Subgroups — C₄×#11745274 #16828259

Order 3712.

500 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×#11745274	C₂	1	1	yes	maximal, characteristic
2	C₄×#90807	C₂	1	2	yes	maximal, complemented
3	C₄×Dic₁₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#14271257	C₂	1	1	yes	maximal, characteristic, complemented
5	#14271254	C₂	1	2	yes	maximal, complemented
6	C₄×#90783	C₂	1	2	yes	maximal, complemented
7	D₈×Dic₂₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#14271200	C₂	1	1	yes	maximal, characteristic, complemented
9	#14271156	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₂₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₈×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#90807	C₂²	1	2	yes	—
13	C₂×D₈×C₂₉	C₂²	1	1	yes	characteristic
14	C₄×D₄×C₂₉	C₂²	1	2	yes	—
15	D₄×Dic₂₉	C₂²	1	2	yes	—
16	C₂×#90783	C₂²	1	2	yes	—
17	C₄×C₂²⋊D₂₉	—	2	2	no	—
18	C₂×C₄×Dic₂₉	—	2	2	no	—
19	#11745274	C₄	1	8	yes	direct product factor
20	C₂×Dic₁₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₂₉	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₂₉	C₂²	1	1	yes	characteristic
23	#11745254	C₂²	1	2	yes	—
24	#11745249	C₂²	1	2	yes	—
25	#11745243	—	2	2	no	—
26	#11745239	—	2	2	no	—
27	C₄²×D₂₉	C₂²	1	1	yes	characteristic
28	C₄×Dic₅₈	C₂²	1	2	yes	—
29	C₂₉×#107	C₂²	1	1	yes	characteristic
30	C₂₉×#102	C₂²	1	2	yes	—
31	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
32	#11745180	C₂²	1	2	yes	—
33	#11745166	C₂²	1	1	yes	characteristic
34	#11745164	C₂²	1	2	yes	—
35	C₈×Dic₂₉	C₂²	1	1	yes	characteristic
36	C₈×Dic₂₉	C₂²	1	1	yes	characteristic
37	#11745158	C₂²	1	2	yes	—
38	#11745156	C₂²	1	2	yes	—
39	C₄×#6597	C₂²	1	1	yes	characteristic
40	C₂×D₄×C₂₉	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₉	—	2	2	no	—
42	C₂×C₂²⋊D₂₉	—	2	2	no	—
43	C₂²×Dic₂₉	—	2	2	no	—
44	C₂²×Dic₂₉	—	2	2	no	—
45	#90807	—	2	4	no	—
46	#90807	C₂×C₄	1	8	yes	complemented
47	C₂×C₄×D₂₉	D₄	1	1	yes	characteristic
48	C₂×C₄×D₂₉	D₄	1	1	yes	characteristic
49	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
50	C₂×Dic₅₈	C₂³	1	2	yes	—
51	D₈×C₂₉	C₂×C₄	1	4	yes	complemented
52	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₂₉	C₂³	1	2	yes	—
55	C₂²⋊C₄×C₂₉	—	2	2	no	—
56	C₄²×C₂₉	C₂³	1	1	yes	characteristic
57	C₂²⋊Dic₂₉	—	2	2	no	—
58	#90783	C₂×C₄	1	8	yes	complemented
59	#90781	—	2	2	no	—
60	C₄⋊Dic₂₉	C₂³	1	2	yes	—
61	#90779	—	2	2	no	—
62	#90779	—	2	2	no	—
63	C₄×Dic₂₉	—	2	2	no	—
64	C₄×Dic₂₉	—	2	2	no	—
65	C₄×Dic₂₉	—	2	2	no	—
66	C₄×Dic₂₉	D₄	1	1	yes	characteristic
67	C₄×Dic₂₉	D₄	1	1	yes	characteristic
68	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
69	C₂×#6597	C₂³	1	1	yes	characteristic
70	C₂×#6597	C₂³	1	1	yes	characteristic
71	Dic₁₁₆	C₂×C₄	1	4	yes	complemented
72	C₈×D₂₉	—	2	2	no	—
73	C₈×D₂₉	C₂×C₄	1	4	yes	complemented
74	C₂³×C₂₉	—	2	2	no	—
75	C₂²×D₂₉	C₂×D₄	1	1	yes	characteristic
76	D₄×C₂₉	—	2	2	no	—
77	D₄×C₂₉	C₂²×C₄	1	4	yes	—
78	C₂×C₄×C₂₉	—	2	2	no	—
79	C₂×C₄×C₂₉	—	4	4	no	—
