Subgroups — C₁₃₇×#28678 #46878064

Order 35072.

718 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#499×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#2584×C₁₃₇	C₂	1	2	yes	maximal
4	#2087×C₁₃₇	C₂	1	2	yes	maximal
5	#2086×C₁₃₇	C₂	1	2	yes	maximal
6	#1989×C₁₃₇	C₂	1	2	yes	maximal
7	#1941×C₁₃₇	C₂	1	4	yes	maximal
8	C₄×#106×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#125×C₁₃₇	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
11	#499×C₁₃₇	—	2	4	no	—
12	C₈×Q₈×C₁₃₇	—	2	4	no	—
13	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
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	2	yes	—
18	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
21	#398×C₁₃₇	—	2	2	no	—
22	#395×C₁₃₇	—	2	2	no	—
23	#389×C₁₃₇	—	2	4	no	—
24	#389×C₁₃₇	C₂²	1	4	yes	—
25	#389×C₁₃₇	C₂²	1	4	yes	—
26	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
27	#340×C₁₃₇	C₂²	1	2	yes	—
28	#340×C₁₃₇	C₂²	1	4	yes	—
29	#340×C₁₃₇	C₂²	1	4	yes	—
30	C₂²×Q₈×C₁₃₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
32	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
34	#125×C₁₃₇	—	2	4	no	—
35	#125×C₁₃₇	—	2	4	no	—
36	C₄×Q₈×C₁₃₇	—	2	2	no	—
37	C₄×Q₈×C₁₃₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
40	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
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	2	no	—
51	#107×C₁₃₇	—	2	2	no	—
52	#106×C₁₃₇	—	2	4	no	—
53	#106×C₁₃₇	Q₈	1	8	yes	complemented
54	C₄⋊C₈×C₁₃₇	—	2	4	no	—
55	C₄⋊C₈×C₁₃₇	—	2	2	no	—
56	C₄×C₈×C₁₃₇	—	2	4	no	—
57	C₄×C₈×C₁₃₇	D₄	1	4	yes	—
58	#95×C₁₃₇	—	2	2	no	—
59	#95×C₁₃₇	—	2	4	no	—
60	#95×C₁₃₇	—	2	4	no	—
61	C₂×Q₈×C₁₃₇	—	2	2	no	—
62	C₂×Q₈×C₁₃₇	—	2	4	no	—
63	C₂²×C₄×C₁₃₇	—	2	4	no	—
64	C₂²×C₄×C₁₃₇	—	2	2	no	—
65	C₂²×C₄×C₁₃₇	—	2	2	no	—
66	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
71	C₂×C₈×C₁₃₇	—	2	2	no	—
72	C₂×C₈×C₁₃₇	—	2	2	no	—
73	C₂×C₈×C₁₃₇	—	2	4	no	—
74	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
77	C₄⋊C₄×C₁₃₇	—	2	4	no	—
78	C₄⋊C₄×C₁₃₇	—	2	2	no	—
79	C₄⋊C₄×C₁₃₇	—	2	4	no	—
80	C₄⋊C₄×C₁₃₇	—	2	2	no	—
81	C₄⋊C₄×C₁₃₇	—	2	2	no	—
82	C₄⋊C₄×C₁₃₇	—	2	2	no	—
83	C₄⋊C₄×C₁₃₇	—	2	2	no	—
84	C₄⋊C₄×C₁₃₇	—	4	4	no	—
85	C₄⋊C₄×C₁₃₇	—	4	4	no	—
86	C₄⋊C₄×C₁₃₇	—	4	8	no	—
87	C₄⋊C₄×C₁₃₇	—	8	4	no	—
88	C₄⋊C₄×C₁₃₇	C₂×Q₈	1	8	yes	—
89	C₄²×C₁₃₇	—	2	2	no	—
90	C₄²×C₁₃₇	—	2	2	no	—
91	C₄²×C₁₃₇	—	8	2	no	—
92	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
93	C₄²×C₁₃₇	#41	1	2	yes	—
94	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
95	Q₈×C₁₃₇	—	2	8	no	—
96	C₂×C₄×C₁₃₇	—	2	2	no	—
97	C₂×C₄×C₁₃₇	—	2	2	no	—
98	C₂×C₄×C₁₃₇	—	2	2	no	—
99	C₂×C₄×C₁₃₇	—	2	2	no	—
100	C₂×C₄×C₁₃₇	—	2	2	no	—
101	C₂×C₄×C₁₃₇	—	2	8	no	—
102	C₂×C₄×C₁₃₇	—	2	4	no	—
103	C₂×C₄×C₁₃₇	—	4	4	no	—
104	C₂×C₄×C₁₃₇	—	4	4	no	—
