Subgroups — C₁₃₇×#22674 #46872060

Order 35072.

914 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#441×C₁₃₇	C₂	1	2	yes	maximal
2	C₂×C₄×Q₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₄⋊C₄×C₁₃₇	C₂	1	2	yes	maximal
4	#1943×C₁₃₇	C₂	1	2	yes	maximal
5	#1865×C₁₃₇	C₂	1	4	yes	maximal
6	#1844×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	#1803×C₁₃₇	C₂	1	2	yes	maximal
8	#1770×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	#441×C₁₃₇	—	2	8	no	—
13	C₄×Q₁₆×C₁₃₇	C₄	1	8	yes	complemented
14	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
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×#97×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	#384×C₁₃₇	—	2	2	no	—
25	#384×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	2	no	—
43	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
44	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
45	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
48	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
49	C₂×C₄²×C₁₃₇	—	2	2	no	—
50	C₂×C₄²×C₁₃₇	—	2	4	no	—
51	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
52	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
53	#107×C₁₃₇	C₂×C₄	1	4	yes	—
54	#106×C₁₃₇	—	2	4	no	—
55	#106×C₁₃₇	—	4	2	no	—
56	C₄⋊C₈×C₁₃₇	—	2	2	no	—
57	#103×C₁₃₇	—	2	8	no	—
58	#103×C₁₃₇	C₂×C₄	1	8	yes	—
59	#97×C₁₃₇	—	2	4	no	—
60	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
61	#95×C₁₃₇	—	2	4	no	—
62	#95×C₁₃₇	D₄	1	2	yes	—
63	C₂×Q₈×C₁₃₇	—	2	2	no	—
64	C₂×Q₈×C₁₃₇	—	2	2	no	—
65	C₂×Q₈×C₁₃₇	—	2	2	no	—
66	C₂×Q₈×C₁₃₇	C₂²×C₄	1	4	yes	—
67	C₂×Q₈×C₁₃₇	C₂×D₄	1	4	yes	—
68	C₂×Q₈×C₁₃₇	C₂×Q₈	1	4	yes	—
69	C₂²×C₄×C₁₃₇	—	2	2	no	—
70	C₂²×C₄×C₁₃₇	—	2	2	no	—
71	C₂²×C₄×C₁₃₇	—	2	4	no	—
72	C₂²×C₄×C₁₃₇	C₂×D₄	1	4	yes	—
73	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
76	Q₁₆×C₁₃₇	C₄⋊C₄	1	16	yes	complemented
77	C₂×C₈×C₁₃₇	—	2	1	no	—
78	C₂×C₈×C₁₃₇	—	2	1	no	—
79	C₂×C₈×C₁₃₇	—	2	2	no	—
80	C₂×C₈×C₁₃₇	—	2	2	no	—
81	C₂×C₈×C₁₃₇	—	2	2	no	—
82	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
86	C₄⋊C₄×C₁₃₇	—	2	8	no	—
87	C₄⋊C₄×C₁₃₇	—	2	4	no	—
88	C₄⋊C₄×C₁₃₇	—	2	2	no	—
89	C₄⋊C₄×C₁₃₇	—	2	4	no	—
90	C₄⋊C₄×C₁₃₇	—	2	4	no	—
91	C₄⋊C₄×C₁₃₇	—	4	8	no	—
92	C₄⋊C₄×C₁₃₇	—	4	4	no	—
93	C₄⋊C₄×C₁₃₇	—	8	2	no	—
94	C₄⋊C₄×C₁₃₇	—	8	4	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	4	no	—
109	C₂×C₄×C₁₃₇	—	2	4	no	—
110	C₂×C₄×C₁₃₇	—	2	2	no	—
111	C₂×C₄×C₁₃₇	—	2	4	no	—
112	C₂×C₄×C₁₃₇	—	2	4	no	—
113	C₂×C₄×C₁₃₇	—	2	4	no	—
114	C₂×C₄×C₁₃₇	—	2	4	no	—
115	C₂×C₄×C₁₃₇	—	4	2	no	—
116	C₂×C₄×C₁₃₇	—	4	2	no	—
117	C₂×C₄×C₁₃₇	—	4	4	no	—
118	C₂×C₄×C₁₃₇	—	4	4	no	—
119	C₂×C₄×C₁₃₇	—	4	4	no	—
120	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₁₃₇	#137	1	4	yes	—
123	C₂×C₄×C₁₃₇	C₂³×C₄	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	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
129	C₈×C₁₃₇	—	2	2	no	—
130	C₈×C₁₃₇	—	4	2	no	—
131	C₈×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
132	C₂²×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
