Subgroups — C₂×#14272129 #16836054

Order 3712.

592 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₂₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#11745345	C₂	1	1	yes	maximal, characteristic
3	#14272129	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11745208	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×#1181	C₂	1	2	yes	maximal, complemented
6	C₂×M₁₆×D₂₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄×D₂₉	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊#1181	C₄	1	2	yes	—
9	C₂²×#6599	C₄	1	2	yes	complemented
10	#11745345	C₄	1	4	yes	—
11	#11745345	C₂²	1	4	yes	—
12	C₂×C₄⋊#1181	C₄	1	2	yes	—
13	C₂×C₄×#1181	C₂²	1	2	yes	—
14	C₂×D₂₉⋊C₈	C₄	1	4	yes	complemented
15	C₂×D₂₉⋊C₈	C₄	1	2	yes	complemented
16	C₂×D₂₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×D₂₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×D₂₉⋊C₈	C₂²	1	4	yes	—
19	C₂×M₁₆×C₂₉	C₄	1	1	yes	characteristic, complemented
20	C₂×#90777	C₄	1	1	yes	characteristic, complemented
21	M₁₆×D₂₉	C₄	1	4	yes	complemented
22	M₁₆×D₂₉	C₂²	1	4	yes	—
23	C₂×#90772	C₄	1	2	yes	complemented
24	C₂×C₈×D₂₉	C₂²	1	2	yes	—
25	#11745208	C₂²	1	8	yes	complemented
26	C₈×#1181	C₂²	1	8	yes	complemented
27	C₂³×D₂₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×#1181	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×Dic₂₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×D₂₉	C₂×C₄	1	2	yes	—
33	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₂₉	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₂₉	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
38	C₂×C₄×D₂₉	C₂³	1	2	yes	—
39	C₂²⋊#1181	C₂×C₄	1	8	yes	—
40	C₂×#6599	C₂×C₄	1	2	yes	—
41	C₂×#6599	C₂×C₄	1	2	yes	—
42	C₂×#6599	C₂×C₄	1	8	yes	complemented
43	C₄⋊#1181	C₂×C₄	1	8	yes	—
44	C₄×#1181	C₂×C₄	1	4	yes	—
45	C₄×#1181	C₂³	1	4	yes	—
46	D₂₉⋊C₈	C₂×C₄	1	2	yes	—
47	D₂₉⋊C₈	C₂×C₄	1	2	yes	—
48	D₂₉⋊C₈	C₂×C₄	1	8	yes	complemented
49	D₂₉⋊C₈	C₂³	1	2	yes	—
50	D₂₉⋊C₈	C₂³	1	2	yes	—
51	M₁₆×C₂₉	C₂×C₄	1	4	yes	complemented
52	C₂×C₈×C₂₉	C₂×C₄	1	2	yes	—
53	#90777	C₂×C₄	1	4	yes	complemented
54	C₂×#6597	C₂×C₄	1	2	yes	—
55	#90772	C₂×C₄	1	8	yes	complemented
56	C₈×D₂₉	C₂×C₄	1	4	yes	—
57	C₈×D₂₉	C₂³	1	4	yes	—
58	C₂³×C₂₉	C₄²	1	1	yes	characteristic
59	C₂²×D₂₉	—	2	4	no	—
60	C₂²×D₂₉	C₄²	1	1	yes	characteristic
61	C₂²×D₂₉	C₄²	1	2	yes	—
62	C₂²×D₂₉	C₂²×C₄	1	2	yes	—
63	C₂²×D₂₉	C₂²×C₄	1	1	yes	characteristic
64	C₂×#1181	—	2	8	no	—
65	C₂×#1181	C₂²×C₄	1	8	yes	—
66	C₂×C₄×C₂₉	C₄²	1	2	yes	—
67	C₂×C₄×C₂₉	C₄²	1	1	yes	characteristic
68	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
70	C₂×Dic₂₉	C₄²	1	2	yes	—
71	C₂×Dic₂₉	C₄²	1	1	yes	characteristic
