Subgroups — #3299×C₁₃₇ #31778310

Order 17536.

388 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#554×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	D₄×Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#487×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#484×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#477×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#467×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#462×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#451×C₁₃₇	C₂	1	2	yes	maximal, complemented
9	C₄×Q₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×D₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
11	C₂×#103×C₁₃₇	C₂	1	2	yes	maximal, complemented
12	C₂²×Q₈×C₁₃₇	C₂²	1	2	yes	—
13	C₂×Q₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×Q₁₆×C₁₃₇	C₂²	1	2	yes	—
15	C₂²×C₈×C₁₃₇	C₂²	1	2	yes	—
16	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
17	C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
19	#125×C₁₃₇	—	2	1	no	—
20	C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
21	C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
22	C₂²⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
23	C₂²⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
24	C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
25	C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₁₃₇	—	2	1	no	—
27	C₄×D₄×C₁₃₇	—	2	1	no	—
28	C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
30	#107×C₁₃₇	C₂²	1	1	yes	characteristic
31	#107×C₁₃₇	C₂²	1	2	yes	—
32	C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
33	#103×C₁₃₇	C₂²	1	1	yes	characteristic
34	#103×C₁₃₇	C₂²	1	2	yes	—
35	#103×C₁₃₇	C₂²	1	2	yes	—
36	#103×C₁₃₇	C₂²	1	2	yes	—
37	#103×C₁₃₇	C₂²	1	1	yes	characteristic
38	#103×C₁₃₇	C₂²	1	2	yes	—
39	C₂²⋊C₈×C₁₃₇	C₂²	1	2	yes	—
40	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₁₃₇	—	2	2	no	—
42	C₂×Q₈×C₁₃₇	—	2	1	no	—
43	C₂×Q₈×C₁₃₇	—	2	4	no	—
44	C₂×Q₈×C₁₃₇	—	4	2	no	—
45	C₂×Q₈×C₁₃₇	C₂³	1	2	yes	—
46	C₂×Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
47	C₂×Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₁₃₇	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃₇	—	2	2	no	—
50	C₂²×C₄×C₁₃₇	—	2	2	no	—
51	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
52	Q₁₆×C₁₃₇	—	2	2	no	—
53	Q₁₆×C₁₃₇	—	4	2	no	—
54	C₂×C₈×C₁₃₇	—	4	2	no	—
55	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
58	C₄⋊C₄×C₁₃₇	—	2	2	no	—
59	C₄⋊C₄×C₁₃₇	—	2	1	no	—
60	C₄⋊C₄×C₁₃₇	—	2	1	no	—
61	C₄⋊C₄×C₁₃₇	—	2	1	no	—
62	C₄⋊C₄×C₁₃₇	—	2	1	no	—
63	C₄⋊C₄×C₁₃₇	—	2	1	no	—
64	C₄⋊C₄×C₁₃₇	—	2	2	no	—
65	C₄⋊C₄×C₁₃₇	—	2	2	no	—
66	C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
67	C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
69	C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
70	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
71	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
72	C₂²⋊C₄×C₁₃₇	D₄	1	2	yes	—
73	C₄²×C₁₃₇	—	2	1	no	—
74	C₄²×C₁₃₇	—	2	1	no	—
75	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
