Subgroups — C₁₃₇×#33843 #46883229

Order 35072.

668 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#510×C₁₃₇	C₂	1	2	yes	maximal, complemented
2	#3285×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	#2249×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#2220×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#2211×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#2188×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#2186×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	C₄×Q₃₂×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#2169×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#135×C₁₃₇	C₂²	1	2	yes	—
11	C₂²×Q₁₆×C₁₃₇	C₂²	1	2	yes	—
12	#548×C₁₃₇	—	2	2	no	—
13	#510×C₁₃₇	—	2	8	no	—
14	C₂×Q₃₂×C₁₃₇	C₂²	1	2	yes	—
15	C₂×Q₃₂×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×Q₃₂×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×SD₃₂×C₁₃₇	C₂²	1	2	yes	—
18	C₂×SD₃₂×C₁₃₇	C₂²	1	2	yes	—
19	C₂×D₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₂×M₃₂×C₁₃₇	C₂²	1	2	yes	—
21	#495×C₁₃₇	C₂²	1	2	yes	—
22	#467×C₁₃₇	C₂²	1	2	yes	—
23	#466×C₁₃₇	C₂²	1	2	yes	—
24	#464×C₁₃₇	—	2	2	no	—
25	#452×C₁₃₇	—	2	2	no	—
26	#448×C₁₃₇	—	2	2	no	—
27	C₄×Q₁₆×C₁₃₇	C₂²	1	2	yes	—
28	C₈×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
29	#366×C₁₃₇	C₂²	1	1	yes	characteristic
30	C₄⋊C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
31	#358×C₁₃₇	C₂²	1	2	yes	—
32	#358×C₁₃₇	C₂²	1	2	yes	—
33	#357×C₁₃₇	C₂²	1	2	yes	—
34	C₂²⋊C₁₆×C₁₃₇	C₂²	1	2	yes	—
35	C₄×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
36	C₂×#41×C₁₃₇	—	2	2	no	—
37	C₂²×Q₈×C₁₃₇	—	2	2	no	—
38	#135×C₁₃₇	—	2	4	no	—
39	#135×C₁₃₇	—	4	2	no	—
40	C₂×Q₁₆×C₁₃₇	—	2	4	no	—
41	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
42	C₂×Q₁₆×C₁₃₇	—	4	2	no	—
43	C₂×Q₁₆×C₁₃₇	C₂³	1	2	yes	—
44	C₂×Q₁₆×C₁₃₇	C₂³	1	2	yes	—
45	C₂×SD₁₆×C₁₃₇	—	2	2	no	—
46	C₂×SD₁₆×C₁₃₇	—	2	2	no	—
47	C₂×D₈×C₁₃₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
49	#124×C₁₃₇	—	2	2	no	—
50	#124×C₁₃₇	—	4	2	no	—
51	C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
52	C₂²⋊Q₈×C₁₃₇	—	4	2	no	—
53	#121×C₁₃₇	—	2	2	no	—
54	#121×C₁₃₇	—	4	2	no	—
55	C₄×Q₈×C₁₃₇	—	2	2	no	—
56	C₄×D₄×C₁₃₇	—	4	2	no	—
57	C₄×D₄×C₁₃₇	D₄	1	1	yes	characteristic
58	Q₃₂×C₁₃₇	—	2	4	no	—
59	Q₃₂×C₁₃₇	—	4	1	no	—
60	Q₃₂×C₁₃₇	D₄	1	4	yes	complemented
61	SD₃₂×C₁₃₇	—	2	4	no	—
62	SD₃₂×C₁₃₇	—	2	4	no	—
63	D₁₆×C₁₃₇	—	4	1	no	—
64	M₃₂×C₁₃₇	—	2	4	no	—
65	C₂×C₁₆×C₁₃₇	C₂³	1	2	yes	—
66	C₂×C₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
67	C₂×C₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
68	#107×C₁₃₇	C₂³	1	2	yes	—
69	C₄⋊C₈×C₁₃₇	D₄	1	1	yes	characteristic
70	#103×C₁₃₇	—	2	2	no	—
71	#103×C₁₃₇	—	2	2	no	—
72	#102×C₁₃₇	—	2	2	no	—
73	C₂²⋊C₈×C₁₃₇	D₄	1	2	yes	—
74	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
