Subgroups — C₁₃₇×#11354 #46860740

Order 35072.

774 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#384×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#343×C₁₃₇	C₂	1	4	yes	maximal
3	#1728×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1728×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×M₁₆×C₁₃₇	C₂²	1	2	yes	—
6	C₂²×M₁₆×C₁₃₇	C₂²	1	2	yes	—
7	C₂²×C₄⋊C₄×C₁₃₇	C₄	1	4	yes	—
8	C₂²×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
9	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
10	#384×C₁₃₇	—	2	1	no	—
11	#384×C₁₃₇	—	2	2	no	—
12	#384×C₁₃₇	—	2	1	no	—
13	#343×C₁₃₇	—	2	8	no	—
14	#336×C₁₃₇	—	2	1	no	—
15	#336×C₁₃₇	—	2	2	no	—
16	#336×C₁₃₇	—	2	1	no	—
17	C₂³×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₁₃₇	C₂×C₄	1	4	yes	—
19	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₁₃₇	—	2	2	no	—
21	C₂×M₁₆×C₁₃₇	—	2	2	no	—
22	C₂×M₁₆×C₁₃₇	—	2	4	no	—
23	C₂×M₁₆×C₁₃₇	—	2	4	no	—
24	C₂×M₁₆×C₁₃₇	—	2	4	no	—
25	C₂×M₁₆×C₁₃₇	—	2	4	no	—
26	C₂×M₁₆×C₁₃₇	D₄	1	4	yes	—
27	C₂×M₁₆×C₁₃₇	Q₈	1	4	yes	—
28	C₂²×C₈×C₁₃₇	—	2	2	no	—
29	C₂²×C₈×C₁₃₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
31	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
32	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
33	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
35	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
36	C₂×C₄⋊C₄×C₁₃₇	—	4	8	no	—
37	C₂×C₄²×C₁₃₇	—	2	4	no	—
38	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
40	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
41	C₂⁴×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃₇	—	2	4	no	—
43	C₂²×C₄×C₁₃₇	—	2	4	no	—
44	C₂²×C₄×C₁₃₇	—	2	4	no	—
45	C₂²×C₄×C₁₃₇	—	2	4	no	—
46	C₂²×C₄×C₁₃₇	—	2	8	no	—
47	C₂²×C₄×C₁₃₇	—	4	8	no	—
48	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
54	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
55	C₂²×C₄×C₁₃₇	#41	1	4	yes	—
56	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
57	M₁₆×C₁₃₇	—	2	8	no	—
58	M₁₆×C₁₃₇	—	2	8	no	—
59	M₁₆×C₁₃₇	—	4	4	no	—
60	M₁₆×C₁₃₇	—	4	4	no	—
61	C₂×C₈×C₁₃₇	—	2	4	no	—
62	C₂×C₈×C₁₃₇	—	2	4	no	—
63	C₂×C₈×C₁₃₇	—	4	2	no	—
64	C₂×C₈×C₁₃₇	—	4	2	no	—
65	C₂×C₈×C₁₃₇	—	4	2	no	—
66	C₂×C₈×C₁₃₇	—	4	2	no	—
67	C₄⋊C₄×C₁₃₇	—	4	8	no	—
68	C₄⋊C₄×C₁₃₇	—	4	8	no	—
69	C₄⋊C₄×C₁₃₇	—	4	4	no	—
70	C₄⋊C₄×C₁₃₇	—	4	2	no	—
71	C₄⋊C₄×C₁₃₇	—	4	2	no	—
72	C₄²×C₁₃₇	—	2	2	no	—
73	C₄²×C₁₃₇	—	2	2	no	—
74	C₄²×C₁₃₇	—	2	4	no	—
75	C₂³×C₁₃₇	—	2	2	no	—
76	C₂³×C₁₃₇	—	2	2	no	—
77	C₂³×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂³×C₁₃₇	#117	1	2	yes	—
79	C₂³×C₁₃₇	C₄×D₄	1	2	yes	—
80	C₂³×C₁₃₇	C₄×Q₈	1	2	yes	—
81	C₂×C₄×C₁₃₇	—	2	16	no	—
82	C₂×C₄×C₁₃₇	—	2	2	no	—
83	C₂×C₄×C₁₃₇	—	2	2	no	—
84	C₂×C₄×C₁₃₇	—	2	2	no	—
85	C₂×C₄×C₁₃₇	—	2	2	no	—
86	C₂×C₄×C₁₃₇	—	4	4	no	—
87	C₂×C₄×C₁₃₇	—	4	8	no	—
88	C₂×C₄×C₁₃₇	—	8	8	no	—
89	C₂×C₄×C₁₃₇	#100	1	4	yes	—
