Subgroups — C₂×C₄×#108×C₁₃₇ #46869842

Order 35072.

1086 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#108×C₁₃₇	C₂	1	2	yes	maximal
2	C₂×C₄×M₁₆×C₁₃₇	C₂	1	2	yes	maximal
3	C₂²×C₄×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₄×#108×C₁₃₇	C₂	1	8	yes	maximal, direct product factor
5	C₂×#341×C₁₃₇	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₁₃₇	C₂²	1	4	yes	—
7	C₂³×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×#108×C₁₃₇	C₄	1	16	yes	direct product factor
11	C₂×#108×C₁₃₇	C₂²	1	8	yes	—
12	C₄×M₁₆×C₁₃₇	—	2	4	no	—
13	C₄×M₁₆×C₁₃₇	C₂²	1	8	yes	—
14	C₂×#97×C₁₃₇	—	2	2	no	—
15	C₂×C₄×C₈×C₁₃₇	—	2	2	no	—
16	C₂×C₄×C₈×C₁₃₇	C₄	1	4	yes	—
17	C₂×C₄×C₈×C₁₃₇	C₄	1	4	yes	—
18	C₂×C₄×C₈×C₁₃₇	C₂²	1	4	yes	—
19	#341×C₁₃₇	C₂²	1	8	yes	—
20	C₂³×C₄×C₁₃₇	Q₈	1	2	yes	—
21	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₁₃₇	—	2	4	no	—
23	C₂×M₁₆×C₁₃₇	—	2	8	no	—
24	C₂×M₁₆×C₁₃₇	C₂×C₄	1	16	yes	—
25	C₂×M₁₆×C₁₃₇	C₂³	1	8	yes	—
26	C₂²×C₈×C₁₃₇	—	2	4	no	—
27	C₂²×C₈×C₁₃₇	—	2	4	no	—
28	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₃₇	C₂×C₄	1	8	yes	—
33	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
35	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
39	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃₇	Q₈	1	2	yes	—
41	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
42	#108×C₁₃₇	C₂×C₄	1	32	yes	direct product factor
43	#97×C₁₃₇	—	2	8	no	—
44	C₄×C₈×C₁₃₇	—	2	8	no	—
45	C₄×C₈×C₁₃₇	C₂×C₄	1	16	yes	—
46	C₂⁴×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₄×C₁₃₇	—	2	8	no	—
48	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
50	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
54	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
57	C₂²×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₃₇	#41	1	4	yes	—
59	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
60	M₁₆×C₁₃₇	—	2	16	no	—
61	M₁₆×C₁₃₇	—	2	8	no	—
62	M₁₆×C₁₃₇	C₂²×C₄	1	16	yes	—
63	C₂×C₈×C₁₃₇	—	2	4	no	—
64	C₂×C₈×C₁₃₇	—	2	4	no	—
65	C₂×C₈×C₁₃₇	—	2	16	no	—
66	C₂×C₈×C₁₃₇	—	2	8	no	—
67	C₂×C₈×C₁₃₇	—	2	4	no	—
68	C₂×C₈×C₁₃₇	C₄²	1	8	yes	—
69	C₂×C₈×C₁₃₇	C₄²	1	8	yes	—
70	C₂×C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
71	C₂×C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
72	C₂×C₈×C₁₃₇	C₂²×C₄	1	8	yes	—
73	C₄²×C₁₃₇	C₄⋊C₄	1	2	yes	—
74	C₄²×C₁₃₇	C₄⋊C₄	1	4	yes	—
75	C₄²×C₁₃₇	C₄⋊C₄	1	2	yes	—
76	C₄²×C₁₃₇	C₂²×C₄	1	4	yes	—
77	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
78	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
79	C₂³×C₁₃₇	—	2	4	no	—
80	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂³×C₁₃₇	C₄×Q₈	1	4	yes	—
83	C₂³×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
84	C₂×C₄×C₁₃₇	—	2	4	no	—
85	C₂×C₄×C₁₃₇	—	2	8	no	—
86	C₂×C₄×C₁₃₇	—	2	8	no	—
87	C₂×C₄×C₁₃₇	#108	1	8	yes	direct product factor
88	C₂×C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
89	C₂×C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
90	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
91	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
94	C₂×C₄×C₁₃₇	#117	1	2	yes	—
95	C₂×C₄×C₁₃₇	#117	1	2	yes	—
96	C₂×C₄×C₁₃₇	C₄×D₄	1	4	yes	—
97	C₂×C₄×C₁₃₇	C₄×D₄	1	4	yes	—
98	C₂×C₄×C₁₃₇	C₄×Q₈	1	4	yes	—
99	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃₇	C₂³×C₄	1	2	yes	—
101	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
103	C₈×C₁₃₇	—	2	4	no	—
104	C₈×C₁₃₇	—	2	16	no	—
105	C₈×C₁₃₇	C₂×C₄²	1	8	yes	—
