Subgroups — C₁₃₇×#33693 #46883079

Order 35072.

512 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3177×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#2955×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#2239×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	#2231×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#2214×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#2191×C₁₃₇	C₂	1	2	yes	maximal
7	#2180×C₁₃₇	C₂	1	4	yes	maximal, complemented
8	#2166×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#575×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×#131×C₁₃₇	—	2	1	no	—
11	#541×C₁₃₇	—	2	1	no	—
12	#510×C₁₃₇	C₂²	1	4	yes	—
13	#499×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
14	#494×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
15	#491×C₁₃₇	—	2	1	no	—
16	#470×C₁₃₇	C₂²	1	2	yes	—
17	#467×C₁₃₇	—	2	1	no	—
18	#444×C₁₃₇	—	2	1	no	—
19	#444×C₁₃₇	C₂²	1	2	yes	—
20	#443×C₁₃₇	—	2	1	no	—
21	#443×C₁₃₇	C₂²	1	2	yes	—
22	#441×C₁₃₇	C₂²	1	2	yes	complemented
23	C₄×Q₁₆×C₁₃₇	—	2	1	no	—
24	#430×C₁₃₇	C₂²	1	1	yes	characteristic
25	#421×C₁₃₇	—	2	1	no	—
26	#405×C₁₃₇	C₂²	1	1	yes	characteristic
27	#365×C₁₃₇	C₂²	1	4	yes	complemented
28	#364×C₁₃₇	C₂²	1	2	yes	complemented
29	#362×C₁₃₇	C₂²	1	4	yes	complemented
30	#360×C₁₃₇	C₂²	1	4	yes	—
31	#349×C₁₃₇	C₂²	1	2	yes	—
32	#347×C₁₃₇	C₂²	1	2	yes	complemented
33	C₂×#41×C₁₃₇	—	2	1	no	—
34	#137×C₁₃₇	—	2	2	no	—
35	AGL(1,ℤ/8)×C₁₃₇	—	2	2	no	—
36	#135×C₁₃₇	—	2	1	no	—
37	#135×C₁₃₇	C₂³	1	2	yes	—
38	C₂×Q₁₆×C₁₃₇	—	2	1	no	—
39	C₂×Q₁₆×C₁₃₇	C₂³	1	2	yes	—
40	#131×C₁₃₇	—	2	2	no	—
41	#131×C₁₃₇	—	2	2	no	—
42	#131×C₁₃₇	—	4	1	no	—
43	#131×C₁₃₇	—	4	1	no	—
44	C₂×M₁₆×C₁₃₇	—	2	1	no	—
45	C₂×M₁₆×C₁₃₇	—	4	1	no	—
46	C₂×M₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₁₃₇	—	4	1	no	—
48	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
49	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
50	#126×C₁₃₇	—	4	2	no	—
51	#125×C₁₃₇	—	2	1	no	—
52	#125×C₁₃₇	—	4	2	no	—
53	C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
54	C₄×Q₈×C₁₃₇	—	2	2	no	—
55	#117×C₁₃₇	D₄	1	1	yes	characteristic
56	Q₃₂×C₁₃₇	—	2	4	no	—
57	SD₃₂×C₁₃₇	—	2	4	no	—
58	M₃₂×C₁₃₇	C₂³	1	4	yes	—
59	#108×C₁₃₇	—	2	1	no	—
60	#108×C₁₃₇	C₂³	1	2	yes	—
61	#107×C₁₃₇	—	2	1	no	—
62	#106×C₁₃₇	C₂³	1	2	yes	—
63	#104×C₁₃₇	—	2	2	no	—
64	#104×C₁₃₇	—	2	2	no	—
65	#103×C₁₃₇	—	2	2	no	—
66	#103×C₁₃₇	—	2	2	no	—
67	#97×C₁₃₇	D₄	1	1	yes	characteristic
68	#97×C₁₃₇	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
70	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
71	#41×C₁₃₇	—	2	2	no	—
72	#41×C₁₃₇	—	2	2	no	—
73	#41×C₁₃₇	—	4	1	no	—
74	#41×C₁₃₇	—	4	1	no	—
75	C₂×Q₈×C₁₃₇	—	2	1	no	—