105	C₂×C₄×C₁₃₇	—	4	2	no	—
106	C₂×C₄×C₁₃₇	—	4	2	no	—
107	C₂×C₄×C₁₃₇	—	4	4	no	—
108	C₂×C₄×C₁₃₇	—	4	2	no	—
109	C₂×C₄×C₁₃₇	—	4	2	no	—
110	C₂×C₄×C₁₃₇	#121	1	2	yes	—
111	C₂×C₄×C₁₃₇	#121	1	2	yes	—
112	C₂×C₄×C₁₃₇	#135	1	4	yes	—
113	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃₇	C₂²×Q₈	1	2	yes	—
115	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
116	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
117	C₈×C₁₃₇	—	2	2	no	—
118	C₈×C₁₃₇	—	4	2	no	—
119	C₈×C₁₃₇	C₄⋊Q₈	1	4	yes	—
120	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
121	C₂²×C₁₃₇	D₄×Q₈	1	2	yes	—
122	C₂²×C₁₃₇	C₂×#135	1	2	yes	—
123	C₂²×C₁₃₇	#577	1	2	yes	—
124	C₄×C₁₃₇	—	2	2	no	—
125	C₄×C₁₃₇	—	2	4	no	—
126	C₄×C₁₃₇	—	2	2	no	—
127	C₄×C₁₃₇	—	4	8	no	—
128	C₄×C₁₃₇	—	4	2	no	—
129	C₄×C₁₃₇	—	8	2	no	—
130	C₄×C₁₃₇	—	8	2	no	—
131	C₄×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
132	C₄×C₁₃₇	#536	1	1	yes	characteristic
133	C₄×C₁₃₇	#557	1	2	yes	—
134	C₂×C₁₃₇	#2546	1	1	yes	characteristic
135	C₂×C₁₃₇	#3050	1	1	yes	characteristic
136	C₂×C₁₃₇	#3060	1	1	yes	characteristic
137	C₂×C₁₃₇	#3381	1	4	yes	—
138	#28678	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₁₃₇	#28678	1	1	yes	characteristic, direct product factor, Sylow
140	C₂×#499	C₂×C₁₃₇	1	1	yes	characteristic
141	C₂×C₈×Q₈	C₂×C₁₃₇	1	1	yes	characteristic
142	#2584	C₂×C₁₃₇	1	2	yes	—
143	#2087	C₂×C₁₃₇	1	2	yes	—
144	#2086	C₂×C₁₃₇	1	2	yes	—
145	#1989	C₂×C₁₃₇	1	2	yes	—
146	#1941	C₂×C₁₃₇	1	4	yes	—
147	C₄×#106	C₂×C₁₃₇	1	1	yes	characteristic
148	C₂×#125	C₂²×C₁₃₇	1	2	yes	—
149	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
150	#499	—	2	4	no	—
151	C₈×Q₈	—	2	4	no	—
152	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
153	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
154	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
155	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
156	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
157	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
159	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
160	#398	—	2	2	no	—
161	#395	—	2	2	no	—
162	#389	—	2	4	no	—
163	#389	C₂²×C₁₃₇	1	4	yes	—
164	#389	C₂²×C₁₃₇	1	4	yes	—
165	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
166	#340	C₂²×C₁₃₇	1	2	yes	—
167	#340	C₂²×C₁₃₇	1	4	yes	—
168	#340	C₂²×C₁₃₇	1	4	yes	—
169	C₂²×Q₈	D₄×C₁₃₇	1	1	yes	characteristic
170	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
171	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
172	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
173	#125	—	2	4	no	—
174	#125	—	2	4	no	—
175	C₄×Q₈	—	2	2	no	—
176	C₄×Q₈	—	2	2	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	2	4	no	—
180	C₂×C₄⋊C₄	—	2	4	no	—
