Subgroups — #2664×C₁₄₉ #34041577

Order 19072.

434 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#126×C₁₄₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#125×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#400×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#399×C₁₄₉	C₂	1	1	yes	maximal, characteristic
5	#398×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#398×C₁₄₉	C₂	1	2	yes	maximal, complemented
7	#397×C₁₄₉	C₂	1	1	yes	maximal, characteristic
8	#396×C₁₄₉	C₂	1	2	yes	maximal
9	#389×C₁₄₉	C₂	1	2	yes	maximal, complemented
10	#388×C₁₄₉	C₂	1	2	yes	maximal
11	C₄×C₄⋊C₄×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#126×C₁₄₉	—	2	4	no	—
13	#125×C₁₄₉	—	2	4	no	—
14	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₁₄₉	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₄×C₁₄₉	C₂²	1	2	yes	—
29	C₂×C₄²×C₁₄₉	C₂²	1	2	yes	—
30	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
31	#95×C₁₄₉	C₂²	1	1	yes	characteristic
32	#95×C₁₄₉	C₂²	1	1	yes	characteristic
33	#95×C₁₄₉	C₂²	1	2	yes	—
34	#95×C₁₄₉	C₂²	1	2	yes	—
35	#95×C₁₄₉	C₂²	1	2	yes	—
36	#95×C₁₄₉	C₂²	1	2	yes	—
37	C₂⁴×C₁₄₉	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₁₄₉	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₁₄₉	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₄₉	C₂³	1	2	yes	—
44	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₁₄₉	C₂³	1	2	yes	—
48	C₄⋊C₄×C₁₄₉	—	2	4	no	—
49	C₄⋊C₄×C₁₄₉	—	2	4	no	—
50	C₄⋊C₄×C₁₄₉	—	2	2	no	—
51	C₄⋊C₄×C₁₄₉	—	2	2	no	—
52	C₄⋊C₄×C₁₄₉	—	2	2	no	—
53	C₄⋊C₄×C₁₄₉	—	2	2	no	—
54	C₄⋊C₄×C₁₄₉	—	2	2	no	—
55	C₄⋊C₄×C₁₄₉	—	2	4	no	—
56	C₄⋊C₄×C₁₄₉	—	2	4	no	—
57	C₄⋊C₄×C₁₄₉	—	4	2	no	—
58	C₂²⋊C₄×C₁₄₉	—	2	2	no	—
59	C₂²⋊C₄×C₁₄₉	—	2	2	no	—
60	C₂²⋊C₄×C₁₄₉	—	2	2	no	—
61	C₂²⋊C₄×C₁₄₉	—	2	4	no	—
62	C₂²⋊C₄×C₁₄₉	—	4	2	no	—
63	C₄²×C₁₄₉	—	4	2	no	—
64	C₄²×C₁₄₉	D₄	1	4	yes	—
65	C₂³×C₁₄₉	—	2	1	no	—
66	C₂³×C₁₄₉	—	2	2	no	—
67	C₂³×C₁₄₉	—	2	1	no	—
68	C₂³×C₁₄₉	—	2	2	no	—
69	C₂³×C₁₄₉	—	2	1	no	—
70	C₂³×C₁₄₉	C₂⁴	1	1	yes	characteristic
71	C₂×C₄×C₁₄₉	—	2	2	no	—
72	C₂×C₄×C₁₄₉	—	2	1	no	—
73	C₂×C₄×C₁₄₉	—	2	2	no	—
74	C₂×C₄×C₁₄₉	—	2	2	no	—
75	C₂×C₄×C₁₄₉	—	2	1	no	—
76	C₂×C₄×C₁₄₉	—	2	2	no	—
77	C₂×C₄×C₁₄₉	—	2	1	no	—
78	C₂×C₄×C₁₄₉	—	2	1	no	—
79	C₂×C₄×C₁₄₉	—	2	1	no	—
80	C₂×C₄×C₁₄₉	—	2	2	no	—
81	C₂×C₄×C₁₄₉	—	2	2	no	—
82	C₂×C₄×C₁₄₉	—	2	2	no	—
83	C₂×C₄×C₁₄₉	—	2	2	no	—
84	C₂×C₄×C₁₄₉	—	2	2	no	—
85	C₂×C₄×C₁₄₉	—	2	2	no	—
86	C₂×C₄×C₁₄₉	—	2	2	no	—
87	C₂×C₄×C₁₄₉	—	2	2	no	—
88	C₂×C₄×C₁₄₉	—	2	1	no	—
89	C₂×C₄×C₁₄₉	—	2	2	no	—
90	C₂×C₄×C₁₄₉	—	2	1	no	—