72	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
74	C₄×D₂₉	C₄²	1	2	yes	—
75	C₄×D₂₉	C₄²	1	2	yes	—
76	C₄×D₂₉	C₄²	1	2	yes	—
77	C₄×D₂₉	C₂²×C₄	1	1	yes	characteristic
78	C₄×D₂₉	C₂²×C₄	1	2	yes	—
79	C₄×D₂₉	C₂²×C₄	1	1	yes	characteristic
80	C₄×D₂₉	C₂²×C₄	1	2	yes	—
81	C₄×D₂₉	C₂²×C₄	1	1	yes	characteristic
82	C₄×D₂₉	C₂²×C₄	1	2	yes	—
83	C₄×D₂₉	C₂⁴	1	1	yes	characteristic
84	#6599	C₂²×C₄	1	4	yes	—
85	#6599	C₂²×C₄	1	4	yes	—
86	C₈×C₂₉	C₂²×C₄	1	4	yes	—
87	#6597	C₂²×C₄	1	4	yes	—
88	C₂×#405	—	29	1	no	maximal, Sylow
89	C₂²×C₂₉	—	2	2	no	—
90	C₂²×C₂₉	C₂×C₄²	1	2	yes	—
91	C₂²×C₂₉	C₂×C₄²	1	1	yes	characteristic
92	C₂×D₂₉	—	2	2	no	—
93	C₂×D₂₉	—	2	4	no	—
94	C₂×D₂₉	—	2	4	no	—
95	C₂×D₂₉	C₂×C₄²	1	2	yes	—
96	C₂×D₂₉	C₂×C₄²	1	1	yes	characteristic
97	C₂×D₂₉	#131	1	4	yes	—
98	C₂×D₂₉	C₂³×C₄	1	1	yes	characteristic
99	#1181	—	2	8	no	—
100	C₄×C₂₉	C₂×C₄²	1	2	yes	—
101	C₄×C₂₉	C₂×C₄²	1	1	yes	characteristic
102	C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
103	Dic₂₉	C₂×C₄²	1	1	yes	characteristic
104	Dic₂₉	C₂×C₄²	1	2	yes	—
105	Dic₂₉	C₂³×C₄	1	1	yes	characteristic
106	C₂²×M₁₆	—	29	1	no	—
107	C₂³×C₈	—	29	1	no	—
108	C₂×#117	—	29	1	no	—
109	#405	—	29	8	no	—
110	C₂×#97	—	29	2	no	—
111	C₂×C₄×C₈	—	29	2	no	—
112	C₂×C₂₉	—	2	2	no	—
113	C₂×C₂₉	#405	1	2	yes	complemented
114	C₂×C₂₉	C₂²×C₄²	1	1	yes	characteristic
115	D₂₉	—	2	2	no	—
116	D₂₉	#405	1	2	yes	complemented
117	D₂₉	C₂×#131	1	2	yes	—
118	C₂³×C₄	—	29	1	no	—
119	C₂×M₁₆	—	29	1	no	—
120	C₂×M₁₆	—	29	4	no	—
121	C₂×M₁₆	—	29	4	no	—
122	C₂×M₁₆	—	29	2	no	—
123	C₂×M₁₆	#1181	1	1	yes	characteristic, complemented
124	C₂²×C₈	—	29	4	no	—
125	C₂²×C₈	—	29	2	no	—
126	C₂²×C₈	—	29	1	no	—
127	C₂²×C₈	—	29	1	no	—
128	C₂²×C₈	—	29	4	no	—
129	C₂²×C₈	—	29	2	no	—
130	C₂²×C₈	—	29	2	no	—
131	#117	—	29	4	no	—
132	#117	—	29	4	no	—
133	C₂×C₄⋊C₄	—	29	2	no	—
134	C₂×C₂²⋊C₄	—	29	2	no	—
135	C₂×C₄²	—	29	2	no	—
136	#97	—	29	8	no	—
137	C₄×C₈	—	29	8	no	—
138	C₂₉	C₂×#405	1	1	yes	characteristic, complemented, Sylow
139	C₂⁴	—	29	1	no	—
140	C₂²×C₄	—	29	1	no	—
141	C₂²×C₄	—	29	1	no	—
142	C₂²×C₄	—	29	2	no	—
143	C₂²×C₄	—	29	2	no	—
144	C₂²×C₄	—	29	1	no	—
145	C₂²×C₄	—	29	2	no	—
146	C₂²×C₄	—	29	2	no	—
147	C₂²×C₄	—	29	1	no	—
148	C₂²×C₄	—	29	1	no	—
149	C₂²×C₄	—	29	4	no	—
150	C₂²×C₄	C₂×#1181	1	1	yes	characteristic
151	M₁₆	—	29	4	no	—
152	M₁₆	—	29	8	no	—
153	M₁₆	C₂×#1181	1	4	yes	complemented
154	C₂×C₈	—	29	2	no	—
155	C₂×C₈	—	29	2	no	—
156	C₂×C₈	—	29	2	no	—
157	C₂×C₈	—	29	2	no	—
158	C₂×C₈	—	29	8	no	—
159	C₂×C₈	—	29	2	no	—
160	C₂×C₈	—	29	2	no	—
161	C₂×C₈	—	29	8	no	—
162	C₂×C₈	—	29	2	no	—
163	C₂×C₈	—	29	4	no	—
164	C₂×C₈	—	29	4	no	—
165	C₂×C₈	C₂×#1181	1	2	yes	—