90	C₂×C₄×C₁₃₇	#101	1	4	yes	—
91	C₂×C₄×C₁₃₇	#121	1	2	yes	—
92	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	4	yes	—
93	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
94	C₂×C₄×C₁₃₇	#123	1	2	yes	—
95	C₂×C₄×C₁₃₇	#123	1	4	yes	—
96	C₂×C₄×C₁₃₇	#123	1	2	yes	—
97	C₂×C₄×C₁₃₇	#124	1	1	yes	characteristic
98	C₂×C₄×C₁₃₇	#124	1	1	yes	characteristic
99	C₂×C₄×C₁₃₇	#125	1	4	yes	—
100	C₂×C₄×C₁₃₇	#126	1	4	yes	—
101	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
102	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
103	C₈×C₁₃₇	—	4	4	no	—
104	C₈×C₁₃₇	—	4	4	no	—
105	C₂²×C₁₃₇	—	2	4	no	—
106	C₂²×C₁₃₇	—	2	2	no	—
107	C₂²×C₁₃₇	—	2	2	no	—
108	C₂²×C₁₃₇	—	2	4	no	—
109	C₂²×C₁₃₇	#381	1	1	yes	characteristic
110	C₂²×C₁₃₇	#387	1	4	yes	—
111	C₂²×C₁₃₇	#390	1	1	yes	characteristic
112	C₂²×C₁₃₇	#391	1	1	yes	characteristic
113	C₂²×C₁₃₇	C₂×#100	1	2	yes	—
114	C₂²×C₁₃₇	C₂×#101	1	2	yes	—
115	C₄×C₁₃₇	—	2	4	no	—
116	C₄×C₁₃₇	—	2	4	no	—
117	C₄×C₁₃₇	—	8	4	no	—
118	C₄×C₁₃₇	#397	1	2	yes	—
119	C₄×C₁₃₇	#398	1	2	yes	—
120	C₄×C₁₃₇	#399	1	2	yes	—
121	C₄×C₁₃₇	#400	1	2	yes	—
122	C₂×C₁₃₇	—	2	4	no	—
123	C₂×C₁₃₇	#1445	1	1	yes	characteristic
124	C₂×C₁₃₇	#1821	1	2	yes	—
125	C₂×C₁₃₇	#1965	1	1	yes	characteristic
126	C₂×C₁₃₇	#1966	1	1	yes	characteristic
127	C₂×C₁₃₇	#1997	1	2	yes	—
128	#11354	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₁₃₇	#11354	1	1	yes	characteristic, direct product factor, Sylow
130	C₂×#384	C₂×C₁₃₇	1	1	yes	characteristic
131	C₂×#343	C₂×C₁₃₇	1	4	yes	—
132	#1728	C₂×C₁₃₇	1	1	yes	characteristic
133	#1728	C₂×C₁₃₇	1	1	yes	characteristic
134	C₂²×M₁₆	C₂²×C₁₃₇	1	2	yes	—
135	C₂²×M₁₆	C₂²×C₁₃₇	1	2	yes	—
136	C₂²×C₄⋊C₄	C₄×C₁₃₇	1	4	yes	—
137	C₂²×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
138	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
139	#384	—	2	1	no	—
140	#384	—	2	2	no	—
141	#384	—	2	1	no	—
142	#343	—	2	8	no	—
143	#336	—	2	1	no	—
144	#336	—	2	2	no	—
145	#336	—	2	1	no	—
146	C₂³×C₄	C₂×C₄×C₁₃₇	1	2	yes	—
147	C₂³×C₄	C₂×C₄×C₁₃₇	1	4	yes	—
148	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
149	C₂×M₁₆	—	2	2	no	—
150	C₂×M₁₆	—	2	2	no	—
151	C₂×M₁₆	—	2	4	no	—
152	C₂×M₁₆	—	2	4	no	—
153	C₂×M₁₆	—	2	4	no	—
154	C₂×M₁₆	—	2	4	no	—
155	C₂×M₁₆	D₄×C₁₃₇	1	4	yes	—
156	C₂×M₁₆	Q₈×C₁₃₇	1	4	yes	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	2	2	no	—
159	C₂×C₄⋊C₄	—	2	4	no	—
160	C₂×C₄⋊C₄	—	2	4	no	—
161	C₂×C₄⋊C₄	—	2	4	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	2	4	no	—
165	C₂×C₄⋊C₄	—	4	8	no	—
166	C₂×C₄²	—	2	4	no	—
167	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
168	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
169	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
170	C₂⁴	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	2	4	no	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	—	2	8	no	—
176	C₂²×C₄	—	4	8	no	—
