Subgroups — D₄×#90772 #16828223

Order 3712.

544 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#90772	C₂	1	2	yes	maximal, complemented
2	C₄×D₄×D₂₉	C₂	1	1	yes	maximal, characteristic
3	#14271242	C₂	1	1	yes	maximal, characteristic, complemented
4	#14271240	C₂	1	1	yes	maximal, characteristic, complemented
5	#14271234	C₂	1	2	yes	maximal, complemented
6	#14271231	C₂	1	2	yes	maximal, complemented
7	D₄×#6597	C₂	1	1	yes	maximal, characteristic, complemented
8	#14271185	C₂	1	1	yes	maximal, characteristic, complemented
9	#14271125	C₂	1	2	yes	maximal, complemented
10	C₄×#90772	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₂₉	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₂₉	C₂²	1	2	yes	—
14	C₂²×C₈×C₂₉	C₂²	1	2	yes	complemented
15	C₄×D₄×C₂₉	C₂²	1	1	yes	characteristic
16	D₄×Dic₂₉	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊D₂₉	C₂²	1	2	yes	—
18	C₂²×#6597	C₂²	1	2	yes	complemented
19	C₂×#90772	—	2	8	no	—
20	C₂×#90772	C₂²	1	1	yes	characteristic
21	C₂×#90772	C₂²	1	2	yes	—
22	C₂×#90772	C₂²	1	1	yes	characteristic
23	C₂×#90772	C₂²	1	2	yes	complemented
24	C₂×#90772	C₂²	1	2	yes	complemented
25	#11745255	C₄	1	1	yes	characteristic
26	C₄⋊C₄×D₂₉	C₄	1	1	yes	characteristic
27	#11745243	C₄	1	2	yes	—
28	C₂²⋊C₄×D₂₉	C₄	1	2	yes	—
29	C₄×D₁₁₆	C₂²	1	1	yes	characteristic
30	C₄²×D₂₉	C₂²	1	1	yes	characteristic
31	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
32	C₂₉×C₂²⋊C₈	C₂²	1	2	yes	complemented
33	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic, complemented
34	C₂²⋊#6597	C₂²	1	2	yes	complemented
35	#11745168	C₂²	1	2	yes	complemented
36	#11745168	C₂²	1	2	yes	complemented
37	#11745163	C₂²	1	1	yes	characteristic, complemented
38	#11745163	C₂²	1	1	yes	characteristic, complemented
39	#11745162	C₂²	1	2	yes	complemented
40	C₄⋊#6597	C₂²	1	1	yes	characteristic, complemented
41	C₄×#6597	C₂²	1	1	yes	characteristic, complemented
42	C₂³×D₂₉	C₂×C₄	1	2	yes	—
43	C₂×D₄×C₂₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
45	C₂×C₂²⋊D₂₉	C₂×C₄	1	2	yes	—
46	C₂²×Dic₂₉	C₂×C₄	1	2	yes	—
47	D₄×D₂₉	—	2	4	no	—
48	C₂×D₁₁₆	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₂₉	—	2	4	no	—
50	C₂×C₄×D₂₉	—	2	4	no	—
51	C₂×C₄×D₂₉	C₂×C₄	1	2	yes	—
52	C₂×C₄×D₂₉	C₂×C₄	1	2	yes	—
53	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₂₉	C₂³	1	2	yes	—
56	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₂₉	—	2	4	no	—
58	C₂×C₈×C₂₉	C₂³	1	2	yes	—
59	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
62	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
63	C₄²×C₂₉	C₂³	1	1	yes	characteristic