75	#41×C₁₃₇	—	4	4	no	—
76	#41×C₁₃₇	—	8	2	no	—
77	C₂×Q₈×C₁₃₇	—	2	2	no	—
78	C₂×Q₈×C₁₃₇	—	2	2	no	—
79	C₂×Q₈×C₁₃₇	—	4	4	no	—
80	C₂×Q₈×C₁₃₇	—	4	2	no	—
81	C₂×Q₈×C₁₃₇	—	8	2	no	—
82	C₂×D₄×C₁₃₇	—	2	2	no	—
83	C₂×D₄×C₁₃₇	—	4	2	no	—
84	C₂×D₄×C₁₃₇	—	4	2	no	—
85	C₂×D₄×C₁₃₇	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₁₃₇	—	4	2	no	—
87	C₂²×C₄×C₁₃₇	—	4	2	no	—
88	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
89	Q₁₆×C₁₃₇	—	2	2	no	—
90	Q₁₆×C₁₃₇	—	2	2	no	—
91	Q₁₆×C₁₃₇	—	2	2	no	—
92	Q₁₆×C₁₃₇	—	4	4	no	—
93	Q₁₆×C₁₃₇	C₂×D₄	1	4	yes	—
94	SD₁₆×C₁₃₇	—	4	4	no	—
95	SD₁₆×C₁₃₇	—	8	2	no	—
96	D₈×C₁₃₇	—	2	2	no	—
97	D₈×C₁₃₇	—	2	2	no	—
98	C₂×C₈×C₁₃₇	—	2	2	no	—
99	C₂×C₈×C₁₃₇	—	2	2	no	—
100	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₁₃₇	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₁₃₇	—	2	2	no	—
104	C₄⋊C₄×C₁₃₇	—	4	2	no	—
105	C₄⋊C₄×C₁₃₇	—	4	2	no	—
106	C₄⋊C₄×C₁₃₇	D₈	1	1	yes	characteristic
107	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
108	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
109	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
110	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
111	C₂²⋊C₄×C₁₃₇	D₈	1	2	yes	—
112	C₄²×C₁₃₇	—	4	2	no	—
113	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
114	C₁₆×C₁₃₇	—	2	2	no	—
115	C₁₆×C₁₃₇	—	2	1	no	—
116	C₁₆×C₁₃₇	C₂×D₄	1	2	yes	—
117	C₂³×C₁₃₇	—	4	2	no	—
118	C₂³×C₁₃₇	C₂×D₈	1	2	yes	—
119	Q₈×C₁₃₇	—	2	4	no	—
120	Q₈×C₁₃₇	—	4	2	no	—
121	Q₈×C₁₃₇	—	4	2	no	—
122	Q₈×C₁₃₇	—	4	2	no	—
123	Q₈×C₁₃₇	—	8	4	no	—
124	D₄×C₁₃₇	—	4	1	no	—
125	D₄×C₁₃₇	—	4	2	no	—
126	D₄×C₁₃₇	—	4	2	no	—
127	D₄×C₁₃₇	—	8	4	no	—
128	D₄×C₁₃₇	—	8	4	no	—
129	C₂×C₄×C₁₃₇	—	2	2	no	—
130	C₂×C₄×C₁₃₇	—	2	2	no	—
131	C₂×C₄×C₁₃₇	—	4	2	no	—
132	C₂×C₄×C₁₃₇	—	4	2	no	—
133	C₂×C₄×C₁₃₇	—	4	2	no	—
134	C₂×C₄×C₁₃₇	—	8	4	no	—
135	C₂×C₄×C₁₃₇	—	8	4	no	—
136	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₁₃₇	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
140	C₈×C₁₃₇	—	2	2	no	—
141	C₈×C₁₃₇	—	2	1	no	—
142	C₈×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
143	C₈×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
144	C₂²×C₁₃₇	—	2	2	no	—
145	C₂²×C₁₃₇	—	2	2	no	—
146	C₂²×C₁₃₇	—	2	2	no	—
147	C₂²×C₁₃₇	—	8	2	no	—
148	C₂²×C₁₃₇	—	8	2	no	—
149	C₂²×C₁₃₇	—	8	2	no	—
150	C₂²×C₁₃₇	C₂²×D₈	1	1	yes	characteristic
151	C₄×C₁₃₇	—	2	1	no	—
152	C₄×C₁₃₇	—	2	1	no	—
153	C₄×C₁₃₇	—	2	2	no	—
154	C₄×C₁₃₇	—	4	4	no	—
155	C₄×C₁₃₇	—	8	2	no	—
156	C₄×C₁₃₇	—	8	2	no	—
157	C₄×C₁₃₇	D₄²	1	1	yes	characteristic
158	C₄×C₁₃₇	#576	1	1	yes	characteristic
159	C₂×C₁₃₇	—	4	2	no	—
160	C₂×C₁₃₇	—	16	2	no	—
161	C₂×C₁₃₇	D₄×D₈	1	1	yes	characteristic
162	C₂×C₁₃₇	#3414	1	1	yes	characteristic
163	C₂×C₁₃₇	#3417	1	1	yes	characteristic
164	#33843	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
