Subgroups — C₁₃₇×#28736 #46878122

Order 35072.

1022 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#493×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#2548×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	#2054×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#1982×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#1970×C₁₃₇	C₂	1	4	yes	maximal, complemented
7	C₄×#107×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₁₃₇	C₂²	1	2	yes	—
9	C₂²×D₈×C₁₃₇	C₂²	1	2	yes	—
10	C₂×#127×C₁₃₇	C₂²	1	2	yes	—
11	C₂×#124×C₁₃₇	—	2	2	no	—
12	C₂×C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
13	#493×C₁₃₇	—	2	4	no	—
14	C₈×Q₈×C₁₃₇	—	2	4	no	—
15	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×#102×C₁₃₇	C₂²	1	8	yes	—
19	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
21	#394×C₁₃₇	—	2	4	no	—
22	#390×C₁₃₇	—	2	2	no	—
23	#390×C₁₃₇	C₂²	1	8	yes	—
24	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
25	#340×C₁₃₇	C₂²	1	2	yes	—
26	C₂²×Q₈×C₁₃₇	D₄	1	1	yes	characteristic
27	C₂²×D₄×C₁₃₇	—	2	2	no	—
28	C₂²×D₄×C₁₃₇	—	2	2	no	—
29	C₂²×D₄×C₁₃₇	C₂³	1	4	yes	—
30	C₂×D₈×C₁₃₇	—	2	4	no	—
31	C₂×D₈×C₁₃₇	—	2	4	no	—
32	C₂×D₈×C₁₃₇	—	2	8	no	—
33	C₂×D₈×C₁₃₇	—	4	4	no	—
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	#127×C₁₃₇	—	2	4	no	—
38	#127×C₁₃₇	—	2	4	no	—
39	#124×C₁₃₇	—	4	8	no	—
40	C₄×Q₈×C₁₃₇	—	2	2	no	—
41	C₄×Q₈×C₁₃₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
45	C₂×C₂²⋊C₄×C₁₃₇	—	2	8	no	—
46	C₂×C₂²⋊C₄×C₁₃₇	—	2	8	no	—
47	C₂×C₄²×C₁₃₇	—	2	2	no	—
48	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
49	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
50	#107×C₁₃₇	—	2	4	no	—
51	C₄⋊C₈×C₁₃₇	—	2	4	no	—
52	C₄⋊C₈×C₁₃₇	—	2	2	no	—
53	#102×C₁₃₇	—	4	8	no	—
54	C₄×C₈×C₁₃₇	—	2	4	no	—
55	C₄×C₈×C₁₃₇	D₄	1	4	yes	—
56	#95×C₁₃₇	—	2	2	no	—
57	C₂⁴×C₁₃₇	—	2	4	no	—
58	C₂×Q₈×C₁₃₇	—	2	2	no	—
59	C₂×Q₈×C₁₃₇	—	2	2	no	—
60	C₂×Q₈×C₁₃₇	D₈	1	4	yes	complemented
61	C₂×D₄×C₁₃₇	—	2	4	no	—
62	C₂×D₄×C₁₃₇	—	2	4	no	—
63	C₂×D₄×C₁₃₇	—	2	8	no	—
64	C₂×D₄×C₁₃₇	—	2	8	no	—
65	C₂×D₄×C₁₃₇	—	4	4	no	—
66	C₂×D₄×C₁₃₇	—	4	4	no	—
67	C₂×D₄×C₁₃₇	—	4	4	no	—
68	C₂×D₄×C₁₃₇	—	4	4	no	—
69	C₂×D₄×C₁₃₇	—	8	4	no	—
70	C₂²×C₄×C₁₃₇	—	2	4	no	—
71	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
76	D₈×C₁₃₇	—	4	8	no	—
77	D₈×C₁₃₇	—	4	8	no	—
78	C₂×C₈×C₁₃₇	—	2	2	no	—
79	C₂×C₈×C₁₃₇	—	2	2	no	—
80	C₂×C₈×C₁₃₇	—	2	4	no	—
81	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₁₃₇	#41	1	4	yes	—
84	C₄⋊C₄×C₁₃₇	—	2	4	no	—
85	C₄⋊C₄×C₁₃₇	—	2	2	no	—
86	C₄⋊C₄×C₁₃₇	—	2	4	no	—
87	C₄⋊C₄×C₁₃₇	—	2	4	no	—