64	C₂²⋊Dic₂₉	C₂×C₄	1	2	yes	—
65	#90781	C₂×C₄	1	2	yes	—
66	#90781	C₂×C₄	1	2	yes	—
67	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
68	#90779	C₂×C₄	1	2	yes	—
69	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
72	C₂×#6597	—	2	4	no	—
73	C₂×#6597	C₂³	1	2	yes	—
74	C₂×#6597	C₂³	1	1	yes	characteristic
75	C₂×#6597	C₂³	1	1	yes	characteristic
76	#90772	—	2	4	no	—
77	#90772	—	2	2	no	—
78	#90772	—	2	4	no	—
79	#90772	—	2	4	no	—
80	#90772	D₄	1	4	yes	direct product factor
81	C₂³×C₂₉	C₂²×C₄	1	2	yes	—
82	C₂²×D₂₉	—	2	4	no	—
83	C₂²×D₂₉	—	2	4	no	—
84	C₂²×D₂₉	—	4	2	no	—
85	C₂²×D₂₉	C₂²×C₄	1	2	yes	—
86	C₂²×D₂₉	C₂²×C₄	1	1	yes	characteristic
87	D₄×C₂₉	M₁₆	1	4	yes	complemented
88	C₂×C₄×C₂₉	—	2	2	no	—
89	C₂×C₄×C₂₉	—	2	2	no	—
90	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
94	C₂²⋊D₂₉	—	2	4	no	—
95	C₂×Dic₂₉	—	2	4	no	—
96	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
98	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
99	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
100	D₁₁₆	—	2	2	no	—
101	C₄×D₂₉	—	2	4	no	—
102	C₄×D₂₉	—	2	2	no	—
103	C₄×D₂₉	—	2	2	no	—
104	C₄×D₂₉	—	2	2	no	—
105	C₄×D₂₉	—	2	4	no	—
106	C₄×D₂₉	—	4	1	no	—
107	C₄×D₂₉	C₂×D₄	1	2	yes	—
108	C₄×D₂₉	#41	1	2	yes	—
109	C₈×C₂₉	—	2	2	no	—
110	C₈×C₂₉	—	2	1	no	—
111	C₈×C₂₉	C₂×D₄	1	2	yes	complemented
112	#6597	—	2	2	no	—
113	#6597	—	2	1	no	—
114	#6597	C₂×D₄	1	2	yes	complemented
115	D₄×M₁₆	—	29	1	no	maximal, Sylow
116	C₂²×C₂₉	—	2	2	no	—
117	C₂²×C₂₉	—	2	2	no	—
118	C₂²×C₂₉	C₂×M₁₆	1	4	yes	complemented
119	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
120	C₂×D₂₉	—	2	1	no	—
121	C₂×D₂₉	—	2	1	no	—
122	C₂×D₂₉	—	2	2	no	—
123	C₂×D₂₉	—	2	2	no	—
124	C₂×D₂₉	—	2	2	no	—
125	C₂×D₂₉	—	4	8	no	—
126	C₂×D₂₉	C₄×D₄	1	2	yes	—
127	C₄×C₂₉	—	2	2	no	—
128	C₄×C₂₉	—	2	1	no	—
129	C₄×C₂₉	C₂×M₁₆	1	2	yes	complemented
130	C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
131	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
132	Dic₂₉	—	2	2	no	—
133	Dic₂₉	—	2	1	no	—
134	Dic₂₉	—	2	1	no	—
135	Dic₂₉	C₄×D₄	1	2	yes	—
136	C₂²×M₁₆	—	29	2	no	—
137	C₂×C₄×D₄	—	29	1	no	—
138	#436	—	29	1	no	—
139	#436	—	29	1	no	—
140	#435	—	29	2	no	—
141	#435	—	29	2	no	—
142	C₈×D₄	—	29	1	no	—
143	C₈×D₄	D₂₉	1	1	yes	characteristic, complemented
144	#423	—	29	1	no	—
145	#407	—	29	2	no	—
146	C₄×M₁₆	—	29	1	no	—
147	C₂×C₂₉	—	2	4	no	—
148	C₂×C₂₉	C₂×C₄×D₄	1	1	yes	characteristic
149	C₂×C₂₉	C₂²×M₁₆	1	1	yes	characteristic
150	C₂×C₂₉	#568	1	1	yes	characteristic
151	D₂₉	—	2	2	no	—
152	D₂₉	—	4	2	no	—
153	C₂²×D₄	—	29	1	no	—
154	C₂³×C₄	—	29	2	no	—
155	C₂×M₁₆	—	29	1	no	—
