Subgroups — C₂×#1822×C₁₃₇ #46870079

Order 35072.

1430 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#117×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#407×C₁₃₇	C₂	1	2	yes	maximal
3	#1822×C₁₃₇	C₂	1	8	yes	maximal, direct product factor
4	C₂×#343×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
6	C₂²×M₁₆×C₁₃₇	C₂²	1	4	yes	—
7	C₂×#117×C₁₃₇	—	2	4	no	—
8	C₂×#117×C₁₃₇	—	2	4	no	—
9	C₂×#117×C₁₃₇	C₄	1	4	yes	—
10	C₂×#117×C₁₃₇	C₂²	1	4	yes	—
11	C₂²×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
12	C₂²×C₂²⋊C₄×C₁₃₇	C₄	1	2	yes	—
13	C₂²×C₄²×C₁₃₇	C₄	1	2	yes	—
14	#407×C₁₃₇	—	2	4	no	—
15	#407×C₁₃₇	C₂²	1	8	yes	—
16	C₂×C₂²⋊C₈×C₁₃₇	—	2	4	no	—
17	#343×C₁₃₇	C₂²	1	16	yes	complemented
18	C₂⁵×C₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₁₃₇	—	2	8	no	—
20	C₂³×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₁₃₇	C₂×C₄	1	4	yes	—
23	C₂³×C₄×C₁₃₇	D₄	1	4	yes	—
24	C₂³×C₄×C₁₃₇	D₄	1	4	yes	—
25	C₂³×C₄×C₁₃₇	C₂³	1	2	yes	—
26	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
27	C₂×M₁₆×C₁₃₇	—	2	4	no	—
28	C₂×M₁₆×C₁₃₇	—	2	16	no	—
29	C₂×M₁₆×C₁₃₇	C₂³	1	8	yes	—
30	C₂²×C₈×C₁₃₇	—	2	4	no	—
31	#117×C₁₃₇	—	2	16	no	—
32	#117×C₁₃₇	—	2	16	no	—
33	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
34	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
35	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
36	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
37	C₂×C₂²⋊C₄×C₁₃₇	—	2	4	no	—
38	C₂×C₂²⋊C₄×C₁₃₇	—	2	4	no	—
39	C₂×C₂²⋊C₄×C₁₃₇	C₂×C₄	1	8	yes	—
40	C₂×C₄²×C₁₃₇	—	2	4	no	—
41	C₂×C₄²×C₁₃₇	—	2	4	no	—
42	C₂×C₄²×C₁₃₇	C₂×C₄	1	8	yes	—
43	C₂²⋊C₈×C₁₃₇	—	2	16	no	—
44	C₂⁴×C₁₃₇	—	2	4	no	—
45	C₂⁴×C₁₃₇	—	4	4	no	—
46	C₂⁴×C₁₃₇	C₂²⋊C₄	1	2	yes	—
47	C₂⁴×C₁₃₇	C₂²⋊C₄	1	2	yes	—
48	C₂⁴×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
49	C₂⁴×C₁₃₇	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₁₃₇	—	2	4	no	—
51	C₂²×C₄×C₁₃₇	—	2	8	no	—
52	C₂²×C₄×C₁₃₇	—	2	16	no	—
53	C₂²×C₄×C₁₃₇	—	2	16	no	—
54	C₂²×C₄×C₁₃₇	—	2	8	no	—
55	C₂²×C₄×C₁₃₇	—	2	4	no	—
56	C₂²×C₄×C₁₃₇	—	2	4	no	—
57	C₂²×C₄×C₁₃₇	—	4	8	no	—
58	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	8	yes	—
60	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₁₃₇	C₂²×C₄	1	8	yes	—
63	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₇	C₂×D₄	1	8	yes	—
65	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₁₃₇	#41	1	8	yes	—
68	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
69	M₁₆×C₁₃₇	—	2	16	no	—
70	M₁₆×C₁₃₇	—	4	8	no	—
71	C₂×C₈×C₁₃₇	—	2	8	no	—
72	C₂×C₈×C₁₃₇	—	4	8	no	—
73	C₄⋊C₄×C₁₃₇	—	2	8	no	—
74	C₄⋊C₄×C₁₃₇	—	2	8	no	—
75	C₂²⋊C₄×C₁₃₇	—	2	16	no	—
76	C₄²×C₁₃₇	—	2	16	no	—
77	C₂³×C₁₃₇	—	2	4	no	—
78	C₂³×C₁₃₇	—	2	4	no	—
79	C₂³×C₁₃₇	—	2	4	no	—
80	C₂³×C₁₃₇	—	2	4	no	—
81	C₂³×C₁₃₇	—	2	8	no	—
82	C₂³×C₁₃₇	—	4	4	no	—
83	C₂³×C₁₃₇	—	4	4	no	—
84	C₂³×C₁₃₇	—	4	16	no	—
85	C₂³×C₁₃₇	C₂×C₂²⋊C₄	1	4	yes	—
86	C₂³×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂³×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂³×C₁₃₇	#117	1	4	yes	—
89	C₂³×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃₇	—	2	4	no	—
