Subgroups — #16828444

Order 3712.

388 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×D₂₉	C₂	1	1	yes	maximal, characteristic
2	#14271294	C₂	1	1	yes	maximal, characteristic, complemented
3	#14271292	C₂	1	2	yes	maximal, complemented
4	#14271291	C₂	1	2	yes	maximal, complemented
5	C₈×Dic₅₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#14271287	C₂	1	1	yes	maximal, characteristic, complemented
7	#14271185	C₂	1	1	yes	maximal, characteristic
8	#14271185	C₂	1	2	yes	maximal
9	#14271115	C₂	1	2	yes	maximal
10	C₄×#90772	C₂	1	1	yes	maximal, characteristic
11	C₂₉×#446	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₂₉	C₄	1	1	yes	characteristic
13	C₄×Q₈×C₂₉	C₂²	1	1	yes	characteristic
14	Q₈×Dic₂₉	C₄	1	1	yes	characteristic
15	C₂×#90772	C₂²	1	1	yes	characteristic
16	C₂×#90772	C₂²	1	2	yes	—
17	C₂×#90772	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×D₂₉	C₄	1	1	yes	characteristic
19	C₄⋊C₄×D₂₉	C₄	1	2	yes	—
20	#11745249	C₄	1	2	yes	—
21	#11745249	C₄	1	1	yes	characteristic
22	C₄²×D₂₉	C₂²	1	1	yes	characteristic
23	C₄²×D₂₉	C₂²	1	2	yes	—
24	C₄×Dic₅₈	C₂²	1	1	yes	characteristic
25	C₄×Dic₅₈	C₂²	1	2	yes	—
26	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₂₉×C₄⋊C₈	C₂²	1	2	yes	—
28	C₂₉×#97	C₂²	1	2	yes	—
29	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
30	#11745163	C₂²	1	1	yes	characteristic
31	#11745163	C₂²	1	1	yes	characteristic
32	#11745162	C₂²	1	2	yes	—
33	#11745162	C₂²	1	2	yes	—
34	#11745162	C₂²	1	2	yes	—
35	C₈×Dic₂₉	C₂²	1	2	yes	—
36	C₈×Dic₂₉	C₂²	1	2	yes	—
37	C₄⋊#6597	C₂²	1	1	yes	characteristic
38	C₄⋊#6597	C₂²	1	2	yes	—
39	#11745151	C₂²	1	2	yes	—
40	C₄×#6597	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₂₉	C₂×C₄	1	1	yes	characteristic
42	Q₈×D₂₉	—	2	4	no	—
43	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₂₉	C₂×C₄	1	2	yes	—
45	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₂₉	C₂×C₄	1	2	yes	—
47	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
48	C₂×Dic₅₈	C₂×C₄	1	2	yes	—
49	C₂×Dic₅₈	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×C₂₉	C₂³	1	2	yes	—
51	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
53	C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
54	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
55	C₄²×C₂₉	C₂³	1	1	yes	characteristic
56	C₄²×C₂₉	C₂³	1	2	yes	—
57	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
58	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
59	#90779	C₂×C₄	1	2	yes	—
60	#90779	C₂×C₄	1	2	yes	—
61	#90779	C₂×C₄	1	2	yes	—
62	C₄×Dic₂₉	C₂×C₄	1	2	yes	—
63	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₂₉	C₂×C₄	1	2	yes	—
66	C₄×Dic₂₉	C₂³	1	2	yes	—
67	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
