Subgroups — C₂×#1574×C₁₃₇ #46867191

Order 35072.

714 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#431×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#431×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#423×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1574×C₁₃₇	C₂	1	8	yes	maximal, direct product factor
5	C₂×#335×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	C₂×#334×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	C₂²×M₁₆×C₁₃₇	—	2	1	no	—
8	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
9	#431×C₁₃₇	—	2	2	no	—
10	#431×C₁₃₇	C₄	1	4	yes	—
11	#431×C₁₃₇	C₂²	1	4	yes	—
12	#431×C₁₃₇	C₂²	1	4	yes	—
13	#423×C₁₃₇	—	2	2	no	—
14	#423×C₁₃₇	C₂²	1	4	yes	—
15	C₂×C₄⋊C₈×C₁₃₇	—	2	1	no	—
16	C₂×C₄⋊C₈×C₁₃₇	C₄	1	2	yes	—
17	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₈×C₁₃₇	—	2	1	no	—
21	C₂×C₂²⋊C₈×C₁₃₇	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₁₃₇	—	2	1	no	—
23	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
24	#335×C₁₃₇	C₂²	1	8	yes	complemented
25	#334×C₁₃₇	C₂²	1	8	yes	complemented
26	C₂³×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₃₇	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₃₇	Q₈	1	1	yes	characteristic
29	C₂×M₁₆×C₁₃₇	—	2	4	no	—
30	C₂×M₁₆×C₁₃₇	—	4	4	no	—
31	C₂²×C₈×C₁₃₇	—	2	1	no	—
32	C₂²×C₈×C₁₃₇	—	2	1	no	—
33	C₂²×C₈×C₁₃₇	—	2	2	no	—
34	C₂²×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
38	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃₇	Q₈	1	1	yes	characteristic
40	C₂×C₄²×C₁₃₇	Q₈	1	2	yes	—
41	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
42	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₁₃₇	—	2	4	no	—
44	C₄⋊C₈×C₁₃₇	—	2	2	no	—
45	C₄⋊C₈×C₁₃₇	—	2	1	no	—
46	C₄⋊C₈×C₁₃₇	—	2	1	no	—
47	C₄⋊C₈×C₁₃₇	C₂×C₄	1	8	yes	—
48	C₄⋊C₈×C₁₃₇	C₂³	1	4	yes	—
49	C₄⋊C₈×C₁₃₇	C₂³	1	2	yes	—
50	C₄⋊C₈×C₁₃₇	C₂³	1	2	yes	—
51	C₂²⋊C₈×C₁₃₇	—	2	4	no	—
52	C₂²⋊C₈×C₁₃₇	C₂×C₄	1	8	yes	—
53	C₄×C₈×C₁₃₇	—	2	4	no	—
54	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
55	C₄×C₈×C₁₃₇	C₂³	1	4	yes	—
56	C₂⁴×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃₇	—	2	4	no	—
58	C₂²×C₄×C₁₃₇	—	2	4	no	—
59	C₂²×C₄×C₁₃₇	—	2	2	no	—
60	C₂²×C₄×C₁₃₇	—	2	2	no	—
61	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
73	M₁₆×C₁₃₇	—	4	8	no	—
74	C₂×C₈×C₁₃₇	—	2	8	no	—
75	C₂×C₈×C₁₃₇	—	2	2	no	—
76	C₂×C₈×C₁₃₇	—	2	2	no	—
77	C₂×C₈×C₁₃₇	—	2	4	no	—
78	C₂×C₈×C₁₃₇	—	4	2	no	—
79	C₂×C₈×C₁₃₇	—	4	4	no	—
80	C₂×C₈×C₁₃₇	—	4	2	no	—
81	C₂×C₈×C₁₃₇	C₂²×C₄	1	8	yes	—
82	C₄²×C₁₃₇	C₄⋊C₄	1	2	yes	—
83	C₄²×C₁₃₇	C₄⋊C₄	1	2	yes	—
84	C₄²×C₁₃₇	C₄⋊C₄	1	2	yes	—
85	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
87	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
88	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
90	C₄²×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
91	C₄²×C₁₃₇	C₂⁴	1	1	yes	characteristic
92	C₂³×C₁₃₇	—	2	2	no	—
93	C₂³×C₁₃₇	—	2	2	no	—
94	C₂³×C₁₃₇	—	2	2	no	—
95	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
96	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃₇	—	2	2	no	—
98	C₂×C₄×C₁₃₇	—	2	2	no	—
99	C₂×C₄×C₁₃₇	—	2	2	no	—
100	C₂×C₄×C₁₃₇	—	2	4	no	—
101	C₂×C₄×C₁₃₇	—	2	2	no	—
102	C₂×C₄×C₁₃₇	—	2	2	no	—
103	C₂×C₄×C₁₃₇	—	2	4	no	—
104	C₂×C₄×C₁₃₇	—	2	4	no	—
105	C₂×C₄×C₁₃₇	—	2	4	no	—
106	C₂×C₄×C₁₃₇	—	2	4	no	—
107	C₂×C₄×C₁₃₇	#108	1	4	yes	—
108	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₁₃₇	#131	1	4	yes	—
115	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
