Subgroups — #3386×C₁₃₇ #31778397

Order 17536.

360 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#557×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#501×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	#500×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#478×C₁₃₇	C₂	1	2	yes	maximal
6	#477×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#475×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#474×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	#446×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#440×C₁₃₇	C₂	1	2	yes	maximal
11	C₄×SD₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₈×C₁₃₇	—	2	1	no	—
13	C₂×SD₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
15	C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
18	#125×C₁₃₇	—	2	2	no	—
19	#125×C₁₃₇	C₂²	1	2	yes	—
20	C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
21	C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
22	C₄×Q₈×C₁₃₇	—	2	1	no	—
23	C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₁₃₇	C₂²	1	2	yes	—
25	C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₁₃₇	C₂²	1	2	yes	—
28	#107×C₁₃₇	C₂²	1	2	yes	—
29	#107×C₁₃₇	C₂²	1	2	yes	—
30	#107×C₁₃₇	C₂²	1	2	yes	—
31	#106×C₁₃₇	C₂²	1	1	yes	characteristic
32	#106×C₁₃₇	C₂²	1	2	yes	—
33	C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
34	C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
35	#103×C₁₃₇	C₂²	1	1	yes	characteristic
36	#103×C₁₃₇	C₂²	1	2	yes	—
37	#102×C₁₃₇	C₂²	1	1	yes	characteristic
38	#102×C₁₃₇	C₂²	1	2	yes	—
39	#97×C₁₃₇	C₂²	1	2	yes	—
40	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₁₃₇	—	2	1	no	—
42	C₂×Q₈×C₁₃₇	—	2	2	no	—
43	C₂×Q₈×C₁₃₇	—	2	1	no	—
44	C₂×Q₈×C₁₃₇	—	4	4	no	—
45	C₂×Q₈×C₁₃₇	D₄	1	1	yes	characteristic
46	C₂×Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
47	C₂×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₃₇	—	2	2	no	—
49	C₂²×C₄×C₁₃₇	—	2	1	no	—
50	SD₁₆×C₁₃₇	Q₈	1	4	yes	complemented
51	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
53	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₁₃₇	—	2	2	no	—
55	C₄⋊C₄×C₁₃₇	—	2	1	no	—
56	C₄⋊C₄×C₁₃₇	—	2	2	no	—
57	C₄⋊C₄×C₁₃₇	—	2	2	no	—
58	C₄⋊C₄×C₁₃₇	—	2	1	no	—
59	C₄⋊C₄×C₁₃₇	—	2	1	no	—
60	C₄⋊C₄×C₁₃₇	—	2	2	no	—
61	C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
63	C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
65	C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
67	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
68	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
69	C₄²×C₁₃₇	—	2	2	no	—
70	C₄²×C₁₃₇	—	2	1	no	—
71	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
72	C₄²×C₁₃₇	C₂³	1	2	yes	—
73	C₂³×C₁₃₇	—	2	1	no	—
74	Q₈×C₁₃₇	—	2	2	no	—
75	Q₈×C₁₃₇	—	2	1	no	—
76	Q₈×C₁₃₇	—	4	4	no	—
77	Q₈×C₁₃₇	—	4	2	no	—
78	Q₈×C₁₃₇	—	4	2	no	—
79	Q₈×C₁₃₇	C₂×Q₈	1	2	yes	complemented
80	D₄×C₁₃₇	—	2	1	no	—
81	D₄×C₁₃₇	C₂×Q₈	1	2	yes	—
82	C₂×C₄×C₁₃₇	—	2	1	no	—
83	C₂×C₄×C₁₃₇	—	2	1	no	—
84	C₂×C₄×C₁₃₇	—	2	2	no	—
85	C₂×C₄×C₁₃₇	—	2	2	no	—
86	C₂×C₄×C₁₃₇	—	2	1	no	—
87	C₂×C₄×C₁₃₇	—	4	4	no	—
88	C₂×C₄×C₁₃₇	—	4	2	no	—
89	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
90	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
94	C₈×C₁₃₇	—	2	1	no	—
95	C₈×C₁₃₇	—	2	2	no	—
96	C₈×C₁₃₇	C₂×Q₈	1	2	yes	complemented
97	C₂²×C₁₃₇	—	2	2	no	—
98	C₂²×C₁₃₇	—	4	1	no	—
99	C₂²×C₁₃₇	—	4	1	no	—
100	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
101	C₄×C₁₃₇	—	2	2	no	—
102	C₄×C₁₃₇	—	2	2	no	—
103	C₄×C₁₃₇	—	2	2	no	—
104	C₄×C₁₃₇	—	2	1	no	—
105	C₄×C₁₃₇	—	4	2	no	—
106	C₄×C₁₃₇	—	4	1	no	—
107	C₄×C₁₃₇	—	4	2	no	—
108	C₄×C₁₃₇	—	4	1	no	—
109	C₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
110	C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
111	C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
112	C₂×C₁₃₇	—	4	2	no	—
113	C₂×C₁₃₇	D₄×Q₈	1	1	yes	characteristic
114	C₂×C₁₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
115	C₂×C₁₃₇	#578	1	1	yes	characteristic
116	C₁₃₇	#3386	1	1	yes	characteristic, direct product factor, Sylow
117	#3386	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	#557	C₂×C₁₃₇	1	1	yes	characteristic, complemented
119	D₄×Q₈	C₂×C₁₃₇	1	1	yes	characteristic
120	#501	C₂×C₁₃₇	1	2	yes	complemented
121	#500	C₂×C₁₃₇	1	1	yes	characteristic, complemented
122	#478	C₂×C₁₃₇	1	2	yes	—
123	#477	C₂×C₁₃₇	1	2	yes	complemented
124	#475	C₂×C₁₃₇	1	1	yes	characteristic, complemented
