Subgroups — #16828508

Order 3712.

402 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#90807	C₂	1	1	yes	maximal, characteristic
2	C₈×Dic₅₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#14271287	C₂	1	1	yes	maximal, characteristic, complemented
4	#14271241	C₂	1	1	yes	maximal, characteristic, complemented
5	#14271239	C₂	1	2	yes	maximal, complemented
6	#14271238	C₂	1	2	yes	maximal, complemented
7	#14271219	C₂	1	1	yes	maximal, characteristic, complemented
8	#14271217	C₂	1	2	yes	maximal, complemented
9	M₁₆×Dic₂₉	C₂	1	2	yes	maximal, complemented
10	C₄×#90772	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₉×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#90807	C₄	1	1	yes	characteristic
13	C₂×M₁₆×C₂₉	C₂²	1	2	yes	—
14	C₄×D₄×C₂₉	C₂²	1	1	yes	characteristic
15	D₄×Dic₂₉	C₄	1	1	yes	characteristic
16	C₄×C₂²⋊D₂₉	C₂²	1	2	yes	—
17	C₂×C₄×Dic₂₉	C₂²	1	2	yes	—
18	C₂×#90777	C₂²	1	2	yes	—
19	C₂×#90772	C₂²	1	1	yes	characteristic
20	C₂×#90772	C₂²	1	1	yes	characteristic
21	#11745254	C₄	1	1	yes	characteristic
22	#11745249	C₄	1	1	yes	characteristic
23	#11745243	C₄	1	2	yes	—
24	#11745239	C₄	1	2	yes	—
25	C₄²×D₂₉	C₂²	1	1	yes	characteristic
26	C₄×Dic₅₈	C₂²	1	1	yes	characteristic
27	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₂₉×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
30	C₂²⋊#6597	C₂²	1	2	yes	—
31	#11745168	C₂²	1	2	yes	—
32	#11745163	C₂²	1	1	yes	characteristic
33	#11745163	C₂²	1	1	yes	characteristic
34	#11745163	C₂²	1	2	yes	—
35	#11745162	C₂²	1	2	yes	—
36	#11745162	C₂²	1	2	yes	—
37	C₈×Dic₂₉	C₂²	1	2	yes	—
38	C₈×Dic₂₉	C₂²	1	2	yes	—
39	C₄⋊#6597	C₂²	1	1	yes	characteristic
40	C₄×#6597	C₂²	1	1	yes	characteristic
41	C₂×D₄×C₂₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
43	C₂×C₂²⋊D₂₉	C₂×C₄	1	2	yes	—
44	C₂²×Dic₂₉	C₂×C₄	1	2	yes	—
45	C₂²×Dic₂₉	C₂×C₄	1	2	yes	—
46	#90807	—	2	4	no	—
47	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
50	C₂×Dic₅₈	C₂×C₄	1	1	yes	characteristic
51	M₁₆×C₂₉	—	2	4	no	—
52	C₂×C₈×C₂₉	C₂³	1	2	yes	—
53	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
55	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
56	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
57	C₄²×C₂₉	C₂³	1	1	yes	characteristic
58	C₂²⋊Dic₂₉	C₂×C₄	1	2	yes	—
59	#90781	C₂×C₄	1	2	yes	—
60	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
61	#90779	C₂×C₄	1	2	yes	—
62	#90779	C₂×C₄	1	2	yes	—
63	C₄×Dic₂₉	C₂×C₄	1	2	yes	—
64	C₄×Dic₂₉	C₂×C₄	1	2	yes	—
65	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
67	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
68	C₄×Dic₂₉	C₂³	1	2	yes	—
69	#90777	—	2	4	no	—
