Subgroups — C₄×#441×C₁₃₇ #46869888

Order 35072.

1086 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#441×C₁₃₇	C₂	1	2	yes	maximal
2	C₂×C₄×Q₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₄²×Q₈×C₁₃₇	C₂	1	2	yes	maximal
4	#1776×C₁₃₇	C₂	1	2	yes	maximal
5	C₄×#106×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	#1763×C₁₃₇	C₂	1	4	yes	maximal
7	C₄×#103×C₁₃₇	C₂	1	2	yes	maximal
8	C₄×#97×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₁₃₇	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₁₃₇	C₂²	1	4	yes	—
12	#441×C₁₃₇	C₄	1	16	yes	direct product factor
13	C₄×Q₁₆×C₁₃₇	C₄	1	8	yes	complemented
14	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
16	C₂×#103×C₁₃₇	C₂²	1	4	yes	—
17	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
18	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
19	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₁₃₇	—	2	4	no	—
23	C₄×C₄⋊C₄×C₁₃₇	—	2	2	no	—
24	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	4	yes	—
26	C₄³×C₁₃₇	—	2	2	no	—
27	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
28	#340×C₁₃₇	C₂²	1	4	yes	—
29	#340×C₁₃₇	C₂²	1	2	yes	—
30	C₂²×Q₈×C₁₃₇	C₂³	1	2	yes	—
31	C₂×Q₁₆×C₁₃₇	C₂×C₄	1	4	yes	—
32	C₂×Q₁₆×C₁₃₇	C₂×C₄	1	8	yes	—
33	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
36	C₄×Q₈×C₁₃₇	—	2	4	no	—
37	C₄×Q₈×C₁₃₇	—	2	8	no	—
38	C₄×Q₈×C₁₃₇	C₂×C₄	1	8	yes	—
39	C₄×Q₈×C₁₃₇	C₂×C₄	1	16	yes	—
40	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
43	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
44	C₂×C₄²×C₁₃₇	—	2	4	no	—
45	C₂×C₄²×C₁₃₇	—	2	2	no	—
46	C₂×C₄²×C₁₃₇	—	2	4	no	—
47	C₂×C₄²×C₁₃₇	—	2	2	no	—
48	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
49	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
50	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
51	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
52	#107×C₁₃₇	C₂×C₄	1	4	yes	—
53	#106×C₁₃₇	C₂×C₄	1	8	yes	—
54	#103×C₁₃₇	C₂×C₄	1	8	yes	—
55	#103×C₁₃₇	C₂×C₄	1	16	yes	—
56	#97×C₁₃₇	—	2	4	no	—
57	#97×C₁₃₇	C₂×C₄	1	8	yes	—
58	C₄×C₈×C₁₃₇	—	2	2	no	—
59	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
60	#95×C₁₃₇	—	2	2	no	—
61	#95×C₁₃₇	—	2	4	no	—
62	C₂×Q₈×C₁₃₇	—	2	2	no	—
63	C₂×Q₈×C₁₃₇	—	2	4	no	—
64	C₂×Q₈×C₁₃₇	C₂²×C₄	1	4	yes	—
65	C₂×Q₈×C₁₃₇	C₂²×C₄	1	8	yes	—
66	C₂²×C₄×C₁₃₇	—	2	2	no	—
67	C₂²×C₄×C₁₃₇	—	2	4	no	—
68	C₂²×C₄×C₁₃₇	—	2	2	no	—
69	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃₇	C₂×D₄	1	4	yes	—
71	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
73	Q₁₆×C₁₃₇	C₄²	1	16	yes	complemented
74	C₂×C₈×C₁₃₇	—	2	2	no	—
75	C₂×C₈×C₁₃₇	—	2	2	no	—
76	C₂×C₈×C₁₃₇	—	2	2	no	—
77	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
79	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
81	C₄⋊C₄×C₁₃₇	—	2	16	no	—
82	C₄⋊C₄×C₁₃₇	—	2	8	no	—
83	C₄⋊C₄×C₁₃₇	—	2	4	no	—
84	C₄⋊C₄×C₁₃₇	—	2	2	no	—
85	C₄⋊C₄×C₁₃₇	C₂²×C₄	1	8	yes	—
86	C₄⋊C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
87	C₄²×C₁₃₇	—	2	16	no	—
88	C₄²×C₁₃₇	—	2	8	no	—
89	C₄²×C₁₃₇	—	2	4	no	—
90	C₄²×C₁₃₇	—	4	2	no	—
91	C₄²×C₁₃₇	—	4	8	no	—
92	C₄²×C₁₃₇	—	4	4	no	—
93	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
94	C₄²×C₁₃₇	C₂²×C₄	1	4	yes	—
95	C₄²×C₁₃₇	#41	1	2	yes	—
96	C₄²×C₁₃₇	#41	1	4	yes	—
97	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
98	Q₈×C₁₃₇	—	2	4	no	—
99	Q₈×C₁₃₇	—	2	8	no	—
100	Q₈×C₁₃₇	C₂×C₄²	1	8	yes	—
101	C₂×C₄×C₁₃₇	—	2	4	no	—