165	C₁₃₇	#33843	1	1	yes	characteristic, direct product factor, Sylow
166	C₂×#510	C₂×C₁₃₇	1	2	yes	complemented
167	#3285	C₂×C₁₃₇	1	2	yes	complemented
168	#2249	C₂×C₁₃₇	1	1	yes	characteristic, complemented
169	#2220	C₂×C₁₃₇	1	2	yes	complemented
170	#2211	C₂×C₁₃₇	1	2	yes	complemented
171	#2188	C₂×C₁₃₇	1	2	yes	complemented
172	#2186	C₂×C₁₃₇	1	2	yes	complemented
173	C₄×Q₃₂	C₂×C₁₃₇	1	1	yes	characteristic, complemented
174	#2169	C₂×C₁₃₇	1	1	yes	characteristic, complemented
175	C₂×#135	C₂²×C₁₃₇	1	2	yes	—
176	C₂²×Q₁₆	C₂²×C₁₃₇	1	2	yes	—
177	#548	—	2	2	no	—
178	#510	—	2	8	no	—
179	C₂×Q₃₂	C₂²×C₁₃₇	1	2	yes	—
180	C₂×Q₃₂	C₂²×C₁₃₇	1	1	yes	characteristic
181	C₂×Q₃₂	C₂²×C₁₃₇	1	1	yes	characteristic
182	C₂×SD₃₂	C₂²×C₁₃₇	1	2	yes	—
183	C₂×SD₃₂	C₂²×C₁₃₇	1	2	yes	—
184	C₂×D₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
185	C₂×M₃₂	C₂²×C₁₃₇	1	2	yes	—
186	#495	C₂²×C₁₃₇	1	2	yes	—
187	#467	C₂²×C₁₃₇	1	2	yes	—
188	#466	C₂²×C₁₃₇	1	2	yes	—
189	#464	—	2	2	no	—
190	#452	—	2	2	no	—
191	#448	—	2	2	no	—
192	C₄×Q₁₆	C₂²×C₁₃₇	1	2	yes	—
193	C₈×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
194	#366	C₂²×C₁₃₇	1	1	yes	characteristic
195	C₄⋊C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
196	#358	C₂²×C₁₃₇	1	2	yes	—
197	#358	C₂²×C₁₃₇	1	2	yes	—
198	#357	C₂²×C₁₃₇	1	2	yes	—
199	C₂²⋊C₁₆	C₂²×C₁₃₇	1	2	yes	—
200	C₄×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
201	C₂×#41	—	2	2	no	—
202	C₂²×Q₈	—	2	2	no	—
203	#135	—	2	4	no	—
204	#135	—	4	2	no	—
205	C₂×Q₁₆	—	2	4	no	—
206	C₂×Q₁₆	—	2	2	no	—
207	C₂×Q₁₆	—	4	2	no	—
208	C₂×Q₁₆	C₂³×C₁₃₇	1	2	yes	—
209	C₂×Q₁₆	C₂³×C₁₃₇	1	2	yes	—
210	C₂×SD₁₆	—	2	2	no	—
211	C₂×SD₁₆	—	2	2	no	—
212	C₂×D₈	C₂³×C₁₃₇	1	2	yes	—
213	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
214	#124	—	2	2	no	—
215	#124	—	4	2	no	—
216	C₂²⋊Q₈	—	2	2	no	—
217	C₂²⋊Q₈	—	4	2	no	—
218	#121	—	2	2	no	—
219	#121	—	4	2	no	—
220	C₄×Q₈	—	2	2	no	—
221	C₄×D₄	—	4	2	no	—
222	C₄×D₄	D₄×C₁₃₇	1	1	yes	characteristic
223	Q₃₂	—	2	4	no	—
224	Q₃₂	—	4	1	no	—
225	Q₃₂	D₄×C₁₃₇	1	4	yes	complemented
226	SD₃₂	—	2	4	no	—
227	SD₃₂	—	2	4	no	—
228	D₁₆	—	4	1	no	—
229	M₃₂	—	2	4	no	—
230	C₂×C₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
231	C₂×C₁₆	C₂³×C₁₃₇	1	2	yes	—
232	C₂×C₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
233	#107	C₂³×C₁₃₇	1	2	yes	—
234	C₄⋊C₈	D₄×C₁₃₇	1	1	yes	characteristic
235	#103	—	2	2	no	—
236	#103	—	2	2	no	—
237	#102	—	2	2	no	—
238	C₂²⋊C₈	D₄×C₁₃₇	1	2	yes	—
239	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
240	#41	—	4	4	no	—
241	#41	—	8	2	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	4	4	no	—
245	C₂×Q₈	—	4	2	no	—
246	C₂×Q₈	—	8	2	no	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	—	4	2	no	—
249	C₂×D₄	—	4	2	no	—
250	C₂×D₄	D₈×C₁₃₇	1	1	yes	characteristic
251	C₂²×C₄	—	4	2	no	—
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
