Subgroups — #31780527

Order 17536.

388 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14555494	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984047	C₂	1	1	yes	maximal, characteristic
3	#22577171	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14555494	—	2	2	no	—
6	C₂×#14555494	—	2	1	no	—
7	C₂×#14555494	C₄	1	4	yes	complemented
8	C₂×#14555494	C₂²	1	2	yes	—
9	C₂²×#11764040	—	2	1	no	—
10	#17984047	—	2	1	no	—
11	#17984047	—	2	1	no	—
12	#17984047	—	2	1	no	—
13	#17984047	—	2	1	no	—
14	C₂×C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
15	C₂×C₂²⋊Dic₁₃₇	C₄	1	2	yes	—
16	C₂×C₄×Dic₁₃₇	C₄	1	2	yes	—
17	C₂³×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₁₃₇	—	2	2	no	—
19	C₂×M₁₆×C₁₃₇	—	2	1	no	—
20	C₂×M₁₆×C₁₃₇	C₄	1	4	yes	complemented
21	C₂×M₁₆×C₁₃₇	C₂²	1	2	yes	—
22	C₂²×C₈×C₁₃₇	—	2	1	no	—
23	C₂²×Dic₁₃₇	C₂×C₄	1	4	yes	—
24	#14555494	—	2	4	no	—
25	#14555494	—	2	2	no	—
26	#14555494	C₂×C₄	1	4	yes	—
27	C₂×#11764040	—	2	4	no	—
28	C₂×#11764040	—	2	2	no	—
29	C₄⋊Dic₁₃₇	—	2	2	no	—
30	C₄⋊Dic₁₃₇	—	2	2	no	—
31	C₂²⋊Dic₁₃₇	—	2	2	no	—
32	C₂²⋊Dic₁₃₇	—	2	2	no	—
33	C₄×Dic₁₃₇	—	2	2	no	—
34	C₄×Dic₁₃₇	—	2	2	no	—
35	C₂⁴×C₁₃₇	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₁₃₇	D₄	1	2	yes	—
40	C₂²×C₄×C₁₃₇	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃₇	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃₇	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃₇	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃₇	Q₈	1	2	yes	—
45	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
46	M₁₆×C₁₃₇	—	2	4	no	—
47	M₁₆×C₁₃₇	—	2	2	no	—
48	M₁₆×C₁₃₇	C₂×C₄	1	4	yes	—
49	C₂×C₈×C₁₃₇	—	2	4	no	—
50	C₂×C₈×C₁₃₇	—	2	2	no	—
51	C₂×Dic₁₃₇	—	2	4	no	—
52	C₂×Dic₁₃₇	—	4	4	no	—
53	#11764040	—	2	4	no	—
54	C₂³×C₁₃₇	—	4	2	no	—
55	C₂³×C₁₃₇	C₄²	1	2	yes	—
56	C₂³×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂³×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂³×C₁₃₇	C₄⋊C₄	1	2	yes	—
59	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₃₇	C₄²	1	2	yes	—
61	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
63	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
64	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
66	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
67	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
68	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
69	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
70	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
71	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
78	C₈×C₁₃₇	—	2	4	no	—
79	C₂²×C₁₃₇	—	2	2	no	—
80	C₂²×C₁₃₇	—	2	2	no	—
81	C₂²×C₁₃₇	—	2	2	no	—
82	C₂²×C₁₃₇	—	4	2	no	—
83	C₂²×C₁₃₇	—	4	2	no	—
84	C₂²×C₁₃₇	#95	1	2	yes	—
85	C₂²×C₁₃₇	#95	1	2	yes	—
86	C₂²×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
87	C₂²×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₄×C₁₃₇	#95	1	2	yes	—
90	C₄×C₁₃₇	#95	1	2	yes	—
91	C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
95	Dic₁₃₇	—	4	4	no	—
96	C₂×C₁₃₇	—	2	2	no	—
97	C₂×C₁₃₇	—	2	2	no	—
98	C₂×C₁₃₇	—	2	1	no	—
99	C₂×C₁₃₇	—	2	1	no	—
100	C₂×C₁₃₇	C₂×#95	1	1	yes	characteristic
101	C₂×C₁₃₇	#413	1	2	yes	—
102	C₁₃₇	#1742	1	1	yes	characteristic, complemented, Sylow
103	#1742	—	137	1	no	maximal, Sylow
104	C₂²×M₁₆	—	137	1	no	—
105	C₂²×M₁₆	D₁₃₇	1	1	yes	characteristic
106	C₂×#117	—	137	1	no	—
107	#343	—	137	4	no	—
108	C₂³×C₄	C₂×D₁₃₇	1	1	yes	characteristic
109	C₂×M₁₆	—	2	2	no	—
110	C₂×M₁₆	—	2	1	no	—
111	C₂×M₁₆	—	274	2	no	—
