Subgroups — C₂×#14271235 #16828172

Order 3712.

564 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	C₂²×#90777	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×#6597	C₂	1	1	yes	maximal, characteristic, complemented
3	#14271235	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊#6597	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂²⋊#6597	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂²⋊#6597	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11745151	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₄×C₂₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×D₄×C₂₉	C₄	1	1	yes	characteristic
10	C₂³×C₄×C₂₉	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₉	C₂²	1	1	yes	characteristic
12	C₄×D₄×C₂₉	C₄	1	4	yes	—
13	C₄×D₄×C₂₉	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₂₉	C₄	1	1	yes	characteristic
15	C₂×C₂²⋊C₄×C₂₉	C₄	1	1	yes	characteristic
16	C₂×C₂²⋊C₄×C₂₉	C₄	1	1	yes	characteristic
17	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
18	C₂×#90777	—	2	4	no	—
19	C₂×#90777	C₂²	1	4	yes	complemented
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	1	yes	characteristic
24	C₂²×#6597	C₂²	1	1	yes	characteristic
25	C₂²×#6597	C₂²	1	4	yes	complemented
26	C₂²⋊#6597	C₂²	1	4	yes	complemented
27	C₂²⋊#6597	C₂²	1	4	yes	complemented
28	C₄⋊#6597	C₂²	1	4	yes	complemented
29	#11745151	C₂²	1	4	yes	complemented
30	C₂⁴×C₂₉	C₂×C₄	1	1	yes	characteristic
31	C₂⁴×C₂₉	C₂×C₄	1	1	yes	characteristic
32	C₂×D₄×C₂₉	—	2	4	no	—
33	C₂×D₄×C₂₉	C₂×C₄	1	4	yes	—
34	C₂²×C₄×C₂₉	—	2	2	no	—
35	C₂²×C₄×C₂₉	—	2	2	no	—
36	C₂²×C₄×C₂₉	—	2	2	no	—
37	C₂²×C₄×C₂₉	—	2	2	no	—
38	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₉	C₂×C₄	1	2	yes	—
43	C₂²×C₄×C₂₉	C₂×C₄	1	2	yes	—
44	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
45	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
46	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₂₉	C₂×C₄	1	4	yes	—
48	C₂²⋊C₄×C₂₉	C₂×C₄	1	4	yes	—
49	C₂²⋊C₄×C₂₉	C₂×C₄	1	4	yes	—
50	C₄²×C₂₉	C₂×C₄	1	2	yes	—
51	C₄²×C₂₉	C₂³	1	2	yes	—
52	#90777	—	2	8	no	—
53	C₂×#6597	—	2	2	no	—
54	C₂×#6597	—	2	2	no	—
55	C₂×#6597	—	2	2	no	—
56	C₂×#6597	—	2	8	no	—
57	C₂×#6597	D₄	1	4	yes	complemented
58	C₂×#6597	C₂³	1	2	yes	—
59	C₂×#6597	C₂³	1	2	yes	—
60	C₂×#6597	C₂³	1	2	yes	—
61	C₂×#6597	C₂³	1	2	yes	—
62	C₂³×C₂₉	—	2	2	no	—
63	C₂³×C₂₉	—	2	2	no	—
64	C₂³×C₂₉	—	2	2	no	—
65	C₂³×C₂₉	—	2	2	no	—
66	C₂³×C₂₉	—	2	2	no	—
67	C₂³×C₂₉	M₁₆	1	4	yes	complemented
68	C₂³×C₂₉	C₂²×C₄	1	2	yes	—
69	C₂³×C₂₉	C₂²×C₄	1	2	yes	—
70	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
71	D₄×C₂₉	—	2	8	no	—
72	C₂×C₄×C₂₉	—	2	2	no	—
73	C₂×C₄×C₂₉	—	2	2	no	—
74	C₂×C₄×C₂₉	—	2	2	no	—
75	C₂×C₄×C₂₉	—	2	2	no	—
76	C₂×C₄×C₂₉	—	2	4	no	—