91	C₂×C₄×C₁₃₇	—	2	4	no	—
92	C₂×C₄×C₁₃₇	—	2	4	no	—
93	C₂×C₄×C₁₃₇	—	2	16	no	—
94	C₂×C₄×C₁₃₇	—	2	16	no	—
95	C₂×C₄×C₁₃₇	—	2	8	no	—
96	C₂×C₄×C₁₃₇	—	4	8	no	—
97	C₂×C₄×C₁₃₇	—	4	8	no	—
98	C₂×C₄×C₁₃₇	—	4	8	no	—
99	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	4	yes	—
100	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	2	yes	—
101	C₂×C₄×C₁₃₇	#117	1	4	yes	—
102	C₂×C₄×C₁₃₇	#123	1	8	yes	—
103	C₂×C₄×C₁₃₇	#123	1	8	yes	—
104	C₂×C₄×C₁₃₇	C₂²×D₄	1	2	yes	—
105	C₂×C₄×C₁₃₇	C₂×#41	1	4	yes	—
106	C₈×C₁₃₇	—	4	8	no	—
107	C₂²×C₁₃₇	—	2	8	no	—
108	C₂²×C₁₃₇	—	2	2	no	—
109	C₂²×C₁₃₇	—	2	2	no	—
110	C₂²×C₁₃₇	—	2	8	no	—
111	C₂²×C₁₃₇	—	2	4	no	—
112	C₂²×C₁₃₇	—	4	8	no	—
113	C₂²×C₁₃₇	—	4	8	no	—
114	C₂²×C₁₃₇	—	4	8	no	—
115	C₂²×C₁₃₇	#381	1	2	yes	—
116	C₂²×C₁₃₇	#381	1	2	yes	—
117	C₂²×C₁₃₇	#413	1	4	yes	—
118	C₂²×C₁₃₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
119	C₂²×C₁₃₇	C₂×#117	1	2	yes	—
120	C₄×C₁₃₇	—	2	4	no	—
121	C₄×C₁₃₇	—	4	8	no	—
122	C₄×C₁₃₇	#381	1	4	yes	—
123	C₄×C₁₃₇	C₂×#123	1	4	yes	—
124	C₂×C₁₃₇	—	2	2	no	—
125	C₂×C₁₃₇	—	2	2	no	—
126	C₂×C₁₃₇	—	4	4	no	—
127	C₂×C₁₃₇	#1822	1	4	yes	direct product factor
128	C₂×C₁₃₇	C₂×#381	1	1	yes	characteristic
129	C₂×C₁₃₇	C₂×#413	1	2	yes	—
130	C₂×#1822	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₁₃₇	C₂×#1822	1	1	yes	characteristic, direct product factor, Sylow
132	C₂²×#117	C₂×C₁₃₇	1	1	yes	characteristic
133	C₂×#407	C₂×C₁₃₇	1	2	yes	—
134	#1822	C₂×C₁₃₇	1	8	yes	direct product factor
135	C₂×#343	C₂×C₁₃₇	1	4	yes	complemented
136	C₂⁴×C₄	C₂²×C₁₃₇	1	1	yes	characteristic
137	C₂²×M₁₆	C₂²×C₁₃₇	1	4	yes	—
138	C₂×#117	—	2	4	no	—
139	C₂×#117	—	2	4	no	—
140	C₂×#117	C₄×C₁₃₇	1	4	yes	—
141	C₂×#117	C₂²×C₁₃₇	1	4	yes	—
142	C₂²×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
143	C₂²×C₂²⋊C₄	C₄×C₁₃₇	1	2	yes	—
144	C₂²×C₄²	C₄×C₁₃₇	1	2	yes	—
145	#407	—	2	4	no	—
146	#407	C₂²×C₁₃₇	1	8	yes	—
147	C₂×C₂²⋊C₈	—	2	4	no	—
148	#343	C₂²×C₁₃₇	1	16	yes	complemented
149	C₂⁵	C₂×C₄×C₁₃₇	1	1	yes	characteristic
150	C₂³×C₄	—	2	8	no	—
151	C₂³×C₄	C₂×C₄×C₁₃₇	1	2	yes	—
152	C₂³×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
153	C₂³×C₄	C₂×C₄×C₁₃₇	1	4	yes	—
154	C₂³×C₄	D₄×C₁₃₇	1	4	yes	—
155	C₂³×C₄	D₄×C₁₃₇	1	4	yes	—
156	C₂³×C₄	C₂³×C₁₃₇	1	2	yes	—
157	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
158	C₂×M₁₆	—	2	4	no	—
159	C₂×M₁₆	—	2	16	no	—
160	C₂×M₁₆	C₂³×C₁₃₇	1	8	yes	—
161	C₂²×C₈	—	2	4	no	—
162	#117	—	2	16	no	—
163	#117	—	2	16	no	—
164	C₂×C₄⋊C₄	—	2	4	no	—
165	C₂×C₄⋊C₄	—	2	4	no	—
166	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	4	yes	—
167	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
168	C₂×C₂²⋊C₄	—	2	4	no	—
169	C₂×C₂²⋊C₄	—	2	4	no	—
170	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₇	1	8	yes	—
171	C₂×C₄²	—	2	4	no	—
172	C₂×C₄²	—	2	4	no	—
173	C₂×C₄²	C₂×C₄×C₁₃₇	1	8	yes	—
174	C₂²⋊C₈	—	2	16	no	—
175	C₂⁴	—	2	4	no	—
176	C₂⁴	—	4	4	no	—
177	C₂⁴	C₂²⋊C₄×C₁₃₇	1	2	yes	—
178	C₂⁴	C₂²⋊C₄×C₁₃₇	1	2	yes	—
179	C₂⁴	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
180	C₂⁴	C₂²×C₄×C₁₃₇	1	2	yes	—
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	—	2	8	no	—