125	#474	C₂×C₁₃₇	1	1	yes	characteristic
126	#446	C₂×C₁₃₇	1	1	yes	characteristic, complemented
127	#440	C₂×C₁₃₇	1	2	yes	—
128	C₄×SD₁₆	C₂×C₁₃₇	1	1	yes	characteristic
129	C₂²×Q₈	—	2	1	no	—
130	C₂×SD₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
131	C₄⋊Q₈	—	2	1	no	—
132	C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
133	C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
134	C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
135	#125	—	2	2	no	—
136	#125	C₂²×C₁₃₇	1	2	yes	—
137	C₂²⋊Q₈	—	2	1	no	—
138	C₂²⋊Q₈	—	2	2	no	—
139	C₄×Q₈	—	2	1	no	—
140	C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
141	C₄×Q₈	C₂²×C₁₃₇	1	2	yes	—
142	C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
143	C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
144	C₄×D₄	C₂²×C₁₃₇	1	2	yes	—
145	#107	C₂²×C₁₃₇	1	2	yes	—
146	#107	C₂²×C₁₃₇	1	2	yes	—
147	#107	C₂²×C₁₃₇	1	2	yes	—
148	#106	C₂²×C₁₃₇	1	1	yes	characteristic
149	#106	C₂²×C₁₃₇	1	2	yes	—
150	C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
151	C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
152	#103	C₂²×C₁₃₇	1	1	yes	characteristic
153	#103	C₂²×C₁₃₇	1	2	yes	—
154	#102	C₂²×C₁₃₇	1	1	yes	characteristic
155	#102	C₂²×C₁₃₇	1	2	yes	—
156	#97	C₂²×C₁₃₇	1	2	yes	—
157	C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
158	C₂×Q₈	—	2	1	no	—
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	—	2	1	no	—
161	C₂×Q₈	—	4	4	no	—
162	C₂×Q₈	D₄×C₁₃₇	1	1	yes	characteristic
163	C₂×Q₈	C₂³×C₁₃₇	1	1	yes	characteristic
164	C₂×D₄	C₂³×C₁₃₇	1	1	yes	characteristic
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	1	no	—
167	SD₁₆	Q₈×C₁₃₇	1	4	yes	complemented
168	C₂×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
169	C₂×C₈	C₂³×C₁₃₇	1	2	yes	—
170	C₂×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	1	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	1	no	—
176	C₄⋊C₄	—	2	1	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
179	C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
180	C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
181	C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
182	C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
183	C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	—	2	1	no	—
186	C₄²	—	2	2	no	—
187	C₄²	—	2	1	no	—
188	C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
189	C₄²	C₂³×C₁₃₇	1	2	yes	—
190	C₂³	—	2	1	no	—
191	Q₈	—	2	2	no	—
192	Q₈	—	2	1	no	—
193	Q₈	—	4	4	no	—
194	Q₈	—	4	2	no	—
195	Q₈	—	4	2	no	—
196	Q₈	C₂×Q₈×C₁₃₇	1	2	yes	complemented
197	D₄	—	2	1	no	—
198	D₄	C₂×Q₈×C₁₃₇	1	2	yes	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
207	C₂×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
208	C₂×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
209	C₂×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
210	C₂×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
211	C₈	—	2	1	no	—
212	C₈	—	2	2	no	—
213	C₈	C₂×Q₈×C₁₃₇	1	2	yes	complemented
214	C₂²	—	2	2	no	—
215	C₂²	—	4	1	no	—
216	C₂²	—	4	1	no	—
217	C₂²	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	2	1	no	—
222	C₄	—	4	2	no	—
223	C₄	—	4	1	no	—
224	C₄	—	4	2	no	—
225	C₄	—	4	1	no	—
226	C₄	AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
227	C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
228	C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
229	C₂	—	4	2	no	—
230	C₂	D₄×Q₈×C₁₃₇	1	1	yes	characteristic
231	C₂	C₂×AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
232	C₂	#578×C₁₃₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. #557×C₁₃₇	C₂	semidirect
3. #501×C₁₃₇	C₂	semidirect
4. #500×C₁₃₇	C₂	semidirect
6. #477×C₁₃₇	C₂	semidirect
7. #475×C₁₃₇	C₂	semidirect
9. #446×C₁₃₇	C₂	semidirect
50. SD₁₆×C₁₃₇	Q₈	semidirect
79. Q₈×C₁₃₇	C₂×Q₈	semidirect
96. C₈×C₁₃₇	C₂×Q₈	semidirect
116. C₁₃₇	#3386	direct (N × H)
117. #3386	C₁₃₇	direct (N × H)
118. #557	C₂×C₁₃₇	semidirect
120. #501	C₂×C₁₃₇	semidirect
121. #500	C₂×C₁₃₇	semidirect
123. #477	C₂×C₁₃₇	semidirect
124. #475	C₂×C₁₃₇	semidirect
126. #446	C₂×C₁₃₇	semidirect
167. SD₁₆	Q₈×C₁₃₇	semidirect
196. Q₈	C₂×Q₈×C₁₃₇	semidirect
213. C₈	C₂×Q₈×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#557×C₁₃₇	2
D₄×Q₈×C₁₃₇	2
#501×C₁₃₇	2
#500×C₁₃₇	2
#478×C₁₃₇	2
#477×C₁₃₇	2
#475×C₁₃₇	2
#474×C₁₃₇	2
#446×C₁₃₇	2
#440×C₁₃₇	2
C₄×SD₁₆×C₁₃₇	2
#3386	137

Subgroups — #3386×C137 #31778397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3386×C₁₃₇ #31778397