77	C₂×C₄×C₂₉	—	2	4	no	—
78	C₂×C₄×C₂₉	—	2	2	no	—
79	C₂×C₄×C₂₉	—	2	2	no	—
80	C₂×C₄×C₂₉	—	2	2	no	—
81	C₂×C₄×C₂₉	—	2	2	no	—
82	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₂₉	#41	1	2	yes	—
89	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
90	#6597	—	2	2	no	—
91	#6597	—	2	2	no	—
92	#6597	—	2	2	no	—
93	#6597	C₂×D₄	1	4	yes	complemented
94	C₂×#435	—	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	—	2	2	no	—
99	C₂²×C₂₉	—	2	2	no	—
100	C₂²×C₂₉	—	2	2	no	—
101	C₂²×C₂₉	—	4	4	no	—
102	C₂²×C₂₉	C₄×D₄	1	2	yes	—
103	C₂²×C₂₉	C₂×M₁₆	1	2	yes	—
104	C₂²×C₂₉	C₂×M₁₆	1	4	yes	complemented
105	C₂²×C₂₉	#131	1	2	yes	—
106	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
107	C₄×C₂₉	—	2	2	no	—
108	C₄×C₂₉	—	2	2	no	—
109	C₄×C₂₉	—	2	2	no	—
110	C₄×C₂₉	—	2	2	no	—
111	C₄×C₂₉	C₄×D₄	1	2	yes	—
112	C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
113	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
114	C₂²×M₁₆	—	29	1	no	—
115	C₂³×C₈	—	29	1	no	—
116	C₂×C₄×D₄	D₂₉	1	1	yes	characteristic
117	#435	—	29	8	no	—
118	C₂×C₄⋊C₈	—	29	1	no	—
119	C₂×C₂²⋊C₈	—	29	1	no	—
120	C₂×C₂²⋊C₈	—	29	1	no	—
121	C₂×#97	—	29	1	no	—
122	C₂×C₂₉	—	2	4	no	—
123	C₂×C₂₉	—	4	2	no	—
124	C₂×C₂₉	#435	1	4	yes	complemented
125	C₂×C₂₉	C₂×C₄×D₄	1	1	yes	characteristic
126	C₂×C₂₉	C₂²×M₁₆	1	1	yes	characteristic
127	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
128	C₂²×D₄	Dic₂₉	1	1	yes	characteristic
129	C₂³×C₄	C₂×D₂₉	1	1	yes	characteristic
130	C₂³×C₄	C₂×D₂₉	1	1	yes	characteristic
131	C₂×M₁₆	—	29	4	no	—
132	C₂×M₁₆	—	58	4	no	—
133	C₂²×C₈	—	29	1	no	—
134	C₂²×C₈	—	29	4	no	—
135	C₂²×C₈	—	29	1	no	—
136	C₂²×C₈	—	29	1	no	—
137	C₂²×C₈	—	29	1	no	—
138	C₂²×C₈	—	58	4	no	—
139	C₄×D₄	Dic₂₉	1	4	yes	—
140	C₄×D₄	C₂×D₂₉	1	4	yes	—
141	C₂×C₄⋊C₄	Dic₂₉	1	1	yes	characteristic
142	C₂×C₂²⋊C₄	Dic₂₉	1	1	yes	characteristic
143	C₂×C₂²⋊C₄	Dic₂₉	1	1	yes	characteristic
144	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
145	C₄⋊C₈	—	29	4	no	—
146	C₂²⋊C₈	—	29	4	no	—
147	C₂²⋊C₈	—	29	4	no	—
148	#97	—	29	4	no	—
149	C₂₉	C₂×#435	1	1	yes	characteristic, complemented, Sylow
150	C₂⁴	C₂×Dic₂₉	1	1	yes	characteristic
151	C₂⁴	C₂×Dic₂₉	1	1	yes	characteristic
152	C₂×D₄	—	2	4	no	—
153	C₂×D₄	C₂×Dic₂₉	1	4	yes	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	C₂×Dic₂₉	1	1	yes	characteristic
159	C₂²×C₄	C₂×Dic₂₉	1	1	yes	characteristic
160	C₂²×C₄	C₂×Dic₂₉	1	1	yes	characteristic
161	C₂²×C₄	C₂×Dic₂₉	1	1	yes	characteristic
162	C₂²×C₄	C₂×Dic₂₉	1	2	yes	—
163	C₂²×C₄	C₂×Dic₂₉	1	2	yes	—
164	C₂²×C₄	C₂²×D₂₉	1	2	yes	—
165	C₂²×C₄	C₂²×D₂₉	1	2	yes	—
166	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
167	M₁₆	—	58	8	no	—
168	C₂×C₈	—	29	4	no	—
169	C₂×C₈	—	29	2	no	—
170	C₂×C₈	—	29	2	no	—
171	C₂×C₈	—	29	2	no	—
