Subgroups — #2609×C₁₃₇ #31777620

Order 17536.

540 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#117×C₁₃₇	C₂	1	2	yes	maximal, complemented
2	#398×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#396×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	#394×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#388×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#387×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#386×C₁₃₇	C₂	1	2	yes	maximal
8	#385×C₁₃₇	C₂	1	2	yes	maximal, complemented
9	#382×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₃₇	C₂²	1	2	yes	—
12	#117×C₁₃₇	—	2	8	no	—
13	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
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×C₂²⋊C₄×C₁₃₇	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₁₃₇	C₂²	1	2	yes	—
24	C₂×C₄²×C₁₃₇	C₂²	1	2	yes	—
25	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
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	#95×C₁₃₇	C₂²	1	2	yes	—
32	C₂⁴×C₁₃₇	C₂³	1	2	yes	—
33	C₂×D₄×C₁₃₇	—	2	4	no	—
34	C₂×D₄×C₁₃₇	—	4	1	no	—
35	C₂²×C₄×C₁₃₇	—	2	4	no	—
36	C₂²×C₄×C₁₃₇	—	2	4	no	—
37	C₂²×C₄×C₁₃₇	—	2	4	no	—
38	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
44	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
46	C₄⋊C₄×C₁₃₇	—	2	4	no	—
47	C₄⋊C₄×C₁₃₇	—	2	4	no	—
48	C₄⋊C₄×C₁₃₇	—	2	4	no	—
49	C₄⋊C₄×C₁₃₇	—	4	1	no	—
50	C₄⋊C₄×C₁₃₇	—	4	1	no	—
51	C₄⋊C₄×C₁₃₇	—	4	1	no	—
52	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
53	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
54	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
55	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
56	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
57	C₄²×C₁₃₇	—	2	4	no	—
58	C₄²×C₁₃₇	D₄	1	4	yes	complemented
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	—	2	2	no	—
65	C₂³×C₁₃₇	#41	1	4	yes	—
66	C₂³×C₁₃₇	C₂⁴	1	1	yes	characteristic
67	D₄×C₁₃₇	—	4	4	no	—
68	C₂×C₄×C₁₃₇	—	2	16	no	—
69	C₂×C₄×C₁₃₇	—	2	2	no	—
70	C₂×C₄×C₁₃₇	—	2	2	no	—
71	C₂×C₄×C₁₃₇	—	2	2	no	—
72	C₂×C₄×C₁₃₇	—	2	2	no	—
73	C₂×C₄×C₁₃₇	—	2	1	no	—
74	C₂×C₄×C₁₃₇	—	2	1	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	2	no	—
84	C₂×C₄×C₁₃₇	—	2	2	no	—
85	C₂×C₄×C₁₃₇	—	2	1	no	—
86	C₂×C₄×C₁₃₇	—	2	1	no	—
87	C₂×C₄×C₁₃₇	—	2	2	no	—
88	C₂×C₄×C₁₃₇	C₂×D₄	1	4	yes	—
89	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
90	C₂×C₄×C₁₃₇	#41	1	4	yes	—
91	C₂²×C₁₃₇	—	2	4	no	—
92	C₂²×C₁₃₇	—	2	4	no	—
93	C₂²×C₁₃₇	—	2	4	no	—
94	C₂²×C₁₃₇	—	4	2	no	—
95	C₂²×C₁₃₇	—	4	2	no	—
96	C₂²×C₁₃₇	—	4	2	no	—
97	C₂²×C₁₃₇	—	4	2	no	—
98	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
99	C₂²×C₁₃₇	C₂×#41	1	2	yes	—
100	C₂²×C₁₃₇	C₂×#41	1	2	yes	—
101	C₂²×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
102	C₂²×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
103	C₄×C₁₃₇	—	2	4	no	—
104	C₄×C₁₃₇	—	2	4	no	—
105	C₄×C₁₃₇	—	2	2	no	—
106	C₄×C₁₃₇	—	4	2	no	—
107	C₄×C₁₃₇	—	4	2	no	—
108	C₄×C₁₃₇	—	4	1	no	—
109	C₄×C₁₃₇	—	4	1	no	—
110	C₄×C₁₃₇	—	4	2	no	—
111	C₂×C₁₃₇	—	4	4	no	—
112	C₂×C₁₃₇	#525	1	2	yes	—
113	C₂×C₁₃₇	#537	1	1	yes	characteristic
114	C₂×C₁₃₇	#551	1	1	yes	characteristic
115	C₂×C₁₃₇	#552	1	2	yes	—
116	C₂×C₁₃₇	#555	1	1	yes	characteristic
117	C₁₃₇	#2609	1	1	yes	characteristic, direct product factor, Sylow
118	#2609	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂×#117	C₂×C₁₃₇	1	2	yes	complemented
120	#398	C₂×C₁₃₇	1	1	yes	characteristic, complemented
121	#396	C₂×C₁₃₇	1	2	yes	complemented
122	#394	C₂×C₁₃₇	1	1	yes	characteristic
123	#388	C₂×C₁₃₇	1	2	yes	complemented
124	#387	C₂×C₁₃₇	1	2	yes	complemented
125	#386	C₂×C₁₃₇	1	2	yes	—
126	#385	C₂×C₁₃₇	1	2	yes	complemented
127	#382	C₂×C₁₃₇	1	1	yes	characteristic, complemented
128	C₂²×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