70	C₂×#6597	C₂³	1	2	yes	—
71	C₂×#6597	C₂³	1	1	yes	characteristic
72	C₂×#6597	C₂³	1	1	yes	characteristic
73	#90772	—	2	2	no	—
74	#90772	—	2	2	no	—
75	C₂³×C₂₉	C₂²×C₄	1	2	yes	—
76	C₂²×D₂₉	C₂²×C₄	1	1	yes	characteristic
77	D₄×C₂₉	—	2	2	no	—
78	C₂×C₄×C₂₉	—	2	2	no	—
79	C₂×C₄×C₂₉	—	2	2	no	—
80	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
84	C₂²⋊D₂₉	—	2	4	no	—
85	C₂×Dic₂₉	—	2	4	no	—
86	C₂×Dic₂₉	—	4	2	no	—
87	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
90	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
91	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
92	C₄×D₂₉	—	2	2	no	—
93	C₄×D₂₉	—	2	1	no	—
94	C₄×D₂₉	—	2	1	no	—
95	C₄×D₂₉	—	4	1	no	—
96	Dic₅₈	M₁₆	1	4	yes	complemented
97	C₈×C₂₉	—	2	1	no	—
98	C₈×C₂₉	—	2	2	no	—
99	C₈×C₂₉	#41	1	2	yes	complemented
100	#6597	—	2	1	no	—
101	#6597	—	2	2	no	—
102	#6597	#41	1	2	yes	complemented
103	#2971	—	29	1	no	maximal, Sylow
104	C₂²×C₂₉	—	2	2	no	—
105	C₂²×C₂₉	—	2	2	no	—
106	C₂²×C₂₉	—	2	2	no	—
107	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
108	C₂×D₂₉	—	2	1	no	—
109	C₂×D₂₉	—	2	1	no	—
110	C₂×D₂₉	—	2	1	no	—
111	C₄×C₂₉	—	2	2	no	—
112	C₄×C₂₉	—	2	1	no	—
113	C₄×C₂₉	C₂×M₁₆	1	2	yes	—
114	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
115	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
116	Dic₂₉	—	2	1	no	—
117	Dic₂₉	—	2	1	no	—
118	Dic₂₉	—	2	1	no	—
119	Dic₂₉	—	2	2	no	—
120	Dic₂₉	C₂×M₁₆	1	4	yes	—
121	C₄×#41	—	29	1	no	—
122	C₈×Q₈	—	29	1	no	—
123	C₈×Q₈	—	29	1	no	—
124	#436	—	29	2	no	—
125	#436	—	29	1	no	—
126	#436	—	29	2	no	—
127	#436	D₂₉	1	1	yes	characteristic, complemented
128	#432	—	29	1	no	—
129	#432	—	29	2	no	—
130	C₄×M₁₆	—	29	2	no	—
131	C₄×M₁₆	—	29	1	no	—
132	C₂×C₂₉	—	4	2	no	—
133	C₂×C₂₉	C₄×#41	1	1	yes	characteristic
134	C₂×C₂₉	C₂²×M₁₆	1	1	yes	characteristic
135	C₂×C₂₉	#568	1	1	yes	characteristic
136	D₂₉	—	4	1	no	—
137	C₂×#41	—	29	1	no	—
138	C₂×M₁₆	—	29	2	no	—
139	C₂×M₁₆	—	29	1	no	—
140	C₂×M₁₆	—	29	1	no	—
141	C₂×M₁₆	C₂×D₂₉	1	2	yes	—
142	C₄×Q₈	—	29	1	no	—
143	C₄×Q₈	—	29	1	no	—
144	C₄×D₄	—	29	1	no	—
145	C₄×D₄	—	29	2	no	—
146	C₄×D₄	—	29	2	no	—
147	C₄×D₄	C₂×D₂₉	1	1	yes	characteristic
148	#117	—	29	1	no	—
149	#117	—	29	2	no	—
150	C₂×C₄²	—	29	2	no	—
151	C₂×C₄²	—	29	1	no	—
152	C₄⋊C₈	—	29	2	no	—
153	C₄⋊C₈	—	29	2	no	—
154	C₄⋊C₈	—	29	1	no	—
155	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic
156	C₂²⋊C₈	—	29	2	no	—
157	C₂²⋊C₈	—	29	2	no	—
158	C₂²⋊C₈	C₂×D₂₉	1	2	yes	—
159	#97	—	29	1	no	—
160	#97	—	29	2	no	—
161	#97	—	29	1	no	—
162	C₄×C₈	—	29	2	no	—
163	C₄×C₈	—	29	2	no	—
164	C₄×C₈	—	29	1	no	—
165	C₄×C₈	C₂×D₂₉	1	1	yes	characteristic
