Subgroups — C₄×#11745281 #16828264

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×#11745281	C₂	1	1	yes	maximal, characteristic
2	C₄×#90809	C₂	1	2	yes	maximal, complemented
3	Q₁₆×Dic₂₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#90784	C₂	1	2	yes	maximal, complemented
5	#14271248	C₂	1	2	yes	maximal, complemented
6	C₄×D₂₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	#14271245	C₂	1	1	yes	maximal, characteristic, complemented
8	#14271200	C₂	1	1	yes	maximal, characteristic, complemented
9	#14271158	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₂₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Q₁₆×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#90809	C₂²	1	2	yes	—
13	C₂×Q₁₆×C₂₉	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₂₉	C₂²	1	2	yes	—
15	Q₈×Dic₂₉	C₂²	1	2	yes	—
16	C₂×#90784	C₂²	1	2	yes	—
17	#11745281	C₄	1	8	yes	direct product factor
18	C₂×D₂₃₂	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₂₉	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₂₉	C₂²	1	1	yes	characteristic
21	#11745255	—	2	2	no	—
22	#11745255	C₂²	1	2	yes	—
23	#11745254	—	2	2	no	—
24	#11745254	C₂²	1	2	yes	—
25	C₄×D₁₁₆	—	2	2	no	—
26	C₄×D₁₁₆	C₂²	1	2	yes	—
27	C₄²×D₂₉	—	2	2	no	—
28	C₄²×D₂₉	C₂²	1	1	yes	characteristic
29	C₂₉×#107	C₂²	1	1	yes	characteristic
30	C₂₉×#103	C₂²	1	2	yes	—
31	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
32	#11745183	C₂²	1	2	yes	—
33	#11745169	C₂²	1	2	yes	—
34	#11745166	C₂²	1	1	yes	characteristic
35	C₈×Dic₂₉	C₂²	1	1	yes	characteristic
36	C₈×Dic₂₉	C₂²	1	1	yes	characteristic
37	#11745157	C₂²	1	2	yes	—
38	#11745156	C₂²	1	2	yes	—
39	C₄×#6597	C₂²	1	1	yes	characteristic
40	C₂×Q₈×C₂₉	C₂³	1	2	yes	—
41	#90809	—	2	4	no	—
42	#90809	C₂×C₄	1	8	yes	complemented
43	C₂×D₁₁₆	—	2	2	no	—
44	C₂×D₁₁₆	C₂³	1	2	yes	—
45	C₂×C₄×D₂₉	—	2	2	no	—
46	C₂×C₄×D₂₉	—	2	2	no	—
47	C₂×C₄×D₂₉	—	2	2	no	—
48	C₂×C₄×D₂₉	D₄	1	1	yes	characteristic
49	C₂×C₄×D₂₉	D₄	1	1	yes	characteristic
50	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
51	Q₁₆×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	—	2	2	no	—
55	C₄⋊C₄×C₂₉	C₂³	1	2	yes	—
56	C₄²×C₂₉	—	2	2	no	—
57	C₄²×C₂₉	C₂³	1	1	yes	characteristic
58	#90784	C₂×C₄	1	8	yes	complemented
59	#90781	—	2	2	no	—
60	#90781	—	2	2	no	—
61	#90781	—	2	2	no	—
62	C₄⋊Dic₂₉	—	2	2	no	—
63	C₄⋊Dic₂₉	C₂³	1	2	yes	—
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	D₂₃₂	C₂×C₄	1	4	yes	complemented
72	C₈×D₂₉	—	2	2	no	—
73	C₈×D₂₉	C₂×C₄	1	4	yes	complemented
74	C₂²×D₂₉	—	2	2	no	—
75	C₂²×D₂₉	C₂×D₄	1	1	yes	characteristic
76	Q₈×C₂₉	—	2	2	no	—
77	Q₈×C₂₉	C₂²×C₄	1	4	yes	—
78	C₂×C₄×C₂₉	—	2	2	no	—
79	C₂×C₄×C₂₉	—	2	2	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	C₂×Dic₂₉	—	2	2	no	—
84	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
85	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
86	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
87	D₁₁₆	—	2	8	no	—
88	D₁₁₆	—	2	2	no	—
89	D₁₁₆	C₂²×C₄	1	4	yes	—
90	C₄×D₂₉	—	2	2	no	—
91	C₄×D₂₉	—	2	8	no	—
92	C₄×D₂₉	—	2	2	no	—
93	C₄×D₂₉	—	4	4	no	—
94	C₄×D₂₉	—	4	4	no	—
95	C₄×D₂₉	C₂²×C₄	1	2	yes	—
96	C₄×D₂₉	#41	1	2	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	C₂²×D₄	1	1	yes	characteristic
103	C₂×D₂₉	—	2	1	no	—
104	C₂×D₂₉	—	2	1	no	—
105	C₂×D₂₉	—	2	4	no	—
106	C₂×D₂₉	—	4	2	no	—
107	C₂×D₂₉	—	4	2	no	—
108	C₂×D₂₉	C₄×D₄	1	2	yes	—
109	C₄×C₂₉	—	2	4	no	—
110	C₄×C₂₉	—	2	1	no	—
111	C₄×C₂₉	—	4	2	no	—
112	C₄×C₂₉	#135	1	2	yes	complemented
113	C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
114	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
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	D₂₉	1	1	yes	characteristic, complemented
