Subgroups — C₂×#2608×C₁₃₇ #46884293

Order 35072.

2270 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2608×C₁₃₇	C₂	1	16	yes	maximal, direct product factor
2	C₂×#396×C₁₃₇	C₂	1	4	yes	maximal
3	C₂×#389×C₁₃₇	C₂	1	4	yes	maximal, complemented
4	C₂×#387×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	C₂×#381×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₂²⋊C₄×C₁₃₇	C₂	1	2	yes	maximal
7	C₂⁴×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊C₄×C₁₃₇	C₂²	1	4	yes	—
9	C₂²×C₄⋊C₄×C₁₃₇	C₂²	1	4	yes	—
10	C₂²×C₄⋊C₄×C₁₃₇	C₂²	1	4	yes	—
11	C₂²×C₂²⋊C₄×C₁₃₇	C₂²	1	4	yes	—
12	C₂²×C₂²⋊C₄×C₁₃₇	C₂²	1	2	yes	—
13	C₂²×C₄²×C₁₃₇	C₂²	1	2	yes	—
14	C₂²×C₄²×C₁₃₇	C₂²	1	2	yes	—
15	#396×C₁₃₇	C₂²	1	32	yes	—
16	#389×C₁₃₇	C₂²	1	32	yes	complemented
17	#387×C₁₃₇	C₂²	1	32	yes	complemented
18	#381×C₁₃₇	C₂²	1	8	yes	—
19	C₄×C₂²⋊C₄×C₁₃₇	C₂²	1	16	yes	—
20	C₂×#95×C₁₃₇	C₂²	1	4	yes	—
21	C₂×#95×C₁₃₇	C₂²	1	4	yes	—
22	C₂×#95×C₁₃₇	C₂²	1	4	yes	—
23	C₂⁵×C₁₃₇	C₂³	1	1	yes	characteristic
24	C₂³×C₄×C₁₃₇	—	4	4	no	—
25	C₂³×C₄×C₁₃₇	Q₈	1	8	yes	—
26	C₂³×C₄×C₁₃₇	C₂³	1	4	yes	—
27	C₂³×C₄×C₁₃₇	C₂³	1	4	yes	—
28	C₂³×C₄×C₁₃₇	C₂³	1	2	yes	—
29	C₂³×C₄×C₁₃₇	C₂³	1	4	yes	—
30	C₂³×C₄×C₁₃₇	C₂³	1	4	yes	—
31	C₂×C₄⋊C₄×C₁₃₇	—	2	16	no	—
32	C₂×C₄⋊C₄×C₁₃₇	—	2	16	no	—
33	C₂×C₄⋊C₄×C₁₃₇	—	4	8	no	—
34	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	16	yes	—
35	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	16	yes	—
36	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	16	yes	—
37	C₂×C₂²⋊C₄×C₁₃₇	—	2	16	no	—
38	C₂×C₂²⋊C₄×C₁₃₇	—	4	4	no	—
39	C₂×C₂²⋊C₄×C₁₃₇	C₂³	1	16	yes	—
40	C₂×C₂²⋊C₄×C₁₃₇	C₂³	1	8	yes	—
41	C₂×C₄²×C₁₃₇	—	4	4	no	—
42	C₂×C₄²×C₁₃₇	—	4	4	no	—
43	C₂×C₄²×C₁₃₇	C₂³	1	8	yes	—
44	C₂×C₄²×C₁₃₇	C₂³	1	8	yes	—
45	#95×C₁₃₇	C₂³	1	16	yes	—
46	#95×C₁₃₇	C₂³	1	16	yes	—
47	#95×C₁₃₇	C₂³	1	16	yes	—
48	C₂⁴×C₁₃₇	—	2	8	no	—
49	C₂⁴×C₁₃₇	—	2	4	no	—
50	C₂⁴×C₁₃₇	C₂×Q₈	1	4	yes	—
51	C₂⁴×C₁₃₇	C₂⁴	1	2	yes	—
52	C₂⁴×C₁₃₇	C₂⁴	1	1	yes	characteristic
53	C₂²×C₄×C₁₃₇	—	2	8	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	—	2	8	no	—
58	C₂²×C₄×C₁₃₇	—	2	8	no	—