76	C₂×Q₈×C₁₃₇	—	2	2	no	—
77	C₂×D₄×C₁₃₇	—	2	1	no	—
78	C₂×D₄×C₁₃₇	—	4	1	no	—
79	C₂²×C₄×C₁₃₇	—	4	1	no	—
80	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
81	Q₁₆×C₁₃₇	—	2	1	no	—
82	Q₁₆×C₁₃₇	—	2	4	no	—
83	Q₁₆×C₁₃₇	—	4	1	no	—
84	Q₁₆×C₁₃₇	—	4	1	no	—
85	Q₁₆×C₁₃₇	#41	1	2	yes	—
86	SD₁₆×C₁₃₇	—	2	2	no	—
87	SD₁₆×C₁₃₇	—	2	2	no	—
88	D₈×C₁₃₇	—	2	1	no	—
89	D₈×C₁₃₇	#41	1	2	yes	—
90	M₁₆×C₁₃₇	—	2	2	no	—
91	M₁₆×C₁₃₇	—	2	2	no	—
92	M₁₆×C₁₃₇	—	4	2	no	—
93	M₁₆×C₁₃₇	—	4	1	no	—
94	M₁₆×C₁₃₇	—	4	2	no	—
95	C₂×C₈×C₁₃₇	—	2	1	no	—
96	C₂×C₈×C₁₃₇	—	2	1	no	—
97	C₂×C₈×C₁₃₇	—	4	2	no	—
98	C₂×C₈×C₁₃₇	—	4	1	no	—
99	C₂×C₈×C₁₃₇	—	4	2	no	—
100	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₁₃₇	C₂⁴	1	1	yes	characteristic
104	C₄⋊C₄×C₁₃₇	—	2	1	no	—
105	C₄⋊C₄×C₁₃₇	—	2	2	no	—
106	C₄⋊C₄×C₁₃₇	—	4	2	no	—
107	C₄⋊C₄×C₁₃₇	—	4	2	no	—
108	C₄⋊C₄×C₁₃₇	—	4	2	no	—
109	C₄⋊C₄×C₁₃₇	—	4	2	no	—
110	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
111	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
112	C₄²×C₁₃₇	—	4	2	no	—
113	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
115	C₁₆×C₁₃₇	—	2	4	no	—
116	C₂³×C₁₃₇	—	4	1	no	—
117	C₂³×C₁₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
118	Q₈×C₁₃₇	—	2	2	no	—
119	Q₈×C₁₃₇	—	4	1	no	—
120	Q₈×C₁₃₇	—	4	2	no	—
121	Q₈×C₁₃₇	—	4	2	no	—
122	D₄×C₁₃₇	—	2	2	no	—
123	D₄×C₁₃₇	—	4	1	no	—
124	D₄×C₁₃₇	—	4	2	no	—
125	D₄×C₁₃₇	—	4	2	no	—
126	C₂×C₄×C₁₃₇	—	2	1	no	—
127	C₂×C₄×C₁₃₇	—	2	1	no	—
128	C₂×C₄×C₁₃₇	—	2	1	no	—
129	C₂×C₄×C₁₃₇	—	2	1	no	—
130	C₂×C₄×C₁₃₇	—	4	2	no	—
131	C₂×C₄×C₁₃₇	—	4	1	no	—
132	C₂×C₄×C₁₃₇	—	4	2	no	—
133	C₂×C₄×C₁₃₇	—	4	2	no	—
134	C₂×C₄×C₁₃₇	—	4	2	no	—
135	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
137	C₈×C₁₃₇	—	2	1	no	—
138	C₈×C₁₃₇	—	2	1	no	—
139	C₈×C₁₃₇	—	4	2	no	—
140	C₈×C₁₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₈×C₁₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₈×C₁₃₇	C₂×#41	1	2	yes	—
143	C₂²×C₁₃₇	—	2	1	no	—
144	C₂²×C₁₃₇	—	4	1	no	—
145	C₂²×C₁₃₇	—	4	2	no	—
146	C₂²×C₁₃₇	—	8	2	no	—
147	C₂²×C₁₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₄×C₁₃₇	—	2	1	no	—
149	C₄×C₁₃₇	—	4	1	no	—
150	C₄×C₁₃₇	—	4	1	no	—
151	C₄×C₁₃₇	—	4	2	no	—
152	C₄×C₁₃₇	—	8	2	no	—
153	C₄×C₁₃₇	—	8	2	no	—
154	C₄×C₁₃₇	#525	1	1	yes	characteristic
155	C₄×C₁₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
156	C₂×C₁₃₇	—	2	1	no	—
157	C₂×C₁₃₇	—	4	1	no	—
158	C₂×C₁₃₇	—	8	2	no	—
159	C₂×C₁₃₇	#3109	1	1	yes	characteristic
160	#33693	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₁₃₇	#33693	1	1	yes	characteristic, direct product factor, Sylow
162	#3177	C₂×C₁₃₇	1	1	yes	characteristic, complemented
163	#2955	C₂×C₁₃₇	1	1	yes	characteristic
164	#2239	C₂×C₁₃₇	1	2	yes	complemented