102	C₂×C₄×C₁₃₇	—	2	8	no	—
103	C₂×C₄×C₁₃₇	—	2	4	no	—
104	C₂×C₄×C₁₃₇	—	2	4	no	—
105	C₂×C₄×C₁₃₇	—	2	2	no	—
106	C₂×C₄×C₁₃₇	—	2	4	no	—
107	C₂×C₄×C₁₃₇	—	2	8	no	—
108	C₂×C₄×C₁₃₇	—	4	4	no	—
109	C₂×C₄×C₁₃₇	—	4	4	no	—
110	C₂×C₄×C₁₃₇	—	4	4	no	—
111	C₂×C₄×C₁₃₇	C₄×D₄	1	8	yes	—
112	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₁₃₇	#137	1	4	yes	—
115	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃₇	C₂³×C₄	1	2	yes	—
117	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
119	C₈×C₁₃₇	—	2	4	no	—
120	C₈×C₁₃₇	—	2	2	no	—
121	C₈×C₁₃₇	C₂×C₄²	1	4	yes	—
122	C₂²×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
123	C₂²×C₁₃₇	C₂×C₄×D₄	1	2	yes	—
124	C₂²×C₁₃₇	C₂×#137	1	2	yes	—
125	C₂²×C₁₃₇	#575	1	2	yes	—
126	C₄×C₁₃₇	—	2	2	no	—
127	C₄×C₁₃₇	—	2	8	no	—
128	C₄×C₁₃₇	—	2	8	no	—
129	C₄×C₁₃₇	—	4	4	no	—
130	C₄×C₁₃₇	—	4	8	no	—
131	C₄×C₁₃₇	#441	1	4	yes	direct product factor
132	C₄×C₁₃₇	C₂²×C₄²	1	1	yes	characteristic
133	C₄×C₁₃₇	C₄×#41	1	2	yes	—
134	C₄×C₁₃₇	C₄×#41	1	1	yes	characteristic
135	C₂×C₁₃₇	C₄²×D₄	1	1	yes	characteristic
136	C₂×C₁₃₇	C₂×#441	1	1	yes	characteristic
137	C₂×C₁₃₇	#2943	1	1	yes	characteristic
138	C₂×C₁₃₇	C₄×#137	1	4	yes	—
139	C₄×#441	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₁₃₇	C₄×#441	1	1	yes	characteristic, direct product factor, Sylow
141	C₂×#441	C₂×C₁₃₇	1	2	yes	—
142	C₂×C₄×Q₁₆	C₂×C₁₃₇	1	1	yes	characteristic
143	C₄²×Q₈	C₂×C₁₃₇	1	2	yes	—
144	#1776	C₂×C₁₃₇	1	2	yes	—
145	C₄×#106	C₂×C₁₃₇	1	1	yes	characteristic
146	#1763	C₂×C₁₃₇	1	4	yes	—
147	C₄×#103	C₂×C₁₃₇	1	2	yes	—
148	C₄×#97	C₂×C₁₃₇	1	1	yes	characteristic
149	C₂²×Q₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
150	C₂×C₄×Q₈	C₂²×C₁₃₇	1	2	yes	—
151	C₂×C₄×Q₈	C₂²×C₁₃₇	1	4	yes	—
152	#441	C₄×C₁₃₇	1	16	yes	direct product factor
153	C₄×Q₁₆	C₄×C₁₃₇	1	8	yes	complemented
154	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
155	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
156	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
157	C₂×#103	C₂²×C₁₃₇	1	4	yes	—
158	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
159	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
160	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
161	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
162	C₄×C₄⋊C₄	—	2	2	no	—
163	C₄×C₄⋊C₄	—	2	4	no	—
164	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
165	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	4	yes	—
166	C₄³	—	2	2	no	—
167	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
168	#340	C₂²×C₁₃₇	1	4	yes	—
169	#340	C₂²×C₁₃₇	1	2	yes	—
170	C₂²×Q₈	C₂³×C₁₃₇	1	2	yes	—
171	C₂×Q₁₆	C₂×C₄×C₁₃₇	1	4	yes	—
172	C₂×Q₁₆	C₂×C₄×C₁₃₇	1	8	yes	—
173	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
174	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
175	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
176	C₄×Q₈	—	2	4	no	—
177	C₄×Q₈	—	2	8	no	—
178	C₄×Q₈	C₂×C₄×C₁₃₇	1	8	yes	—
179	C₄×Q₈	C₂×C₄×C₁₃₇	1	16	yes	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	4	no	—
182	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
183	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
184	C₂×C₄²	—	2	4	no	—
185	C₂×C₄²	—	2	2	no	—
186	C₂×C₄²	—	2	4	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
189	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
190	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