68	C₂×#6597	C₂³	1	1	yes	characteristic
69	C₂×#6597	C₂³	1	2	yes	—
70	C₂×#6597	C₂³	1	1	yes	characteristic
71	#90772	—	2	2	no	—
72	#90772	—	2	4	no	—
73	#90772	Q₈	1	4	yes	complemented
74	C₂²×D₂₉	C₂²×C₄	1	1	yes	characteristic
75	Q₈×C₂₉	—	2	2	no	—
76	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
81	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
82	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
86	C₄×D₂₉	—	2	2	no	—
87	C₄×D₂₉	—	2	4	no	—
88	C₄×D₂₉	—	4	1	no	—
89	C₄×D₂₉	—	4	2	no	—
90	C₄×D₂₉	C₂×Q₈	1	2	yes	—
91	C₄×D₂₉	#41	1	2	yes	—
92	Dic₅₈	—	2	4	no	—
93	Dic₅₈	M₁₆	1	4	yes	complemented
94	C₈×C₂₉	—	2	2	no	—
95	C₈×C₂₉	—	2	1	no	—
96	C₈×C₂₉	C₂×Q₈	1	2	yes	complemented
97	#6597	—	2	1	no	—
98	#6597	—	2	2	no	—
99	#6597	C₂×Q₈	1	2	yes	complemented
100	Q₈×M₁₆	—	29	1	no	maximal, Sylow
101	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
102	C₂×D₂₉	—	2	1	no	—
103	C₂×D₂₉	—	2	1	no	—
104	C₂×D₂₉	C₄×Q₈	1	2	yes	—
105	C₄×C₂₉	—	2	2	no	—
106	C₄×C₂₉	—	2	1	no	—
107	C₄×C₂₉	—	2	2	no	—
108	C₄×C₂₉	C₂×M₁₆	1	2	yes	—
109	C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
110	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
111	Dic₂₉	—	2	1	no	—
112	Dic₂₉	—	2	2	no	—
113	Dic₂₉	—	2	1	no	—
114	Dic₂₉	C₄×Q₈	1	2	yes	—
115	Dic₂₉	C₂×M₁₆	1	4	yes	—
116	C₂×C₄×Q₈	—	29	1	no	—
117	#446	—	29	1	no	—
118	#446	—	29	2	no	—
119	#446	—	29	2	no	—
120	#446	D₂₉	1	1	yes	characteristic, complemented
121	C₈×Q₈	—	29	1	no	—
122	C₈×Q₈	—	29	1	no	—
123	#423	—	29	2	no	—
124	#423	—	29	1	no	—
125	C₄×M₁₆	—	29	2	no	—
126	C₄×M₁₆	—	29	1	no	—
127	C₂×C₂₉	C₂×C₄×Q₈	1	1	yes	characteristic
128	C₂×C₂₉	C₂²×M₁₆	1	1	yes	characteristic
129	C₂×C₂₉	#568	1	1	yes	characteristic
130	D₂₉	—	2	2	no	—
131	C₂²×Q₈	—	29	1	no	—
132	C₂×M₁₆	—	29	1	no	—
133	C₂×M₁₆	—	29	2	no	—
134	C₂×M₁₆	—	29	1	no	—
135	C₄×Q₈	—	29	1	no	—
136	C₄×Q₈	—	29	1	no	—
137	C₄×Q₈	—	29	2	no	—
138	C₄×Q₈	—	29	1	no	—
139	C₄×Q₈	—	29	2	no	—
140	C₄×Q₈	C₂×D₂₉	1	1	yes	characteristic
141	C₂×C₄⋊C₄	—	29	2	no	—
142	C₂×C₄⋊C₄	—	29	1	no	—
143	C₂×C₄²	—	29	2	no	—
144	C₂×C₄²	—	29	1	no	—
145	C₄⋊C₈	—	29	2	no	—
146	C₄⋊C₈	—	29	2	no	—
147	C₄⋊C₈	—	29	2	no	—
148	C₄⋊C₈	—	29	1	no	—
149	C₄⋊C₈	—	29	2	no	—
150	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic
151	C₄⋊C₈	C₂×D₂₉	1	2	yes	—
152	#97	—	29	1	no	—
153	#97	—	29	1	no	—
154	#97	—	29	2	no	—
155	#97	C₂×D₂₉	1	2	yes	—
156	C₄×C₈	—	29	2	no	—
157	C₄×C₈	—	29	2	no	—
158	C₄×C₈	—	29	1	no	—
159	C₄×C₈	C₂×D₂₉	1	1	yes	characteristic
160	C₂₉	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
161	C₂×Q₈	—	29	1	no	—
162	C₂×Q₈	—	29	2	no	—
163	C₂×Q₈	—	58	4	no	—
164	C₂×Q₈	C₄×D₂₉	1	1	yes	characteristic
165	C₂²×C₄	—	29	1	no	—
