Subgroups — #2593×C₁₃₇ #31777604

Order 17536.

440 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#400×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#398×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#394×C₁₃₇	C₂	1	2	yes	maximal
4	#390×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#388×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#388×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#388×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	#388×C₁₃₇	C₂	1	2	yes	maximal, complemented
9	C₄×C₄⋊C₄×C₁₃₇	C₂	1	2	yes	maximal, complemented
10	C₂²×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊C₄×C₁₃₇	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₄×C₁₃₇	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₁₃₇	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₁₃₇	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₁₃₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₁₃₇	C₂²	1	2	yes	—
22	C₂×C₄²×C₁₃₇	C₂²	1	2	yes	—
23	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₁₃₇	C₂²	1	2	yes	—
25	#95×C₁₃₇	C₂²	1	2	yes	—
26	#95×C₁₃₇	C₂²	1	2	yes	—
27	#95×C₁₃₇	C₂²	1	2	yes	—
28	#95×C₁₃₇	C₂²	1	1	yes	characteristic
29	#95×C₁₃₇	C₂²	1	2	yes	—
30	#95×C₁₃₇	C₂²	1	1	yes	characteristic
31	C₂⁴×C₁₃₇	C₂³	1	2	yes	—
32	C₂×D₄×C₁₃₇	—	2	2	no	—
33	C₂×D₄×C₁₃₇	—	4	1	no	—
34	C₂×D₄×C₁₃₇	—	4	1	no	—
35	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
36	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
43	C₄⋊C₄×C₁₃₇	—	2	4	no	—
44	C₄⋊C₄×C₁₃₇	—	2	4	no	—
45	C₄⋊C₄×C₁₃₇	—	4	1	no	—
46	C₄⋊C₄×C₁₃₇	—	4	1	no	—
47	C₄⋊C₄×C₁₃₇	—	4	1	no	—
48	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
49	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
50	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
51	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
52	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
53	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
54	C₄²×C₁₃₇	—	2	4	no	—
55	C₄²×C₁₃₇	—	4	1	no	—
56	C₄²×C₁₃₇	—	4	2	no	—
57	C₂³×C₁₃₇	—	2	2	no	—
58	C₂³×C₁₃₇	—	2	2	no	—
59	C₂³×C₁₃₇	—	2	2	no	—
60	C₂³×C₁₃₇	—	2	2	no	—
61	C₂³×C₁₃₇	—	2	2	no	—
62	C₂³×C₁₃₇	—	2	2	no	—
63	C₂³×C₁₃₇	—	2	2	no	—
64	C₂³×C₁₃₇	C₂⁴	1	1	yes	characteristic
65	D₄×C₁₃₇	—	4	4	no	—
66	C₂×C₄×C₁₃₇	—	2	2	no	—
67	C₂×C₄×C₁₃₇	—	2	2	no	—
68	C₂×C₄×C₁₃₇	—	2	2	no	—
69	C₂×C₄×C₁₃₇	—	2	2	no	—
70	C₂×C₄×C₁₃₇	—	2	1	no	—
71	C₂×C₄×C₁₃₇	—	2	1	no	—
72	C₂×C₄×C₁₃₇	—	2	2	no	—
73	C₂×C₄×C₁₃₇	—	2	2	no	—
74	C₂×C₄×C₁₃₇	—	2	2	no	—
75	C₂×C₄×C₁₃₇	—	2	2	no	—
76	C₂×C₄×C₁₃₇	—	2	2	no	—