181	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
182	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
183	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
184	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
185	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
188	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
189	#107	—	2	2	no	—
190	#107	—	2	2	no	—
191	#106	—	2	4	no	—
192	#106	Q₈×C₁₃₇	1	8	yes	complemented
193	C₄⋊C₈	—	2	4	no	—
194	C₄⋊C₈	—	2	2	no	—
195	C₄×C₈	—	2	4	no	—
196	C₄×C₈	D₄×C₁₃₇	1	4	yes	—
197	#95	—	2	2	no	—
198	#95	—	2	4	no	—
199	#95	—	2	4	no	—
200	C₂×Q₈	—	2	2	no	—
201	C₂×Q₈	—	2	4	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
206	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
209	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
214	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
215	C₂×C₈	C₂×Q₈×C₁₃₇	1	4	yes	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	4	4	no	—
224	C₄⋊C₄	—	4	4	no	—
225	C₄⋊C₄	—	4	8	no	—
226	C₄⋊C₄	—	8	4	no	—
227	C₄⋊C₄	C₂×Q₈×C₁₃₇	1	8	yes	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	8	2	no	—
231	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
232	C₄²	#41×C₁₃₇	1	2	yes	—
233	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
234	Q₈	—	2	8	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	#121×C₁₃₇	1	2	yes	—
250	C₂×C₄	#121×C₁₃₇	1	2	yes	—
251	C₂×C₄	#135×C₁₃₇	1	4	yes	—
252	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
253	C₂×C₄	C₂²×Q₈×C₁₃₇	1	2	yes	—
254	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
255	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
256	C₈	—	2	2	no	—
257	C₈	—	4	2	no	—
258	C₈	C₄⋊Q₈×C₁₃₇	1	4	yes	—
259	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
260	C₂²	D₄×Q₈×C₁₃₇	1	2	yes	—
261	C₂²	C₂×#135×C₁₃₇	1	2	yes	—
262	C₂²	#577×C₁₃₇	1	2	yes	—
263	C₄	—	2	2	no	—
264	C₄	—	2	4	no	—
265	C₄	—	2	2	no	—
266	C₄	—	4	2	no	—
267	C₄	—	4	8	no	—
268	C₄	—	8	2	no	—
269	C₄	—	8	2	no	—
270	C₄	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
271	C₄	#536×C₁₃₇	1	1	yes	characteristic
272	C₄	#557×C₁₃₇	1	2	yes	—
273	C₂	#2546×C₁₃₇	1	1	yes	characteristic
274	C₂	#3050×C₁₃₇	1	1	yes	characteristic
275	C₂	#3060×C₁₃₇	1	1	yes	characteristic
276	C₂	#3381×C₁₃₇	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
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
53. #106×C₁₃₇	Q₈	semidirect
138. #28678	C₁₃₇	direct (N × H)
139. C₁₃₇	#28678	direct (N × H)
192. #106	Q₈×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#499×C₁₃₇	2
C₂×C₈×Q₈×C₁₃₇	2
#2584×C₁₃₇	2
#2087×C₁₃₇	2
#2086×C₁₃₇	2
#1989×C₁₃₇	2
#1941×C₁₃₇	2
C₄×#106×C₁₃₇	2
#28678	137

Subgroups — C137×#28678 #46878064

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28678 #46878064