Subgroups — C₄×#135×C₁₃₇ #31777950

Order 17536.

500 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#135×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₄×#41×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	C₄×Q₁₆×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	C₄×SD₁₆×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	C₄×D₈×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#427×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#416×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₁₃₇	C₂²	1	2	yes	—
10	#135×C₁₃₇	C₄	1	8	yes	direct product factor
11	C₂×Q₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×SD₁₆×C₁₃₇	C₂²	1	2	yes	—
13	C₂×D₈×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₄×Q₈×C₁₃₇	C₂²	1	4	yes	—
17	C₄×D₄×C₁₃₇	—	2	4	no	—
18	C₄×D₄×C₁₃₇	C₂²	1	4	yes	—
19	#117×C₁₃₇	—	2	2	no	—
20	#117×C₁₃₇	C₂²	1	2	yes	—
21	C₂×C₄²×C₁₃₇	—	2	2	no	—
22	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
23	#107×C₁₃₇	C₂²	1	2	yes	—
24	#106×C₁₃₇	C₂²	1	2	yes	—
25	#103×C₁₃₇	C₂²	1	4	yes	—
26	#102×C₁₃₇	C₂²	1	4	yes	—
27	C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
28	C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
29	#41×C₁₃₇	—	2	4	no	—
30	#41×C₁₃₇	C₂×C₄	1	8	yes	complemented
31	C₂×Q₈×C₁₃₇	C₂³	1	2	yes	—
32	C₂×D₄×C₁₃₇	—	2	2	no	—
33	C₂×D₄×C₁₃₇	C₂³	1	2	yes	—
34	C₂²×C₄×C₁₃₇	—	2	2	no	—
35	C₂²×C₄×C₁₃₇	—	2	4	no	—
36	C₂²×C₄×C₁₃₇	D₄	1	2	yes	—
37	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
38	Q₁₆×C₁₃₇	C₂×C₄	1	4	yes	complemented
39	SD₁₆×C₁₃₇	C₂×C₄	1	8	yes	complemented
40	D₈×C₁₃₇	C₂×C₄	1	4	yes	complemented
41	C₂×C₈×C₁₃₇	—	2	2	no	—
42	C₂×C₈×C₁₃₇	C₂×C₄	1	4	yes	complemented
43	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
44	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₁₃₇	—	2	4	no	—
47	C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
48	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
49	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
50	C₄²×C₁₃₇	—	2	2	no	—
51	C₄²×C₁₃₇	—	2	4	no	—
52	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
53	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
54	C₄²×C₁₃₇	C₂³	1	2	yes	—
55	C₂³×C₁₃₇	—	2	2	no	—
56	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
57	Q₈×C₁₃₇	—	2	2	no	—
58	Q₈×C₁₃₇	C₂²×C₄	1	4	yes	—
59	D₄×C₁₃₇	—	2	8	no	—
60	D₄×C₁₃₇	—	2	2	no	—
61	D₄×C₁₃₇	C₂²×C₄	1	4	yes	—
62	C₂×C₄×C₁₃₇	—	2	2	no	—
63	C₂×C₄×C₁₃₇	—	2	8	no	—
64	C₂×C₄×C₁₃₇	—	2	4	no	—
65	C₂×C₄×C₁₃₇	—	2	4	no	—
66	C₂×C₄×C₁₃₇	—	4	8	no	—
67	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₁₃₇	#41	1	2	yes	—
72	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
73	C₈×C₁₃₇	—	2	2	no	—
74	C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
75	C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
76	C₂²×C₁₃₇	—	2	2	no	—
77	C₂²×C₁₃₇	—	2	4	no	—
78	C₂²×C₁₃₇	—	4	4	no	—
79	C₂²×C₁₃₇	C₄×D₄	1	2	yes	—
80	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
81	C₄×C₁₃₇	—	2	4	no	—
82	C₄×C₁₃₇	—	2	2	no	—
83	C₄×C₁₃₇	—	4	4	no	—
84	C₄×C₁₃₇	C₄×D₄	1	2	yes	—
85	C₄×C₁₃₇	#135	1	4	yes	direct product factor
86	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
87	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
88	C₂×C₁₃₇	—	2	2	no	—
89	C₂×C₁₃₇	—	4	4	no	—
90	C₂×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
91	C₂×C₁₃₇	C₂×#135	1	2	yes	—
92	C₁₃₇	C₄×#135	1	1	yes	characteristic, direct product factor, Sylow