91	C₂×C₄×C₁₄₉	—	2	2	no	—
92	C₂×C₄×C₁₄₉	—	2	2	no	—
93	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₁₄₉	#41	1	4	yes	—
97	C₂²×C₁₄₉	—	4	1	no	—
98	C₂²×C₁₄₉	—	4	1	no	—
99	C₂²×C₁₄₉	—	4	1	no	—
100	C₂²×C₁₄₉	—	4	2	no	—
101	C₂²×C₁₄₉	—	4	2	no	—
102	C₂²×C₁₄₉	C₂²×D₄	1	1	yes	characteristic
103	C₂²×C₁₄₉	C₂×#41	1	2	yes	—
104	C₂²×C₁₄₉	H(5,𝔽₂)	1	1	yes	characteristic
105	C₂²×C₁₄₉	2₋¹⁺⁴	1	2	yes	—
106	C₂²×C₁₄₉	2₋¹⁺⁴	1	1	yes	characteristic
107	C₄×C₁₄₉	—	2	2	no	—
108	C₄×C₁₄₉	—	2	2	no	—
109	C₄×C₁₄₉	—	2	2	no	—
110	C₄×C₁₄₉	—	2	4	no	—
111	C₄×C₁₄₉	—	4	1	no	—
112	C₄×C₁₄₉	—	4	2	no	—
113	C₄×C₁₄₉	—	4	1	no	—
114	C₄×C₁₄₉	—	4	1	no	—
115	C₄×C₁₄₉	—	4	2	no	—
116	C₄×C₁₄₉	—	4	2	no	—
117	C₂×C₁₄₉	—	8	1	no	—
118	C₂×C₁₄₉	#532	1	1	yes	characteristic
119	C₂×C₁₄₉	#536	1	1	yes	characteristic
120	C₂×C₁₄₉	#537	1	1	yes	characteristic
121	C₂×C₁₄₉	#539	1	2	yes	—
122	C₂×C₁₄₉	#541	1	2	yes	—
123	C₁₄₉	#2664	1	1	yes	characteristic, direct product factor, Sylow
124	#2664	C₁₄₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂×#126	C₂×C₁₄₉	1	1	yes	characteristic
126	C₂×#125	C₂×C₁₄₉	1	1	yes	characteristic, complemented
127	#400	C₂×C₁₄₉	1	1	yes	characteristic, complemented
128	#399	C₂×C₁₄₉	1	1	yes	characteristic
129	#398	C₂×C₁₄₉	1	1	yes	characteristic, complemented
130	#398	C₂×C₁₄₉	1	2	yes	complemented
131	#397	C₂×C₁₄₉	1	1	yes	characteristic
132	#396	C₂×C₁₄₉	1	2	yes	—
133	#389	C₂×C₁₄₉	1	2	yes	complemented
134	#388	C₂×C₁₄₉	1	2	yes	—
135	C₄×C₄⋊C₄	C₂×C₁₄₉	1	1	yes	characteristic, complemented
136	#126	—	2	4	no	—
137	#125	—	2	4	no	—
138	C₂×C₄⋊C₄	C₂²×C₁₄₉	1	2	yes	—
139	C₂×C₄⋊C₄	C₂²×C₁₄₉	1	2	yes	—
140	C₂×C₄⋊C₄	C₂²×C₁₄₉	1	1	yes	characteristic
141	C₂×C₄⋊C₄	C₂²×C₁₄₉	1	1	yes	characteristic
142	C₂×C₄⋊C₄	C₂²×C₁₄₉	1	1	yes	characteristic
143	C₂×C₄⋊C₄	C₂²×C₁₄₉	1	1	yes	characteristic
144	C₂×C₄⋊C₄	C₂²×C₁₄₉	1	1	yes	characteristic
145	C₂×C₄⋊C₄	C₂²×C₁₄₉	1	2	yes	—
146	C₂×C₄⋊C₄	C₂²×C₁₄₉	1	2	yes	—
147	C₂×C₄⋊C₄	C₂²×C₁₄₉	1	2	yes	—
148	C₂×C₂²⋊C₄	C₂²×C₁₄₉	1	1	yes	characteristic
149	C₂×C₂²⋊C₄	C₂²×C₁₄₉	1	1	yes	characteristic
150	C₂×C₂²⋊C₄	C₂²×C₁₄₉	1	1	yes	characteristic
151	C₂×C₂²⋊C₄	C₂²×C₁₄₉	1	2	yes	—
152	C₂×C₂²⋊C₄	C₂²×C₁₄₉	1	2	yes	—
153	C₂×C₄²	C₂²×C₁₄₉	1	2	yes	—
154	C₂×C₄²	C₂²×C₁₄₉	1	1	yes	characteristic
155	#95	C₂²×C₁₄₉	1	1	yes	characteristic
156	#95	C₂²×C₁₄₉	1	1	yes	characteristic
157	#95	C₂²×C₁₄₉	1	2	yes	—
158	#95	C₂²×C₁₄₉	1	2	yes	—
159	#95	C₂²×C₁₄₉	1	2	yes	—
160	#95	C₂²×C₁₄₉	1	2	yes	—
161	C₂⁴	C₂³×C₁₄₉	1	1	yes	characteristic
162	C₂²×C₄	C₂³×C₁₄₉	1	2	yes	—
