Subgroups — C₄×#11745304 #16828311

Order 3712.

484 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11745304	C₂	1	1	yes	maximal, characteristic
2	C₄×#90805	C₂	1	1	yes	maximal, characteristic
3	#14271268	C₂	1	2	yes	maximal, complemented
4	#14271261	C₂	1	2	yes	maximal, complemented
5	C₄×#90785	C₂	1	2	yes	maximal, complemented
6	C₄×#90784	C₂	1	2	yes	maximal, complemented
7	#14271203	C₂	1	1	yes	maximal, characteristic, complemented
8	#14271172	C₂	1	1	yes	maximal, characteristic, complemented
9	#14271162	C₂	1	1	yes	maximal, characteristic
10	C₄×#90777	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×Q₈×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₂₉	C₂²	1	1	yes	characteristic
13	C₂×#90805	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₂₉	—	2	1	no	—
15	C₄×Q₈×C₂₉	—	2	1	no	—
16	C₄×Q₈×C₂₉	C₂²	1	2	yes	—
17	C₄×Q₈×C₂₉	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₂₉	—	2	1	no	—
19	C₂×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₂₉	—	2	1	no	—
21	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
22	C₂×#90785	C₂²	1	2	yes	—
23	#11745304	C₄	1	8	yes	direct product factor
24	C₂×#90784	C₂²	1	2	yes	—
25	C₄×C₂²⋊D₂₉	—	2	1	no	—
26	C₄×C₂²⋊D₂₉	—	2	1	no	—
27	#11745288	C₂²	1	1	yes	characteristic
28	C₂×#90777	C₂²	1	1	yes	characteristic
29	C₂×#90777	C₂²	1	1	yes	characteristic
30	C₄×D₁₁₆	C₂²	1	1	yes	characteristic
31	C₄×D₁₁₆	C₂²	1	1	yes	characteristic
32	#11745234	—	2	1	no	—
33	C₄²×D₂₉	—	2	1	no	—
34	C₄×Dic₅₈	C₂²	1	1	yes	characteristic
35	C₄×Dic₅₈	C₂²	1	1	yes	characteristic
36	#11745183	C₂²	1	2	yes	—
37	#11745183	C₂²	1	2	yes	—
38	#11745158	C₂²	1	2	yes	—
39	#11745157	C₂²	1	2	yes	—
40	#11745156	C₂²	1	2	yes	—
41	#11745155	C₂²	1	2	yes	—
42	#11745151	C₂²	1	2	yes	—
43	C₄×#6597	C₂²	1	2	yes	—
44	C₂×Q₈×C₂₉	—	2	1	no	—
45	C₂×Q₈×C₂₉	—	2	2	no	—
46	C₂×Q₈×C₂₉	C₂×C₄	1	4	yes	complemented
47	C₂×Q₈×C₂₉	C₂³	1	2	yes	—
48	C₂²×C₄×C₂₉	—	2	1	no	—
49	C₂²×C₄×C₂₉	—	2	1	no	—
50	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
53	C₂×C₂²⋊D₂₉	—	2	1	no	—
54	#90805	—	2	2	no	—
55	#90805	C₂×C₄	1	4	yes	—
56	C₂×D₁₁₆	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₂₉	—	2	1	no	—
58	C₂×C₄×D₂₉	—	2	1	no	—
59	C₂×C₄×D₂₉	—	2	1	no	—
60	C₂×Dic₅₈	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₂₉	—	2	2	no	—
62	C₄⋊C₄×C₂₉	—	2	2	no	—
63	C₄⋊C₄×C₂₉	—	2	1	no	—
64	C₄⋊C₄×C₂₉	C₂³	1	2	yes	—
65	C₄²×C₂₉	—	2	2	no	—
66	C₄²×C₂₉	—	2	2	no	—
67	C₄²×C₂₉	D₄	1	1	yes	characteristic
68	C₄²×C₂₉	D₄	1	1	yes	characteristic
69	C₄²×C₂₉	C₂³	1	1	yes	characteristic
70	C₄²×C₂₉	C₂³	1	1	yes	characteristic
71	C₂²⋊Dic₂₉	—	2	1	no	—
72	#90785	C₂×C₄	1	8	yes	complemented
73	#90784	C₂×C₄	1	8	yes	complemented
74	#90781	—	2	1	no	—
75	#90781	—	2	1	no	—
76	C₄⋊Dic₂₉	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₂₉	C₂³	1	1	yes	characteristic
78	#90779	—	2	1	no	—
79	#90779	—	2	1	no	—
80	C₄×Dic₂₉	—	2	1	no	—
81	C₄×Dic₂₉	—	2	1	no	—
82	C₄×Dic₂₉	—	2	1	no	—
83	#90777	—	2	2	no	—
84	#90777	C₂×C₄	1	4	yes	complemented
85	C₂×#6597	C₂³	1	2	yes	—
86	C₂×#6597	C₂³	1	2	yes	—
87	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
88	C₂²×D₂₉	—	2	1	no	—
89	Q₈×C₂₉	—	2	4	no	—
90	Q₈×C₂₉	—	2	2	no	—
