Subgroups — C₂×#344×C₁₃₇ #31776754

Order 17536.

388 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×M₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#117×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#344×C₁₃₇	C₂	1	4	yes	maximal, direct product factor
5	C₂³×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
6	C₂×M₁₆×C₁₃₇	—	2	1	no	—
7	C₂×M₁₆×C₁₃₇	—	2	1	no	—
8	C₂×M₁₆×C₁₃₇	—	2	1	no	—
9	C₂×M₁₆×C₁₃₇	—	2	1	no	—
10	C₂×M₁₆×C₁₃₇	—	2	1	no	—
11	C₂×M₁₆×C₁₃₇	—	2	1	no	—
12	C₂×M₁₆×C₁₃₇	—	2	1	no	—
13	C₂×M₁₆×C₁₃₇	C₄	1	2	yes	complemented
14	C₂×M₁₆×C₁₃₇	C₄	1	2	yes	complemented
15	C₂×M₁₆×C₁₃₇	C₄	1	2	yes	complemented
16	C₂×M₁₆×C₁₃₇	C₂²	1	2	yes	—
17	C₂×M₁₆×C₁₃₇	C₂²	1	2	yes	—
18	C₂²×C₈×C₁₃₇	—	2	1	no	—
19	C₂²×C₈×C₁₃₇	C₄	1	2	yes	complemented
20	#117×C₁₃₇	—	2	1	no	—
21	#117×C₁₃₇	—	2	1	no	—
22	#117×C₁₃₇	C₄	1	2	yes	—
23	#117×C₁₃₇	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
25	C₂×C₂²⋊C₄×C₁₃₇	C₄	1	2	yes	—
26	C₂×C₄²×C₁₃₇	—	2	1	no	—
27	C₂⁴×C₁₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₃₇	—	2	2	no	—
29	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₃₇	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃₇	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₃₇	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃₇	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃₇	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₃₇	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃₇	Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₃₇	Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
43	M₁₆×C₁₃₇	—	2	2	no	—
44	M₁₆×C₁₃₇	—	2	2	no	—
45	M₁₆×C₁₃₇	—	2	2	no	—
46	M₁₆×C₁₃₇	—	2	2	no	—
47	M₁₆×C₁₃₇	—	2	2	no	—
48	M₁₆×C₁₃₇	—	2	2	no	—
49	M₁₆×C₁₃₇	—	2	2	no	—
50	M₁₆×C₁₃₇	C₂×C₄	1	4	yes	complemented
51	C₂×C₈×C₁₃₇	—	2	2	no	—
52	C₂×C₈×C₁₃₇	—	2	2	no	—
53	C₂×C₈×C₁₃₇	—	2	2	no	—
54	C₂×C₈×C₁₃₇	—	2	2	no	—
55	C₂×C₈×C₁₃₇	—	2	2	no	—
56	C₂×C₈×C₁₃₇	C₂×C₄	1	4	yes	complemented
57	C₄⋊C₄×C₁₃₇	—	2	2	no	—
58	C₄⋊C₄×C₁₃₇	—	2	2	no	—
59	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
60	C₂²⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
61	C₄²×C₁₃₇	—	2	2	no	—
62	C₄²×C₁₃₇	—	2	2	no	—
63	C₂³×C₁₃₇	—	4	2	no	—
64	C₂³×C₁₃₇	C₄²	1	1	yes	characteristic
65	C₂³×C₁₃₇	C₄²	1	1	yes	characteristic
66	C₂³×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂³×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂³×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
69	C₂³×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
70	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₃₇	—	2	4	no	—
72	C₂×C₄×C₁₃₇	—	4	4	no	—
73	C₂×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
74	C₂×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
75	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
101	C₈×C₁₃₇	—	2	4	no	—
102	C₈×C₁₃₇	—	2	4	no	—
103	C₂²×C₁₃₇	—	2	2	no	—
104	C₂²×C₁₃₇	—	2	2	no	—
105	C₂²×C₁₃₇	—	2	2	no	—
106	C₂²×C₁₃₇	—	4	1	no	—
107	C₂²×C₁₃₇	—	4	1	no	—
108	C₂²×C₁₃₇	—	4	1	no	—
109	C₂²×C₁₃₇	—	4	1	no	—
110	C₂²×C₁₃₇	#95	1	1	yes	characteristic
111	C₂²×C₁₃₇	#95	1	1	yes	characteristic
112	C₂²×C₁₃₇	#95	1	1	yes	characteristic
113	C₂²×C₁₃₇	#95	1	1	yes	characteristic
114	C₂²×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
115	C₂²×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₄×C₁₃₇	—	4	4	no	—