77	C₂×C₄×C₁₃₇	—	2	2	no	—
78	C₂×C₄×C₁₃₇	—	2	2	no	—
79	C₂×C₄×C₁₃₇	—	2	2	no	—
80	C₂×C₄×C₁₃₇	—	2	2	no	—
81	C₂×C₄×C₁₃₇	—	2	2	no	—
82	C₂×C₄×C₁₃₇	—	2	2	no	—
83	C₂×C₄×C₁₃₇	—	2	1	no	—
84	C₂×C₄×C₁₃₇	—	2	1	no	—
85	C₂×C₄×C₁₃₇	#41	1	4	yes	—
86	C₂×C₄×C₁₃₇	#41	1	4	yes	—
87	C₂×C₄×C₁₃₇	#41	1	2	yes	—
88	C₂²×C₁₃₇	—	4	2	no	—
89	C₂²×C₁₃₇	—	4	2	no	—
90	C₂²×C₁₃₇	—	4	2	no	—
91	C₂²×C₁₃₇	—	4	2	no	—
92	C₂²×C₁₃₇	—	4	2	no	—
93	C₂²×C₁₃₇	—	4	2	no	—
94	C₂²×C₁₃₇	—	4	2	no	—
95	C₂²×C₁₃₇	C₂×#41	1	1	yes	characteristic
96	C₂²×C₁₃₇	C₂×#41	1	2	yes	—
97	C₂²×C₁₃₇	C₂×#41	1	2	yes	—
98	C₂²×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
99	C₂²×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
100	C₄×C₁₃₇	—	2	4	no	—
101	C₄×C₁₃₇	—	2	2	no	—
102	C₄×C₁₃₇	—	2	4	no	—
103	C₄×C₁₃₇	—	4	2	no	—
104	C₄×C₁₃₇	—	4	2	no	—
105	C₄×C₁₃₇	—	4	2	no	—
106	C₄×C₁₃₇	—	4	1	no	—
107	C₄×C₁₃₇	—	4	1	no	—
108	C₂×C₁₃₇	—	8	2	no	—
109	C₂×C₁₃₇	#529	1	2	yes	—
110	C₂×C₁₃₇	#538	1	1	yes	characteristic
111	C₂×C₁₃₇	#553	1	2	yes	—
112	C₂×C₁₃₇	#555	1	1	yes	characteristic
113	C₂×C₁₃₇	#559	1	1	yes	characteristic
114	C₁₃₇	#2593	1	1	yes	characteristic, direct product factor, Sylow
115	#2593	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	#400	C₂×C₁₃₇	1	1	yes	characteristic, complemented
117	#398	C₂×C₁₃₇	1	1	yes	characteristic, complemented
118	#394	C₂×C₁₃₇	1	2	yes	—
119	#390	C₂×C₁₃₇	1	1	yes	characteristic
120	#388	C₂×C₁₃₇	1	2	yes	complemented
121	#388	C₂×C₁₃₇	1	2	yes	complemented
122	#388	C₂×C₁₃₇	1	2	yes	complemented
123	#388	C₂×C₁₃₇	1	2	yes	complemented
124	C₄×C₄⋊C₄	C₂×C₁₃₇	1	2	yes	complemented
125	C₂²×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
126	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
127	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
128	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
129	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
130	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
131	C₂×C₂²⋊C₄	C₂²×C₁₃₇	1	2	yes	—
132	C₂×C₂²⋊C₄	C₂²×C₁₃₇	1	2	yes	—
133	C₂×C₂²⋊C₄	C₂²×C₁₃₇	1	2	yes	—
134	C₂×C₂²⋊C₄	C₂²×C₁₃₇	1	2	yes	—
135	C₂×C₂²⋊C₄	C₂²×C₁₃₇	1	2	yes	—
136	C₂×C₂²⋊C₄	C₂²×C₁₃₇	1	2	yes	—
137	C₂×C₄²	C₂²×C₁₃₇	1	2	yes	—
138	C₂×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
139	C₂×C₄²	C₂²×C₁₃₇	1	2	yes	—
140	#95	C₂²×C₁₃₇	1	2	yes	—
141	#95	C₂²×C₁₃₇	1	2	yes	—
142	#95	C₂²×C₁₃₇	1	2	yes	—
143	#95	C₂²×C₁₃₇	1	1	yes	characteristic
144	#95	C₂²×C₁₃₇	1	2	yes	—
145	#95	C₂²×C₁₃₇	1	1	yes	characteristic
146	C₂⁴	C₂³×C₁₃₇	1	2	yes	—
147	C₂×D₄	—	2	2	no	—
148	C₂×D₄	—	4	1	no	—