112	C₂×M₁₆	—	274	1	no	—
113	C₂×M₁₆	—	137	4	no	—
114	C₂×M₁₆	—	137	2	no	—
115	C₂×M₁₆	Dic₁₃₇	1	4	yes	complemented
116	C₂×M₁₆	C₂×D₁₃₇	1	2	yes	—
117	C₂²×C₈	—	2	1	no	—
118	C₂²×C₈	—	274	1	no	—
119	#117	—	274	1	no	—
120	#117	—	274	1	no	—
121	#117	—	274	1	no	—
122	#117	—	274	1	no	—
123	C₂×C₄⋊C₄	—	137	2	no	—
124	C₂×C₂²⋊C₄	—	137	2	no	—
125	C₂×C₄²	—	137	2	no	—
126	C₂⁴	C₄×D₁₃₇	1	1	yes	characteristic
127	C₂²×C₄	—	137	4	no	—
128	C₂²×C₄	C₄×D₁₃₇	1	1	yes	characteristic
129	C₂²×C₄	C₄×D₁₃₇	1	2	yes	—
130	C₂²×C₄	C₂×Dic₁₃₇	1	2	yes	—
131	C₂²×C₄	C₂²⋊D₁₃₇	1	2	yes	—
132	C₂²×C₄	C₂²⋊D₁₃₇	1	1	yes	characteristic
133	C₂²×C₄	C₂²⋊D₁₃₇	1	1	yes	characteristic
134	C₂²×C₄	D₅₄₈	1	1	yes	characteristic
135	C₂²×C₄	D₅₄₈	1	1	yes	characteristic
136	C₂²×C₄	Dic₂₇₄	1	2	yes	—
137	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
138	M₁₆	—	2	4	no	—
139	M₁₆	—	2	2	no	—
140	M₁₆	—	274	4	no	—
141	M₁₆	—	274	2	no	—
142	M₁₆	—	137	4	no	—
143	M₁₆	C₂×Dic₁₃₇	1	4	yes	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	274	4	no	—
147	C₂×C₈	—	274	2	no	—
148	C₄⋊C₄	—	274	2	no	—
149	C₄⋊C₄	—	274	2	no	—
150	C₂²⋊C₄	—	274	2	no	—
151	C₂²⋊C₄	—	274	2	no	—
152	C₄²	—	274	2	no	—
153	C₄²	—	274	2	no	—
154	C₂³	—	4	2	no	—
155	C₂³	C₄×Dic₁₃₇	1	2	yes	—
156	C₂³	#14555487	1	1	yes	characteristic
157	C₂³	#14555487	1	1	yes	characteristic
158	C₂³	#14555489	1	2	yes	—
159	C₂³	C₂×C₄×D₁₃₇	1	1	yes	characteristic
160	C₂×C₄	—	274	4	no	—
161	C₂×C₄	—	548	4	no	—
162	C₂×C₄	C₄×Dic₁₃₇	1	2	yes	—
163	C₂×C₄	#14555487	1	2	yes	—
164	C₂×C₄	#14555487	1	2	yes	—
165	C₂×C₄	#14555487	1	1	yes	characteristic
166	C₂×C₄	#14555487	1	1	yes	characteristic
167	C₂×C₄	C₂²⋊Dic₁₃₇	1	2	yes	—
168	C₂×C₄	C₂²⋊Dic₁₃₇	1	2	yes	—
169	C₂×C₄	#14555489	1	2	yes	—
170	C₂×C₄	#14555489	1	2	yes	—
171	C₂×C₄	#14555489	1	2	yes	—
172	C₂×C₄	C₄⋊Dic₁₃₇	1	2	yes	—
173	C₂×C₄	C₄⋊Dic₁₃₇	1	2	yes	—
174	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
175	C₂×C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
176	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
177	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
178	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
179	C₂×C₄	C₂×Dic₂₇₄	1	1	yes	characteristic
180	C₈	—	2	4	no	—
181	C₈	—	274	4	no	—
182	C₂²	—	2	2	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	4	2	no	—
186	C₂²	—	4	2	no	—
187	C₂²	#17983995	1	2	yes	—
188	C₂²	#17983995	1	2	yes	—
189	C₂²	C₂×C₄×Dic₁₃₇	1	1	yes	characteristic
190	C₂²	C₂×#14555487	1	1	yes	characteristic
191	C₂²	C₂×#14555489	1	1	yes	characteristic
192	C₄	—	548	4	no	—
193	C₄	#17983995	1	2	yes	—
194	C₄	#17983995	1	2	yes	—
195	C₄	C₂×#14555487	1	1	yes	characteristic
196	C₄	C₂×C₂²⋊Dic₁₃₇	1	1	yes	characteristic
197	C₄	C₂×#14555489	1	1	yes	characteristic
198	C₄	C₂×C₄⋊Dic₁₃₇	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	—	2	2	no	—
202	C₂	—	2	2	no	—
203	C₂	C₂×#17983995	1	1	yes	characteristic
204	C₂	#22577385	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14555487
commutator	C₂²×C₁₃₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₃₇
Fitting	C₂²×M₁₆×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₃₇	C₂×C₄
cosocle	C₂²×C₄×C₁₃₇	C₂³
hypercenter	C₂²×M₁₆	D₁₃₇
hypocenter	C₁₃₇	#1742
LCS	G ▷ C₂²×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #22577171	C₂	semidirect
7. C₂×#14555494	C₄	semidirect
20. C₂×M₁₆×C₁₃₇	C₄	semidirect
102. C₁₃₇	#1742	semidirect
115. C₂×M₁₆	Dic₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14555494	2
C₂×#17984047	2
#22577171	2
C₂²×M₁₆×C₁₃₇	2
#1742	137