Subgroups — #2830×C₁₃₇ #31777841

Order 17536.

408 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#398×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#397×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	#397×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	#396×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#395×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#394×C₁₃₇	C₂	1	2	yes	maximal
7	#393×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	#390×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	#387×C₁₃₇	C₂	1	2	yes	maximal, complemented
10	C₂²×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
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	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×C₂²⋊C₄×C₁₃₇	C₂²	1	2	yes	—
23	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
24	#95×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	2	yes	—
29	#95×C₁₃₇	C₂²	1	1	yes	characteristic
30	#95×C₁₃₇	C₂²	1	1	yes	characteristic
31	C₂⁴×C₁₃₇	C₂³	1	2	yes	—
32	C₂×Q₈×C₁₃₇	—	4	1	no	—
33	C₂×Q₈×C₁₃₇	—	4	2	no	—
34	C₂×D₄×C₁₃₇	—	2	2	no	—
35	C₂×D₄×C₁₃₇	—	4	1	no	—
36	C₂×D₄×C₁₃₇	—	4	1	no	—
37	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
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	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₁₃₇	—	2	4	no	—
46	C₄⋊C₄×C₁₃₇	—	4	1	no	—
47	C₄⋊C₄×C₁₃₇	—	4	2	no	—
48	C₄⋊C₄×C₁₃₇	—	4	1	no	—
49	C₄⋊C₄×C₁₃₇	—	4	2	no	—
50	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
51	C₂²⋊C₄×C₁₃₇	—	4	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	C₄²×C₁₃₇	—	4	1	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	Q₈×C₁₃₇	—	4	4	no	—
66	D₄×C₁₃₇	—	4	4	no	—
67	C₂×C₄×C₁₃₇	—	2	1	no	—
68	C₂×C₄×C₁₃₇	—	2	1	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	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	1	no	—
83	C₂×C₄×C₁₃₇	—	2	1	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	2	no	—
88	C₂×C₄×C₁₃₇	#41	1	2	yes	—
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	—	4	2	no	—
96	C₂²×C₁₃₇	C₂×#41	1	1	yes	characteristic
97	C₂²×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
98	C₂²×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
99	C₂²×C₁₃₇	H(5,𝔽₂)	1	2	yes	—
100	C₂²×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
101	C₄×C₁₃₇	—	2	2	no	—
102	C₄×C₁₃₇	—	4	2	no	—
103	C₄×C₁₃₇	—	4	2	no	—
104	C₄×C₁₃₇	—	4	1	no	—
105	C₄×C₁₃₇	—	4	2	no	—
106	C₄×C₁₃₇	—	4	1	no	—
107	C₄×C₁₃₇	—	4	2	no	—
108	C₄×C₁₃₇	—	4	2	no	—
109	C₂×C₁₃₇	—	8	2	no	—
110	C₂×C₁₃₇	#538	1	1	yes	characteristic
111	C₂×C₁₃₇	#542	1	2	yes	—
112	C₂×C₁₃₇	#560	1	1	yes	characteristic
113	C₂×C₁₃₇	#562	1	2	yes	—
114	C₂×C₁₃₇	#563	1	1	yes	characteristic
115	C₁₃₇	#2830	1	1	yes	characteristic, direct product factor, Sylow
116	#2830	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	#398	C₂×C₁₃₇	1	1	yes	characteristic, complemented
118	#397	C₂×C₁₃₇	1	2	yes	complemented
119	#397	C₂×C₁₃₇	1	2	yes	complemented
120	#396	C₂×C₁₃₇	1	2	yes	complemented
121	#395	C₂×C₁₃₇	1	1	yes	characteristic, complemented
122	#394	C₂×C₁₃₇	1	2	yes	—
123	#393	C₂×C₁₃₇	1	2	yes	complemented
124	#390	C₂×C₁₃₇	1	1	yes	characteristic
125	#387	C₂×C₁₃₇	1	2	yes	complemented
126	C₂²×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
127	C₂²×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
128	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
129	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
130	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
131	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
132	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
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×C₂²⋊C₄	C₂²×C₁₃₇	1	2	yes	—
138	C₂×C₂²⋊C₄	C₂²×C₁₃₇	1	2	yes	—
139	C₂×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
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	2	yes	—
144	#95	C₂²×C₁₃₇	1	2	yes	—
145	#95	C₂²×C₁₃₇	1	1	yes	characteristic
146	#95	C₂²×C₁₃₇	1	1	yes	characteristic
147	C₂⁴	C₂³×C₁₃₇	1	2	yes	—
148	C₂×Q₈	—	4	1	no	—
149	C₂×Q₈	—	4	2	no	—
150	C₂×D₄	—	2	2	no	—
151	C₂×D₄	—	4	1	no	—
152	C₂×D₄	—	4	1	no	—
153	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
154	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
155	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
156	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
157	C₂²×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
158	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
159	C₂²×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
160	C₂²×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
161	C₄⋊C₄	—	2	4	no	—
162	C₄⋊C₄	—	4	1	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	—	4	1	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₂²⋊C₄	—	4	2	no	—
167	C₂²⋊C₄	—	4	2	no	—
168	C₂²⋊C₄	—	4	2	no	—
169	C₂²⋊C₄	—	4	2	no	—
170	C₂²⋊C₄	—	4	2	no	—
171	C₂²⋊C₄	—	4	2	no	—
172	C₄²	—	4	1	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	—	2	2	no	—
180	C₂³	C₂⁴×C₁₃₇	1	1	yes	characteristic
181	Q₈	—	4	4	no	—
182	D₄	—	4	4	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	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	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	#41×C₁₃₇	1	2	yes	—
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	—	4	2	no	—
211	C₂²	—	4	2	no	—
212	C₂²	C₂×#41×C₁₃₇	1	1	yes	characteristic
213	C₂²	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
214	C₂²	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
215	C₂²	H(5,𝔽₂)×C₁₃₇	1	2	yes	—
216	C₂²	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
217	C₄	—	2	2	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	C₄	—	4	2	no	—
224	C₄	—	4	2	no	—
225	C₂	—	8	2	no	—
226	C₂	#538×C₁₃₇	1	1	yes	characteristic
227	C₂	#542×C₁₃₇	1	2	yes	—
228	C₂	#560×C₁₃₇	1	1	yes	characteristic
229	C₂	#562×C₁₃₇	1	2	yes	—
230	C₂	#563×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. #398×C₁₃₇	C₂	semidirect
2. #397×C₁₃₇	C₂	semidirect
3. #397×C₁₃₇	C₂	semidirect
4. #396×C₁₃₇	C₂	semidirect
5. #395×C₁₃₇	C₂	semidirect
7. #393×C₁₃₇	C₂	semidirect
9. #387×C₁₃₇	C₂	semidirect
115. C₁₃₇	#2830	direct (N × H)
116. #2830	C₁₃₇	direct (N × H)
117. #398	C₂×C₁₃₇	semidirect
118. #397	C₂×C₁₃₇	semidirect
119. #397	C₂×C₁₃₇	semidirect
120. #396	C₂×C₁₃₇	semidirect
121. #395	C₂×C₁₃₇	semidirect
123. #393	C₂×C₁₃₇	semidirect
125. #387	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#398×C₁₃₇	2
#397×C₁₃₇	2
#396×C₁₃₇	2
#395×C₁₃₇	2
#394×C₁₃₇	2
#393×C₁₃₇	2
#390×C₁₃₇	2
#387×C₁₃₇	2
#2830	137

Subgroups — #2830×C137 #31777841

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2830×C₁₃₇ #31777841