Subgroups — C₂×#14271226 #16828162

Order 3712.

616 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#6597	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₈×D₂₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₂²⋊D₂₉	C₂	1	1	yes	maximal, characteristic
4	#14271226	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11745162	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11745168	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈×Dic₂₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₉×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₄×C₂₉	C₂²	1	1	yes	characteristic
10	C₂²×C₂²⋊D₂₉	C₄	1	1	yes	characteristic
11	C₂²×C₄×D₂₉	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₂₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₂₉	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊Dic₂₉	C₄	1	1	yes	characteristic
15	C₄×C₂²⋊D₂₉	C₄	1	4	yes	—
16	C₄×C₂²⋊D₂₉	C₂²	1	4	yes	—
17	C₂×#90781	C₄	1	1	yes	characteristic
18	C₂×#90779	C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
20	C₂²×#6597	—	2	4	no	—
21	C₂²×#6597	C₂²	1	1	yes	characteristic
22	C₂²×#6597	C₂²	1	1	yes	characteristic
23	C₂²×#6597	C₂²	1	4	yes	complemented
24	C₂×C₈×D₂₉	—	2	4	no	—
25	C₂×C₈×D₂₉	C₂²	1	4	yes	complemented
26	C₂₉×C₂²⋊C₈	C₂²	1	4	yes	complemented
27	#11745168	C₂²	1	4	yes	complemented
28	#11745162	C₂²	1	4	yes	complemented
29	C₈×Dic₂₉	C₂²	1	4	yes	complemented
30	C₂⁴×C₂₉	C₂×C₄	1	1	yes	characteristic
31	C₂³×D₂₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₉	—	2	2	no	—
33	C₂²×C₄×C₂₉	—	2	2	no	—
34	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₂₉	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
37	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
38	C₂×C₂²⋊D₂₉	C₈	1	8	yes	complemented
39	C₂×C₂²⋊D₂₉	C₂×C₄	1	4	yes	—
40	C₂²×Dic₂₉	C₂×C₄	1	1	yes	characteristic
41	C₂²×Dic₂₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×Dic₂₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₂₉	—	2	2	no	—
44	C₂×C₄×D₂₉	—	2	2	no	—
45	C₂×C₄×D₂₉	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₂₉	C₂³	1	2	yes	—
47	C₂×C₈×C₂₉	—	2	2	no	—
48	C₂×C₈×C₂₉	—	2	2	no	—
49	C₂×C₈×C₂₉	C₂³	1	2	yes	—
50	C₂×C₈×C₂₉	C₂³	1	2	yes	—
51	C₂²⋊Dic₂₉	C₂×C₄	1	4	yes	—
52	#90781	C₂×C₄	1	4	yes	—
53	#90779	C₂×C₄	1	4	yes	—
54	C₄×Dic₂₉	C₂×C₄	1	2	yes	—
55	C₄×Dic₂₉	C₂³	1	2	yes	—
56	C₂×#6597	—	2	2	no	—
57	C₂×#6597	—	2	8	no	—
58	C₂×#6597	D₄	1	4	yes	complemented
59	C₂×#6597	C₂³	1	2	yes	—
60	C₂×#6597	C₂³	1	2	yes	—
61	C₈×D₂₉	—	2	8	no	—
62	C₂³×C₂₉	—	2	2	no	—
63	C₂³×C₂₉	—	2	2	no	—
64	C₂³×C₂₉	C₂×C₈	1	4	yes	complemented
65	C₂³×C₂₉	C₂²×C₄	1	2	yes	—
66	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×D₂₉	—	2	2	no	—
68	C₂²×D₂₉	—	2	2	no	—
69	C₂²×D₂₉	C₂×C₈	1	4	yes	complemented
70	C₂²×D₂₉	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₂₉	—	2	2	no	—
72	C₂×C₄×C₂₉	—	2	4	no	—
73	C₂×C₄×C₂₉	—	2	2	no	—
74	C₂×C₄×C₂₉	—	2	2	no	—
75	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₂₉	#41	1	2	yes	—
