄×C₁₃₇	—	2	2	no	—
113	C₂×C₄×C₁₃₇	—	2	2	no	—
114	C₂×C₄×C₁₃₇	—	2	4	no	—
115	C₂×C₄×C₁₃₇	—	2	2	no	—
116	C₂×C₄×C₁₃₇	—	2	2	no	—
117	C₂×C₄×C₁₃₇	—	2	2	no	—
118	C₂×C₄×C₁₃₇	—	4	4	no	—
119	C₂×C₄×C₁₃₇	—	4	4	no	—
120	C₂×C₄×C₁₃₇	—	4	4	no	—
121	C₂×C₄×C₁₃₇	—	4	4	no	—
122	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₁₃₇	#135	1	4	yes	—
125	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
129	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
130	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
131	C₈×C₁₃₇	—	2	2	no	—
132	C₈×C₁₃₇	—	4	2	no	—
133	C₈×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
134	C₂²×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
135	C₂²×C₁₃₇	D₄²	1	1	yes	characteristic
136	C₂²×C₁₃₇	D₄×Q₈	1	1	yes	characteristic
137	C₂²×C₁₃₇	C₂×#135	1	2	yes	—
138	C₂²×C₁₃₇	#578	1	2	yes	—
139	C₄×C₁₃₇	—	2	8	no	—
140	C₄×C₁₃₇	—	2	2	no	—
141	C₄×C₁₃₇	—	2	2	no	—
142	C₄×C₁₃₇	—	2	4	no	—
143	C₄×C₁₃₇	—	4	4	no	—
144	C₄×C₁₃₇	—	4	2	no	—
145	C₄×C₁₃₇	—	8	4	no	—
146	C₄×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
147	C₄×C₁₃₇	#520	1	1	yes	characteristic
148	C₄×C₁₃₇	#547	1	1	yes	characteristic
149	C₄×C₁₃₇	#554	1	1	yes	characteristic
150	C₂×C₁₃₇	D₄×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₁₃₇	C₄×#135	1	1	yes	characteristic
152	C₂×C₁₃₇	#2950	1	1	yes	characteristic
153	C₂×C₁₃₇	#3285	1	2	yes	—
154	C₂×C₁₃₇	#3383	1	2	yes	—
155	#22685	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₁₃₇	#22685	1	1	yes	characteristic, direct product factor, Sylow
157	C₂×C₄×Q₁₆	C₂×C₁₃₇	1	2	yes	—
158	C₂×C₄×Q₁₆	C₂×C₁₃₇	1	1	yes	characteristic
159	#2351	C₂×C₁₃₇	1	2	yes	—
160	#1942	C₂×C₁₃₇	1	2	yes	—
161	#1865	C₂×C₁₃₇	1	4	yes	—
162	#1846	C₂×C₁₃₇	1	1	yes	characteristic
163	#1805	C₂×C₁₃₇	1	2	yes	—
164	C₈×C₄⋊C₄	C₂×C₁₃₇	1	1	yes	characteristic
165	C₂²×Q₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
166	C₂×C₄×Q₈	C₂²×C₁₃₇	1	4	yes	—
167	C₂×C₄×Q₈	C₂²×C₁₃₇	1	2	yes	—
168	C₄×Q₁₆	—	2	8	no	—
169	C₄×Q₁₆	C₄×C₁₃₇	1	8	yes	complemented
170	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
171	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
172	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
173	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
174	C₂×#103	C₂²×C₁₃₇	1	4	yes	—
175	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
176	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
177	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
178	#389	—	2	4	no	—
179	#389	C₂²×C₁₃₇	1	4	yes	—
180	#382	—	2	2	no	—
181	#382	C₂²×C₁₃₇	1	2	yes	—
182	C₄×C₄⋊C₄	—	2	2	no	—
183	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
184	#340	C₂²×C₁₃₇	1	4	yes	—
185	#336	C₂²×C₁₃₇	1	2	yes	—
186	C₂²×Q₈	C₂³×C₁₃₇	1	2	yes	—
187	C₂×Q₁₆	C₂×C₄×C₁₃₇	1	4	yes	—
188	C₂×Q₁₆	D₄×C₁₃₇	1	4	yes	—
189	C₂×Q₁₆	Q₈×C₁₃₇	1	4	yes	—
190	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
191	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
192	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
193	C₄×Q₈	—	2	4	no	—
194	C₄×Q₈	—	2	8	no	—
195	C₄×Q₈	—	4	4	no	—
196	C₄×Q₈	C₂×C₄×C₁₃₇	1	8	yes	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	4	no	—
199	C₂×C₄⋊C₄	—	2	4	no	—
200	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
201	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
202	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	—	2	4	no	—
205	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