59	C₂²×C₄×C₁₃₇	—	2	8	no	—
60	C₂²×C₄×C₁₃₇	—	2	8	no	—
61	C₂²×C₄×C₁₃₇	—	2	8	no	—
62	C₂²×C₄×C₁₃₇	—	2	8	no	—
63	C₂²×C₄×C₁₃₇	—	4	16	no	—
64	C₂²×C₄×C₁₃₇	—	4	8	no	—
65	C₂²×C₄×C₁₃₇	C₂×Q₈	1	8	yes	—
66	C₂²×C₄×C₁₃₇	C₂×Q₈	1	16	yes	—
67	C₂²×C₄×C₁₃₇	#41	1	16	yes	—
68	C₂²×C₄×C₁₃₇	C₂⁴	1	8	yes	—
69	C₂²×C₄×C₁₃₇	C₂⁴	1	8	yes	—
70	C₂²×C₄×C₁₃₇	C₂⁴	1	4	yes	—
71	C₂²×C₄×C₁₃₇	C₂⁴	1	8	yes	—
72	C₄⋊C₄×C₁₃₇	—	2	32	no	—
73	C₄⋊C₄×C₁₃₇	—	2	32	no	—
74	C₄⋊C₄×C₁₃₇	—	4	16	no	—
75	C₂²⋊C₄×C₁₃₇	—	2	32	no	—
76	C₂²⋊C₄×C₁₃₇	—	4	8	no	—
77	C₄²×C₁₃₇	—	4	8	no	—
78	C₄²×C₁₃₇	—	4	8	no	—
79	C₂³×C₁₃₇	—	2	16	no	—
80	C₂³×C₁₃₇	—	2	8	no	—
81	C₂³×C₁₃₇	—	2	8	no	—
82	C₂³×C₁₃₇	—	2	4	no	—
83	C₂³×C₁₃₇	—	4	16	no	—
84	C₂³×C₁₃₇	C₂²×Q₈	1	4	yes	—
85	C₂³×C₁₃₇	C₂²×Q₈	1	2	yes	—
86	C₂³×C₁₃₇	C₂×#41	1	8	yes	—
87	C₂³×C₁₃₇	H(5,𝔽₂)	1	4	yes	—
88	C₂³×C₁₃₇	C₂⁵	1	1	yes	characteristic
89	C₂×C₄×C₁₃₇	—	2	16	no	—
90	C₂×C₄×C₁₃₇	—	2	32	no	—
91	C₂×C₄×C₁₃₇	—	2	8	no	—
92	C₂×C₄×C₁₃₇	—	2	8	no	—
93	C₂×C₄×C₁₃₇	—	2	4	no	—
94	C₂×C₄×C₁₃₇	—	2	4	no	—
95	C₂×C₄×C₁₃₇	—	2	8	no	—
96	C₂×C₄×C₁₃₇	—	2	8	no	—
97	C₂×C₄×C₁₃₇	—	2	8	no	—
98	C₂×C₄×C₁₃₇	—	2	8	no	—
99	C₂×C₄×C₁₃₇	—	2	8	no	—
100	C₂×C₄×C₁₃₇	—	2	8	no	—
101	C₂×C₄×C₁₃₇	—	4	16	no	—
102	C₂×C₄×C₁₃₇	—	4	16	no	—
103	C₂×C₄×C₁₃₇	—	4	16	no	—
104	C₂×C₄×C₁₃₇	C₂²×Q₈	1	8	yes	—
105	C₂×C₄×C₁₃₇	C₂×#41	1	16	yes	—
106	C₂²×C₁₃₇	—	2	8	no	—
107	C₂²×C₁₃₇	—	2	4	no	—
108	C₂²×C₁₃₇	—	4	16	no	—
109	C₂²×C₁₃₇	—	4	8	no	—
110	C₂²×C₁₃₇	#533	1	8	yes	—
111	C₂²×C₁₃₇	#543	1	4	yes	—
112	C₂²×C₁₃₇	#553	1	16	yes	—
113	C₂²×C₁₃₇	C₂³×Q₈	1	1	yes	characteristic
114	C₂²×C₁₃₇	C₂²×#41	1	4	yes	—
115	C₂²×C₁₃₇	C₂×H(5,𝔽₂)	1	2	yes	—
116	C₄×C₁₃₇	—	2	8	no	—
117	C₄×C₁₃₇	—	2	16	no	—
118	C₄×C₁₃₇	—	4	8	no	—
119	C₄×C₁₃₇	—	4	8	no	—
120	C₂×C₁₃₇	—	4	4	no	—
121	C₂×C₁₃₇	#2608	1	8	yes	direct product factor
122	C₂×C₁₃₇	C₂×#533	1	2	yes	—
123	C₂×C₁₃₇	C₂×#543	1	1	yes	characteristic
124	C₂×C₁₃₇	C₂×#553	1	4	yes	—
125	C₂×#2608	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₁₃₇	C₂×#2608	1	1	yes	characteristic, direct product factor, Sylow
127	#2608	C₂×C₁₃₇	1	16	yes	direct product factor
128	C₂×#396	C₂×C₁₃₇	1	4	yes	—