79	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
80	C₂²⋊D₂₉	C₂×C₈	1	16	yes	complemented
81	C₂×Dic₂₉	—	2	2	no	—
82	C₂×Dic₂₉	—	2	2	no	—
83	C₂×Dic₂₉	C₂×C₈	1	4	yes	complemented
84	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
85	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
86	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
87	C₄×D₂₉	—	2	4	no	—
88	C₄×D₂₉	—	2	2	no	—
89	C₄×D₂₉	—	2	2	no	—
90	C₈×C₂₉	—	2	2	no	—
91	C₈×C₂₉	—	2	2	no	—
92	#6597	—	2	2	no	—
93	#6597	C₂×D₄	1	4	yes	complemented
94	C₂×C₈×D₄	—	29	1	no	maximal, Sylow
95	C₂²×C₂₉	—	2	2	no	—
96	C₂²×C₂₉	—	2	2	no	—
97	C₂²×C₂₉	—	2	8	no	—
98	C₂²×C₂₉	C₄×D₄	1	2	yes	—
99	C₂²×C₂₉	C₂²×C₈	1	2	yes	—
100	C₂²×C₂₉	C₂²×C₈	1	4	yes	complemented
101	C₂²×C₂₉	#131	1	2	yes	—
102	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
103	C₂×D₂₉	—	2	2	no	—
104	C₂×D₂₉	—	2	2	no	—
105	C₂×D₂₉	—	2	8	no	—
106	C₂×D₂₉	C₂²×C₈	1	4	yes	complemented
107	C₄×C₂₉	—	2	2	no	—
108	C₄×C₂₉	C₄×D₄	1	2	yes	—
109	C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
110	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
111	Dic₂₉	—	2	2	no	—
112	Dic₂₉	—	2	2	no	—
113	Dic₂₉	C₂²×C₈	1	4	yes	complemented
114	C₂³×C₈	—	29	1	no	—
115	C₂³×C₈	—	29	1	no	—
116	C₂×C₄×D₄	—	29	1	no	—
117	C₈×D₄	—	29	8	no	—
118	C₂×C₄⋊C₈	—	29	1	no	—
119	C₂×C₂²⋊C₈	—	29	1	no	—
120	C₂×C₂²⋊C₈	D₂₉	1	1	yes	characteristic, complemented
121	C₂×C₄×C₈	—	29	1	no	—
122	C₂×C₂₉	—	2	4	no	—
123	C₂×C₂₉	C₈×D₄	1	4	yes	complemented
124	C₂×C₂₉	C₂×C₄×D₄	1	1	yes	characteristic
125	C₂×C₂₉	C₂³×C₈	1	1	yes	characteristic
126	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
127	D₂₉	—	2	4	no	—
128	C₂²×D₄	—	29	1	no	—
129	C₂³×C₄	—	29	1	no	—
130	C₂³×C₄	C₂×D₂₉	1	1	yes	characteristic
131	C₂²×C₈	—	29	1	no	—
132	C₂²×C₈	—	29	4	no	—
133	C₂²×C₈	—	29	1	no	—
134	C₂²×C₈	—	29	4	no	—
135	C₂²×C₈	—	58	4	no	—
136	C₂²×C₈	—	58	4	no	—
137	C₂²×C₈	C₂×D₂₉	1	1	yes	characteristic
138	C₂²×C₈	C₂×D₂₉	1	1	yes	characteristic
139	C₄×D₄	—	29	4	no	—
140	C₄×D₄	—	29	4	no	—
141	C₂×C₄⋊C₄	—	29	1	no	—
142	C₂×C₂²⋊C₄	—	29	1	no	—
143	C₂×C₂²⋊C₄	—	29	1	no	—
144	C₂×C₄²	—	29	1	no	—
145	C₄⋊C₈	—	29	4	no	—
146	C₂²⋊C₈	—	29	4	no	—
147	C₂²⋊C₈	C₂×D₂₉	1	4	yes	complemented
148	C₄×C₈	—	29	4	no	—
149	C₂₉	C₂×C₈×D₄	1	1	yes	characteristic, complemented, Sylow
150	C₂⁴	—	29	1	no	—
151	C₂⁴	C₄×D₂₉	1	1	yes	characteristic
152	C₂×D₄	—	29	8	no	—
153	C₂×D₄	—	29	4	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	29	1	no	—
157	C₂²×C₄	—	29	1	no	—
158	C₂²×C₄	—	29	1	no	—
159	C₂²×C₄	—	29	2	no	—
160	C₂²×C₄	—	29	2	no	—
161	C₂²×C₄	—	58	2	no	—
162	C₂²×C₄	—	58	2	no	—
163	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
164	C₂²×C₄	C₄×D₂₉	1	2	yes	—
165	C₂²×C₄	C₂²×D₂₉	1	2	yes	—
166	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	29	4	no	—
170	C₂×C₈	—	29	2	no	—
171	C₂×C₈	—	29	2	no	—
172	C₂×C₈	—	58	8	no	—
173	C₂×C₈	—	58	8	no	—
174	C₂×C₈	—	58	2	no	—
175	C₂×C₈	C₂²×D₂₉	1	2	yes	—
176	C₂×C₈	C₂²×D₂₉	1	2	yes	—
177	C₄⋊C₄	—	29	4	no	—
178	C₂²⋊C₄	—	29	4	no	—
179	C₂²⋊C₄	—	29	4	no	—
180	C₄²	—	29	2	no	—
