Subgroups — #2826×C₁₃₇ #31777837

Order 17536.

464 conjugacy classes of subgroups.

The subgroup lattice has 252 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	2	yes	maximal, complemented
2	#399×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#397×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	#396×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#394×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#394×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#392×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	#386×C₁₃₇	C₂	1	2	yes	maximal, complemented
9	#385×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₄×C₁₃₇	C₂²	1	2	yes	—
11	C₂²×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
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×C₂²⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊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×C₂²⋊C₄×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	C₂⁴×C₁₃₇	C₂³	1	1	yes	characteristic
33	C₂×D₄×C₁₃₇	—	2	4	no	—
34	C₂×D₄×C₁₃₇	—	4	2	no	—
35	C₂×D₄×C₁₃₇	—	4	1	no	—
36	C₂×D₄×C₁₃₇	—	4	2	no	—
37	C₂×D₄×C₁₃₇	—	4	2	no	—
38	C₂²×C₄×C₁₃₇	—	4	1	no	—
39	C₂²×C₄×C₁₃₇	—	4	2	no	—
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	1	yes	characteristic
44	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
46	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₁₃₇	—	4	1	no	—
48	C₄⋊C₄×C₁₃₇	—	4	2	no	—
49	C₄⋊C₄×C₁₃₇	—	4	2	no	—
50	C₄⋊C₄×C₁₃₇	—	4	2	no	—
51	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
52	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
53	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
54	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
55	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
56	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
57	C₂²⋊C₄×C₁₃₇	—	4	1	no	—
58	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
59	C₂³×C₁₃₇	—	2	1	no	—
60	C₂³×C₁₃₇	—	2	2	no	—
61	C₂³×C₁₃₇	—	2	2	no	—
62	C₂³×C₁₃₇	—	2	1	no	—
63	C₂³×C₁₃₇	—	2	2	no	—
64	C₂³×C₁₃₇	—	2	2	no	—
65	C₂³×C₁₃₇	—	2	2	no	—
66	C₂³×C₁₃₇	—	2	2	no	—
67	C₂³×C₁₃₇	—	2	2	no	—
68	C₂³×C₁₃₇	—	2	2	no	—
69	C₂³×C₁₃₇	—	2	2	no	—
70	C₂³×C₁₃₇	#41	1	2	yes	—
71	C₂³×C₁₃₇	C₂⁴	1	1	yes	characteristic
72	D₄×C₁₃₇	—	4	8	no	—
73	D₄×C₁₃₇	—	4	4	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	1	no	—
83	C₂×C₄×C₁₃₇	—	2	2	no	—
84	C₂×C₄×C₁₃₇	—	2	2	no	—
85	C₂×C₄×C₁₃₇	—	2	2	no	—
86	C₂×C₄×C₁₃₇	—	2	2	no	—
87	C₂×C₄×C₁₃₇	—	2	1	no	—
88	C₂×C₄×C₁₃₇	—	2	2	no	—
89	C₂×C₄×C₁₃₇	—	2	2	no	—
90	C₂×C₄×C₁₃₇	—	2	2	no	—
91	C₂×C₄×C₁₃₇	—	2	2	no	—
92	C₂×C₄×C₁₃₇	—	2	2	no	—
93	C₂×C₄×C₁₃₇	—	4	4	no	—