91	Q₈×C₂₉	C₂²×C₄	1	4	yes	—
92	C₂×C₄×C₂₉	—	2	4	no	—
93	C₂×C₄×C₂₉	—	2	2	no	—
94	C₂×C₄×C₂₉	—	2	2	no	—
95	C₂×C₄×C₂₉	—	2	2	no	—
96	C₂×C₄×C₂₉	—	4	1	no	—
97	C₂×C₄×C₂₉	—	4	2	no	—
98	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₂₉	#41	1	2	yes	—
105	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
106	C₂²⋊D₂₉	—	2	4	no	—
107	C₂×Dic₂₉	—	2	1	no	—
108	C₂×Dic₂₉	—	2	1	no	—
109	C₂×Dic₂₉	—	2	1	no	—
110	D₁₁₆	—	2	1	no	—
111	D₁₁₆	C₂²×C₄	1	2	yes	—
112	C₄×D₂₉	—	2	4	no	—
113	C₄×D₂₉	—	4	2	no	—
114	C₄×D₂₉	—	4	2	no	—
115	Dic₅₈	—	2	1	no	—
116	Dic₅₈	C₂²×C₄	1	2	yes	—
117	#6597	—	2	2	no	—
118	#6597	C₂²×C₄	1	4	yes	—
119	C₄×#137	—	29	1	no	maximal, Sylow
120	C₂²×C₂₉	—	2	1	no	—
121	C₂²×C₂₉	—	2	1	no	—
122	C₂²×C₂₉	C₄×D₄	1	2	yes	—
123	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
124	C₂×D₂₉	—	2	2	no	—
125	C₂×D₂₉	—	4	1	no	—
126	C₂×D₂₉	—	4	1	no	—
127	C₄×C₂₉	—	2	4	no	—
128	C₄×C₂₉	—	2	1	no	—
129	C₄×C₂₉	—	2	1	no	—
130	C₄×C₂₉	—	2	1	no	—
131	C₄×C₂₉	—	4	2	no	—
132	C₄×C₂₉	C₄×D₄	1	2	yes	—
133	C₄×C₂₉	#137	1	2	yes	complemented
134	C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
135	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
136	Dic₂₉	—	2	2	no	—
137	Dic₂₉	—	4	1	no	—
138	Dic₂₉	—	4	1	no	—
139	C₂×#137	—	29	1	no	—
140	C₄×#41	—	29	1	no	—
141	C₂×C₄×Q₈	D₂₉	1	1	yes	characteristic, complemented
142	#441	—	29	2	no	—
143	#440	—	29	2	no	—
144	C₄×Q₁₆	—	29	2	no	—
145	C₄×SD₁₆	—	29	2	no	—
146	#428	—	29	1	no	—
147	#419	—	29	1	no	—
148	#417	—	29	1	no	—
149	C₄×M₁₆	—	29	1	no	—
150	C₂×C₂₉	—	2	2	no	—
151	C₂×C₂₉	C₂×C₄×D₄	1	1	yes	characteristic
152	C₂×C₂₉	C₂×#137	1	1	yes	characteristic
153	C₂×C₂₉	#575	1	1	yes	characteristic
154	D₂₉	—	4	2	no	—
155	C₂×#41	—	29	1	no	—
156	C₂²×Q₈	C₂×D₂₉	1	1	yes	characteristic
157	#137	—	29	8	no	—
158	C₂×Q₁₆	—	29	2	no	—
159	C₂×SD₁₆	—	29	2	no	—
160	C₂×M₁₆	—	29	1	no	—
161	C₂×M₁₆	—	29	1	no	—
162	C₄×Q₈	—	2	1	no	—
163	C₄×Q₈	—	2	1	no	—
164	C₄×Q₈	—	29	1	no	—
165	C₄×Q₈	—	29	1	no	—
166	C₄×Q₈	C₂×D₂₉	1	2	yes	—
167	C₄×Q₈	C₂×D₂₉	1	2	yes	—
168	C₄×D₄	—	29	1	no	—
169	C₄×D₄	—	29	1	no	—
170	C₄×D₄	—	58	1	no	—
171	C₄×D₄	—	58	1	no	—
172	#117	—	29	1	no	—
173	#117	—	58	1	no	—
174	C₂×C₄⋊C₄	—	2	1	no	—
175	C₂×C₄⋊C₄	C₂×D₂₉	1	1	yes	characteristic
176	C₂×C₄²	—	2	1	no	—
177	C₂×C₄²	—	58	1	no	—
178	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
179	#107	—	29	2	no	—
180	#106	—	29	2	no	—
181	#103	—	29	2	no	—
182	#103	—	29	2	no	—
183	#103	—	29	2	no	—
184	#102	—	29	2	no	—
185	#97	—	29	2	no	—
186	C₄×C₈	—	29	2	no	—
187	C₂₉	C₄×#137	1	1	yes	characteristic, complemented, Sylow
188	#41	—	29	4	no	—
189	#41	—	58	2	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	—	2	1	no	—
192	C₂×Q₈	—	29	1	no	—
193	C₂×Q₈	C₄×D₂₉	1	4	yes	complemented
194	C₂×Q₈	C₂²×D₂₉	1	2	yes	—
195	C₂×D₄	—	29	1	no	—
196	C₂×D₄	—	58	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	58	1	no	—
200	C₂²×C₄	—	58	1	no	—
201	C₂²×C₄	—	58	1	no	—
202	C₂²×C₄	C₂²⋊D₂₉	1	1	yes	characteristic
203	C₂²×C₄	C₂²⋊D₂₉	1	1	yes	characteristic
204	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
205	Q₁₆	—	29	8	no	—
206	SD₁₆	—	29	8	no	—
207	M₁₆	—	29	4	no	—
208	M₁₆	—	58	2	no	—
209	C₂×C₈	—	29	2	no	—