166	C₄⋊C₄	—	29	8	no	—
167	C₂²⋊C₄	—	29	8	no	—
168	C₄²	—	29	4	no	—
169	C₄²	—	29	4	no	—
170	C₂³	—	29	1	no	—
171	C₂³	—	29	2	no	—
172	C₂³	—	29	2	no	—
173	C₂³	—	29	1	no	—
174	C₂³	—	58	4	no	—
175	C₂³	C₄×#1181	1	1	yes	characteristic
176	C₂×C₄	—	29	2	no	—
177	C₂×C₄	—	29	1	no	—
178	C₂×C₄	—	29	2	no	—
179	C₂×C₄	—	29	1	no	—
180	C₂×C₄	—	29	1	no	—
181	C₂×C₄	—	29	2	no	—
182	C₂×C₄	—	29	2	no	—
183	C₂×C₄	—	29	1	no	—
184	C₂×C₄	—	29	2	no	—
185	C₂×C₄	—	29	2	no	—
186	C₂×C₄	—	29	2	no	—
187	C₂×C₄	—	29	1	no	—
188	C₂×C₄	—	29	1	no	—
189	C₂×C₄	—	29	2	no	—
190	C₂×C₄	—	29	8	no	—
191	C₂×C₄	—	58	8	no	—
192	C₂×C₄	C₄×#1181	1	1	yes	characteristic
193	C₂×C₄	C₄×#1181	1	2	yes	—
194	C₂×C₄	C₂²×#1181	1	2	yes	—
195	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
196	C₈	—	29	4	no	—
197	C₈	—	29	4	no	—
198	C₈	—	29	4	no	—
199	C₈	C₂²×#1181	1	4	yes	—
200	C₂²	—	2	2	no	—
201	C₂²	—	29	4	no	—
202	C₂²	—	29	1	no	—
203	C₂²	—	29	1	no	—
204	C₂²	—	29	2	no	—
205	C₂²	—	58	4	no	—
206	C₂²	—	58	4	no	—
207	C₂²	—	58	2	no	—
208	C₂²	C₂×C₄×#1181	1	2	yes	—
209	C₂²	C₂×C₄×#1181	1	1	yes	characteristic
210	C₄	—	29	1	no	—
211	C₄	—	29	1	no	—
212	C₄	—	29	2	no	—
213	C₄	—	58	8	no	—
214	C₄	C₂×C₄×#1181	1	2	yes	—
215	C₄	C₂×C₄×#1181	1	1	yes	characteristic
216	C₄	C₂³×#1181	1	1	yes	characteristic
217	C₂	—	2	2	no	—
218	C₂	—	29	2	no	—
219	C₂	—	29	2	no	—
220	C₂	—	58	2	no	—
221	C₂	#14272129	1	2	yes	direct product factor
222	C₂	C₂²×C₄×#1181	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#1181
commutator	C₂×C₂₉	C₂²×C₄²
Frattini	C₄	C₂³×#1181
Fitting	C₂×M₁₆×C₂₉	C₄
socle	C₂²×C₂₉	C₂×C₄²
prime radical	C₂³×D₂₉	C₂×C₄
cosocle	C₄×D₂₉	C₂⁴
hypercenter	C₂×M₁₆	#1181
hypocenter	C₂₉	C₂×#405
LCS	G ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #14272129	C₂	direct (N × H)
4. C₂×#11745208	C₂	semidirect
5. C₂×C₈×#1181	C₂	semidirect
9. C₂²×#6599	C₄	semidirect
14. C₂×D₂₉⋊C₈	C₄	semidirect
15. C₂×D₂₉⋊C₈	C₄	semidirect
19. C₂×M₁₆×C₂₉	C₄	semidirect
20. C₂×#90777	C₄	semidirect
21. M₁₆×D₂₉	C₄	semidirect
23. C₂×#90772	C₄	semidirect
25. #11745208	C₂²	semidirect
26. C₈×#1181	C₂²	semidirect
42. C₂×#6599	C₂×C₄	semidirect
48. D₂₉⋊C₈	C₂×C₄	semidirect
51. M₁₆×C₂₉	C₂×C₄	semidirect
53. #90777	C₂×C₄	semidirect
55. #90772	C₂×C₄	semidirect
113. C₂×C₂₉	#405	semidirect
116. D₂₉	#405	semidirect
123. C₂×M₁₆	#1181	semidirect
138. C₂₉	C₂×#405	semidirect
153. M₁₆	C₂×#1181	semidirect
221. C₂	#14272129	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×D₂₉⋊C₈	2
C₂×#11745345	2
#14272129	2
C₂×#11745208	2
C₂×C₈×#1181	2
C₂×M₁₆×D₂₉	2
C₂×#405	29

Subgroups — C2×#14272129 #16836054

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14272129 #16836054