191	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
192	#107	C₂×C₄×C₁₃₇	1	4	yes	—
193	#106	C₂×C₄×C₁₃₇	1	8	yes	—
194	#103	C₂×C₄×C₁₃₇	1	8	yes	—
195	#103	C₂×C₄×C₁₃₇	1	16	yes	—
196	#97	—	2	4	no	—
197	#97	C₂×C₄×C₁₃₇	1	8	yes	—
198	C₄×C₈	—	2	2	no	—
199	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
200	#95	—	2	2	no	—
201	#95	—	2	4	no	—
202	C₂×Q₈	—	2	2	no	—
203	C₂×Q₈	—	2	4	no	—
204	C₂×Q₈	C₂²×C₄×C₁₃₇	1	4	yes	—
205	C₂×Q₈	C₂²×C₄×C₁₃₇	1	8	yes	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₄×C₁₃₇	1	4	yes	—
211	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
212	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
213	Q₁₆	C₄²×C₁₃₇	1	16	yes	complemented
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
218	C₂×C₈	C₂²×C₄×C₁₃₇	1	4	yes	—
219	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
220	C₂×C₈	C₂²×C₄×C₁₃₇	1	4	yes	—
221	C₄⋊C₄	—	2	16	no	—
222	C₄⋊C₄	—	2	8	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	C₂²×C₄×C₁₃₇	1	8	yes	—
226	C₄⋊C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
227	C₄²	—	2	16	no	—
228	C₄²	—	2	8	no	—
229	C₄²	—	2	4	no	—
230	C₄²	—	4	2	no	—
231	C₄²	—	4	8	no	—
232	C₄²	—	4	4	no	—
233	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
234	C₄²	C₂²×C₄×C₁₃₇	1	4	yes	—
235	C₄²	#41×C₁₃₇	1	2	yes	—
236	C₄²	#41×C₁₃₇	1	4	yes	—
237	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
238	Q₈	—	2	4	no	—
239	Q₈	—	2	8	no	—
240	Q₈	C₂×C₄²×C₁₃₇	1	8	yes	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	C₄×D₄×C₁₃₇	1	8	yes	—
252	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
253	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
254	C₂×C₄	#137×C₁₃₇	1	4	yes	—
255	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
256	C₂×C₄	C₂³×C₄×C₁₃₇	1	2	yes	—
257	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
258	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
259	C₈	—	2	4	no	—
260	C₈	—	2	2	no	—
261	C₈	C₂×C₄²×C₁₃₇	1	4	yes	—
262	C₂²	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
263	C₂²	C₂×C₄×D₄×C₁₃₇	1	2	yes	—
264	C₂²	C₂×#137×C₁₃₇	1	2	yes	—
265	C₂²	#575×C₁₃₇	1	2	yes	—
266	C₄	—	2	8	no	—
267	C₄	—	2	2	no	—
268	C₄	—	2	8	no	—
269	C₄	—	4	4	no	—
270	C₄	—	4	8	no	—
271	C₄	#441×C₁₃₇	1	4	yes	direct product factor
272	C₄	C₂²×C₄²×C₁₃₇	1	1	yes	characteristic
273	C₄	C₄×#41×C₁₃₇	1	2	yes	—
274	C₄	C₄×#41×C₁₃₇	1	1	yes	characteristic
275	C₂	C₄²×D₄×C₁₃₇	1	1	yes	characteristic
276	C₂	C₂×#441×C₁₃₇	1	1	yes	characteristic
277	C₂	#2943×C₁₃₇	1	1	yes	characteristic
278	C₂	C₄×#137×C₁₃₇	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃₇	C₂×D₄
commutator	C₄	C₂²×C₄²×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
socle, prime radical	C₂³×C₁₃₇	C₂²×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₇ ◁ C₄³×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
12. #441×C₁₃₇	C₄	direct (N × H)
13. C₄×Q₁₆×C₁₃₇	C₄	semidirect
73. Q₁₆×C₁₃₇	C₄²	semidirect
131. C₄×C₁₃₇	#441	direct (N × H)
139. C₄×#441	C₁₃₇	direct (N × H)
140. C₁₃₇	C₄×#441	direct (N × H)
152. #441	C₄×C₁₃₇	direct (N × H)
153. C₄×Q₁₆	C₄×C₁₃₇	semidirect
213. Q₁₆	C₄²×C₁₃₇	semidirect
271. C₄	#441×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×#441×C₁₃₇	2
C₂×C₄×Q₁₆×C₁₃₇	2
C₄²×Q₈×C₁₃₇	2
#1776×C₁₃₇	2
C₄×#106×C₁₃₇	2
#1763×C₁₃₇	2
C₄×#103×C₁₃₇	2
C₄×#97×C₁₃₇	2
C₄×#441	137

Subgroups — C4×#441×C137 #46869888

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#441×C₁₃₇ #46869888