106	C₂²×C₁₃₇	—	2	2	no	—
107	C₂²×C₁₃₇	—	2	2	no	—
108	C₂²×C₁₃₇	—	2	8	no	—
109	C₂²×C₁₃₇	C₄×C₄⋊C₄	1	2	yes	—
110	C₂²×C₁₃₇	C₄×C₄⋊C₄	1	2	yes	—
111	C₂²×C₁₃₇	C₂×#108	1	4	yes	—
112	C₂²×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₁₃₇	C₂×C₄×Q₈	1	2	yes	—
114	C₄×C₁₃₇	—	2	4	no	—
115	C₄×C₁₃₇	C₄×C₄⋊C₄	1	2	yes	—
116	C₄×C₁₃₇	C₄×C₄⋊C₄	1	2	yes	—
117	C₄×C₁₃₇	C₂×#108	1	8	yes	direct product factor
118	C₄×C₁₃₇	C₂²×C₄²	1	1	yes	characteristic
119	C₄×C₁₃₇	C₂×#117	1	1	yes	characteristic
120	C₄×C₁₃₇	C₂×C₄×D₄	1	2	yes	—
121	C₂×C₁₃₇	—	2	2	no	—
122	C₂×C₁₃₇	—	2	2	no	—
123	C₂×C₁₃₇	C₄×#108	1	4	yes	direct product factor
124	C₂×C₁₃₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₁₃₇	C₂²×#108	1	2	yes	—
126	C₂×C₄×#108	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₁₃₇	C₂×C₄×#108	1	1	yes	characteristic, direct product factor, Sylow
128	C₂²×#108	C₂×C₁₃₇	1	2	yes	—
129	C₂×C₄×M₁₆	C₂×C₁₃₇	1	2	yes	—
130	C₂²×C₄×C₈	C₂×C₁₃₇	1	1	yes	characteristic
131	C₄×#108	C₂×C₁₃₇	1	8	yes	direct product factor
132	C₂×#341	C₂×C₁₃₇	1	2	yes	—
133	C₂²×M₁₆	C₂²×C₁₃₇	1	4	yes	—
134	C₂³×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
135	C₂³×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
136	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
137	C₂×#108	C₄×C₁₃₇	1	16	yes	direct product factor
138	C₂×#108	C₂²×C₁₃₇	1	8	yes	—
139	C₄×M₁₆	—	2	4	no	—
140	C₄×M₁₆	C₂²×C₁₃₇	1	8	yes	—
141	C₂×#97	—	2	2	no	—
142	C₂×C₄×C₈	—	2	2	no	—
143	C₂×C₄×C₈	C₄×C₁₃₇	1	4	yes	—
144	C₂×C₄×C₈	C₄×C₁₃₇	1	4	yes	—
145	C₂×C₄×C₈	C₂²×C₁₃₇	1	4	yes	—
146	#341	C₂²×C₁₃₇	1	8	yes	—
147	C₂³×C₄	Q₈×C₁₃₇	1	2	yes	—
148	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
149	C₂×M₁₆	—	2	4	no	—
150	C₂×M₁₆	—	2	8	no	—
151	C₂×M₁₆	C₂×C₄×C₁₃₇	1	16	yes	—
152	C₂×M₁₆	C₂³×C₁₃₇	1	8	yes	—
153	C₂²×C₈	—	2	4	no	—
154	C₂²×C₈	—	2	4	no	—
155	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
156	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
157	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
158	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
159	C₂²×C₈	C₂×C₄×C₁₃₇	1	8	yes	—
160	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
161	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
162	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
163	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
164	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
165	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
166	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
167	C₂×C₄²	Q₈×C₁₃₇	1	2	yes	—
168	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
169	#108	C₂×C₄×C₁₃₇	1	32	yes	direct product factor
170	#97	—	2	8	no	—
171	C₄×C₈	—	2	8	no	—
172	C₄×C₈	C₂×C₄×C₁₃₇	1	16	yes	—
173	C₂⁴	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
174	C₂²×C₄	—	2	8	no	—
175	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	2	yes	—
176	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	4	yes	—
177	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	2	yes	—
178	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
179	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
180	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
181	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
183	C₂²×C₄	C₂×Q₈×C₁₃₇	1	4	yes	—
184	C₂²×C₄	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
185	C₂²×C₄	#41×C₁₃₇	1	4	yes	—
186	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