94	C₂²×C₁₃₇	—	2	4	no	—
95	C₂²×C₁₃₇	—	2	2	no	—
96	C₂²×C₁₃₇	—	4	1	no	—
97	C₂²×C₁₃₇	—	4	1	no	—
98	C₂²×C₁₃₇	—	4	2	no	—
99	C₂²×C₁₃₇	—	4	2	no	—
100	C₂²×C₁₃₇	—	4	2	no	—
101	C₂²×C₁₃₇	—	4	2	no	—
102	C₂²×C₁₃₇	—	4	2	no	—
103	C₂²×C₁₃₇	—	4	2	no	—
104	C₂²×C₁₃₇	—	4	2	no	—
105	C₂²×C₁₃₇	—	4	2	no	—
106	C₂²×C₁₃₇	C₂×#41	1	1	yes	characteristic
107	C₂²×C₁₃₇	H(5,𝔽₂)	1	2	yes	—
108	C₂²×C₁₃₇	H(5,𝔽₂)	1	2	yes	—
109	C₂²×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
110	C₂²×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
111	C₄×C₁₃₇	—	4	2	no	—
112	C₄×C₁₃₇	—	4	2	no	—
113	C₄×C₁₃₇	—	4	2	no	—
114	C₄×C₁₃₇	—	4	1	no	—
115	C₄×C₁₃₇	—	4	2	no	—
116	C₄×C₁₃₇	—	4	1	no	—
117	C₄×C₁₃₇	—	4	2	no	—
118	C₂×C₁₃₇	—	4	2	no	—
119	C₂×C₁₃₇	—	8	2	no	—
120	C₂×C₁₃₇	#538	1	2	yes	—
121	C₂×C₁₃₇	#539	1	1	yes	characteristic
122	C₂×C₁₃₇	#560	1	1	yes	characteristic
123	C₂×C₁₃₇	#560	1	1	yes	characteristic
124	C₂×C₁₃₇	#562	1	2	yes	—
125	C₁₃₇	#2826	1	1	yes	characteristic, direct product factor, Sylow
126	#2826	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	#400	C₂×C₁₃₇	1	2	yes	complemented
128	#399	C₂×C₁₃₇	1	1	yes	characteristic, complemented
129	#397	C₂×C₁₃₇	1	2	yes	complemented
130	#396	C₂×C₁₃₇	1	2	yes	complemented
131	#394	C₂×C₁₃₇	1	2	yes	complemented
132	#394	C₂×C₁₃₇	1	2	yes	complemented
133	#392	C₂×C₁₃₇	1	1	yes	characteristic
134	#386	C₂×C₁₃₇	1	2	yes	complemented
135	#385	C₂×C₁₃₇	1	1	yes	characteristic, complemented
136	C₂²×D₄	C₂²×C₁₃₇	1	2	yes	—
137	C₂²×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
138	C₂³×C₄	C₂²×C₁₃₇	1	1	yes	characteristic
139	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
140	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
141	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
142	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
143	C₂×C₂²⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
144	C₂×C₂²⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
145	C₂×C₂²⋊C₄	C₂²×C₁₃₇	1	2	yes	—
146	C₂×C₂²⋊C₄	C₂²×C₁₃₇	1	2	yes	—
147	C₂×C₂²⋊C₄	C₂²×C₁₃₇	1	2	yes	—
148	C₂×C₂²⋊C₄	C₂²×C₁₃₇	1	2	yes	—
149	C₂×C₂²⋊C₄	C₂²×C₁₃₇	1	2	yes	—
150	C₂×C₂²⋊C₄	C₂²×C₁₃₇	1	2	yes	—
151	#95	C₂²×C₁₃₇	1	2	yes	—
152	#95	C₂²×C₁₃₇	1	2	yes	—
153	#95	C₂²×C₁₃₇	1	2	yes	—
154	#95	C₂²×C₁₃₇	1	1	yes	characteristic
155	#95	C₂²×C₁₃₇	1	2	yes	—
156	#95	C₂²×C₁₃₇	1	1	yes	characteristic
157	C₂⁴	C₂³×C₁₃₇	1	2	yes	—
158	C₂⁴	C₂³×C₁₃₇	1	1	yes	characteristic
159	C₂×D₄	—	2	4	no	—
160	C₂×D₄	—	4	2	no	—
161	C₂×D₄	—	4	1	no	—
162	C₂×D₄	—	4	2	no	—
163	C₂×D₄	—	4	2	no	—
164	C₂²×C₄	—	4	1	no	—
165	C₂²×C₄	—	4	2	no	—
166	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
167	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