133	C₂²×C₁₃₇	D₄²	1	1	yes	characteristic
134	C₂²×C₁₃₇	D₄×Q₈	1	1	yes	characteristic
135	C₂²×C₁₃₇	C₂×#137	1	2	yes	—
136	C₂²×C₁₃₇	#577	1	2	yes	—
137	C₄×C₁₃₇	—	2	8	no	—
138	C₄×C₁₃₇	—	2	2	no	—
139	C₄×C₁₃₇	—	4	4	no	—
140	C₄×C₁₃₇	—	4	4	no	—
141	C₄×C₁₃₇	—	8	4	no	—
142	C₄×C₁₃₇	#441	1	4	yes	complemented
143	C₄×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₄×C₁₃₇	#520	1	1	yes	characteristic
145	C₄×C₁₃₇	#547	1	1	yes	characteristic
146	C₄×C₁₃₇	#554	1	1	yes	characteristic
147	C₂×C₁₃₇	D₄×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₁₃₇	C₂×#441	1	1	yes	characteristic
149	C₂×C₁₃₇	#2952	1	1	yes	characteristic
150	C₂×C₁₃₇	#3278	1	2	yes	—
151	C₂×C₁₃₇	#3387	1	2	yes	—
152	#22674	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₁₃₇	#22674	1	1	yes	characteristic, direct product factor, Sylow
154	C₂×#441	C₂×C₁₃₇	1	2	yes	—
155	C₂×C₄×Q₁₆	C₂×C₁₃₇	1	1	yes	characteristic
156	Q₈×C₄⋊C₄	C₂×C₁₃₇	1	2	yes	—
157	#1943	C₂×C₁₃₇	1	2	yes	—
158	#1865	C₂×C₁₃₇	1	4	yes	—
159	#1844	C₂×C₁₃₇	1	1	yes	characteristic
160	#1803	C₂×C₁₃₇	1	2	yes	—
161	#1770	C₂×C₁₃₇	1	1	yes	characteristic
162	C₂²×Q₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
163	C₂×C₄×Q₈	C₂²×C₁₃₇	1	4	yes	—
164	C₂×C₄×Q₈	C₂²×C₁₃₇	1	2	yes	—
165	#441	—	2	8	no	—
166	C₄×Q₁₆	C₄×C₁₃₇	1	8	yes	complemented
167	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
168	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
169	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
170	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
171	C₂×#103	C₂²×C₁₃₇	1	4	yes	—
172	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
173	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
174	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
175	#389	—	2	4	no	—
176	#389	C₂²×C₁₃₇	1	4	yes	—
177	#384	—	2	2	no	—
178	#384	C₂²×C₁₃₇	1	2	yes	—
179	C₄×C₄⋊C₄	—	2	2	no	—
180	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
181	#340	C₂²×C₁₃₇	1	4	yes	—
182	#336	C₂²×C₁₃₇	1	2	yes	—
183	C₂²×Q₈	C₂³×C₁₃₇	1	2	yes	—
184	C₂×Q₁₆	C₂×C₄×C₁₃₇	1	4	yes	—
185	C₂×Q₁₆	D₄×C₁₃₇	1	4	yes	—
186	C₂×Q₁₆	Q₈×C₁₃₇	1	4	yes	—
187	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
188	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
189	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
190	C₄×Q₈	—	2	4	no	—
191	C₄×Q₈	—	2	8	no	—
192	C₄×Q₈	—	4	4	no	—
193	C₄×Q₈	C₂×C₄×C₁₃₇	1	8	yes	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	—	2	4	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	4	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
200	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
201	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	—	2	4	no	—
204	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
205	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
206	#107	C₂×C₄×C₁₃₇	1	4	yes	—
207	#106	—	2	4	no	—
208	#106	—	4	2	no	—
209	C₄⋊C₈	—	2	2	no	—
210	#103	—	2	8	no	—
211	#103	C₂×C₄×C₁₃₇	1	8	yes	—
212	#97	—	2	4	no	—
213	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
214	#95	—	2	4	no	—