129	C₂×#389	C₂×C₁₃₇	1	4	yes	complemented
130	C₂×#387	C₂×C₁₃₇	1	4	yes	complemented
131	C₂×#381	C₂×C₁₃₇	1	1	yes	characteristic
132	C₂×C₄×C₂²⋊C₄	C₂×C₁₃₇	1	2	yes	—
133	C₂⁴×C₄	C₂²×C₁₃₇	1	1	yes	characteristic
134	C₂²×C₄⋊C₄	C₂²×C₁₃₇	1	4	yes	—
135	C₂²×C₄⋊C₄	C₂²×C₁₃₇	1	4	yes	—
136	C₂²×C₄⋊C₄	C₂²×C₁₃₇	1	4	yes	—
137	C₂²×C₂²⋊C₄	C₂²×C₁₃₇	1	4	yes	—
138	C₂²×C₂²⋊C₄	C₂²×C₁₃₇	1	2	yes	—
139	C₂²×C₄²	C₂²×C₁₃₇	1	2	yes	—
140	C₂²×C₄²	C₂²×C₁₃₇	1	2	yes	—
141	#396	C₂²×C₁₃₇	1	32	yes	—
142	#389	C₂²×C₁₃₇	1	32	yes	complemented
143	#387	C₂²×C₁₃₇	1	32	yes	complemented
144	#381	C₂²×C₁₃₇	1	8	yes	—
145	C₄×C₂²⋊C₄	C₂²×C₁₃₇	1	16	yes	—
146	C₂×#95	C₂²×C₁₃₇	1	4	yes	—
147	C₂×#95	C₂²×C₁₃₇	1	4	yes	—
148	C₂×#95	C₂²×C₁₃₇	1	4	yes	—
149	C₂⁵	C₂³×C₁₃₇	1	1	yes	characteristic
150	C₂³×C₄	—	4	4	no	—
151	C₂³×C₄	Q₈×C₁₃₇	1	8	yes	—
152	C₂³×C₄	C₂³×C₁₃₇	1	4	yes	—
153	C₂³×C₄	C₂³×C₁₃₇	1	4	yes	—
154	C₂³×C₄	C₂³×C₁₃₇	1	2	yes	—
155	C₂³×C₄	C₂³×C₁₃₇	1	4	yes	—
156	C₂³×C₄	C₂³×C₁₃₇	1	4	yes	—
157	C₂×C₄⋊C₄	—	2	16	no	—
158	C₂×C₄⋊C₄	—	2	16	no	—
159	C₂×C₄⋊C₄	—	4	8	no	—
160	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	16	yes	—
161	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	16	yes	—
162	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	16	yes	—
163	C₂×C₂²⋊C₄	—	2	16	no	—
164	C₂×C₂²⋊C₄	—	4	4	no	—
165	C₂×C₂²⋊C₄	C₂³×C₁₃₇	1	16	yes	—
166	C₂×C₂²⋊C₄	C₂³×C₁₃₇	1	8	yes	—
167	C₂×C₄²	—	4	4	no	—
168	C₂×C₄²	—	4	4	no	—
169	C₂×C₄²	C₂³×C₁₃₇	1	8	yes	—
170	C₂×C₄²	C₂³×C₁₃₇	1	8	yes	—
171	#95	C₂³×C₁₃₇	1	16	yes	—
172	#95	C₂³×C₁₃₇	1	16	yes	—
173	#95	C₂³×C₁₃₇	1	16	yes	—
174	C₂⁴	—	2	8	no	—
175	C₂⁴	—	2	4	no	—
176	C₂⁴	C₂×Q₈×C₁₃₇	1	4	yes	—
177	C₂⁴	C₂⁴×C₁₃₇	1	2	yes	—
178	C₂⁴	C₂⁴×C₁₃₇	1	1	yes	characteristic
179	C₂²×C₄	—	2	8	no	—
180	C₂²×C₄	—	2	8	no	—
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	8	no	—
184	C₂²×C₄	—	2	8	no	—
185	C₂²×C₄	—	2	8	no	—
186	C₂²×C₄	—	2	8	no	—
187	C₂²×C₄	—	2	8	no	—
188	C₂²×C₄	—	2	8	no	—
189	C₂²×C₄	—	4	16	no	—
190	C₂²×C₄	—	4	8	no	—
191	C₂²×C₄	C₂×Q₈×C₁₃₇	1	8	yes	—
192	C₂²×C₄	C₂×Q₈×C₁₃₇	1	16	yes	—
193	C₂²×C₄	#41×C₁₃₇	1	16	yes	—
