Subgroups — #16836439

Order 3712.

364 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14272213	C₂	1	1	yes	maximal, characteristic, complemented
2	#14272212	C₂	1	1	yes	maximal, characteristic, complemented
3	#14272198	C₂	1	1	yes	maximal, characteristic, complemented
4	#14272194	C₂	1	2	yes	maximal, complemented
5	D₄×#6599	C₂	1	2	yes	maximal, complemented
6	D₄×#6599	C₂	1	1	yes	maximal, characteristic, complemented
7	#14272187	C₂	1	2	yes	maximal, complemented
8	#14272180	C₂	1	1	yes	maximal, characteristic, complemented
9	#14272169	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊#6599	C₂	1	1	yes	maximal, characteristic, complemented
11	#14271694	C₂	1	1	yes	maximal, characteristic
12	C₂×#90800	C₂²	1	2	yes	—
13	C₂²×#6599	C₂²	1	2	yes	—
14	C₂²×#6599	C₂²	1	2	yes	—
15	C₂₉×#121	C₄	1	1	yes	characteristic
16	#11745298	C₄	1	1	yes	characteristic
17	#11745297	C₄	1	2	yes	—
18	D₄×Dic₂₉	C₂²	1	2	yes	—
19	D₄×Dic₂₉	C₂²	1	1	yes	characteristic
20	#11745286	C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
22	#11745252	C₄	1	1	yes	characteristic
23	#11745249	C₂²	1	1	yes	characteristic
24	#11745241	C₄	1	2	yes	—
25	#11745239	C₂²	1	2	yes	—
26	C₂²⋊#6599	C₂²	1	1	yes	characteristic
27	C₂²⋊#6599	C₂²	1	1	yes	characteristic
28	C₂²⋊#6599	C₂²	1	2	yes	—
29	C₂²⋊#6599	C₂²	1	2	yes	—
30	#11745220	C₂²	1	1	yes	characteristic
31	#11745220	C₂²	1	2	yes	—
32	#11745220	C₂²	1	1	yes	characteristic
33	#11745218	C₂²	1	2	yes	—
34	C₄⋊#6599	C₂²	1	2	yes	—
35	C₄⋊#6599	C₂²	1	1	yes	characteristic
36	C₄⋊#6599	C₂²	1	1	yes	characteristic
37	C₄⋊#6599	C₂²	1	2	yes	—
38	C₄×#6599	C₂²	1	1	yes	characteristic
39	C₄×#6599	C₂²	1	1	yes	characteristic
40	C₄×#6599	C₂²	1	2	yes	—
41	C₂×D₄×C₂₉	C₂×C₄	1	2	yes	—
42	C₂×D₄×C₂₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×Dic₂₉	C₂×C₄	1	1	yes	characteristic
45	C₂²×Dic₂₉	C₂³	1	1	yes	characteristic
46	C₂²×Dic₂₉	C₂³	1	2	yes	—
47	C₂×Dic₅₈	C₂×C₄	1	1	yes	characteristic
48	#90800	—	4	2	no	—
49	C₂×#6599	—	4	2	no	—
50	C₂×#6599	—	4	2	no	—
51	C₂×#6599	C₂³	1	2	yes	—
52	C₂×#6599	C₂³	1	1	yes	characteristic
53	C₂×#6599	C₂³	1	2	yes	—
54	C₂×#6599	C₂³	1	1	yes	characteristic
55	C₂×#6599	C₂³	1	1	yes	characteristic
56	C₂×#6599	C₂³	1	1	yes	characteristic
57	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
58	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
59	C₂²⋊Dic₂₉	C₂×C₄	1	2	yes	—
60	C₂²⋊Dic₂₉	C₂×C₄	1	2	yes	—
61	C₂²⋊Dic₂₉	C₂×C₄	1	2	yes	—
62	C₂²⋊Dic₂₉	C₂×C₄	1	2	yes	—
63	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
64	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
65	#90779	C₂×C₄	1	2	yes	—
66	#90779	C₂×C₄	1	2	yes	—
67	#90779	C₂×C₄	1	2	yes	—