165	#2231	C₂×C₁₃₇	1	2	yes	complemented
166	#2214	C₂×C₁₃₇	1	2	yes	complemented
167	#2191	C₂×C₁₃₇	1	2	yes	—
168	#2180	C₂×C₁₃₇	1	4	yes	complemented
169	#2166	C₂×C₁₃₇	1	1	yes	characteristic, complemented
170	#575	C₂²×C₁₃₇	1	1	yes	characteristic
171	C₂×#131	—	2	1	no	—
172	#541	—	2	1	no	—
173	#510	C₂²×C₁₃₇	1	4	yes	—
174	#499	C₂²×C₁₃₇	1	1	yes	characteristic, complemented
175	#494	C₂²×C₁₃₇	1	1	yes	characteristic, complemented
176	#491	—	2	1	no	—
177	#470	C₂²×C₁₃₇	1	2	yes	—
178	#467	—	2	1	no	—
179	#444	—	2	1	no	—
180	#444	C₂²×C₁₃₇	1	2	yes	—
181	#443	—	2	1	no	—
182	#443	C₂²×C₁₃₇	1	2	yes	—
183	#441	C₂²×C₁₃₇	1	2	yes	complemented
184	C₄×Q₁₆	—	2	1	no	—
185	#430	C₂²×C₁₃₇	1	1	yes	characteristic
186	#421	—	2	1	no	—
187	#405	C₂²×C₁₃₇	1	1	yes	characteristic
188	#365	C₂²×C₁₃₇	1	4	yes	complemented
189	#364	C₂²×C₁₃₇	1	2	yes	complemented
190	#362	C₂²×C₁₃₇	1	4	yes	complemented
191	#360	C₂²×C₁₃₇	1	4	yes	—
192	#349	C₂²×C₁₃₇	1	2	yes	—
193	#347	C₂²×C₁₃₇	1	2	yes	complemented
194	C₂×#41	—	2	1	no	—
195	#137	—	2	2	no	—
196	AGL(1,ℤ/8)	—	2	2	no	—
197	#135	—	2	1	no	—
198	#135	C₂³×C₁₃₇	1	2	yes	—
199	C₂×Q₁₆	—	2	1	no	—
200	C₂×Q₁₆	C₂³×C₁₃₇	1	2	yes	—
201	#131	—	2	2	no	—
202	#131	—	2	2	no	—
203	#131	—	4	1	no	—
204	#131	—	4	1	no	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	—	4	1	no	—
207	C₂×M₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
208	C₂²×C₈	—	4	1	no	—
209	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
210	C₄⋊Q₈	—	2	1	no	—
211	#126	—	4	2	no	—
212	#125	—	2	1	no	—
213	#125	—	4	2	no	—
214	C₂²⋊Q₈	—	2	2	no	—
215	C₄×Q₈	—	2	2	no	—
216	#117	D₄×C₁₃₇	1	1	yes	characteristic
217	Q₃₂	—	2	4	no	—
218	SD₃₂	—	2	4	no	—
219	M₃₂	C₂³×C₁₃₇	1	4	yes	—
220	#108	—	2	1	no	—
221	#108	C₂³×C₁₃₇	1	2	yes	—
222	#107	—	2	1	no	—
223	#106	C₂³×C₁₃₇	1	2	yes	—
224	#104	—	2	2	no	—
225	#104	—	2	2	no	—
226	#103	—	2	2	no	—
227	#103	—	2	2	no	—
228	#97	D₄×C₁₃₇	1	1	yes	characteristic
229	#97	C₂³×C₁₃₇	1	1	yes	characteristic
230	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
231	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
232	#41	—	2	2	no	—
233	#41	—	2	2	no	—
234	#41	—	4	1	no	—
235	#41	—	4	1	no	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
242	Q₁₆	—	2	1	no	—
243	Q₁₆	—	2	4	no	—
244	Q₁₆	—	4	1	no	—
245	Q₁₆	—	4	1	no	—
246	Q₁₆	#41×C₁₃₇	1	2	yes	—
247	SD₁₆	—	2	2	no	—
248	SD₁₆	—	2	2	no	—
249	D₈	—	2	1	no	—
250	D₈	#41×C₁₃₇	1	2	yes	—
251	M₁₆	—	2	2	no	—
252	M₁₆	—	2	2	no	—
253	M₁₆	—	4	2	no	—
254	M₁₆	—	4	1	no	—
255	M₁₆	—	4	2	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	4	2	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	4	2	no	—
261	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