118	C₄×C₁₃₇	#95	1	1	yes	characteristic
119	C₄×C₁₃₇	#95	1	1	yes	characteristic
120	C₄×C₁₃₇	#95	1	1	yes	characteristic
121	C₄×C₁₃₇	#95	1	1	yes	characteristic
122	C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₁₃₇	—	2	1	no	—
127	C₂×C₁₃₇	—	2	1	no	—
128	C₂×C₁₃₇	—	2	1	no	—
129	C₂×C₁₃₇	—	2	1	no	—
130	C₂×C₁₃₇	—	2	1	no	—
131	C₂×C₁₃₇	—	2	1	no	—
132	C₂×C₁₃₇	#344	1	2	yes	direct product factor
133	C₂×C₁₃₇	C₂×#95	1	1	yes	characteristic
134	C₁₃₇	C₂×#344	1	1	yes	characteristic, direct product factor, Sylow
135	C₂×#344	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂²×M₁₆	C₂×C₁₃₇	1	1	yes	characteristic
137	C₂²×M₁₆	C₂×C₁₃₇	1	1	yes	characteristic
138	C₂×#117	C₂×C₁₃₇	1	1	yes	characteristic
139	#344	C₂×C₁₃₇	1	4	yes	direct product factor
140	C₂³×C₄	C₂²×C₁₃₇	1	1	yes	characteristic
141	C₂×M₁₆	—	2	1	no	—
142	C₂×M₁₆	—	2	1	no	—
143	C₂×M₁₆	—	2	1	no	—
144	C₂×M₁₆	—	2	1	no	—
145	C₂×M₁₆	—	2	1	no	—
146	C₂×M₁₆	—	2	1	no	—
147	C₂×M₁₆	—	2	1	no	—
148	C₂×M₁₆	C₄×C₁₃₇	1	2	yes	complemented
149	C₂×M₁₆	C₄×C₁₃₇	1	2	yes	complemented
150	C₂×M₁₆	C₄×C₁₃₇	1	2	yes	complemented
151	C₂×M₁₆	C₂²×C₁₃₇	1	2	yes	—
152	C₂×M₁₆	C₂²×C₁₃₇	1	2	yes	—
153	C₂²×C₈	—	2	1	no	—
154	C₂²×C₈	C₄×C₁₃₇	1	2	yes	complemented
155	#117	—	2	1	no	—
156	#117	—	2	1	no	—
157	#117	C₄×C₁₃₇	1	2	yes	—
158	#117	C₂²×C₁₃₇	1	2	yes	—
159	C₂×C₄⋊C₄	—	2	1	no	—
160	C₂×C₂²⋊C₄	C₄×C₁₃₇	1	2	yes	—
161	C₂×C₄²	—	2	1	no	—
162	C₂⁴	C₂×C₄×C₁₃₇	1	1	yes	characteristic
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
169	C₂²×C₄	D₄×C₁₃₇	1	1	yes	characteristic
170	C₂²×C₄	D₄×C₁₃₇	1	1	yes	characteristic
171	C₂²×C₄	D₄×C₁₃₇	1	1	yes	characteristic
172	C₂²×C₄	D₄×C₁₃₇	1	1	yes	characteristic
173	C₂²×C₄	D₄×C₁₃₇	1	1	yes	characteristic
174	C₂²×C₄	D₄×C₁₃₇	1	1	yes	characteristic
175	C₂²×C₄	Q₈×C₁₃₇	1	1	yes	characteristic
176	C₂²×C₄	Q₈×C₁₃₇	1	1	yes	characteristic
177	C₂²×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
178	M₁₆	—	2	2	no	—
179	M₁₆	—	2	2	no	—
180	M₁₆	—	2	2	no	—
181	M₁₆	—	2	2	no	—
182	M₁₆	—	2	2	no	—
183	M₁₆	—	2	2	no	—
184	M₁₆	—	2	2	no	—
185	M₁₆	C₂×C₄×C₁₃₇	1	4	yes	complemented
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	C₂×C₄×C₁₃₇	1	4	yes	complemented
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₂²⋊C₄	—	2	2	no	—
195	C₂²⋊C₄	C₂×C₄×C₁₃₇	1	4	yes	—
196	C₄²	—	2	2	no	—
197	C₄²	—	2	2	no	—
198	C₂³	—	4	2	no	—
199	C₂³	C₄²×C₁₃₇	1	1	yes	characteristic
200	C₂³	C₄²×C₁₃₇	1	1	yes	characteristic
201	C₂³	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
202	C₂³	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
203	C₂³	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
204	C₂³	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
205	C₂³	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	C₄²×C₁₃₇	1	1	yes	characteristic
209	C₂×C₄	C₄²×C₁₃₇	1	1	yes	characteristic
210	C₂×C₄	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
211	C₂×C₄	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
212	C₂×C₄	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
213	C₂×C₄	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
214	C₂×C₄	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
215	C₂×C₄	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
216	C₂×C₄	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
217	C₂×C₄	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