187	M₁₆	—	2	16	no	—
188	M₁₆	—	2	8	no	—
189	M₁₆	C₂²×C₄×C₁₃₇	1	16	yes	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	2	16	no	—
193	C₂×C₈	—	2	8	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	C₄²×C₁₃₇	1	8	yes	—
196	C₂×C₈	C₄²×C₁₃₇	1	8	yes	—
197	C₂×C₈	C₂²×C₄×C₁₃₇	1	4	yes	—
198	C₂×C₈	C₂²×C₄×C₁₃₇	1	4	yes	—
199	C₂×C₈	C₂²×C₄×C₁₃₇	1	8	yes	—
200	C₄²	C₄⋊C₄×C₁₃₇	1	2	yes	—
201	C₄²	C₄⋊C₄×C₁₃₇	1	4	yes	—
202	C₄²	C₄⋊C₄×C₁₃₇	1	2	yes	—
203	C₄²	C₂²×C₄×C₁₃₇	1	4	yes	—
204	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
205	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
206	C₂³	—	2	4	no	—
207	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
208	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
209	C₂³	C₄×Q₈×C₁₃₇	1	4	yes	—
210	C₂³	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	2	8	no	—
213	C₂×C₄	—	2	8	no	—
214	C₂×C₄	#108×C₁₃₇	1	8	yes	direct product factor
215	C₂×C₄	C₂×C₄²×C₁₃₇	1	2	yes	—
216	C₂×C₄	C₂×C₄²×C₁₃₇	1	2	yes	—
217	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
218	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
219	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
220	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
221	C₂×C₄	#117×C₁₃₇	1	2	yes	—
222	C₂×C₄	#117×C₁₃₇	1	2	yes	—
223	C₂×C₄	C₄×D₄×C₁₃₇	1	4	yes	—
224	C₂×C₄	C₄×D₄×C₁₃₇	1	4	yes	—
225	C₂×C₄	C₄×Q₈×C₁₃₇	1	4	yes	—
226	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
227	C₂×C₄	C₂³×C₄×C₁₃₇	1	2	yes	—
228	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
229	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
230	C₈	—	2	4	no	—
231	C₈	—	2	16	no	—
232	C₈	C₂×C₄²×C₁₃₇	1	8	yes	—
233	C₂²	—	2	8	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	C₄×C₄⋊C₄×C₁₃₇	1	2	yes	—
237	C₂²	C₄×C₄⋊C₄×C₁₃₇	1	2	yes	—
238	C₂²	C₂×#108×C₁₃₇	1	4	yes	—
239	C₂²	C₂²×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
240	C₂²	C₂×C₄×Q₈×C₁₃₇	1	2	yes	—
241	C₄	—	2	4	no	—
242	C₄	C₄×C₄⋊C₄×C₁₃₇	1	2	yes	—
243	C₄	C₄×C₄⋊C₄×C₁₃₇	1	2	yes	—
244	C₄	C₂×#108×C₁₃₇	1	8	yes	direct product factor
245	C₄	C₂²×C₄²×C₁₃₇	1	1	yes	characteristic
246	C₄	C₂×#117×C₁₃₇	1	1	yes	characteristic
247	C₄	C₂×C₄×D₄×C₁₃₇	1	2	yes	—
248	C₂	—	2	2	no	—
249	C₂	—	2	2	no	—
250	C₂	C₄×#108×C₁₃₇	1	4	yes	direct product factor
251	C₂	C₂×C₄×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
252	C₂	C₂²×#108×C₁₃₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²×C₁₃₇	D₄
commutator	C₄	C₂²×C₄²×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
socle	C₂³×C₁₃₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃₇	C₂×Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₁₃₇ ◁ C₂²×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄×#108×C₁₃₇	C₂	direct (N × H)
10. C₂×#108×C₁₃₇	C₄	direct (N × H)
42. #108×C₁₃₇	C₂×C₄	direct (N × H)
87. C₂×C₄×C₁₃₇	#108	direct (N × H)
117. C₄×C₁₃₇	C₂×#108	direct (N × H)
123. C₂×C₁₃₇	C₄×#108	direct (N × H)
126. C₂×C₄×#108	C₁₃₇	direct (N × H)
127. C₁₃₇	C₂×C₄×#108	direct (N × H)
131. C₄×#108	C₂×C₁₃₇	direct (N × H)
137. C₂×#108	C₄×C₁₃₇	direct (N × H)
169. #108	C₂×C₄×C₁₃₇	direct (N × H)
214. C₂×C₄	#108×C₁₃₇	direct (N × H)
244. C₄	C₂×#108×C₁₃₇	direct (N × H)
250. C₂	C₄×#108×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×#108×C₁₃₇	2
C₂×C₄×M₁₆×C₁₃₇	2
C₂²×C₄×C₈×C₁₃₇	2
C₄×#108×C₁₃₇	2
C₂×#341×C₁₃₇	2
C₂×C₄×#108	137

Subgroups — C2×C4×#108×C137 #46869842

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#108×C₁₃₇ #46869842