93	C₄×#135	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂×#135	C₂×C₁₃₇	1	1	yes	characteristic
95	C₄×#41	C₂×C₁₃₇	1	2	yes	complemented
96	C₄×Q₁₆	C₂×C₁₃₇	1	2	yes	complemented
97	C₄×SD₁₆	C₂×C₁₃₇	1	4	yes	complemented
98	C₄×D₈	C₂×C₁₃₇	1	2	yes	complemented
99	#427	C₂×C₁₃₇	1	1	yes	characteristic, complemented
100	#416	C₂×C₁₃₇	1	2	yes	complemented
101	C₂×C₄×C₈	C₂×C₁₃₇	1	1	yes	characteristic, complemented
102	C₂×#41	C₂²×C₁₃₇	1	2	yes	—
103	#135	C₄×C₁₃₇	1	8	yes	direct product factor
104	C₂×Q₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
105	C₂×SD₁₆	C₂²×C₁₃₇	1	2	yes	—
106	C₂×D₈	C₂²×C₁₃₇	1	1	yes	characteristic
107	C₂²×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
108	C₂²×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
109	C₄×Q₈	C₂²×C₁₃₇	1	4	yes	—
110	C₄×D₄	—	2	4	no	—
111	C₄×D₄	C₂²×C₁₃₇	1	4	yes	—
112	#117	—	2	2	no	—
113	#117	C₂²×C₁₃₇	1	2	yes	—
114	C₂×C₄²	—	2	2	no	—
115	C₂×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
116	#107	C₂²×C₁₃₇	1	2	yes	—
117	#106	C₂²×C₁₃₇	1	2	yes	—
118	#103	C₂²×C₁₃₇	1	4	yes	—
119	#102	C₂²×C₁₃₇	1	4	yes	—
120	C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
121	C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
122	#41	—	2	4	no	—
123	#41	C₂×C₄×C₁₃₇	1	8	yes	complemented
124	C₂×Q₈	C₂³×C₁₃₇	1	2	yes	—
125	C₂×D₄	—	2	2	no	—
126	C₂×D₄	C₂³×C₁₃₇	1	2	yes	—
127	C₂²×C₄	—	2	2	no	—
128	C₂²×C₄	—	2	4	no	—
129	C₂²×C₄	D₄×C₁₃₇	1	2	yes	—
130	C₂²×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
131	Q₁₆	C₂×C₄×C₁₃₇	1	4	yes	complemented
132	SD₁₆	C₂×C₄×C₁₃₇	1	8	yes	complemented
133	D₈	C₂×C₄×C₁₃₇	1	4	yes	complemented
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	C₂×C₄×C₁₃₇	1	4	yes	complemented
136	C₂×C₈	C₂³×C₁₃₇	1	2	yes	—
137	C₂×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
138	C₂×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
139	C₄⋊C₄	—	2	4	no	—
140	C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
141	C₂²⋊C₄	—	2	2	no	—
142	C₂²⋊C₄	—	2	4	no	—
143	C₄²	—	2	2	no	—
144	C₄²	—	2	4	no	—
145	C₄²	D₄×C₁₃₇	1	1	yes	characteristic
146	C₄²	D₄×C₁₃₇	1	1	yes	characteristic
147	C₄²	C₂³×C₁₃₇	1	2	yes	—
148	C₂³	—	2	2	no	—
149	C₂³	C₂×D₄×C₁₃₇	1	1	yes	characteristic
150	Q₈	—	2	2	no	—
151	Q₈	C₂²×C₄×C₁₃₇	1	4	yes	—
152	D₄	—	2	8	no	—
153	D₄	—	2	2	no	—
154	D₄	C₂²×C₄×C₁₃₇	1	4	yes	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	8	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	4	8	no	—
160	C₂×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
161	C₂×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
162	C₂×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
163	C₂×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
164	C₂×C₄	#41×C₁₃₇	1	2	yes	—
165	C₂×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
166	C₈	—	2	2	no	—
167	C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
168	C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	4	no	—
171	C₂²	—	4	4	no	—
172	C₂²	C₄×D₄×C₁₃₇	1	2	yes	—
173	C₂²	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
174	C₄	—	2	4	no	—
175	C₄	—	2	2	no	—
176	C₄	—	4	4	no	—
177	C₄	C₄×D₄×C₁₃₇	1	2	yes	—
178	C₄	#135×C₁₃₇	1	4	yes	direct product factor
179	C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
180	C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
181	C₂	—	2	2	no	—
182	C₂	—	4	4	no	—
183	C₂	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
184	C₂	C₂×#135×C₁₃₇	1	2	yes	—