76	C₂³×C₁₃₇	C₂×D₄	1	2	yes	—
77	Q₈×C₁₃₇	—	2	1	no	—
78	Q₈×C₁₃₇	—	2	1	no	—
79	Q₈×C₁₃₇	—	2	2	no	—
80	Q₈×C₁₃₇	—	2	1	no	—
81	Q₈×C₁₃₇	—	2	2	no	—
82	Q₈×C₁₃₇	—	4	4	no	—
83	Q₈×C₁₃₇	—	8	1	no	—
84	Q₈×C₁₃₇	#41	1	2	yes	—
85	D₄×C₁₃₇	Q₁₆	1	4	yes	complemented
86	C₂×C₄×C₁₃₇	—	2	1	no	—
87	C₂×C₄×C₁₃₇	—	2	2	no	—
88	C₂×C₄×C₁₃₇	—	2	2	no	—
89	C₂×C₄×C₁₃₇	—	2	1	no	—
90	C₂×C₄×C₁₃₇	—	2	1	no	—
91	C₂×C₄×C₁₃₇	—	2	1	no	—
92	C₂×C₄×C₁₃₇	—	2	2	no	—
93	C₂×C₄×C₁₃₇	—	2	2	no	—
94	C₂×C₄×C₁₃₇	—	4	4	no	—
95	C₂×C₄×C₁₃₇	—	8	2	no	—
96	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
100	C₈×C₁₃₇	—	2	1	no	—
101	C₈×C₁₃₇	—	2	1	no	—
102	C₈×C₁₃₇	—	2	2	no	—
103	C₂²×C₁₃₇	—	2	2	no	—
104	C₂²×C₁₃₇	—	2	2	no	—
105	C₂²×C₁₃₇	C₂×Q₁₆	1	4	yes	—
106	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
107	C₄×C₁₃₇	—	2	2	no	—
108	C₄×C₁₃₇	—	2	2	no	—
109	C₄×C₁₃₇	—	2	1	no	—
110	C₄×C₁₃₇	—	4	2	no	—
111	C₄×C₁₃₇	—	4	1	no	—
112	C₄×C₁₃₇	—	4	1	no	—
113	C₄×C₁₃₇	—	4	1	no	—
114	C₄×C₁₃₇	—	4	2	no	—
115	C₄×C₁₃₇	C₂×Q₁₆	1	2	yes	—
116	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
117	C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂×C₁₃₇	—	2	4	no	—
119	C₂×C₁₃₇	#546	1	1	yes	characteristic
120	C₂×C₁₃₇	C₂²×Q₁₆	1	1	yes	characteristic
121	C₂×C₁₃₇	#577	1	1	yes	characteristic
122	C₁₃₇	#3299	1	1	yes	characteristic, direct product factor, Sylow
123	#3299	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	#554	C₂×C₁₃₇	1	1	yes	characteristic
125	D₄×Q₈	C₂×C₁₃₇	1	1	yes	characteristic
126	#487	C₂×C₁₃₇	1	1	yes	characteristic, complemented
127	#484	C₂×C₁₃₇	1	2	yes	complemented
128	#477	C₂×C₁₃₇	1	1	yes	characteristic, complemented
129	#467	C₂×C₁₃₇	1	2	yes	complemented
130	#462	C₂×C₁₃₇	1	1	yes	characteristic, complemented
131	#451	C₂×C₁₃₇	1	2	yes	complemented
132	C₄×Q₁₆	C₂×C₁₃₇	1	1	yes	characteristic, complemented
133	C₈×D₄	C₂×C₁₃₇	1	1	yes	characteristic
134	C₂×#103	C₂×C₁₃₇	1	2	yes	complemented
135	C₂²×Q₈	C₂²×C₁₃₇	1	2	yes	—
136	C₂×Q₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
137	C₂×Q₁₆	C₂²×C₁₃₇	1	2	yes	—
138	C₂²×C₈	C₂²×C₁₃₇	1	2	yes	—
139	C₄⋊Q₈	—	2	1	no	—
140	C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
141	C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
142	#125	—	2	1	no	—
143	C₂²⋊Q₈	—	2	2	no	—
144	C₂²⋊Q₈	—	2	2	no	—
145	C₂²⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
146	C₂²⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
147	C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
148	C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
149	C₄×D₄	—	2	1	no	—
150	C₄×D₄	—	2	1	no	—
151	C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
152	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
153	#107	C₂²×C₁₃₇	1	1	yes	characteristic
154	#107	C₂²×C₁₃₇	1	2	yes	—
155	C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
156	#103	C₂²×C₁₃₇	1	1	yes	characteristic
157	#103	C₂²×C₁₃₇	1	2	yes	—
158	#103	C₂²×C₁₃₇	1	2	yes	—
159	#103	C₂²×C₁₃₇	1	2	yes	—
160	#103	C₂²×C₁₃₇	1	1	yes	characteristic
161	#103	C₂²×C₁₃₇	1	2	yes	—
162	C₂²⋊C₈	C₂²×C₁₃₇	1	2	yes	—