218	C₂×C₄	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
219	C₂×C₄	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
220	C₂×C₄	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
221	C₂×C₄	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
222	C₂×C₄	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
223	C₂×C₄	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
224	C₂×C₄	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
225	C₂×C₄	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
226	C₂×C₄	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
227	C₂×C₄	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
228	C₂×C₄	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
229	C₂×C₄	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
230	C₂×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
231	C₂×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
232	C₂×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
233	C₂×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
234	C₂×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
235	C₂×C₄	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
236	C₈	—	2	4	no	—
237	C₈	—	2	4	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	4	1	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	—	4	1	no	—
245	C₂²	#95×C₁₃₇	1	1	yes	characteristic
246	C₂²	#95×C₁₃₇	1	1	yes	characteristic
247	C₂²	#95×C₁₃₇	1	1	yes	characteristic
248	C₂²	#95×C₁₃₇	1	1	yes	characteristic
249	C₂²	C₂×C₄²×C₁₃₇	1	1	yes	characteristic
250	C₂²	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
251	C₂²	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
252	C₄	—	4	4	no	—
253	C₄	#95×C₁₃₇	1	1	yes	characteristic
254	C₄	#95×C₁₃₇	1	1	yes	characteristic
255	C₄	#95×C₁₃₇	1	1	yes	characteristic
256	C₄	#95×C₁₃₇	1	1	yes	characteristic
257	C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
258	C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
259	C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
260	C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
261	C₂	—	2	1	no	—
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	—	2	1	no	—
267	C₂	#344×C₁₃₇	1	2	yes	direct product factor
268	C₂	C₂×#95×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃₇	C₂²⋊C₄
commutator	C₂²	C₂×C₄²×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂³×C₁₃₇
socle	C₂²×C₁₃₇	#95
prime radical	C₂⁴×C₁₃₇	C₂×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ C₂³×C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. #344×C₁₃₇	C₂	direct (N × H)
13. C₂×M₁₆×C₁₃₇	C₄	semidirect
14. C₂×M₁₆×C₁₃₇	C₄	semidirect
15. C₂×M₁₆×C₁₃₇	C₄	semidirect
19. C₂²×C₈×C₁₃₇	C₄	semidirect
50. M₁₆×C₁₃₇	C₂×C₄	semidirect
56. C₂×C₈×C₁₃₇	C₂×C₄	semidirect
132. C₂×C₁₃₇	#344	direct (N × H)
134. C₁₃₇	C₂×#344	direct (N × H)
135. C₂×#344	C₁₃₇	direct (N × H)
139. #344	C₂×C₁₃₇	direct (N × H)
148. C₂×M₁₆	C₄×C₁₃₇	semidirect
149. C₂×M₁₆	C₄×C₁₃₇	semidirect
150. C₂×M₁₆	C₄×C₁₃₇	semidirect
154. C₂²×C₈	C₄×C₁₃₇	semidirect
185. M₁₆	C₂×C₄×C₁₃₇	semidirect
191. C₂×C₈	C₂×C₄×C₁₃₇	semidirect
267. C₂	#344×C₁₃₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×M₁₆×C₁₃₇	2
C₂×#117×C₁₃₇	2
#344×C₁₃₇	2
C₂×#344	137

Subgroups — C2×#344×C137 #31776754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#344×C₁₃₇ #31776754