177	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	2	yes	—
178	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	2	yes	—
179	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
181	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
182	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
183	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
184	C₂²×C₄	#41×C₁₃₇	1	4	yes	—
185	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
186	M₁₆	—	2	8	no	—
187	M₁₆	—	2	8	no	—
188	M₁₆	—	4	4	no	—
189	M₁₆	—	4	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	4	2	no	—
196	C₄⋊C₄	—	4	8	no	—
197	C₄⋊C₄	—	4	8	no	—
198	C₄⋊C₄	—	4	4	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	4	no	—
204	C₂³	—	2	2	no	—
205	C₂³	—	2	2	no	—
206	C₂³	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
207	C₂³	#117×C₁₃₇	1	2	yes	—
208	C₂³	C₄×D₄×C₁₃₇	1	2	yes	—
209	C₂³	C₄×Q₈×C₁₃₇	1	2	yes	—
210	C₂×C₄	—	2	16	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	8	no	—
217	C₂×C₄	—	8	8	no	—
218	C₂×C₄	#100×C₁₃₇	1	4	yes	—
219	C₂×C₄	#101×C₁₃₇	1	4	yes	—
220	C₂×C₄	#121×C₁₃₇	1	2	yes	—
221	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	4	yes	—
222	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	2	yes	—
223	C₂×C₄	#123×C₁₃₇	1	2	yes	—
224	C₂×C₄	#123×C₁₃₇	1	4	yes	—
225	C₂×C₄	#123×C₁₃₇	1	2	yes	—
226	C₂×C₄	#124×C₁₃₇	1	1	yes	characteristic
227	C₂×C₄	#124×C₁₃₇	1	1	yes	characteristic
228	C₂×C₄	#125×C₁₃₇	1	4	yes	—
229	C₂×C₄	#126×C₁₃₇	1	4	yes	—
230	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
231	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
232	C₈	—	4	4	no	—
233	C₈	—	4	4	no	—
234	C₂²	—	2	4	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	4	no	—
238	C₂²	#381×C₁₃₇	1	1	yes	characteristic
239	C₂²	#387×C₁₃₇	1	4	yes	—
240	C₂²	#390×C₁₃₇	1	1	yes	characteristic
241	C₂²	#391×C₁₃₇	1	1	yes	characteristic
242	C₂²	C₂×#100×C₁₃₇	1	2	yes	—
243	C₂²	C₂×#101×C₁₃₇	1	2	yes	—
244	C₄	—	2	4	no	—
245	C₄	—	2	4	no	—
246	C₄	—	8	4	no	—
247	C₄	#397×C₁₃₇	1	2	yes	—
248	C₄	#398×C₁₃₇	1	2	yes	—
249	C₄	#399×C₁₃₇	1	2	yes	—
250	C₄	#400×C₁₃₇	1	2	yes	—
251	C₂	—	2	4	no	—
252	C₂	#1445×C₁₃₇	1	1	yes	characteristic
253	C₂	#1821×C₁₃₇	1	2	yes	—
254	C₂	#1965×C₁₃₇	1	1	yes	characteristic
255	C₂	#1966×C₁₃₇	1	1	yes	characteristic
256	C₂	#1997×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂×C₂²⋊C₄
commutator	C₂⁴	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃₇
prime radical	C₂⁴×C₁₃₇	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₂²×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
128. #11354	C₁₃₇	direct (N × H)
129. C₁₃₇	#11354	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#384×C₁₃₇	2
C₂×#343×C₁₃₇	2
#1728×C₁₃₇	2
#11354	137

Subgroups — C137×#11354 #46860740

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#11354 #46860740