156	C₂×M₁₆	—	29	2	no	—
157	C₂×M₁₆	—	29	2	no	—
158	C₂×M₁₆	—	29	2	no	—
159	C₂×M₁₆	—	29	1	no	—
160	C₂×M₁₆	—	58	8	no	—
161	C₂²×C₈	—	29	2	no	—
162	C₂²×C₈	C₂×D₂₉	1	2	yes	complemented
163	C₄×D₄	—	29	1	no	—
164	C₄×D₄	—	29	2	no	—
165	C₄×D₄	—	29	1	no	—
166	C₄×D₄	—	29	2	no	—
167	C₄×D₄	—	29	1	no	—
168	C₄×D₄	C₂×D₂₉	1	1	yes	characteristic
169	C₂×C₄⋊C₄	—	29	1	no	—
170	C₂×C₂²⋊C₄	—	29	2	no	—
171	C₂×C₄²	—	29	1	no	—
172	C₄⋊C₈	—	29	2	no	—
173	C₄⋊C₈	—	29	1	no	—
174	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic, complemented
175	C₂²⋊C₈	—	29	2	no	—
176	C₂²⋊C₈	—	29	2	no	—
177	C₂²⋊C₈	—	29	2	no	—
178	C₂²⋊C₈	C₂×D₂₉	1	2	yes	complemented
179	#97	—	29	1	no	—
180	#97	—	29	1	no	—
181	C₄×C₈	—	29	1	no	—
182	C₄×C₈	C₂×D₂₉	1	1	yes	characteristic, complemented
183	C₂₉	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
184	C₂⁴	—	29	2	no	—
185	C₂×D₄	—	29	2	no	—
186	C₂×D₄	—	29	1	no	—
187	C₂×D₄	—	58	4	no	—
188	C₂×D₄	C₄×D₂₉	1	1	yes	characteristic
189	C₂²×C₄	—	29	1	no	—
190	C₂²×C₄	—	29	2	no	—
191	C₂²×C₄	—	29	1	no	—
192	C₂²×C₄	—	29	2	no	—
193	C₂²×C₄	—	29	2	no	—
194	C₂²×C₄	—	29	2	no	—
195	C₂²×C₄	—	29	1	no	—
196	C₂²×C₄	—	58	4	no	—
197	C₂²×C₄	—	58	4	no	—
198	C₂²×C₄	C₂²×D₂₉	1	2	yes	—
199	M₁₆	—	29	4	no	—
200	M₁₆	—	58	4	no	—
201	M₁₆	—	58	4	no	—
202	M₁₆	—	58	2	no	—
203	M₁₆	—	58	4	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	29	1	no	—
206	C₂×C₈	—	29	1	no	—
207	C₂×C₈	—	29	2	no	—
208	C₂×C₈	—	58	4	no	—
209	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
210	C₂×C₈	C₂²×D₂₉	1	2	yes	—
211	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
212	C₄⋊C₄	—	29	2	no	—
213	C₄⋊C₄	—	29	1	no	—
214	C₄⋊C₄	C₄×D₂₉	1	1	yes	characteristic
215	C₂²⋊C₄	—	29	2	no	—
216	C₂²⋊C₄	—	29	2	no	—
217	C₂²⋊C₄	—	29	2	no	—
218	C₂²⋊C₄	C₄×D₂₉	1	2	yes	—
219	C₄²	—	29	1	no	—
220	C₄²	—	29	1	no	—
221	C₄²	—	29	1	no	—
222	C₄²	C₂²×D₂₉	1	1	yes	characteristic
223	C₂³	—	29	1	no	—
224	C₂³	—	29	2	no	—
225	C₂³	—	58	4	no	—
226	C₂³	—	58	4	no	—
227	C₂³	—	116	2	no	—
228	C₂³	C₂×C₄×D₂₉	1	2	yes	—
229	D₄	—	58	4	no	—
230	D₄	—	58	2	no	—
231	D₄	#90772	1	4	yes	direct product factor
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	29	1	no	—
235	C₂×C₄	—	29	2	no	—
236	C₂×C₄	—	29	2	no	—
237	C₂×C₄	—	29	1	no	—
238	C₂×C₄	—	29	1	no	—
239	C₂×C₄	—	29	2	no	—
240	C₂×C₄	—	58	4	no	—
241	C₂×C₄	—	58	4	no	—
242	C₂×C₄	—	58	4	no	—
243	C₂×C₄	—	58	2	no	—
244	C₂×C₄	—	58	2	no	—
245	C₂×C₄	—	58	2	no	—
246	C₂×C₄	—	116	1	no	—
247	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
249	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