88	C₂²⋊C₄×C₁₃₇	—	4	16	no	—
89	C₂²⋊C₄×C₁₃₇	—	8	8	no	—
90	C₄²×C₁₃₇	—	2	2	no	—
91	C₄²×C₁₃₇	—	2	2	no	—
92	C₄²×C₁₃₇	—	4	4	no	—
93	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
94	C₄²×C₁₃₇	#41	1	2	yes	—
95	C₂³×C₁₃₇	—	4	4	no	—
96	C₂³×C₁₃₇	—	4	8	no	—
97	C₂³×C₁₃₇	—	4	4	no	—
98	C₂³×C₁₃₇	—	4	4	no	—
99	C₂³×C₁₃₇	—	4	8	no	—
100	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
101	Q₈×C₁₃₇	—	2	8	no	—
102	D₄×C₁₃₇	—	4	4	no	—
103	D₄×C₁₃₇	—	4	4	no	—
104	D₄×C₁₃₇	—	4	8	no	—
105	D₄×C₁₃₇	—	8	8	no	—
106	D₄×C₁₃₇	—	8	8	no	—
107	C₂×C₄×C₁₃₇	—	2	2	no	—
108	C₂×C₄×C₁₃₇	—	2	2	no	—
109	C₂×C₄×C₁₃₇	—	2	2	no	—
110	C₂×C₄×C₁₃₇	—	2	2	no	—
111	C₂×C₄×C₁₃₇	—	2	4	no	—
112	C₂×C₄×C₁₃₇	—	4	4	no	—
113	C₂×C₄×C₁₃₇	—	4	4	no	—
114	C₂×C₄×C₁₃₇	—	4	4	no	—
115	C₂×C₄×C₁₃₇	#121	1	2	yes	—
116	C₂×C₄×C₁₃₇	#121	1	2	yes	—
117	C₂×C₄×C₁₃₇	C₂×D₈	1	4	yes	—
118	C₂×C₄×C₁₃₇	C₂×D₈	1	2	yes	—
119	C₂×C₄×C₁₃₇	#135	1	2	yes	—
120	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
122	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
123	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	2	yes	—
124	C₈×C₁₃₇	—	2	2	no	—
125	C₈×C₁₃₇	—	4	2	no	—
126	C₈×C₁₃₇	#124	1	4	yes	complemented
127	C₂²×C₁₃₇	—	8	8	no	—
128	C₂²×C₁₃₇	—	8	8	no	—
129	C₂²×C₁₃₇	—	8	4	no	—
130	C₂²×C₁₃₇	—	8	4	no	—
131	C₂²×C₁₃₇	—	8	4	no	—
132	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
133	C₂²×C₁₃₇	#550	1	2	yes	—
134	C₂²×C₁₃₇	C₂²×D₈	1	1	yes	characteristic
135	C₂²×C₁₃₇	C₂×#135	1	1	yes	characteristic
136	C₂²×C₁₃₇	#576	1	2	yes	—
137	C₄×C₁₃₇	—	2	4	no	—
138	C₄×C₁₃₇	—	2	2	no	—
139	C₄×C₁₃₇	—	4	2	no	—
140	C₄×C₁₃₇	—	8	4	no	—
141	C₄×C₁₃₇	#466	1	4	yes	—
142	C₄×C₁₃₇	C₂×#124	1	1	yes	characteristic
143	C₄×C₁₃₇	#535	1	1	yes	characteristic
144	C₄×C₁₃₇	#559	1	2	yes	—
145	C₂×C₁₃₇	—	16	4	no	—
146	C₂×C₁₃₇	#2548	1	1	yes	characteristic
147	C₂×C₁₃₇	C₂×#466	1	1	yes	characteristic
148	C₂×C₁₃₇	#3058	1	1	yes	characteristic
149	C₂×C₁₃₇	#3391	1	2	yes	—
150	C₂×C₁₃₇	#3396	1	2	yes	—
151	#28736	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₁₃₇	#28736	1	1	yes	characteristic, direct product factor, Sylow
153	C₂×#493	C₂×C₁₃₇	1	1	yes	characteristic
154	C₂×C₈×Q₈	C₂×C₁₃₇	1	1	yes	characteristic, complemented
155	#2548	C₂×C₁₃₇	1	2	yes	complemented
156	#2054	C₂×C₁₃₇	1	4	yes	complemented
157	#1982	C₂×C₁₃₇	1	2	yes	complemented
158	#1970	C₂×C₁₃₇	1	4	yes	complemented
159	C₄×#107	C₂×C₁₃₇	1	1	yes	characteristic, complemented
160	C₂²×D₈	C₂²×C₁₃₇	1	2	yes	—
161	C₂²×D₈	C₂²×C₁₃₇	1	2	yes	—
162	C₂×#127	C₂²×C₁₃₇	1	2	yes	—
163	C₂×#124	—	2	2	no	—