123	C₄×SD₁₆	—	29	2	no	—
124	C₄×SD₁₆	—	29	2	no	—
125	C₄×D₈	—	29	1	no	—
126	C₄×D₈	—	29	1	no	—
127	#427	—	29	1	no	—
128	#416	—	29	2	no	—
129	C₂×C₄×C₈	—	29	1	no	—
130	C₂×C₂₉	C₂×C₄×D₄	1	1	yes	characteristic
131	C₂×C₂₉	C₂×#135	1	1	yes	characteristic
132	C₂×C₂₉	C₂×#135	1	1	yes	characteristic
133	D₂₉	—	2	2	no	—
134	D₂₉	—	4	4	no	—
135	C₂×#41	—	29	2	no	—
136	#135	—	29	8	no	—
137	C₂×Q₁₆	C₂×D₂₉	1	1	yes	characteristic
138	C₂×SD₁₆	—	29	2	no	—
139	C₂×D₈	—	29	1	no	—
140	C₂²×C₈	—	29	1	no	—
141	C₂²×C₈	—	29	1	no	—
142	C₄×Q₈	—	29	2	no	—
143	C₄×Q₈	C₂×D₂₉	1	2	yes	—
144	C₄×D₄	—	29	2	no	—
145	C₄×D₄	—	29	2	no	—
146	C₄×D₄	—	58	2	no	—
147	C₄×D₄	—	58	2	no	—
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	C₂×D₂₉	1	2	yes	—
157	#102	—	29	2	no	—
158	#102	—	29	2	no	—
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	C₂²×D₂₉	1	2	yes	—
167	C₂×D₄	—	29	2	no	—
168	C₂×D₄	—	58	2	no	—
169	C₂²×C₄	—	29	1	no	—
170	C₂²×C₄	—	29	1	no	—
171	C₂²×C₄	—	29	1	no	—
172	C₂²×C₄	—	58	2	no	—
173	C₂²×C₄	—	58	2	no	—
174	C₂²×C₄	—	58	2	no	—
175	Q₁₆	C₄×D₂₉	1	4	yes	complemented
176	SD₁₆	—	29	8	no	—
177	D₈	—	29	4	no	—
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	—	2	2	no	—
185	C₄⋊C₄	—	29	2	no	—
186	C₄⋊C₄	—	58	2	no	—
187	C₄⋊C₄	C₂²×D₂₉	1	2	yes	—
188	C₂²⋊C₄	—	58	2	no	—
189	C₂²⋊C₄	—	58	2	no	—
190	C₂²⋊C₄	—	58	2	no	—
191	C₄²	—	2	2	no	—
192	C₄²	—	29	1	no	—
193	C₄²	—	29	1	no	—
194	C₄²	—	29	1	no	—
195	C₄²	—	58	2	no	—
196	C₄²	—	58	2	no	—
197	C₄²	C₂²×D₂₉	1	1	yes	characteristic
198	C₂³	—	29	1	no	—
199	C₂³	—	58	2	no	—
200	Q₈	—	2	2	no	—
201	Q₈	C₂×C₄×D₂₉	1	4	yes	—
202	D₄	—	29	4	no	—
203	D₄	—	58	8	no	—
204	D₄	—	58	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	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	—	116	4	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	—	29	2	no	—
226	C₂²	—	58	1	no	—
227	C₂²	—	58	1	no	—
228	C₂²	—	58	4	no	—
229	C₂²	—	116	2	no	—
230	C₂²	—	116	2	no	—
231	C₂²	C₂×D₄×D₂₉	1	1	yes	characteristic
232	C₄	—	2	4	no	—
233	C₄	—	2	1	no	—
234	C₄	—	4	2	no	—
235	C₄	—	29	2	no	—
236	C₄	—	29	2	no	—
237	C₄	—	58	1	no	—
238	C₄	—	116	2	no	—
239	C₄	#11745281	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	—	58	2	no	—
243	C₂	—	116	4	no	—
244	C₂	C₄×D₄×D₂₉	1	1	yes	characteristic
245	C₂	#135×D₂₉	1	1	yes	characteristic
246	C₂	C₂×#11745281	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₂₉
commutator	C₄×C₂₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₉
Fitting	C₄×Q₁₆×C₂₉	C₂
socle	C₂²×C₂₉	C₂²×D₄
prime radical	C₂×#11745281	C₂
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	C₄×Q₁₆	D₂₉
hypocenter	C₂₉	C₄×#135
LCS	G ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. C₄×#90809	C₂	semidirect
3. Q₁₆×Dic₂₉	C₂	semidirect
4. C₄×#90784	C₂	semidirect
5. #14271248	C₂	semidirect
6. C₄×D₂₃₂	C₂	semidirect
7. #14271245	C₂	semidirect
8. #14271200	C₂	semidirect
9. #14271158	C₂	semidirect
10. C₄×C₈×D₂₉	C₂	semidirect
11. C₄×Q₁₆×C₂₉	C₂	semidirect
17. #11745281	C₄	direct (N × H)
42. #90809	C₂×C₄	semidirect
51. Q₁₆×C₂₉	C₂×C₄	semidirect
58. #90784	C₂×C₄	semidirect
71. D₂₃₂	C₂×C₄	semidirect
73. C₈×D₂₉	C₂×C₄	semidirect
112. C₄×C₂₉	#135	semidirect
118. Dic₂₉	#135	semidirect
122. C₄×Q₁₆	D₂₉	semidirect
163. C₂₉	C₄×#135	semidirect
175. Q₁₆	C₄×D₂₉	semidirect
239. C₄	#11745281	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×#11745281	2
C₄×#90809	2
Q₁₆×Dic₂₉	2
C₄×#90784	2
#14271248	2
C₄×D₂₃₂	2
#14271245	2
#14271200	2
#14271158	2
C₄×C₈×D₂₉	2
C₄×Q₁₆×C₂₉	2
C₄×#135	29

Subgroups — C4×#11745281 #16828264

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11745281 #16828264