116	C₂×C₄×C₁₃₇	#137	1	2	yes	—
117	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
120	C₈×C₁₃₇	—	2	8	no	—
121	C₈×C₁₃₇	—	4	2	no	—
122	C₈×C₁₃₇	—	4	4	no	—
123	C₈×C₁₃₇	—	4	2	no	—
124	C₂²×C₁₃₇	—	2	2	no	—
125	C₂²×C₁₃₇	—	2	2	no	—
126	C₂²×C₁₃₇	—	2	2	no	—
127	C₂²×C₁₃₇	—	4	4	no	—
128	C₂²×C₁₃₇	#424	1	2	yes	—
129	C₂²×C₁₃₇	#428	1	2	yes	—
130	C₂²×C₁₃₇	C₂×#108	1	2	yes	—
131	C₂²×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
132	C₄×C₁₃₇	—	2	2	no	—
133	C₄×C₁₃₇	—	2	2	no	—
134	C₄×C₁₃₇	—	2	1	no	—
135	C₄×C₁₃₇	—	2	1	no	—
136	C₄×C₁₃₇	#424	1	2	yes	—
137	C₄×C₁₃₇	#428	1	2	yes	—
138	C₄×C₁₃₇	C₂×#108	1	4	yes	—
139	C₄×C₁₃₇	C₂×#131	1	2	yes	—
140	C₄×C₁₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₄×C₁₃₇	C₂×#137	1	1	yes	characteristic
142	C₂×C₁₃₇	—	4	2	no	—
143	C₂×C₁₃₇	#1574	1	4	yes	direct product factor
144	C₂×C₁₃₇	C₂×#424	1	1	yes	characteristic
145	C₂×C₁₃₇	C₂×#428	1	1	yes	characteristic
146	C₂×C₁₃₇	C₂²×#108	1	1	yes	characteristic
147	C₂×#1574	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₁₃₇	C₂×#1574	1	1	yes	characteristic, direct product factor, Sylow
149	C₂×#431	C₂×C₁₃₇	1	1	yes	characteristic
150	C₂×#431	C₂×C₁₃₇	1	1	yes	characteristic
151	C₂×#423	C₂×C₁₃₇	1	1	yes	characteristic
152	#1574	C₂×C₁₃₇	1	8	yes	direct product factor
153	C₂×#335	C₂×C₁₃₇	1	2	yes	complemented
154	C₂×#334	C₂×C₁₃₇	1	2	yes	complemented
155	C₂²×M₁₆	—	2	1	no	—
156	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
157	#431	—	2	2	no	—
158	#431	C₄×C₁₃₇	1	4	yes	—
159	#431	C₂²×C₁₃₇	1	4	yes	—
160	#431	C₂²×C₁₃₇	1	4	yes	—
161	#423	—	2	2	no	—
162	#423	C₂²×C₁₃₇	1	4	yes	—
163	C₂×C₄⋊C₈	—	2	1	no	—
164	C₂×C₄⋊C₈	C₄×C₁₃₇	1	2	yes	—
165	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
166	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
167	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
168	C₂×C₂²⋊C₈	—	2	1	no	—
169	C₂×C₂²⋊C₈	C₄×C₁₃₇	1	2	yes	—
170	C₂×C₄×C₈	—	2	1	no	—
171	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
172	#335	C₂²×C₁₃₇	1	8	yes	complemented
173	#334	C₂²×C₁₃₇	1	8	yes	complemented
174	C₂³×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
175	C₂³×C₄	D₄×C₁₃₇	1	1	yes	characteristic
176	C₂³×C₄	Q₈×C₁₃₇	1	1	yes	characteristic
177	C₂×M₁₆	—	2	4	no	—
178	C₂×M₁₆	—	4	4	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	2	1	no	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
183	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
184	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
185	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
186	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
187	C₂×C₄²	Q₈×C₁₃₇	1	1	yes	characteristic
188	C₂×C₄²	Q₈×C₁₃₇	1	2	yes	—
189	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
190	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
191	C₄⋊C₈	—	2	4	no	—
192	C₄⋊C₈	—	2	2	no	—
193	C₄⋊C₈	—	2	1	no	—
194	C₄⋊C₈	—	2	1	no	—
195	C₄⋊C₈	C₂×C₄×C₁₃₇	1	8	yes	—
196	C₄⋊C₈	C₂³×C₁₃₇	1	2	yes	—
197	C₄⋊C₈	C₂³×C₁₃₇	1	4	yes	—
198	C₄⋊C₈	C₂³×C₁₃₇	1	2	yes	—
199	C₂²⋊C₈	—	2	4	no	—
200	C₂²⋊C₈	C₂×C₄×C₁₃₇	1	8	yes	—
201	C₄×C₈	—	2	4	no	—
202	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
203	C₄×C₈	C₂³×C₁₃₇	1	4	yes	—
204	C₂⁴	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	2	yes	—
210	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
211	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	2	yes	—
212	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