	subgroup	G/H
center	C₄²×C₁₃₇	D₄
commutator	C₄	C₂³×C₄×C₁₃₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃₇
socle	C₂²×C₁₃₇	C₂²×D₄
prime radical	C₂×#135×C₁₃₇	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×#41×C₁₃₇	C₂	semidirect
3. C₄×Q₁₆×C₁₃₇	C₂	semidirect
4. C₄×SD₁₆×C₁₃₇	C₂	semidirect
5. C₄×D₈×C₁₃₇	C₂	semidirect
6. #427×C₁₃₇	C₂	semidirect
7. #416×C₁₃₇	C₂	semidirect
8. C₂×C₄×C₈×C₁₃₇	C₂	semidirect
10. #135×C₁₃₇	C₄	direct (N × H)
30. #41×C₁₃₇	C₂×C₄	semidirect
38. Q₁₆×C₁₃₇	C₂×C₄	semidirect
39. SD₁₆×C₁₃₇	C₂×C₄	semidirect
40. D₈×C₁₃₇	C₂×C₄	semidirect
42. C₂×C₈×C₁₃₇	C₂×C₄	semidirect
85. C₄×C₁₃₇	#135	direct (N × H)
92. C₁₃₇	C₄×#135	direct (N × H)
93. C₄×#135	C₁₃₇	direct (N × H)
95. C₄×#41	C₂×C₁₃₇	semidirect
96. C₄×Q₁₆	C₂×C₁₃₇	semidirect
97. C₄×SD₁₆	C₂×C₁₃₇	semidirect
98. C₄×D₈	C₂×C₁₃₇	semidirect
99. #427	C₂×C₁₃₇	semidirect
100. #416	C₂×C₁₃₇	semidirect
101. C₂×C₄×C₈	C₂×C₁₃₇	semidirect
103. #135	C₄×C₁₃₇	direct (N × H)
123. #41	C₂×C₄×C₁₃₇	semidirect
131. Q₁₆	C₂×C₄×C₁₃₇	semidirect
132. SD₁₆	C₂×C₄×C₁₃₇	semidirect
133. D₈	C₂×C₄×C₁₃₇	semidirect
135. C₂×C₈	C₂×C₄×C₁₃₇	semidirect
178. C₄	#135×C₁₃₇	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×#135×C₁₃₇	2
C₄×#41×C₁₃₇	2
C₄×Q₁₆×C₁₃₇	2
C₄×SD₁₆×C₁₃₇	2
C₄×D₈×C₁₃₇	2
#427×C₁₃₇	2
#416×C₁₃₇	2
C₂×C₄×C₈×C₁₃₇	2
C₄×#135	137

Subgroups — C4×#135×C137 #31777950

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#135×C₁₃₇ #31777950