163	C₂²×C₄	C₂³×C₁₄₉	1	1	yes	characteristic
164	C₂²×C₄	C₂³×C₁₄₉	1	1	yes	characteristic
165	C₂²×C₄	C₂³×C₁₄₉	1	1	yes	characteristic
166	C₂²×C₄	C₂³×C₁₄₉	1	2	yes	—
167	C₂²×C₄	C₂³×C₁₄₉	1	2	yes	—
168	C₂²×C₄	C₂³×C₁₄₉	1	1	yes	characteristic
169	C₂²×C₄	C₂³×C₁₄₉	1	1	yes	characteristic
170	C₂²×C₄	C₂³×C₁₄₉	1	1	yes	characteristic
171	C₂²×C₄	C₂³×C₁₄₉	1	2	yes	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	2	4	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	2	4	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	—	2	4	no	—
186	C₂²⋊C₄	—	4	2	no	—
187	C₄²	—	4	2	no	—
188	C₄²	D₄×C₁₄₉	1	4	yes	—
189	C₂³	—	2	1	no	—
190	C₂³	—	2	2	no	—
191	C₂³	—	2	1	no	—
192	C₂³	—	2	2	no	—
193	C₂³	—	2	1	no	—
194	C₂³	C₂⁴×C₁₄₉	1	1	yes	characteristic
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	C₂×D₄×C₁₄₉	1	2	yes	—
218	C₂×C₄	C₂×D₄×C₁₄₉	1	2	yes	—
219	C₂×C₄	C₂×D₄×C₁₄₉	1	2	yes	—
220	C₂×C₄	#41×C₁₄₉	1	4	yes	—
221	C₂²	—	4	1	no	—
222	C₂²	—	4	1	no	—
223	C₂²	—	4	1	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	C₂²×D₄×C₁₄₉	1	1	yes	characteristic
227	C₂²	C₂×#41×C₁₄₉	1	2	yes	—
228	C₂²	H(5,𝔽₂)×C₁₄₉	1	1	yes	characteristic
229	C₂²	2₋¹⁺⁴×C₁₄₉	1	2	yes	—
230	C₂²	2₋¹⁺⁴×C₁₄₉	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	4	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	1	no	—
238	C₄	—	4	1	no	—
239	C₄	—	4	2	no	—
240	C₄	—	4	2	no	—
241	C₂	—	8	1	no	—
242	C₂	#532×C₁₄₉	1	1	yes	characteristic
243	C₂	#536×C₁₄₉	1	1	yes	characteristic
244	C₂	#537×C₁₄₉	1	1	yes	characteristic
245	C₂	#539×C₁₄₉	1	2	yes	—
246	C₂	#541×C₁₄₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₄₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₄₉
prime radical	C₂⁴×C₁₄₉	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₄₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#125×C₁₄₉	C₂	semidirect
3. #400×C₁₄₉	C₂	semidirect
5. #398×C₁₄₉	C₂	semidirect
6. #398×C₁₄₉	C₂	semidirect
9. #389×C₁₄₉	C₂	semidirect
11. C₄×C₄⋊C₄×C₁₄₉	C₂	semidirect
123. C₁₄₉	#2664	direct (N × H)
124. #2664	C₁₄₉	direct (N × H)
126. C₂×#125	C₂×C₁₄₉	semidirect
127. #400	C₂×C₁₄₉	semidirect
129. #398	C₂×C₁₄₉	semidirect
130. #398	C₂×C₁₄₉	semidirect
133. #389	C₂×C₁₄₉	semidirect
135. C₄×C₄⋊C₄	C₂×C₁₄₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#126×C₁₄₉	2
C₂×#125×C₁₄₉	2
#400×C₁₄₉	2
#399×C₁₄₉	2
#398×C₁₄₉	2
#397×C₁₄₉	2
#396×C₁₄₉	2
#389×C₁₄₉	2
#388×C₁₄₉	2
C₄×C₄⋊C₄×C₁₄₉	2
#2664	149

Subgroups — #2664×C149 #34041577

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2664×C₁₄₉ #34041577