68	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
71	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
72	C₄×Dic₂₉	C₂³	1	2	yes	—
73	C₂³×C₂₉	C₂²×C₄	1	2	yes	—
74	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
75	D₄×C₂₉	—	2	4	no	—
76	D₄×C₂₉	—	2	2	no	—
77	C₂×C₄×C₂₉	—	2	2	no	—
78	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₂₉	—	2	2	no	—
82	C₂×Dic₂₉	—	2	1	no	—
83	C₂×Dic₂₉	—	2	2	no	—
84	C₂×Dic₂₉	—	2	1	no	—
85	C₂×Dic₂₉	—	2	2	no	—
86	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
90	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
91	C₂×Dic₂₉	C₂⁴	1	1	yes	characteristic
92	Dic₅₈	—	2	2	no	—
93	#6599	—	2	1	no	—
94	#6599	—	2	2	no	—
95	#6599	—	2	2	no	—
96	#6599	—	2	1	no	—
97	#6599	—	2	1	no	—
98	#6599	—	2	1	no	—
99	#2994	—	29	1	no	maximal, Sylow
100	C₂²×C₂₉	—	2	1	no	—
101	C₂²×C₂₉	—	2	1	no	—
102	C₂²×C₂₉	—	2	2	no	—
103	C₂²×C₂₉	—	2	2	no	—
104	C₂²×C₂₉	—	2	2	no	—
105	C₂²×C₂₉	#131	1	2	yes	—
106	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
107	C₄×C₂₉	—	2	1	no	—
108	C₄×C₂₉	—	2	2	no	—
109	C₄×C₂₉	#131	1	2	yes	—
110	Dic₂₉	—	2	1	no	—
111	Dic₂₉	—	2	2	no	—
112	Dic₂₉	—	2	2	no	—
113	Dic₂₉	—	2	1	no	—
114	Dic₂₉	—	2	1	no	—
115	Dic₂₉	H(5,𝔽₂)	1	1	yes	characteristic
116	Dic₂₉	2₋¹⁺⁴	1	1	yes	characteristic
117	#529	—	29	1	no	—
118	#446	—	29	1	no	—
119	C₈×Q₈	—	29	1	no	—
120	#436	—	29	1	no	—
121	#435	—	29	2	no	—
122	C₈×D₄	—	29	1	no	—
123	C₈×D₄	—	29	2	no	—
124	#433	—	29	2	no	—
125	#431	—	29	1	no	—
126	#424	—	29	2	no	—
127	C₂×C₄⋊C₈	—	29	1	no	—
128	C₂×C₂₉	—	2	2	no	—
129	C₂×C₂₉	—	4	2	no	—
130	C₂×C₂₉	#520	1	1	yes	characteristic
131	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
132	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
133	C₂×M₁₆	—	29	2	no	—
134	C₂²×C₈	—	29	2	no	—
135	C₂²×C₈	—	29	2	no	—
136	#126	—	29	2	no	—
137	#125	—	29	1	no	—
138	#124	—	29	1	no	—
139	#123	—	29	2	no	—
140	C₂²⋊Q₈	—	29	1	no	—
141	#121	#1181	1	1	yes	characteristic
142	C₄×Q₈	—	29	1	no	—
143	C₄×D₄	—	29	2	no	—
144	C₄×D₄	—	29	1	no	—
145	#117	—	29	2	no	—
146	C₂×C₄²	—	29	1	no	—
147	C₄⋊C₈	—	29	2	no	—
148	C₄⋊C₈	—	29	1	no	—
149	C₄⋊C₈	—	29	1	no	—
150	C₄⋊C₈	—	29	1	no	—
151	C₄⋊C₈	—	29	2	no	—
152	C₄⋊C₈	—	29	2	no	—
153	C₄⋊C₈	—	29	1	no	—
154	C₂²⋊C₈	—	29	1	no	—
155	C₂²⋊C₈	—	29	1	no	—
156	C₂²⋊C₈	—	29	2	no	—
157	C₂²⋊C₈	—	29	2	no	—
158	#97	—	29	2	no	—
159	C₄×C₈	—	29	1	no	—
160	C₄×C₈	—	29	1	no	—
161	C₄×C₈	—	29	2	no	—
162	C₂₉	#2994	1	1	yes	characteristic, complemented, Sylow
163	C₂×Q₈	—	29	1	no	—
164	C₂×D₄	C₂×#1181	1	2	yes	—
165	C₂×D₄	C₂×#1181	1	1	yes	characteristic
166	C₂²×C₄	—	29	1	no	—
167	C₂²×C₄	—	29	2	no	—
168	C₂²×C₄	—	29	1	no	—