215	#95	D₄×C₁₃₇	1	2	yes	—
216	C₂×Q₈	—	2	2	no	—
217	C₂×Q₈	—	2	2	no	—
218	C₂×Q₈	—	2	2	no	—
219	C₂×Q₈	C₂²×C₄×C₁₃₇	1	4	yes	—
220	C₂×Q₈	C₂×D₄×C₁₃₇	1	4	yes	—
221	C₂×Q₈	C₂×Q₈×C₁₃₇	1	4	yes	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	C₂×D₄×C₁₃₇	1	4	yes	—
226	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
228	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
229	Q₁₆	C₄⋊C₄×C₁₃₇	1	16	yes	complemented
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
236	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
237	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
238	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
239	C₄⋊C₄	—	2	8	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	4	8	no	—
245	C₄⋊C₄	—	4	4	no	—
246	C₄⋊C₄	—	8	2	no	—
247	C₄⋊C₄	—	8	4	no	—
248	C₄⋊C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
249	C₄²	—	2	8	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	4	8	no	—
253	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
254	C₄²	#41×C₁₃₇	1	2	yes	—
255	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
256	Q₈	—	2	4	no	—
257	Q₈	—	2	4	no	—
258	Q₈	—	2	4	no	—
259	Q₈	C₂×C₄⋊C₄×C₁₃₇	1	8	yes	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
274	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
275	C₂×C₄	#137×C₁₃₇	1	4	yes	—
276	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
277	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
278	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
279	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
280	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
281	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	—	4	2	no	—
284	C₈	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
285	C₂²	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
286	C₂²	D₄²×C₁₃₇	1	1	yes	characteristic
287	C₂²	D₄×Q₈×C₁₃₇	1	1	yes	characteristic
288	C₂²	C₂×#137×C₁₃₇	1	2	yes	—
289	C₂²	#577×C₁₃₇	1	2	yes	—
290	C₄	—	2	8	no	—
291	C₄	—	2	2	no	—
292	C₄	—	4	4	no	—
293	C₄	—	4	4	no	—
294	C₄	—	8	4	no	—
295	C₄	#441×C₁₃₇	1	4	yes	complemented
296	C₄	C₂²×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
297	C₄	#520×C₁₃₇	1	1	yes	characteristic
298	C₄	#547×C₁₃₇	1	1	yes	characteristic
299	C₄	#554×C₁₃₇	1	1	yes	characteristic
300	C₂	D₄×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
301	C₂	C₂×#441×C₁₃₇	1	1	yes	characteristic
302	C₂	#2952×C₁₃₇	1	1	yes	characteristic
303	C₂	#3278×C₁₃₇	1	2	yes	—
304	C₂	#3387×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
76. Q₁₆×C₁₃₇	C₄⋊C₄	semidirect
142. C₄×C₁₃₇	#441	semidirect
152. #22674	C₁₃₇	direct (N × H)
153. C₁₃₇	#22674	direct (N × H)
166. C₄×Q₁₆	C₄×C₁₃₇	semidirect
229. Q₁₆	C₄⋊C₄×C₁₃₇	semidirect
295. C₄	#441×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#441×C₁₃₇	2
C₂×C₄×Q₁₆×C₁₃₇	2
Q₈×C₄⋊C₄×C₁₃₇	2
#1943×C₁₃₇	2
#1865×C₁₃₇	2
#1844×C₁₃₇	2
#1803×C₁₃₇	2
#1770×C₁₃₇	2
#22674	137

Subgroups — C137×#22674 #46872060

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#22674 #46872060