166	C₂²×C₄	—	29	2	no	—
167	C₂²×C₄	—	29	1	no	—
168	C₂²×C₄	—	29	2	no	—
169	C₂²×C₄	—	29	1	no	—
170	M₁₆	—	29	4	no	—
171	M₁₆	—	58	4	no	—
172	M₁₆	—	58	2	no	—
173	C₂×C₈	—	29	1	no	—
174	C₂×C₈	—	29	2	no	—
175	C₂×C₈	—	29	1	no	—
176	C₂×C₈	C₂²×D₂₉	1	2	yes	—
177	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
179	C₄⋊C₄	—	29	2	no	—
180	C₄⋊C₄	—	29	2	no	—
181	C₄⋊C₄	—	29	2	no	—
182	C₄⋊C₄	—	29	2	no	—
183	C₄⋊C₄	—	29	1	no	—
184	C₄⋊C₄	C₄×D₂₉	1	2	yes	—
185	C₄⋊C₄	C₄×D₂₉	1	1	yes	characteristic
186	C₄²	—	29	2	no	—
187	C₄²	—	29	2	no	—
188	C₄²	—	29	1	no	—
189	C₄²	—	29	1	no	—
190	C₄²	—	29	2	no	—
191	C₄²	—	29	1	no	—
192	C₄²	C₂²×D₂₉	1	1	yes	characteristic
193	C₄²	C₂²×D₂₉	1	2	yes	—
194	C₂³	—	29	1	no	—
195	Q₈	—	2	2	no	—
196	Q₈	—	29	4	no	—
197	Q₈	—	58	4	no	—
198	C₂×C₄	—	29	1	no	—
199	C₂×C₄	—	29	2	no	—
200	C₂×C₄	—	29	2	no	—
201	C₂×C₄	—	29	2	no	—
202	C₂×C₄	—	29	2	no	—
203	C₂×C₄	—	29	1	no	—
204	C₂×C₄	—	29	1	no	—
205	C₂×C₄	—	58	4	no	—
206	C₂×C₄	—	58	2	no	—
207	C₂×C₄	—	116	1	no	—
208	C₂×C₄	—	116	2	no	—
209	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
211	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
212	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
213	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
214	C₈	—	2	2	no	—
215	C₈	—	2	1	no	—
216	C₈	—	29	2	no	—
217	C₈	—	58	2	no	—
218	C₈	—	58	1	no	—
219	C₈	Q₈×D₂₉	1	2	yes	complemented
220	C₂²	—	29	2	no	—
221	C₂²	—	58	1	no	—
222	C₂²	—	58	1	no	—
223	C₂²	C₂²×C₄×D₂₉	1	1	yes	characteristic
224	C₄	—	2	2	no	—
225	C₄	—	2	1	no	—
226	C₄	—	2	2	no	—
227	C₄	—	29	2	no	—
228	C₄	—	29	4	no	—
229	C₄	—	58	1	no	—
230	C₄	—	58	2	no	—
231	C₄	—	58	1	no	—
232	C₄	M₁₆×D₂₉	1	2	yes	—
233	C₄	C₂×Q₈×D₂₉	1	1	yes	characteristic
234	C₄	#41×D₂₉	1	1	yes	characteristic
235	C₂	—	58	2	no	—
236	C₂	C₄×Q₈×D₂₉	1	1	yes	characteristic
237	C₂	C₂×M₁₆×D₂₉	1	1	yes	characteristic
238	C₂	#14271928	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₉
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting	C₂₉×#446	C₂
prime radical	C₂²×D₂₉	C₂²×C₄
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	#446	D₂₉
hypocenter	C₂₉	Q₈×M₁₆
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #14271294	C₂	semidirect
3. #14271292	C₂	semidirect
4. #14271291	C₂	semidirect
5. C₈×Dic₅₈	C₂	semidirect
6. #14271287	C₂	semidirect
11. C₂₉×#446	C₂	semidirect
73. #90772	Q₈	semidirect
93. Dic₅₈	M₁₆	semidirect
96. C₈×C₂₉	C₂×Q₈	semidirect
99. #6597	C₂×Q₈	semidirect
120. #446	D₂₉	semidirect
160. C₂₉	Q₈×M₁₆	semidirect
219. C₈	Q₈×D₂₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×D₂₉	2
#14271294	2
#14271292	2
#14271291	2
C₈×Dic₅₈	2
#14271287	2
#14271185	2
#14271115	2
C₄×#90772	2
C₂₉×#446	2
Q₈×M₁₆	29