169	C₂²×C₄	C₂×#1181	1	1	yes	characteristic
170	M₁₆	—	116	2	no	—
171	C₂×C₈	—	29	1	no	—
172	C₂×C₈	—	29	1	no	—
173	C₂×C₈	—	29	1	no	—
174	C₂×C₈	—	29	2	no	—
175	C₂×C₈	—	29	1	no	—
176	C₂×C₈	—	29	2	no	—
177	C₂×C₈	—	116	2	no	—
178	C₂×C₈	—	116	2	no	—
179	C₄⋊C₄	—	29	2	no	—
180	C₄⋊C₄	—	29	2	no	—
181	C₄⋊C₄	—	29	1	no	—
182	C₄⋊C₄	—	29	2	no	—
183	C₄⋊C₄	—	29	2	no	—
184	C₄⋊C₄	C₂×#1181	1	1	yes	characteristic
185	C₂²⋊C₄	—	29	2	no	—
186	C₂²⋊C₄	—	29	2	no	—
187	C₂²⋊C₄	—	29	2	no	—
188	C₂²⋊C₄	—	29	2	no	—
189	C₂²⋊C₄	C₂×#1181	1	2	yes	—
190	C₄²	—	29	1	no	—
191	C₄²	—	29	2	no	—
192	C₄²	—	29	1	no	—
193	C₄²	—	29	1	no	—
194	C₄²	—	29	1	no	—
195	C₂³	C₂²×#1181	1	2	yes	—
196	C₂³	C₂²×#1181	1	1	yes	characteristic
197	Q₈	—	58	2	no	—
198	D₄	—	2	4	no	—
199	D₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	29	1	no	—
202	C₂×C₄	—	29	1	no	—
203	C₂×C₄	—	29	1	no	—
204	C₂×C₄	—	29	1	no	—
205	C₂×C₄	—	29	2	no	—
206	C₂×C₄	—	29	2	no	—
207	C₂×C₄	—	58	1	no	—
208	C₂×C₄	—	58	2	no	—
209	C₂×C₄	—	58	1	no	—
210	C₂×C₄	—	58	2	no	—
211	C₂×C₄	—	58	2	no	—
212	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
213	C₂×C₄	C₂²×#1181	1	2	yes	—
214	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
215	C₈	—	58	1	no	—
216	C₈	—	58	2	no	—
217	C₈	—	58	2	no	—
218	C₈	—	58	1	no	—
219	C₈	—	58	1	no	—
220	C₈	—	58	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	#11745346	1	2	yes	—
227	C₂²	C₂³×#1181	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	2	no	—
230	C₄	—	29	1	no	—
231	C₄	—	29	1	no	—
232	C₄	—	58	1	no	—
233	C₄	—	58	2	no	—
234	C₄	—	58	1	no	—
235	C₄	—	58	1	no	—
236	C₄	—	58	2	no	—
237	C₄	#11745346	1	2	yes	—
238	C₂	—	2	2	no	—
239	C₂	—	4	2	no	—
240	C₂	#14272232	1	1	yes	characteristic
241	C₂	C₂×#11745346	1	1	yes	characteristic
242	C₂	C₂×#11745346	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#1181
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting, prime radical	C₂₉×#121	C₄
cosocle	C₂×Dic₂₉	C₂⁴
hypercenter	#121	#1181
hypocenter	C₂₉	#2994
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ #121

Complements

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

N	H ≅ G/N	type
1. #14272213	C₂	semidirect
2. #14272212	C₂	semidirect
3. #14272198	C₂	semidirect
4. #14272194	C₂	semidirect
5. D₄×#6599	C₂	semidirect
6. D₄×#6599	C₂	semidirect
7. #14272187	C₂	semidirect
8. #14272180	C₂	semidirect
9. #14272169	C₂	semidirect
10. C₂×C₄⋊#6599	C₂	semidirect
162. C₂₉	#2994	semidirect

Maximal subgroups

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

subgroup	index
#14272213	2
#14272212	2
#14272198	2
#14272194	2
D₄×#6599	2
#14272187	2
#14272180	2
#14272169	2
C₂×C₄⋊#6599	2
#14271694	2
#2994	29