168	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
169	C₂²×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
170	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
171	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
172	C₂²×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
173	C₄⋊C₄	—	4	1	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₂²⋊C₄	—	2	2	no	—
178	C₂²⋊C₄	—	4	2	no	—
179	C₂²⋊C₄	—	4	2	no	—
180	C₂²⋊C₄	—	4	2	no	—
181	C₂²⋊C₄	—	4	2	no	—
182	C₂²⋊C₄	—	4	2	no	—
183	C₂²⋊C₄	—	4	1	no	—
184	C₂²⋊C₄	—	4	2	no	—
185	C₂³	—	2	1	no	—
186	C₂³	—	2	2	no	—
187	C₂³	—	2	2	no	—
188	C₂³	—	2	1	no	—
189	C₂³	—	2	2	no	—
190	C₂³	—	2	2	no	—
191	C₂³	—	2	2	no	—
192	C₂³	—	2	2	no	—
193	C₂³	—	2	2	no	—
194	C₂³	—	2	2	no	—
195	C₂³	—	2	2	no	—
196	C₂³	#41×C₁₃₇	1	2	yes	—
197	C₂³	C₂⁴×C₁₃₇	1	1	yes	characteristic
198	D₄	—	4	8	no	—
199	D₄	—	4	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂²	—	2	4	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	4	1	no	—
223	C₂²	—	4	1	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	—	4	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	—	4	2	no	—
230	C₂²	—	4	2	no	—
231	C₂²	—	4	2	no	—
232	C₂²	C₂×#41×C₁₃₇	1	1	yes	characteristic
233	C₂²	H(5,𝔽₂)×C₁₃₇	1	2	yes	—
234	C₂²	H(5,𝔽₂)×C₁₃₇	1	2	yes	—
235	C₂²	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
236	C₂²	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
237	C₄	—	4	2	no	—
238	C₄	—	4	2	no	—
239	C₄	—	4	2	no	—
240	C₄	—	4	1	no	—
241	C₄	—	4	2	no	—
242	C₄	—	4	1	no	—
243	C₄	—	4	2	no	—
244	C₂	—	4	2	no	—
245	C₂	—	8	2	no	—
246	C₂	#538×C₁₃₇	1	2	yes	—
247	C₂	#539×C₁₃₇	1	1	yes	characteristic
248	C₂	#560×C₁₃₇	1	1	yes	characteristic
249	C₂	#560×C₁₃₇	1	1	yes	characteristic
250	C₂	#562×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₃₇
prime radical	#392×C₁₃₇	C₂
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. #399×C₁₃₇	C₂	semidirect
3. #397×C₁₃₇	C₂	semidirect
4. #396×C₁₃₇	C₂	semidirect
5. #394×C₁₃₇	C₂	semidirect
6. #394×C₁₃₇	C₂	semidirect
8. #386×C₁₃₇	C₂	semidirect
9. #385×C₁₃₇	C₂	semidirect
125. C₁₃₇	#2826	direct (N × H)
126. #2826	C₁₃₇	direct (N × H)
127. #400	C₂×C₁₃₇	semidirect
128. #399	C₂×C₁₃₇	semidirect
129. #397	C₂×C₁₃₇	semidirect
130. #396	C₂×C₁₃₇	semidirect
131. #394	C₂×C₁₃₇	semidirect
132. #394	C₂×C₁₃₇	semidirect
134. #386	C₂×C₁₃₇	semidirect
135. #385	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
#399×C₁₃₇	2
#397×C₁₃₇	2
#396×C₁₃₇	2
#394×C₁₃₇	2
#392×C₁₃₇	2
#386×C₁₃₇	2
#385×C₁₃₇	2
#2826	137

Subgroups — #2826×C137 #31777837

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2826×C₁₃₇ #31777837