254	Q₁₆	—	2	2	no	—
255	Q₁₆	—	2	2	no	—
256	Q₁₆	—	2	2	no	—
257	Q₁₆	—	4	4	no	—
258	Q₁₆	C₂×D₄×C₁₃₇	1	4	yes	—
259	SD₁₆	—	4	4	no	—
260	SD₁₆	—	8	2	no	—
261	D₈	—	2	2	no	—
262	D₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
266	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
267	C₂×C₈	C₂⁴×C₁₃₇	1	1	yes	characteristic
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	D₈×C₁₃₇	1	1	yes	characteristic
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	D₈×C₁₃₇	1	2	yes	—
277	C₄²	—	4	2	no	—
278	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
279	C₁₆	—	2	2	no	—
280	C₁₆	—	2	1	no	—
281	C₁₆	C₂×D₄×C₁₃₇	1	2	yes	—
282	C₂³	—	4	2	no	—
283	C₂³	C₂×D₈×C₁₃₇	1	2	yes	—
284	Q₈	—	2	4	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	4	2	no	—
287	Q₈	—	4	2	no	—
288	Q₈	—	8	4	no	—
289	D₄	—	4	1	no	—
290	D₄	—	4	2	no	—
291	D₄	—	4	2	no	—
292	D₄	—	8	4	no	—
293	D₄	—	8	4	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	8	4	no	—
300	C₂×C₄	—	8	4	no	—
301	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
302	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
303	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
304	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
305	C₈	—	2	2	no	—
306	C₈	—	2	1	no	—
307	C₈	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
308	C₈	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	C₂²×D₈×C₁₃₇	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	2	no	—
319	C₄	—	4	4	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	D₄²×C₁₃₇	1	1	yes	characteristic
323	C₄	#576×C₁₃₇	1	1	yes	characteristic
324	C₂	—	4	2	no	—
325	C₂	—	16	2	no	—
326	C₂	D₄×D₈×C₁₃₇	1	1	yes	characteristic
327	C₂	#3414×C₁₃₇	1	1	yes	characteristic
328	C₂	#3417×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₄×D₄×C₁₃₇ ◁ C₈×D₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#510×C₁₃₇	C₂	semidirect
2. #3285×C₁₃₇	C₂	semidirect
3. #2249×C₁₃₇	C₂	semidirect
4. #2220×C₁₃₇	C₂	semidirect
5. #2211×C₁₃₇	C₂	semidirect
6. #2188×C₁₃₇	C₂	semidirect
7. #2186×C₁₃₇	C₂	semidirect
8. C₄×Q₃₂×C₁₃₇	C₂	semidirect
9. #2169×C₁₃₇	C₂	semidirect
60. Q₃₂×C₁₃₇	D₄	semidirect
164. #33843	C₁₃₇	direct (N × H)
165. C₁₃₇	#33843	direct (N × H)
166. C₂×#510	C₂×C₁₃₇	semidirect
167. #3285	C₂×C₁₃₇	semidirect
168. #2249	C₂×C₁₃₇	semidirect
169. #2220	C₂×C₁₃₇	semidirect
170. #2211	C₂×C₁₃₇	semidirect
171. #2188	C₂×C₁₃₇	semidirect
172. #2186	C₂×C₁₃₇	semidirect
173. C₄×Q₃₂	C₂×C₁₃₇	semidirect
174. #2169	C₂×C₁₃₇	semidirect
225. Q₃₂	D₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#510×C₁₃₇	2
#3285×C₁₃₇	2
#2249×C₁₃₇	2
#2220×C₁₃₇	2
#2211×C₁₃₇	2
#2188×C₁₃₇	2
#2186×C₁₃₇	2
C₄×Q₃₂×C₁₃₇	2
#2169×C₁₃₇	2
#33843	137

Subgroups — C137×#33843 #46883229

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#33843 #46883229