181	C₄²	—	29	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	29	2	no	—
185	C₂³	—	29	4	no	—
186	C₂³	—	58	2	no	—
187	C₂³	—	58	2	no	—
188	C₂³	C₈×D₂₉	1	4	yes	complemented
189	C₂³	C₂×C₄×D₂₉	1	2	yes	—
190	C₂³	C₂×C₄×D₂₉	1	1	yes	characteristic
191	D₄	—	29	16	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	29	2	no	—
197	C₂×C₄	—	29	4	no	—
198	C₂×C₄	—	29	2	no	—
199	C₂×C₄	—	29	2	no	—
200	C₂×C₄	—	58	2	no	—
201	C₂×C₄	—	58	2	no	—
202	C₂×C₄	—	58	2	no	—
203	C₂×C₄	—	58	2	no	—
204	C₂×C₄	—	58	4	no	—
205	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
207	C₂×C₄	D₄×D₂₉	1	2	yes	—
208	C₂×C₄	#90807	1	2	yes	—
209	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
210	C₈	—	2	2	no	—
211	C₈	—	2	2	no	—
212	C₈	—	29	4	no	—
213	C₈	—	58	2	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	8	no	—
217	C₂²	—	29	4	no	—
218	C₂²	—	58	2	no	—
219	C₂²	—	58	2	no	—
220	C₂²	—	58	8	no	—
221	C₂²	#11745243	1	2	yes	—
222	C₂²	C₂×C₈×D₂₉	1	2	yes	—
223	C₂²	C₂×C₈×D₂₉	1	4	yes	complemented
224	C₂²	#11745269	1	2	yes	—
225	C₂²	C₂²×C₄×D₂₉	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	29	4	no	—
228	C₄	—	58	2	no	—
229	C₄	—	58	2	no	—
230	C₄	#11745243	1	2	yes	—
231	C₄	C₂×D₄×D₂₉	1	1	yes	characteristic
232	C₄	C₂×#90807	1	1	yes	characteristic
233	C₂	—	2	4	no	—
234	C₂	—	58	4	no	—
235	C₂	#14271226	1	4	yes	direct product factor
236	C₂	C₂×#11745243	1	1	yes	characteristic
237	C₂	C₂²×C₈×D₂₉	1	1	yes	characteristic
238	C₂	C₂×#11745269	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₂₉
commutator	C₂×C₂₉	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₂₉
Fitting	C₂×C₂₉×C₂²⋊C₈	C₂
socle	C₂³×C₂₉	C₂²×C₄
prime radical	C₂²×C₂²⋊D₂₉	C₄
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	C₂×C₂²⋊C₈	D₂₉
hypocenter	C₂₉	C₂×C₈×D₄
LCS	G ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂³×#6597	C₂	semidirect
2. C₂²×C₈×D₂₉	C₂	semidirect
4. #14271226	C₂	direct (N × H)
5. C₂×#11745162	C₂	semidirect
6. C₂×#11745168	C₂	semidirect
7. C₂×C₈×Dic₂₉	C₂	semidirect
8. C₂×C₂₉×C₂²⋊C₈	C₂	semidirect
23. C₂²×#6597	C₂²	semidirect
25. C₂×C₈×D₂₉	C₂²	semidirect
26. C₂₉×C₂²⋊C₈	C₂²	semidirect
27. #11745168	C₂²	semidirect
28. #11745162	C₂²	semidirect
29. C₈×Dic₂₉	C₂²	semidirect
38. C₂×C₂²⋊D₂₉	C₈	semidirect
58. C₂×#6597	D₄	semidirect
64. C₂³×C₂₉	C₂×C₈	semidirect
69. C₂²×D₂₉	C₂×C₈	semidirect
80. C₂²⋊D₂₉	C₂×C₈	semidirect
83. C₂×Dic₂₉	C₂×C₈	semidirect
93. #6597	C₂×D₄	semidirect
100. C₂²×C₂₉	C₂²×C₈	semidirect
106. C₂×D₂₉	C₂²×C₈	semidirect
113. Dic₂₉	C₂²×C₈	semidirect
120. C₂×C₂²⋊C₈	D₂₉	semidirect
123. C₂×C₂₉	C₈×D₄	semidirect
147. C₂²⋊C₈	C₂×D₂₉	semidirect
149. C₂₉	C₂×C₈×D₄	semidirect
188. C₂³	C₈×D₂₉	semidirect
223. C₂²	C₂×C₈×D₂₉	semidirect
235. C₂	#14271226	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#6597	2
C₂²×C₈×D₂₉	2
C₂×C₄×C₂²⋊D₂₉	2
#14271226	2
C₂×#11745162	2
C₂×#11745168	2
C₂×C₈×Dic₂₉	2
C₂×C₂₉×C₂²⋊C₈	2
C₂×C₈×D₄	29

Subgroups — C2×#14271226 #16828162

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14271226 #16828162