164	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
165	#493	—	2	4	no	—
166	C₈×Q₈	—	2	4	no	—
167	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
168	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
169	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
170	C₂×#102	C₂²×C₁₃₇	1	8	yes	—
171	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
172	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
173	#394	—	2	4	no	—
174	#390	—	2	2	no	—
175	#390	C₂²×C₁₃₇	1	8	yes	—
176	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
177	#340	C₂²×C₁₃₇	1	2	yes	—
178	C₂²×Q₈	D₄×C₁₃₇	1	1	yes	characteristic
179	C₂²×D₄	—	2	2	no	—
180	C₂²×D₄	—	2	2	no	—
181	C₂²×D₄	C₂³×C₁₃₇	1	4	yes	—
182	C₂×D₈	—	2	4	no	—
183	C₂×D₈	—	2	4	no	—
184	C₂×D₈	—	2	8	no	—
185	C₂×D₈	—	4	4	no	—
186	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
187	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
188	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
189	#127	—	2	4	no	—
190	#127	—	2	4	no	—
191	#124	—	4	8	no	—
192	C₄×Q₈	—	2	2	no	—
193	C₄×Q₈	—	2	2	no	—
194	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
195	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
196	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
197	C₂×C₂²⋊C₄	—	2	8	no	—
198	C₂×C₂²⋊C₄	—	2	8	no	—
199	C₂×C₄²	—	2	2	no	—
200	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
201	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
202	#107	—	2	4	no	—
203	C₄⋊C₈	—	2	4	no	—
204	C₄⋊C₈	—	2	2	no	—
205	#102	—	4	8	no	—
206	C₄×C₈	—	2	4	no	—
207	C₄×C₈	D₄×C₁₃₇	1	4	yes	—
208	#95	—	2	2	no	—
209	C₂⁴	—	2	4	no	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×Q₈	D₈×C₁₃₇	1	4	yes	complemented
213	C₂×D₄	—	2	4	no	—
214	C₂×D₄	—	2	4	no	—
215	C₂×D₄	—	2	8	no	—
216	C₂×D₄	—	2	8	no	—
217	C₂×D₄	—	4	4	no	—
218	C₂×D₄	—	4	4	no	—
219	C₂×D₄	—	4	4	no	—
220	C₂×D₄	—	4	4	no	—
221	C₂×D₄	—	8	4	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
224	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
225	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
226	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
227	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
228	D₈	—	4	8	no	—
229	D₈	—	4	8	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
234	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
235	C₂×C₈	#41×C₁₃₇	1	4	yes	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₂²⋊C₄	—	4	16	no	—
241	C₂²⋊C₄	—	8	8	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	4	4	no	—
245	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