206	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
207	#107	—	2	4	no	—
208	#107	C₂×C₄×C₁₃₇	1	4	yes	—
209	#106	—	4	2	no	—
210	C₄⋊C₈	—	2	2	no	—
211	#103	—	2	8	no	—
212	#103	C₂×C₄×C₁₃₇	1	8	yes	—
213	C₄×C₈	—	2	4	no	—
214	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
215	#95	—	2	2	no	—
216	#95	—	2	4	no	—
217	#95	—	2	2	no	—
218	#95	D₄×C₁₃₇	1	1	yes	characteristic
219	#95	D₄×C₁₃₇	1	1	yes	characteristic
220	C₂×Q₈	—	2	2	no	—
221	C₂×Q₈	—	2	2	no	—
222	C₂×Q₈	—	2	2	no	—
223	C₂×Q₈	C₂²×C₄×C₁₃₇	1	4	yes	—
224	C₂×Q₈	C₂×D₄×C₁₃₇	1	4	yes	—
225	C₂×Q₈	C₂×Q₈×C₁₃₇	1	4	yes	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
230	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
233	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
234	Q₁₆	C₄⋊C₄×C₁₃₇	1	16	yes	complemented
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
241	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
242	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
243	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
244	C₄⋊C₄	—	2	8	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	4	8	no	—
250	C₄⋊C₄	—	4	8	no	—
251	C₄⋊C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
252	C₄²	—	2	8	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	4	8	no	—
256	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
257	C₄²	#41×C₁₃₇	1	2	yes	—
258	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
259	Q₈	—	2	4	no	—
260	Q₈	—	2	4	no	—
261	Q₈	—	2	4	no	—
262	Q₈	C₂×C₄⋊C₄×C₁₃₇	1	8	yes	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
279	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
280	C₂×C₄	#135×C₁₃₇	1	4	yes	—
281	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
282	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
283	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
284	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
285	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
286	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
287	C₈	—	2	2	no	—
288	C₈	—	4	2	no	—
289	C₈	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
290	C₂²	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
291	C₂²	D₄²×C₁₃₇	1	1	yes	characteristic
292	C₂²	D₄×Q₈×C₁₃₇	1	1	yes	characteristic
293	C₂²	C₂×#135×C₁₃₇	1	2	yes	—
294	C₂²	#578×C₁₃₇	1	2	yes	—
295	C₄	—	2	8	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	4	no	—
299	C₄	—	4	4	no	—
300	C₄	—	4	2	no	—
301	C₄	—	8	4	no	—
302	C₄	C₂²×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
303	C₄	#520×C₁₃₇	1	1	yes	characteristic
304	C₄	#547×C₁₃₇	1	1	yes	characteristic
305	C₄	#554×C₁₃₇	1	1	yes	characteristic
306	C₂	D₄×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
307	C₂	C₄×#135×C₁₃₇	1	1	yes	characteristic
308	C₂	#2950×C₁₃₇	1	1	yes	characteristic
309	C₂	#3285×C₁₃₇	1	2	yes	—
310	C₂	#3383×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
LCS	G ▷ C₂×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
13. C₄×Q₁₆×C₁₃₇	C₄	semidirect
78. Q₁₆×C₁₃₇	C₄⋊C₄	semidirect
155. #22685	C₁₃₇	direct (N × H)
156. C₁₃₇	#22685	direct (N × H)
169. C₄×Q₁₆	C₄×C₁₃₇	semidirect
234. Q₁₆	C₄⋊C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₁₆×C₁₃₇	2
#2351×C₁₃₇	2
#1942×C₁₃₇	2
#1865×C₁₃₇	2
#1846×C₁₃₇	2
#1805×C₁₃₇	2
C₈×C₄⋊C₄×C₁₃₇	2
#22685	137

Subgroups — C137×#22685 #46872071

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#22685 #46872071