194	C₂²×C₄	C₂⁴×C₁₃₇	1	8	yes	—
195	C₂²×C₄	C₂⁴×C₁₃₇	1	8	yes	—
196	C₂²×C₄	C₂⁴×C₁₃₇	1	4	yes	—
197	C₂²×C₄	C₂⁴×C₁₃₇	1	8	yes	—
198	C₄⋊C₄	—	2	32	no	—
199	C₄⋊C₄	—	2	32	no	—
200	C₄⋊C₄	—	4	16	no	—
201	C₂²⋊C₄	—	2	32	no	—
202	C₂²⋊C₄	—	4	8	no	—
203	C₄²	—	4	8	no	—
204	C₄²	—	4	8	no	—
205	C₂³	—	2	16	no	—
206	C₂³	—	2	8	no	—
207	C₂³	—	2	8	no	—
208	C₂³	—	2	4	no	—
209	C₂³	—	4	16	no	—
210	C₂³	C₂²×Q₈×C₁₃₇	1	4	yes	—
211	C₂³	C₂²×Q₈×C₁₃₇	1	2	yes	—
212	C₂³	C₂×#41×C₁₃₇	1	8	yes	—
213	C₂³	H(5,𝔽₂)×C₁₃₇	1	4	yes	—
214	C₂³	C₂⁵×C₁₃₇	1	1	yes	characteristic
215	C₂×C₄	—	2	16	no	—
216	C₂×C₄	—	2	32	no	—
217	C₂×C₄	—	2	8	no	—
218	C₂×C₄	—	2	8	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	—	4	16	no	—
228	C₂×C₄	—	4	16	no	—
229	C₂×C₄	—	4	16	no	—
230	C₂×C₄	C₂²×Q₈×C₁₃₇	1	8	yes	—
231	C₂×C₄	C₂×#41×C₁₃₇	1	16	yes	—
232	C₂²	—	2	8	no	—
233	C₂²	—	2	4	no	—
234	C₂²	—	4	16	no	—
235	C₂²	—	4	8	no	—
236	C₂²	#533×C₁₃₇	1	8	yes	—
237	C₂²	#543×C₁₃₇	1	4	yes	—
238	C₂²	#553×C₁₃₇	1	16	yes	—
239	C₂²	C₂³×Q₈×C₁₃₇	1	1	yes	characteristic
240	C₂²	C₂²×#41×C₁₃₇	1	4	yes	—
241	C₂²	C₂×H(5,𝔽₂)×C₁₃₇	1	2	yes	—
242	C₄	—	2	8	no	—
243	C₄	—	2	16	no	—
244	C₄	—	4	8	no	—
245	C₄	—	4	8	no	—
246	C₂	—	4	4	no	—
247	C₂	#2608×C₁₃₇	1	8	yes	direct product factor
248	C₂	C₂×#533×C₁₃₇	1	2	yes	—
249	C₂	C₂×#543×C₁₃₇	1	1	yes	characteristic
250	C₂	C₂×#553×C₁₃₇	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
1. #2608×C₁₃₇	C₂	direct (N × H)
3. C₂×#389×C₁₃₇	C₂	semidirect
4. C₂×#387×C₁₃₇	C₂	semidirect
16. #389×C₁₃₇	C₂²	semidirect
17. #387×C₁₃₇	C₂²	semidirect
121. C₂×C₁₃₇	#2608	direct (N × H)
125. C₂×#2608	C₁₃₇	direct (N × H)
126. C₁₃₇	C₂×#2608	direct (N × H)
127. #2608	C₂×C₁₃₇	direct (N × H)
129. C₂×#389	C₂×C₁₃₇	semidirect
130. C₂×#387	C₂×C₁₃₇	semidirect
142. #389	C₂²×C₁₃₇	semidirect
143. #387	C₂²×C₁₃₇	semidirect
247. C₂	#2608×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
#2608×C₁₃₇	2
C₂×#396×C₁₃₇	2
C₂×#389×C₁₃₇	2
C₂×#387×C₁₃₇	2
C₂×#381×C₁₃₇	2
C₂×C₄×C₂²⋊C₄×C₁₃₇	2
C₂×#2608	137

Subgroups — C2×#2608×C137 #46884293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#2608×C₁₃₇ #46884293