172	C₂×C₈	—	29	2	no	—
173	C₂×C₈	—	58	8	no	—
174	C₂×C₈	—	58	2	no	—
175	C₂×C₈	—	58	2	no	—
176	C₂×C₈	—	58	2	no	—
177	C₄⋊C₄	C₂×Dic₂₉	1	4	yes	—
178	C₂²⋊C₄	C₂×Dic₂₉	1	4	yes	—
179	C₂²⋊C₄	C₂×Dic₂₉	1	4	yes	—
180	C₄²	C₂×Dic₂₉	1	2	yes	—
181	C₄²	C₂²×D₂₉	1	2	yes	—
182	C₂³	—	2	2	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	2	2	no	—
185	C₂³	—	2	2	no	—
186	C₂³	—	2	2	no	—
187	C₂³	#90777	1	4	yes	complemented
188	C₂³	C₂²×Dic₂₉	1	2	yes	—
189	C₂³	C₂²×Dic₂₉	1	2	yes	—
190	C₂³	C₂²×Dic₂₉	1	1	yes	characteristic
191	D₄	—	2	8	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	D₄×D₂₉	1	2	yes	—
203	C₂×C₄	#90807	1	2	yes	—
204	C₂×C₄	C₂²×Dic₂₉	1	2	yes	—
205	C₂×C₄	C₂²×Dic₂₉	1	2	yes	—
206	C₂×C₄	C₂²×Dic₂₉	1	2	yes	—
207	C₂×C₄	C₂²×Dic₂₉	1	2	yes	—
208	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
209	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
210	C₈	—	29	4	no	—
211	C₈	—	58	2	no	—
212	C₈	—	58	2	no	—
213	C₈	—	58	2	no	—
214	C₂²	—	2	8	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	4	4	no	—
221	C₂²	C₂×#90777	1	2	yes	—
222	C₂²	C₂×#90777	1	4	yes	complemented
223	C₂²	D₄×Dic₂₉	1	2	yes	—
224	C₂²	#11745310	1	2	yes	—
225	C₂²	C₂³×Dic₂₉	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	D₄×Dic₂₉	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	—	4	2	no	—
235	C₂	#14271235	1	4	yes	direct product factor
236	C₂	C₂×D₄×Dic₂₉	1	1	yes	characteristic
237	C₂	C₂²×#90777	1	1	yes	characteristic
238	C₂	C₂×#11745310	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₂₉
commutator	C₂²×C₂₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₉
Fitting	C₂×C₄×D₄×C₂₉	C₂
socle	C₂³×C₂₉	C₂²×C₄
prime radical	C₂²×D₄×C₂₉	C₄
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	C₂×C₄×D₄	D₂₉
hypocenter	C₂₉	C₂×#435
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#90777	C₂	semidirect
2. C₂³×#6597	C₂	semidirect
3. #14271235	C₂	direct (N × H)
4. C₂×C₄⋊#6597	C₂	semidirect
5. C₂×C₂²⋊#6597	C₂	semidirect
6. C₂×C₂²⋊#6597	C₂	semidirect
7. C₂×#11745151	C₂	semidirect
19. C₂×#90777	C₂²	semidirect
25. C₂²×#6597	C₂²	semidirect
26. C₂²⋊#6597	C₂²	semidirect
27. C₂²⋊#6597	C₂²	semidirect
28. C₄⋊#6597	C₂²	semidirect
29. #11745151	C₂²	semidirect
57. C₂×#6597	D₄	semidirect
67. C₂³×C₂₉	M₁₆	semidirect
93. #6597	C₂×D₄	semidirect
104. C₂²×C₂₉	C₂×M₁₆	semidirect
124. C₂×C₂₉	#435	semidirect
149. C₂₉	C₂×#435	semidirect
187. C₂³	#90777	semidirect
222. C₂²	C₂×#90777	semidirect
235. C₂	#14271235	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×#90777	2
C₂³×#6597	2
#14271235	2
C₂×C₄⋊#6597	2
C₂×C₂²⋊#6597	2
C₂×#11745151	2
C₂×C₄×D₄×C₂₉	2
C₂×#435	29

Subgroups — C2×#14271235 #16828172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14271235 #16828172