129	C₂³×C₄	C₂²×C₁₃₇	1	2	yes	—
130	#117	—	2	8	no	—
131	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
132	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
133	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
134	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
135	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
136	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
137	C₂×C₂²⋊C₄	C₂²×C₁₃₇	1	2	yes	—
138	C₂×C₂²⋊C₄	C₂²×C₁₃₇	1	2	yes	—
139	C₂×C₂²⋊C₄	C₂²×C₁₃₇	1	2	yes	—
140	C₂×C₂²⋊C₄	C₂²×C₁₃₇	1	2	yes	—
141	C₂×C₂²⋊C₄	C₂²×C₁₃₇	1	2	yes	—
142	C₂×C₄²	C₂²×C₁₃₇	1	2	yes	—
143	C₂×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
144	#95	C₂²×C₁₃₇	1	2	yes	—
145	#95	C₂²×C₁₃₇	1	2	yes	—
146	#95	C₂²×C₁₃₇	1	1	yes	characteristic
147	#95	C₂²×C₁₃₇	1	2	yes	—
148	#95	C₂²×C₁₃₇	1	1	yes	characteristic
149	#95	C₂²×C₁₃₇	1	2	yes	—
150	C₂⁴	C₂³×C₁₃₇	1	2	yes	—
151	C₂×D₄	—	2	4	no	—
152	C₂×D₄	—	4	1	no	—
153	C₂²×C₄	—	2	4	no	—
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
157	C₂²×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
158	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
159	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
160	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
161	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
162	C₂²×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
163	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	4	1	no	—
168	C₄⋊C₄	—	4	1	no	—
169	C₄⋊C₄	—	4	1	no	—
170	C₂²⋊C₄	—	2	4	no	—
171	C₂²⋊C₄	—	2	4	no	—
172	C₂²⋊C₄	—	2	4	no	—
173	C₂²⋊C₄	—	2	4	no	—
174	C₂²⋊C₄	—	4	2	no	—
175	C₄²	—	2	4	no	—
176	C₄²	D₄×C₁₃₇	1	4	yes	complemented
177	C₂³	—	2	2	no	—
178	C₂³	—	2	2	no	—
179	C₂³	—	2	2	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	#41×C₁₃₇	1	4	yes	—
184	C₂³	C₂⁴×C₁₃₇	1	1	yes	characteristic
185	D₄	—	4	4	no	—
186	C₂×C₄	—	2	16	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	1	no	—
192	C₂×C₄	—	2	1	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	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	C₂×D₄×C₁₃₇	1	4	yes	—
207	C₂×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
208	C₂×C₄	#41×C₁₃₇	1	4	yes	—
209	C₂²	—	2	4	no	—
210	C₂²	—	2	4	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	—	4	2	no	—
215	C₂²	—	4	2	no	—
216	C₂²	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
217	C₂²	C₂×#41×C₁₃₇	1	2	yes	—
218	C₂²	C₂×#41×C₁₃₇	1	2	yes	—
219	C₂²	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
220	C₂²	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
221	C₄	—	2	4	no	—
222	C₄	—	2	4	no	—
223	C₄	—	2	2	no	—
224	C₄	—	4	2	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	1	no	—
227	C₄	—	4	1	no	—
228	C₄	—	4	2	no	—
229	C₂	—	4	4	no	—
230	C₂	#525×C₁₃₇	1	2	yes	—
231	C₂	#537×C₁₃₇	1	1	yes	characteristic
232	C₂	#551×C₁₃₇	1	1	yes	characteristic
233	C₂	#552×C₁₃₇	1	2	yes	—
234	C₂	#555×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. C₂×#117×C₁₃₇	C₂	semidirect
2. #398×C₁₃₇	C₂	semidirect
3. #396×C₁₃₇	C₂	semidirect
5. #388×C₁₃₇	C₂	semidirect
6. #387×C₁₃₇	C₂	semidirect
8. #385×C₁₃₇	C₂	semidirect
9. #382×C₁₃₇	C₂	semidirect
58. C₄²×C₁₃₇	D₄	semidirect
117. C₁₃₇	#2609	direct (N × H)
118. #2609	C₁₃₇	direct (N × H)
119. C₂×#117	C₂×C₁₃₇	semidirect
120. #398	C₂×C₁₃₇	semidirect
121. #396	C₂×C₁₃₇	semidirect
123. #388	C₂×C₁₃₇	semidirect
124. #387	C₂×C₁₃₇	semidirect
126. #385	C₂×C₁₃₇	semidirect
127. #382	C₂×C₁₃₇	semidirect
176. C₄²	D₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#117×C₁₃₇	2
#398×C₁₃₇	2
#396×C₁₃₇	2
#394×C₁₃₇	2
#388×C₁₃₇	2
#387×C₁₃₇	2
#386×C₁₃₇	2
#385×C₁₃₇	2
#382×C₁₃₇	2
#2609	137

Subgroups — #2609×C137 #31777620

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2609×C₁₃₇ #31777620