149	C₂×D₄	—	4	1	no	—
150	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
151	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
152	C₂²×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
153	C₂²×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
154	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
155	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
156	C₂²×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
157	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	4	1	no	—
161	C₄⋊C₄	—	4	1	no	—
162	C₄⋊C₄	—	4	1	no	—
163	C₂²⋊C₄	—	2	4	no	—
164	C₂²⋊C₄	—	2	4	no	—
165	C₂²⋊C₄	—	2	4	no	—
166	C₂²⋊C₄	—	2	4	no	—
167	C₂²⋊C₄	—	4	2	no	—
168	C₂²⋊C₄	—	4	2	no	—
169	C₄²	—	2	4	no	—
170	C₄²	—	4	1	no	—
171	C₄²	—	4	2	no	—
172	C₂³	—	2	2	no	—
173	C₂³	—	2	2	no	—
174	C₂³	—	2	2	no	—
175	C₂³	—	2	2	no	—
176	C₂³	—	2	2	no	—
177	C₂³	—	2	2	no	—
178	C₂³	—	2	2	no	—
179	C₂³	C₂⁴×C₁₃₇	1	1	yes	characteristic
180	D₄	—	4	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	#41×C₁₃₇	1	4	yes	—
201	C₂×C₄	#41×C₁₃₇	1	4	yes	—
202	C₂×C₄	#41×C₁₃₇	1	2	yes	—
203	C₂²	—	4	2	no	—
204	C₂²	—	4	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	—	4	2	no	—
207	C₂²	—	4	2	no	—
208	C₂²	—	4	2	no	—
209	C₂²	—	4	2	no	—
210	C₂²	C₂×#41×C₁₃₇	1	1	yes	characteristic
211	C₂²	C₂×#41×C₁₃₇	1	2	yes	—
212	C₂²	C₂×#41×C₁₃₇	1	2	yes	—
213	C₂²	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
214	C₂²	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
215	C₄	—	2	4	no	—
216	C₄	—	2	2	no	—
217	C₄	—	2	4	no	—
218	C₄	—	4	2	no	—
219	C₄	—	4	2	no	—
220	C₄	—	4	2	no	—
221	C₄	—	4	1	no	—
222	C₄	—	4	1	no	—
223	C₂	—	8	2	no	—
224	C₂	#529×C₁₃₇	1	2	yes	—
225	C₂	#538×C₁₃₇	1	1	yes	characteristic
226	C₂	#553×C₁₃₇	1	2	yes	—
227	C₂	#555×C₁₃₇	1	1	yes	characteristic
228	C₂	#559×C₁₃₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. #400×C₁₃₇	C₂	semidirect
2. #398×C₁₃₇	C₂	semidirect
5. #388×C₁₃₇	C₂	semidirect
6. #388×C₁₃₇	C₂	semidirect
7. #388×C₁₃₇	C₂	semidirect
8. #388×C₁₃₇	C₂	semidirect
9. C₄×C₄⋊C₄×C₁₃₇	C₂	semidirect
114. C₁₃₇	#2593	direct (N × H)
115. #2593	C₁₃₇	direct (N × H)
116. #400	C₂×C₁₃₇	semidirect
117. #398	C₂×C₁₃₇	semidirect
120. #388	C₂×C₁₃₇	semidirect
121. #388	C₂×C₁₃₇	semidirect
122. #388	C₂×C₁₃₇	semidirect
123. #388	C₂×C₁₃₇	semidirect
124. C₄×C₄⋊C₄	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#400×C₁₃₇	2
#398×C₁₃₇	2
#394×C₁₃₇	2
#390×C₁₃₇	2
#388×C₁₃₇	2
C₄×C₄⋊C₄×C₁₃₇	2
#2593	137

Subgroups — #2593×C137 #31777604

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2593×C₁₃₇ #31777604