163	C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
164	C₂×Q₈	—	2	2	no	—
165	C₂×Q₈	—	2	1	no	—
166	C₂×Q₈	—	2	4	no	—
167	C₂×Q₈	—	4	2	no	—
168	C₂×Q₈	C₂³×C₁₃₇	1	2	yes	—
169	C₂×Q₈	C₂³×C₁₃₇	1	1	yes	characteristic
170	C₂×Q₈	C₂³×C₁₃₇	1	1	yes	characteristic
171	C₂×D₄	D₄×C₁₃₇	1	1	yes	characteristic
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
175	Q₁₆	—	2	2	no	—
176	Q₁₆	—	4	2	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
179	C₂×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
180	C₂×C₈	C₂³×C₁₃₇	1	2	yes	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	2	1	no	—
183	C₄⋊C₄	—	2	1	no	—
184	C₄⋊C₄	—	2	1	no	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
190	C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
191	C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
192	C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
193	C₂²⋊C₄	—	2	2	no	—
194	C₂²⋊C₄	—	2	2	no	—
195	C₂²⋊C₄	D₄×C₁₃₇	1	2	yes	—
196	C₄²	—	2	1	no	—
197	C₄²	—	2	1	no	—
198	C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
199	C₂³	C₂×D₄×C₁₃₇	1	2	yes	—
200	Q₈	—	2	1	no	—
201	Q₈	—	2	1	no	—
202	Q₈	—	2	2	no	—
203	Q₈	—	2	1	no	—
204	Q₈	—	2	2	no	—
205	Q₈	—	4	4	no	—
206	Q₈	—	8	1	no	—
207	Q₈	#41×C₁₃₇	1	2	yes	—
208	D₄	Q₁₆×C₁₃₇	1	4	yes	complemented
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	8	2	no	—
219	C₂×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
220	C₂×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
221	C₂×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
222	C₂×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	2	no	—
228	C₂²	C₂×Q₁₆×C₁₃₇	1	4	yes	—
229	C₂²	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	1	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	1	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	1	no	—
238	C₄	C₂×Q₁₆×C₁₃₇	1	2	yes	—
239	C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
240	C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
241	C₂	—	2	4	no	—
242	C₂	#546×C₁₃₇	1	1	yes	characteristic
243	C₂	C₂²×Q₁₆×C₁₃₇	1	1	yes	characteristic
244	C₂	#577×C₁₃₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. #487×C₁₃₇	C₂	semidirect
4. #484×C₁₃₇	C₂	semidirect
5. #477×C₁₃₇	C₂	semidirect
6. #467×C₁₃₇	C₂	semidirect
7. #462×C₁₃₇	C₂	semidirect
8. #451×C₁₃₇	C₂	semidirect
9. C₄×Q₁₆×C₁₃₇	C₂	semidirect
11. C₂×#103×C₁₃₇	C₂	semidirect
85. D₄×C₁₃₇	Q₁₆	semidirect
122. C₁₃₇	#3299	direct (N × H)
123. #3299	C₁₃₇	direct (N × H)
126. #487	C₂×C₁₃₇	semidirect
127. #484	C₂×C₁₃₇	semidirect
128. #477	C₂×C₁₃₇	semidirect
129. #467	C₂×C₁₃₇	semidirect
130. #462	C₂×C₁₃₇	semidirect
131. #451	C₂×C₁₃₇	semidirect
132. C₄×Q₁₆	C₂×C₁₃₇	semidirect
134. C₂×#103	C₂×C₁₃₇	semidirect
208. D₄	Q₁₆×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#554×C₁₃₇	2
D₄×Q₈×C₁₃₇	2
#487×C₁₃₇	2
#484×C₁₃₇	2
#477×C₁₃₇	2
#467×C₁₃₇	2
#462×C₁₃₇	2
#451×C₁₃₇	2
C₄×Q₁₆×C₁₃₇	2
C₈×D₄×C₁₃₇	2
C₂×#103×C₁₃₇	2
#3299	137

Subgroups — #3299×C137 #31778310

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3299×C₁₃₇ #31778310