Subgroups — #16828227

Order 3712.

544 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×M₁₆×D₂₉	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₁₁₆	C₂	1	1	yes	maximal, characteristic
3	#14271240	C₂	1	2	yes	maximal, complemented
4	#14271233	C₂	1	4	yes	maximal, complemented
5	#14271228	C₂	1	2	yes	maximal, complemented
6	#14271126	C₂	1	2	yes	maximal, complemented
7	C₄×#90777	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₉×#423	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₁₁₆	C₄	1	1	yes	characteristic
10	C₂²×C₄×D₂₉	C₂²	1	2	yes	—
11	C₂×M₁₆×C₂₉	C₂²	1	2	yes	—
12	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
13	C₂×#90781	C₄	1	2	yes	—
14	C₂×C₄⋊Dic₂₉	C₄	1	1	yes	characteristic
15	C₂×#90777	C₂²	1	1	yes	characteristic
16	C₂×#90777	C₂²	1	1	yes	characteristic
17	M₁₆×D₂₉	—	2	8	no	—
18	C₂×#90772	C₂²	1	4	yes	complemented
19	C₂×C₈×D₂₉	C₂²	1	4	yes	complemented
20	C₄×D₁₁₆	C₄	1	2	yes	—
21	C₄×D₁₁₆	C₄	1	1	yes	characteristic
22	C₄×D₁₁₆	C₄	1	1	yes	characteristic
23	C₄×D₁₁₆	C₂²	1	2	yes	—
24	C₄×D₁₁₆	C₂²	1	1	yes	characteristic
25	C₄×D₁₁₆	C₂²	1	1	yes	characteristic
26	C₂₉×C₄⋊C₈	C₂²	1	2	yes	complemented
27	C₂₉×C₄⋊C₈	C₂²	1	2	yes	complemented
28	#11745168	C₂²	1	4	yes	complemented
29	#11745168	C₂²	1	4	yes	complemented
30	#11745151	C₂²	1	2	yes	complemented
31	C₄×#6597	C₂²	1	2	yes	complemented
32	C₂³×D₂₉	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
36	C₂²×Dic₂₉	C₂×C₄	1	2	yes	—
37	C₂×D₁₁₆	—	2	4	no	—
38	C₂×D₁₁₆	C₂×C₄	1	2	yes	—
39	C₂×D₁₁₆	C₂×C₄	1	1	yes	characteristic
40	C₂×D₁₁₆	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×D₂₉	—	2	4	no	—
42	C₂×C₄×D₂₉	—	2	4	no	—
43	C₂×C₄×D₂₉	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₂₉	C₂×C₄	1	2	yes	—
45	C₂×C₄×D₂₉	C₂³	1	2	yes	—
46	C₂×C₄×D₂₉	C₂³	1	2	yes	—
47	M₁₆×C₂₉	—	2	4	no	—
48	C₂×C₈×C₂₉	C₂³	1	4	yes	—
49	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
50	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
51	C₄²×C₂₉	C₂³	1	1	yes	characteristic
52	C₄²×C₂₉	C₂³	1	1	yes	characteristic
53	#90781	C₂×C₄	1	2	yes	—
54	#90781	C₂×C₄	1	2	yes	—
55	#90781	C₂×C₄	1	2	yes	—
56	#90781	C₂×C₄	1	2	yes	—
57	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
58	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
59	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
60	#90777	—	2	2	no	—
61	#90777	D₄	1	4	yes	complemented
62	C₂×#6597	C₂³	1	2	yes	—
63	C₂×#6597	C₂³	1	2	yes	—
64	#90772	—	2	8	no	—
65	C₈×D₂₉	—	2	8	no	—
66	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×D₂₉	—	2	4	no	—
68	C₂²×D₂₉	—	2	4	no	—
69	C₂²×D₂₉	—	4	2	no	—
70	C₂²×D₂₉	C₂²×C₄	1	2	yes	—
71	C₂²×D₂₉	C₂²×C₄	1	2	yes	—
72	C₂×C₄×C₂₉	—	2	2	no	—
73	C₂×C₄×C₂₉	—	4	1	no	—
74	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₂₉	#41	1	2	yes	—