246	C₄²	#41×C₁₃₇	1	2	yes	—
247	C₂³	—	4	4	no	—
248	C₂³	—	4	8	no	—
249	C₂³	—	4	4	no	—
250	C₂³	—	4	4	no	—
251	C₂³	—	4	8	no	—
252	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
253	Q₈	—	2	8	no	—
254	D₄	—	4	4	no	—
255	D₄	—	4	4	no	—
256	D₄	—	4	8	no	—
257	D₄	—	8	8	no	—
258	D₄	—	8	8	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	#121×C₁₃₇	1	2	yes	—
268	C₂×C₄	#121×C₁₃₇	1	2	yes	—
269	C₂×C₄	C₂×D₈×C₁₃₇	1	4	yes	—
270	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
271	C₂×C₄	#135×C₁₃₇	1	2	yes	—
272	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
273	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
274	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
275	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	2	yes	—
276	C₈	—	2	2	no	—
277	C₈	—	4	2	no	—
278	C₈	#124×C₁₃₇	1	4	yes	complemented
279	C₂²	—	8	4	no	—
280	C₂²	—	8	4	no	—
281	C₂²	—	8	4	no	—
282	C₂²	—	8	8	no	—
283	C₂²	—	8	8	no	—
284	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
285	C₂²	#550×C₁₃₇	1	2	yes	—
286	C₂²	C₂²×D₈×C₁₃₇	1	1	yes	characteristic
287	C₂²	C₂×#135×C₁₃₇	1	1	yes	characteristic
288	C₂²	#576×C₁₃₇	1	2	yes	—
289	C₄	—	2	4	no	—
290	C₄	—	2	2	no	—
291	C₄	—	4	2	no	—
292	C₄	—	8	4	no	—
293	C₄	#466×C₁₃₇	1	4	yes	—
294	C₄	C₂×#124×C₁₃₇	1	1	yes	characteristic
295	C₄	#535×C₁₃₇	1	1	yes	characteristic
296	C₄	#559×C₁₃₇	1	2	yes	—
297	C₂	—	16	4	no	—
298	C₂	#2548×C₁₃₇	1	1	yes	characteristic
299	C₂	C₂×#466×C₁₃₇	1	1	yes	characteristic
300	C₂	#3058×C₁₃₇	1	1	yes	characteristic
301	C₂	#3391×C₁₃₇	1	2	yes	—
302	C₂	#3396×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂×#493×C₁₃₇	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₂×C₄×Q₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×Q₈×C₁₃₇	C₂	semidirect
3. #2548×C₁₃₇	C₂	semidirect
4. #2054×C₁₃₇	C₂	semidirect
5. #1982×C₁₃₇	C₂	semidirect
6. #1970×C₁₃₇	C₂	semidirect
7. C₄×#107×C₁₃₇	C₂	semidirect
60. C₂×Q₈×C₁₃₇	D₈	semidirect
126. C₈×C₁₃₇	#124	semidirect
151. #28736	C₁₃₇	direct (N × H)
152. C₁₃₇	#28736	direct (N × H)
154. C₂×C₈×Q₈	C₂×C₁₃₇	semidirect
155. #2548	C₂×C₁₃₇	semidirect
156. #2054	C₂×C₁₃₇	semidirect
157. #1982	C₂×C₁₃₇	semidirect
158. #1970	C₂×C₁₃₇	semidirect
159. C₄×#107	C₂×C₁₃₇	semidirect
212. C₂×Q₈	D₈×C₁₃₇	semidirect
278. C₈	#124×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#493×C₁₃₇	2
C₂×C₈×Q₈×C₁₃₇	2
#2548×C₁₃₇	2
#2054×C₁₃₇	2
#1982×C₁₃₇	2
#1970×C₁₃₇	2
C₄×#107×C₁₃₇	2
#28736	137

Subgroups — C137×#28736 #46878122

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28736 #46878122