80	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
83	C₂²⋊D₂₉	—	2	8	no	—
84	C₂×Dic₂₉	—	2	2	no	—
85	C₂×Dic₂₉	—	2	8	no	—
86	C₂×Dic₂₉	—	2	2	no	—
87	C₂×Dic₂₉	—	4	4	no	—
88	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
89	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
90	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
91	C₄×D₂₉	—	2	2	no	—
92	C₄×D₂₉	—	2	2	no	—
93	C₄×D₂₉	C₂²×C₄	1	2	yes	—
94	C₄×D₂₉	#41	1	2	yes	—
95	Dic₅₈	—	2	2	no	—
96	Dic₅₈	C₂²×C₄	1	4	yes	—
97	C₈×C₂₉	—	2	1	no	—
98	C₈×C₂₉	C₂²×C₄	1	2	yes	—
99	#6597	—	2	1	no	—
100	#6597	C₂²×C₄	1	2	yes	—
101	C₄×#135	—	29	1	no	maximal, Sylow
102	C₂²×C₂₉	—	2	4	no	—
103	C₂²×C₂₉	—	4	2	no	—
104	C₂²×C₂₉	—	4	2	no	—
105	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
106	C₂×D₂₉	—	2	1	no	—
107	C₂×D₂₉	—	2	1	no	—
108	C₂×D₂₉	C₄×D₄	1	2	yes	—
109	C₄×C₂₉	—	2	1	no	—
110	C₄×C₂₉	—	4	2	no	—
111	C₄×C₂₉	#135	1	2	yes	complemented
112	C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
113	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
114	Dic₂₉	—	2	4	no	—
115	Dic₂₉	—	2	1	no	—
116	Dic₂₉	—	4	2	no	—
117	Dic₂₉	C₄×D₄	1	2	yes	—
118	Dic₂₉	#135	1	2	yes	complemented
119	C₂×#135	—	29	1	no	—
120	C₄×#41	—	29	2	no	—
121	C₄×Q₁₆	—	29	1	no	—
122	C₄×Q₁₆	—	29	1	no	—
123	C₄×SD₁₆	—	29	2	no	—
124	C₄×SD₁₆	—	29	2	no	—
125	C₄×D₈	—	29	1	no	—
126	C₄×D₈	D₂₉	1	1	yes	characteristic, complemented
127	#427	—	29	1	no	—
128	#416	—	29	2	no	—
129	C₂×C₄×C₈	—	29	1	no	—
130	C₂×C₂₉	—	4	4	no	—
131	C₂×C₂₉	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂×C₂₉	C₂×#135	1	1	yes	characteristic
133	C₂×C₂₉	C₂×#135	1	1	yes	characteristic
134	D₂₉	—	2	2	no	—
135	C₂×#41	—	29	2	no	—
136	#135	—	29	8	no	—
137	C₂×Q₁₆	—	29	1	no	—
138	C₂×SD₁₆	—	29	2	no	—
139	C₂×D₈	C₂×D₂₉	1	1	yes	characteristic
140	C₂²×C₈	—	29	1	no	—
141	C₂²×C₈	—	29	1	no	—
142	C₄×Q₈	—	29	2	no	—
143	C₄×Q₈	—	29	2	no	—
144	C₄×D₄	—	29	2	no	—
145	C₄×D₄	—	58	2	no	—
146	C₄×D₄	—	58	2	no	—
147	C₄×D₄	C₂×D₂₉	1	2	yes	—
148	#117	—	29	2	no	—
149	#117	—	58	2	no	—
150	C₂×C₄²	—	29	1	no	—
151	C₂×C₄²	—	58	2	no	—
152	#107	—	29	1	no	—
153	#107	C₂×D₂₉	1	1	yes	characteristic
154	#106	—	29	2	no	—
155	#103	—	29	2	no	—
156	#103	—	29	2	no	—
157	#102	—	29	2	no	—
158	#102	C₂×D₂₉	1	2	yes	—
159	C₄×C₈	—	29	1	no	—
160	C₄×C₈	—	29	1	no	—
161	C₄×C₈	—	29	1	no	—
162	C₄×C₈	C₂×D₂₉	1	1	yes	characteristic
163	C₂₉	C₄×#135	1	1	yes	characteristic, complemented, Sylow
164	#41	—	29	8	no	—
165	#41	—	58	4	no	—
166	C₂×Q₈	—	29	2	no	—
167	C₂×D₄	—	58	2	no	—
168	C₂×D₄	C₂²×D₂₉	1	2	yes	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	29	1	no	—
171	C₂²×C₄	—	29	1	no	—
172	C₂²×C₄	—	29	1	no	—
173	C₂²×C₄	—	58	2	no	—
174	C₂²×C₄	—	58	2	no	—
175	Q₁₆	—	29	4	no	—
176	SD₁₆	—	29	8	no	—
177	D₈	C₄×D₂₉	1	4	yes	complemented
178	C₂×C₈	—	29	1	no	—
179	C₂×C₈	—	29	4	no	—
180	C₂×C₈	—	29	1	no	—
181	C₂×C₈	—	58	2	no	—
182	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
183	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