81	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
82	C₂×Dic₂₉	—	2	4	no	—
83	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
84	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
85	D₁₁₆	—	2	4	no	—
86	D₁₁₆	—	2	2	no	—
87	D₁₁₆	M₁₆	1	4	yes	complemented
88	C₄×D₂₉	—	2	4	no	—
89	C₄×D₂₉	—	2	4	no	—
90	C₄×D₂₉	—	2	2	no	—
91	C₄×D₂₉	—	2	2	no	—
92	C₄×D₂₉	—	2	2	no	—
93	C₄×D₂₉	—	2	2	no	—
94	C₈×C₂₉	—	2	4	no	—
95	#6597	—	2	2	no	—
96	#6597	C₂×D₄	1	4	yes	complemented
97	D₄×M₁₆	—	29	1	no	maximal, Sylow
98	C₂²×C₂₉	—	2	1	no	—
99	C₂²×C₂₉	—	2	1	no	—
100	C₂²×C₂₉	C₄×D₄	1	2	yes	—
101	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
102	C₂×D₂₉	—	2	2	no	—
103	C₂×D₂₉	—	2	2	no	—
104	C₂×D₂₉	—	2	2	no	—
105	C₂×D₂₉	—	2	2	no	—
106	C₂×D₂₉	—	2	2	no	—
107	C₂×D₂₉	—	4	8	no	—
108	C₂×D₂₉	C₂×M₁₆	1	4	yes	complemented
109	C₄×C₂₉	—	2	1	no	—
110	C₄×C₂₉	—	2	1	no	—
111	C₄×C₂₉	—	2	1	no	—
112	C₄×C₂₉	C₄×D₄	1	2	yes	—
113	C₄×C₂₉	C₂×M₁₆	1	2	yes	complemented
114	C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
115	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
116	Dic₂₉	—	2	2	no	—
117	Dic₂₉	—	2	2	no	—
118	C₂²×M₁₆	—	29	2	no	—
119	C₂×C₄×D₄	—	29	1	no	—
120	#436	—	29	2	no	—
121	#435	—	29	4	no	—
122	C₈×D₄	—	29	2	no	—
123	#423	D₂₉	1	1	yes	characteristic, complemented
124	#407	—	29	2	no	—
125	C₄×M₁₆	—	29	1	no	—
126	C₂×C₂₉	—	2	2	no	—
127	C₂×C₂₉	C₂×C₄×D₄	1	1	yes	characteristic
128	C₂×C₂₉	C₂²×M₁₆	1	1	yes	characteristic
129	C₂×C₂₉	#568	1	1	yes	characteristic
130	D₂₉	—	2	4	no	—
131	D₂₉	—	4	2	no	—
132	C₂²×D₄	—	29	1	no	—
133	C₂³×C₄	—	29	2	no	—
134	C₂×M₁₆	—	29	1	no	—
135	C₂×M₁₆	—	29	1	no	—
136	C₂×M₁₆	—	29	4	no	—
137	C₂×M₁₆	—	58	8	no	—
138	C₂×M₁₆	C₂×D₂₉	1	2	yes	—
139	C₂²×C₈	—	29	4	no	—
140	C₄×D₄	—	29	1	no	—
141	C₄×D₄	—	29	2	no	—
142	C₄×D₄	—	29	1	no	—
143	C₄×D₄	—	29	1	no	—
144	C₄×D₄	—	29	2	no	—
145	C₄×D₄	—	29	1	no	—
146	C₂×C₄⋊C₄	—	29	1	no	—
147	C₂×C₂²⋊C₄	—	29	2	no	—
148	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
149	C₄⋊C₈	C₂×D₂₉	1	2	yes	complemented
150	C₄⋊C₈	C₂×D₂₉	1	2	yes	complemented
151	C₂²⋊C₈	—	29	4	no	—
152	C₂²⋊C₈	—	29	4	no	—
153	#97	—	29	2	no	—
154	C₄×C₈	—	29	2	no	—
155	C₂₉	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
156	C₂⁴	—	29	2	no	—
157	C₂×D₄	—	29	1	no	—
158	C₂×D₄	—	29	2	no	—
159	C₂×D₄	—	29	1	no	—
160	C₂×D₄	—	58	4	no	—
161	C₂²×C₄	—	29	2	no	—
162	C₂²×C₄	—	29	2	no	—
163	C₂²×C₄	—	29	2	no	—
164	C₂²×C₄	—	29	2	no	—
165	C₂²×C₄	—	29	2	no	—
166	C₂²×C₄	—	58	4	no	—
167	C₂²×C₄	—	58	4	no	—
168	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
169	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
171	M₁₆	—	2	4	no	—
172	M₁₆	—	29	4	no	—
173	M₁₆	—	58	8	no	—
174	M₁₆	—	58	2	no	—
175	C₂×C₈	—	29	2	no	—
176	C₂×C₈	—	29	2	no	—
177	C₂×C₈	—	58	8	no	—
178	C₂×C₈	C₂²×D₂₉	1	4	yes	—
179	C₄⋊C₄	—	29	2	no	—
180	C₄⋊C₄	—	29	1	no	—