183	C₂²×C₄	—	2	16	no	—
184	C₂²×C₄	—	2	16	no	—
185	C₂²×C₄	—	2	8	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	4	8	no	—
189	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	2	yes	—
190	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	2	yes	—
191	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	8	yes	—
192	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
193	C₂²×C₄	C₂²×C₄×C₁₃₇	1	8	yes	—
194	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₄×C₁₃₇	1	8	yes	—
196	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
197	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
198	C₂²×C₄	#41×C₁₃₇	1	8	yes	—
199	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
200	M₁₆	—	2	16	no	—
201	M₁₆	—	4	8	no	—
202	C₂×C₈	—	2	8	no	—
203	C₂×C₈	—	4	8	no	—
204	C₄⋊C₄	—	2	8	no	—
205	C₄⋊C₄	—	2	8	no	—
206	C₂²⋊C₄	—	2	16	no	—
207	C₄²	—	2	16	no	—
208	C₂³	—	2	4	no	—
209	C₂³	—	2	4	no	—
210	C₂³	—	2	4	no	—
211	C₂³	—	2	8	no	—
212	C₂³	—	2	4	no	—
213	C₂³	—	4	4	no	—
214	C₂³	—	4	4	no	—
215	C₂³	—	4	16	no	—
216	C₂³	C₂×C₂²⋊C₄×C₁₃₇	1	4	yes	—
217	C₂³	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
218	C₂³	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
219	C₂³	#117×C₁₃₇	1	4	yes	—
220	C₂³	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	16	no	—
225	C₂×C₄	—	2	16	no	—
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	—	4	8	no	—
228	C₂×C₄	—	4	8	no	—
229	C₂×C₄	—	4	8	no	—
230	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	4	yes	—
231	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	2	yes	—
232	C₂×C₄	#117×C₁₃₇	1	4	yes	—
233	C₂×C₄	#123×C₁₃₇	1	8	yes	—
234	C₂×C₄	#123×C₁₃₇	1	8	yes	—
235	C₂×C₄	C₂²×D₄×C₁₃₇	1	2	yes	—
236	C₂×C₄	C₂×#41×C₁₃₇	1	4	yes	—
237	C₈	—	4	8	no	—
238	C₂²	—	2	8	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	2	8	no	—
243	C₂²	—	4	8	no	—
244	C₂²	—	4	8	no	—
245	C₂²	—	4	8	no	—
246	C₂²	#381×C₁₃₇	1	2	yes	—
247	C₂²	#381×C₁₃₇	1	2	yes	—
248	C₂²	#413×C₁₃₇	1	4	yes	—
249	C₂²	C₂²×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
250	C₂²	C₂×#117×C₁₃₇	1	2	yes	—
251	C₄	—	2	4	no	—
252	C₄	—	4	8	no	—
253	C₄	#381×C₁₃₇	1	4	yes	—
254	C₄	C₂×#123×C₁₃₇	1	4	yes	—
255	C₂	—	2	2	no	—
256	C₂	—	2	2	no	—
257	C₂	—	4	4	no	—
258	C₂	#1822×C₁₃₇	1	4	yes	direct product factor
259	C₂	C₂×#381×C₁₃₇	1	1	yes	characteristic
260	C₂	C₂×#413×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂⁵×C₁₃₇	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₂⁴×C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #1822×C₁₃₇	C₂	direct (N × H)
4. C₂×#343×C₁₃₇	C₂	semidirect
17. #343×C₁₃₇	C₂²	semidirect
127. C₂×C₁₃₇	#1822	direct (N × H)
130. C₂×#1822	C₁₃₇	direct (N × H)
131. C₁₃₇	C₂×#1822	direct (N × H)
134. #1822	C₂×C₁₃₇	direct (N × H)
135. C₂×#343	C₂×C₁₃₇	semidirect
148. #343	C₂²×C₁₃₇	semidirect
258. C₂	#1822×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×#117×C₁₃₇	2
C₂×#407×C₁₃₇	2
#1822×C₁₃₇	2
C₂×#343×C₁₃₇	2
C₂×#1822	137

Subgroups — C2×#1822×C137 #46870079

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1822×C₁₃₇ #46870079