184	C₄⋊C₄	—	29	2	no	—
185	C₄⋊C₄	—	58	2	no	—
186	C₄⋊C₄	—	58	2	no	—
187	C₄⋊C₄	C₂²×D₂₉	1	2	yes	—
188	C₂²⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	—	58	2	no	—
190	C₂²⋊C₄	—	58	2	no	—
191	C₄²	—	29	1	no	—
192	C₄²	—	29	1	no	—
193	C₄²	—	29	1	no	—
194	C₄²	—	58	2	no	—
195	C₄²	—	58	2	no	—
196	C₄²	—	58	2	no	—
197	C₄²	C₂²×D₂₉	1	1	yes	characteristic
198	C₂³	—	2	2	no	—
199	C₂³	—	29	1	no	—
200	Q₈	—	29	4	no	—
201	Q₈	—	58	2	no	—
202	D₄	—	2	2	no	—
203	D₄	—	58	8	no	—
204	D₄	C₂×C₄×D₂₉	1	4	yes	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	29	1	no	—
208	C₂×C₄	—	29	1	no	—
209	C₂×C₄	—	29	2	no	—
210	C₂×C₄	—	29	2	no	—
211	C₂×C₄	—	29	1	no	—
212	C₂×C₄	—	58	2	no	—
213	C₂×C₄	—	58	8	no	—
214	C₂×C₄	—	58	2	no	—
215	C₂×C₄	—	58	2	no	—
216	C₂×C₄	—	58	2	no	—
217	C₂×C₄	—	116	4	no	—
218	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
219	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
220	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
221	C₈	—	2	1	no	—
222	C₈	—	29	2	no	—
223	C₈	—	58	1	no	—
224	C₈	C₂×C₄×D₂₉	1	2	yes	—
225	C₂²	—	2	4	no	—
226	C₂²	—	4	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	29	2	no	—
229	C₂²	—	58	1	no	—
230	C₂²	—	58	1	no	—
231	C₂²	C₂×D₄×D₂₉	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	4	2	no	—
234	C₄	—	29	2	no	—
235	C₄	—	29	2	no	—
236	C₄	—	58	4	no	—
237	C₄	—	58	1	no	—
238	C₄	—	116	2	no	—
239	C₄	#11745274	1	2	yes	direct product factor
240	C₄	C₂²×C₄×D₂₉	1	1	yes	characteristic
241	C₄	#41×D₂₉	1	1	yes	characteristic
242	C₂	—	4	4	no	—
243	C₂	—	58	2	no	—
244	C₂	C₄×D₄×D₂₉	1	1	yes	characteristic
245	C₂	#135×D₂₉	1	1	yes	characteristic
246	C₂	C₂×#11745274	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₂₉
commutator	C₄×C₂₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₉
Fitting	C₄×D₈×C₂₉	C₂
socle	C₂²×C₂₉	C₂²×D₄
prime radical	C₂×#11745274	C₂
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	C₄×D₈	D₂₉
hypocenter	C₂₉	C₄×#135
LCS	G ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
2. C₄×#90807	C₂	semidirect
3. C₄×Dic₁₁₆	C₂	semidirect
4. #14271257	C₂	semidirect
5. #14271254	C₂	semidirect
6. C₄×#90783	C₂	semidirect
7. D₈×Dic₂₉	C₂	semidirect
8. #14271200	C₂	semidirect
9. #14271156	C₂	semidirect
10. C₄×C₈×D₂₉	C₂	semidirect
11. C₄×D₈×C₂₉	C₂	semidirect
19. #11745274	C₄	direct (N × H)
46. #90807	C₂×C₄	semidirect
51. D₈×C₂₉	C₂×C₄	semidirect
58. #90783	C₂×C₄	semidirect
71. Dic₁₁₆	C₂×C₄	semidirect
73. C₈×D₂₉	C₂×C₄	semidirect
111. C₄×C₂₉	#135	semidirect
118. Dic₂₉	#135	semidirect
126. C₄×D₈	D₂₉	semidirect
163. C₂₉	C₄×#135	semidirect
177. D₈	C₄×D₂₉	semidirect
239. C₄	#11745274	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11745274	2
C₄×#90807	2
C₄×Dic₁₁₆	2
#14271257	2
#14271254	2
C₄×#90783	2
D₈×Dic₂₉	2
#14271200	2
#14271156	2
C₄×C₈×D₂₉	2
C₄×D₈×C₂₉	2
C₄×#135	29

Subgroups — C4×#11745274 #16828259

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11745274 #16828259