181	C₄⋊C₄	—	29	1	no	—
182	C₂²⋊C₄	—	29	2	no	—
183	C₂²⋊C₄	—	29	2	no	—
184	C₂²⋊C₄	—	29	2	no	—
185	C₂²⋊C₄	—	29	2	no	—
186	C₄²	C₄×D₂₉	1	1	yes	characteristic
187	C₄²	C₄×D₂₉	1	1	yes	characteristic
188	C₄²	C₂²×D₂₉	1	1	yes	characteristic
189	C₄²	C₂²×D₂₉	1	1	yes	characteristic
190	C₂³	—	29	2	no	—
191	C₂³	—	29	2	no	—
192	C₂³	—	58	4	no	—
193	C₂³	—	58	4	no	—
194	C₂³	—	116	2	no	—
195	C₂³	C₂×C₄×D₂₉	1	1	yes	characteristic
196	D₄	—	29	4	no	—
197	D₄	—	58	4	no	—
198	D₄	—	58	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	29	2	no	—
202	C₂×C₄	—	29	2	no	—
203	C₂×C₄	—	58	4	no	—
204	C₂×C₄	—	58	2	no	—
205	C₂×C₄	—	58	2	no	—
206	C₂×C₄	—	58	2	no	—
207	C₂×C₄	—	58	2	no	—
208	C₂×C₄	—	58	4	no	—
209	C₂×C₄	—	58	4	no	—
210	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
214	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
215	C₂×C₄	D₄×D₂₉	1	2	yes	—
216	C₂×C₄	#90809	1	2	yes	—
217	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
218	C₈	—	2	4	no	—
219	C₈	—	29	4	no	—
220	C₈	—	58	2	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	29	4	no	—
224	C₂²	—	58	2	no	—
225	C₂²	—	58	2	no	—
226	C₂²	—	58	2	no	—
227	C₂²	—	58	2	no	—
228	C₂²	—	58	2	no	—
229	C₂²	—	116	8	no	—
230	C₂²	#11745255	1	2	yes	—
231	C₂²	C₂²×C₄×D₂₉	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	58	2	no	—
236	C₄	—	58	2	no	—
237	C₄	#11745255	1	2	yes	—
238	C₄	M₁₆×D₂₉	1	2	yes	complemented
239	C₄	C₂×D₄×D₂₉	1	1	yes	characteristic
240	C₄	C₂×#90809	1	1	yes	characteristic
241	C₂	—	2	2	no	—
242	C₂	—	58	4	no	—
243	C₂	—	116	2	no	—
244	C₂	C₂×#11745255	1	1	yes	characteristic
245	C₂	C₂×M₁₆×D₂₉	1	1	yes	characteristic
246	C₂	#14271929	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₉
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting	C₂₉×#423	C₂
prime radical	C₂²×D₁₁₆	C₄
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	#423	D₂₉
hypocenter	C₂₉	D₄×M₁₆
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
1. C₂×M₁₆×D₂₉	C₂	semidirect
3. #14271240	C₂	semidirect
4. #14271233	C₂	semidirect
5. #14271228	C₂	semidirect
6. #14271126	C₂	semidirect
7. C₄×#90777	C₂	semidirect
8. C₂₉×#423	C₂	semidirect
18. C₂×#90772	C₂²	semidirect
19. C₂×C₈×D₂₉	C₂²	semidirect
26. C₂₉×C₄⋊C₈	C₂²	semidirect
27. C₂₉×C₄⋊C₈	C₂²	semidirect
28. #11745168	C₂²	semidirect
29. #11745168	C₂²	semidirect
30. #11745151	C₂²	semidirect
31. C₄×#6597	C₂²	semidirect
61. #90777	D₄	semidirect
87. D₁₁₆	M₁₆	semidirect
96. #6597	C₂×D₄	semidirect
108. C₂×D₂₉	C₂×M₁₆	semidirect
113. C₄×C₂₉	C₂×M₁₆	semidirect
123. #423	D₂₉	semidirect
149. C₄⋊C₈	C₂×D₂₉	semidirect
150. C₄⋊C₈	C₂×D₂₉	semidirect
155. C₂₉	D₄×M₁₆	semidirect
238. C₄	M₁₆×D₂₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×D₂₉	2
C₂×C₄×D₁₁₆	2
#14271240	2
#14271233	2
#14271228	2
#14271126	2
C₄×#90777	2
C₂₉×#423	2
D₄×M₁₆	29