213	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	2	yes	—
214	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
215	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
216	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
218	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
219	C₂²×C₄	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
220	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
221	M₁₆	—	4	8	no	—
222	C₂×C₈	—	2	8	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	4	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	C₂²×C₄×C₁₃₇	1	8	yes	—
230	C₄²	C₄⋊C₄×C₁₃₇	1	2	yes	—
231	C₄²	C₄⋊C₄×C₁₃₇	1	2	yes	—
232	C₄²	C₄⋊C₄×C₁₃₇	1	2	yes	—
233	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
234	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
235	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
236	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
237	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
238	C₄²	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
239	C₄²	C₂⁴×C₁₃₇	1	1	yes	characteristic
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
244	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	#108×C₁₃₇	1	4	yes	—
256	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
258	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
259	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
260	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
261	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
262	C₂×C₄	#131×C₁₃₇	1	4	yes	—
263	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
264	C₂×C₄	#137×C₁₃₇	1	2	yes	—
265	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
267	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
268	C₈	—	2	8	no	—
269	C₈	—	4	4	no	—
270	C₈	—	4	2	no	—
271	C₈	—	4	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	4	4	no	—
276	C₂²	#424×C₁₃₇	1	2	yes	—
277	C₂²	#428×C₁₃₇	1	2	yes	—
278	C₂²	C₂×#108×C₁₃₇	1	2	yes	—
279	C₂²	C₂²×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	#424×C₁₃₇	1	2	yes	—
285	C₄	#428×C₁₃₇	1	2	yes	—
286	C₄	C₂×#108×C₁₃₇	1	4	yes	—
287	C₄	C₂×#131×C₁₃₇	1	2	yes	—
288	C₄	C₂×AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
289	C₄	C₂×#137×C₁₃₇	1	1	yes	characteristic
290	C₂	—	4	2	no	—
291	C₂	#1574×C₁₃₇	1	4	yes	direct product factor
292	C₂	C₂×#424×C₁₃₇	1	1	yes	characteristic
293	C₂	C₂×#428×C₁₃₇	1	1	yes	characteristic
294	C₂	C₂²×#108×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₃₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃₇
Frattini, cosocle	C₄²	C₂⁴×C₁₃₇
socle	C₂³×C₁₃₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃₇	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₇ ◁ C₂²×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. #1574×C₁₃₇	C₂	direct (N × H)
5. C₂×#335×C₁₃₇	C₂	semidirect
6. C₂×#334×C₁₃₇	C₂	semidirect
24. #335×C₁₃₇	C₂²	semidirect
25. #334×C₁₃₇	C₂²	semidirect
143. C₂×C₁₃₇	#1574	direct (N × H)
147. C₂×#1574	C₁₃₇	direct (N × H)
148. C₁₃₇	C₂×#1574	direct (N × H)
152. #1574	C₂×C₁₃₇	direct (N × H)
153. C₂×#335	C₂×C₁₃₇	semidirect
154. C₂×#334	C₂×C₁₃₇	semidirect
172. #335	C₂²×C₁₃₇	semidirect
173. #334	C₂²×C₁₃₇	semidirect
291. C₂	#1574×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
C₂×#431×C₁₃₇	2
C₂×#423×C₁₃₇	2
#1574×C₁₃₇	2
C₂×#335×C₁₃₇	2
C₂×#334×C₁₃₇	2
C₂×#1574	137

Subgroups — C2×#1574×C137 #46867191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1574×C₁₃₇ #46867191