210	C₂×C₈	—	29	2	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	—	29	1	no	—
215	C₄⋊C₄	—	29	1	no	—
216	C₄⋊C₄	—	58	1	no	—
217	C₄⋊C₄	—	58	1	no	—
218	C₄⋊C₄	C₂²×D₂₉	1	2	yes	—
219	C₂²⋊C₄	—	58	1	no	—
220	C₂²⋊C₄	—	58	1	no	—
221	C₂²⋊C₄	—	58	1	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	58	1	no	—
225	C₄²	—	58	1	no	—
226	C₄²	—	58	1	no	—
227	C₄²	C₂²⋊D₂₉	1	1	yes	characteristic
228	C₄²	C₂²⋊D₂₉	1	1	yes	characteristic
229	C₄²	C₂²×D₂₉	1	1	yes	characteristic
230	C₄²	C₂²×D₂₉	1	1	yes	characteristic
231	C₂³	—	58	1	no	—
232	C₂³	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
233	Q₈	—	2	4	no	—
234	Q₈	—	2	2	no	—
235	Q₈	—	29	2	no	—
236	Q₈	—	58	1	no	—
237	Q₈	C₂×C₄×D₂₉	1	4	yes	—
238	D₄	—	29	2	no	—
239	D₄	—	58	4	no	—
240	D₄	—	58	1	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	58	1	no	—
248	C₂×C₄	—	58	4	no	—
249	C₂×C₄	—	58	1	no	—
250	C₂×C₄	—	58	1	no	—
251	C₂×C₄	—	116	2	no	—
252	C₂×C₄	—	116	2	no	—
253	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
254	C₂×C₄	#90805	1	2	yes	—
255	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
256	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
258	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
259	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
260	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
261	C₈	—	29	4	no	—
262	C₈	—	58	2	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	58	2	no	—
266	C₂²	—	116	1	no	—
267	C₂²	—	116	1	no	—
268	C₂²	C₄×C₂²⋊D₂₉	1	2	yes	—
269	C₂²	C₂²×C₂²⋊D₂₉	1	1	yes	characteristic
270	C₄	—	2	4	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	4	2	no	—
275	C₄	—	58	2	no	—
276	C₄	—	116	1	no	—
277	C₄	—	116	1	no	—
278	C₄	C₄×C₂²⋊D₂₉	1	2	yes	—
279	C₄	#11745304	1	2	yes	direct product factor
280	C₄	C₂²×C₄×D₂₉	1	1	yes	characteristic
281	C₄	C₂×#90805	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	116	2	no	—
284	C₂	C₂×C₄×C₂²⋊D₂₉	1	1	yes	characteristic
285	C₂	C₂×#11745304	1	1	yes	characteristic
286	C₂	#14271961	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₉
commutator	C₄×C₂₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₉
Fitting	C₂×C₄×Q₈×C₂₉	C₂
socle	C₂²×C₂₉	C₂²×D₄
prime radical	C₂×#90805	C₂²
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	C₂×C₄×Q₈	D₂₉
hypocenter	C₂₉	C₄×#137
LCS	G ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
3. #14271268	C₂	semidirect
4. #14271261	C₂	semidirect
5. C₄×#90785	C₂	semidirect
6. C₄×#90784	C₂	semidirect
7. #14271203	C₂	semidirect
8. #14271172	C₂	semidirect
10. C₄×#90777	C₂	semidirect
11. C₂×C₄×Q₈×C₂₉	C₂	semidirect
23. #11745304	C₄	direct (N × H)
46. C₂×Q₈×C₂₉	C₂×C₄	semidirect
72. #90785	C₂×C₄	semidirect
73. #90784	C₂×C₄	semidirect
84. #90777	C₂×C₄	semidirect
133. C₄×C₂₉	#137	semidirect
141. C₂×C₄×Q₈	D₂₉	semidirect
187. C₂₉	C₄×#137	semidirect
193. C₂×Q₈	C₄×D₂₉	semidirect
279. C₄	#11745304	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×#11745304	2
C₄×#90805	2
#14271268	2
#14271261	2
C₄×#90785	2
C₄×#90784	2
#14271203	2
#14271172	2
#14271162	2
C₄×#90777	2
C₂×C₄×Q₈×C₂₉	2
C₄×#137	29

Subgroups — C4×#11745304 #16828311

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11745304 #16828311