262	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
263	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
264	C₂×C₈	C₂⁴×C₁₃₇	1	1	yes	characteristic
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	2	1	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₄²	—	4	2	no	—
274	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
275	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
276	C₁₆	—	2	4	no	—
277	C₂³	—	4	1	no	—
278	C₂³	AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
279	Q₈	—	2	2	no	—
280	Q₈	—	4	1	no	—
281	Q₈	—	4	2	no	—
282	Q₈	—	4	2	no	—
283	D₄	—	2	2	no	—
284	D₄	—	4	1	no	—
285	D₄	—	4	2	no	—
286	D₄	—	4	2	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
297	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	4	2	no	—
301	C₈	AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
302	C₈	AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
303	C₈	C₂×#41×C₁₃₇	1	2	yes	—
304	C₂²	—	2	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	2	no	—
307	C₂²	—	8	2	no	—
308	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	2	no	—
313	C₄	—	8	2	no	—
314	C₄	—	8	2	no	—
315	C₄	#525×C₁₃₇	1	1	yes	characteristic
316	C₄	C₂×AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
317	C₂	—	2	1	no	—
318	C₂	—	4	1	no	—
319	C₂	—	8	2	no	—
320	C₂	#3109×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃₇	#3109
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₇
prime radical	#575×C₁₃₇	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃₇ ◁ C₂×C₄×C₁₃₇ ◁ #405×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. #3177×C₁₃₇	C₂	semidirect
3. #2239×C₁₃₇	C₂	semidirect
4. #2231×C₁₃₇	C₂	semidirect
5. #2214×C₁₃₇	C₂	semidirect
7. #2180×C₁₃₇	C₂	semidirect
8. #2166×C₁₃₇	C₂	semidirect
13. #499×C₁₃₇	C₂²	semidirect
14. #494×C₁₃₇	C₂²	semidirect
22. #441×C₁₃₇	C₂²	semidirect
27. #365×C₁₃₇	C₂²	semidirect
28. #364×C₁₃₇	C₂²	semidirect
29. #362×C₁₃₇	C₂²	semidirect
32. #347×C₁₃₇	C₂²	semidirect
160. #33693	C₁₃₇	direct (N × H)
161. C₁₃₇	#33693	direct (N × H)
162. #3177	C₂×C₁₃₇	semidirect
164. #2239	C₂×C₁₃₇	semidirect
165. #2231	C₂×C₁₃₇	semidirect
166. #2214	C₂×C₁₃₇	semidirect
168. #2180	C₂×C₁₃₇	semidirect
169. #2166	C₂×C₁₃₇	semidirect
174. #499	C₂²×C₁₃₇	semidirect
175. #494	C₂²×C₁₃₇	semidirect
183. #441	C₂²×C₁₃₇	semidirect
188. #365	C₂²×C₁₃₇	semidirect
189. #364	C₂²×C₁₃₇	semidirect
190. #362	C₂²×C₁₃₇	semidirect
193. #347	C₂²×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3177×C₁₃₇	2
#2955×C₁₃₇	2
#2239×C₁₃₇	2
#2231×C₁₃₇	2
#2214×C₁₃₇	2
#2191×C₁₃₇	2
#2180×C₁₃₇	2
#2166×C₁₃₇	2
#33693	137

Subgroups — C137×#33693 #46883079

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#33693 #46883079