250	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	2	1	no	—
253	C₈	—	29	2	no	—
254	C₈	—	58	2	no	—
255	C₈	—	58	1	no	—
256	C₈	D₄×D₂₉	1	2	yes	complemented
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	29	2	no	—
260	C₂²	—	58	1	no	—
261	C₂²	—	58	1	no	—
262	C₂²	—	58	2	no	—
263	C₂²	—	58	2	no	—
264	C₂²	—	58	2	no	—
265	C₂²	—	116	8	no	—
266	C₂²	C₂×#90772	1	4	yes	complemented
267	C₂²	C₂²×C₄×D₂₉	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	—	29	2	no	—
271	C₄	—	58	2	no	—
272	C₄	—	58	1	no	—
273	C₄	—	58	1	no	—
274	C₄	C₂×#90772	1	2	yes	complemented
275	C₄	C₂×D₄×D₂₉	1	1	yes	characteristic
276	C₄	#41×D₂₉	1	1	yes	characteristic
277	C₂	—	2	4	no	—
278	C₂	—	58	2	no	—
279	C₂	—	116	2	no	—
280	C₂	C₄×D₄×D₂₉	1	1	yes	characteristic
281	C₂	C₂²×#90772	1	1	yes	characteristic
282	C₂	#14271928	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₉
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting	C₈×D₄×C₂₉	C₂
prime radical	C₂×D₄×D₂₉	C₄
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	C₈×D₄	D₂₉
hypocenter	C₂₉	D₄×M₁₆
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#90772	C₂	semidirect
3. #14271242	C₂	semidirect
4. #14271240	C₂	semidirect
5. #14271234	C₂	semidirect
6. #14271231	C₂	semidirect
7. D₄×#6597	C₂	semidirect
8. #14271185	C₂	semidirect
9. #14271125	C₂	semidirect
10. C₄×#90772	C₂	semidirect
11. C₈×D₄×C₂₉	C₂	semidirect
14. C₂²×C₈×C₂₉	C₂²	semidirect
18. C₂²×#6597	C₂²	semidirect
23. C₂×#90772	C₂²	semidirect
24. C₂×#90772	C₂²	semidirect
31. C₂₉×C₄⋊C₈	C₂²	semidirect
32. C₂₉×C₂²⋊C₈	C₂²	semidirect
33. C₄×C₈×C₂₉	C₂²	semidirect
34. C₂²⋊#6597	C₂²	semidirect
35. #11745168	C₂²	semidirect
36. #11745168	C₂²	semidirect
37. #11745163	C₂²	semidirect
38. #11745163	C₂²	semidirect
39. #11745162	C₂²	semidirect
40. C₄⋊#6597	C₂²	semidirect
41. C₄×#6597	C₂²	semidirect
80. #90772	D₄	direct (N × H)
87. D₄×C₂₉	M₁₆	semidirect
111. C₈×C₂₉	C₂×D₄	semidirect
114. #6597	C₂×D₄	semidirect
118. C₂²×C₂₉	C₂×M₁₆	semidirect
129. C₄×C₂₉	C₂×M₁₆	semidirect
143. C₈×D₄	D₂₉	semidirect
162. C₂²×C₈	C₂×D₂₉	semidirect
174. C₄⋊C₈	C₂×D₂₉	semidirect
178. C₂²⋊C₈	C₂×D₂₉	semidirect
182. C₄×C₈	C₂×D₂₉	semidirect
183. C₂₉	D₄×M₁₆	semidirect
231. D₄	#90772	direct (N × H)
256. C₈	D₄×D₂₉	semidirect
266. C₂²	C₂×#90772	semidirect
274. C₄	C₂×#90772	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#90772	2
C₄×D₄×D₂₉	2
#14271242	2
#14271240	2
#14271234	2
#14271231	2
D₄×#6597	2
#14271185	2
#14271125	2
C₄×#90772	2
C₈×D₄×C₂₉	2
D₄×M₁₆	29

Subgroups — D4×#90772 #16828223

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#90772 #16828223