166	C₂₉	#2971	1	1	yes	characteristic, complemented, Sylow
167	#41	—	58	4	no	—
168	C₂×Q₈	—	29	1	no	—
169	C₂×D₄	—	29	2	no	—
170	C₂×D₄	C₄×D₂₉	1	1	yes	characteristic
171	C₂²×C₄	—	29	2	no	—
172	C₂²×C₄	—	29	2	no	—
173	C₂²×C₄	—	29	1	no	—
174	C₂²×C₄	—	29	1	no	—
175	C₂²×C₄	—	29	1	no	—
176	C₂²×C₄	C₂²×D₂₉	1	2	yes	—
177	M₁₆	—	2	4	no	—
178	M₁₆	—	58	2	no	—
179	M₁₆	—	58	4	no	—
180	M₁₆	—	58	2	no	—
181	C₂×C₈	—	29	1	no	—
182	C₂×C₈	—	29	1	no	—
183	C₂×C₈	—	29	2	no	—
184	C₂×C₈	C₂²×D₂₉	1	2	yes	—
185	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
186	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
187	C₄⋊C₄	—	29	2	no	—
188	C₄⋊C₄	—	29	2	no	—
189	C₄⋊C₄	—	29	1	no	—
190	C₄⋊C₄	C₄×D₂₉	1	1	yes	characteristic
191	C₂²⋊C₄	—	29	2	no	—
192	C₂²⋊C₄	—	29	2	no	—
193	C₂²⋊C₄	C₄×D₂₉	1	2	yes	—
194	C₄²	—	29	2	no	—
195	C₄²	—	29	1	no	—
196	C₄²	—	29	2	no	—
197	C₄²	—	29	1	no	—
198	C₄²	—	29	2	no	—
199	C₄²	—	29	1	no	—
200	C₄²	C₂²×D₂₉	1	1	yes	characteristic
201	C₂³	—	29	1	no	—
202	C₂³	C₂×C₄×D₂₉	1	2	yes	—
203	Q₈	—	29	4	no	—
204	D₄	—	2	2	no	—
205	D₄	—	58	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	29	2	no	—
209	C₂×C₄	—	29	1	no	—
210	C₂×C₄	—	29	1	no	—
211	C₂×C₄	—	29	1	no	—
212	C₂×C₄	—	29	2	no	—
213	C₂×C₄	—	58	4	no	—
214	C₂×C₄	—	58	1	no	—
215	C₂×C₄	—	58	1	no	—
216	C₂×C₄	—	58	2	no	—
217	C₂×C₄	—	116	1	no	—
218	C₂×C₄	—	116	2	no	—
219	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
220	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
221	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
222	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	2	1	no	—
225	C₈	—	29	2	no	—
226	C₈	—	58	1	no	—
227	C₈	—	58	2	no	—
228	C₈	#90807	1	2	yes	complemented
229	C₂²	—	2	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	58	1	no	—
233	C₂²	—	58	1	no	—
234	C₂²	—	58	1	no	—
235	C₂²	C₂²×C₄×D₂₉	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	2	1	no	—
238	C₄	—	29	4	no	—
239	C₄	—	58	1	no	—
240	C₄	—	58	1	no	—
241	C₄	—	58	1	no	—
242	C₄	—	58	2	no	—
243	C₄	M₁₆×D₂₉	1	2	yes	—
244	C₄	C₂×#90807	1	1	yes	characteristic
245	C₄	#41×D₂₉	1	1	yes	characteristic
246	C₂	—	4	2	no	—
247	C₂	—	116	1	no	—
248	C₂	C₄×#90807	1	1	yes	characteristic
249	C₂	C₂×M₁₆×D₂₉	1	1	yes	characteristic
250	C₂	#14271928	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₉
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting	C₂₉×#436	C₂
prime radical	C₂×#90807	C₄
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	#436	D₂₉
hypocenter	C₂₉	#2971
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
2. C₈×Dic₅₈	C₂	semidirect
3. #14271287	C₂	semidirect
4. #14271241	C₂	semidirect
5. #14271239	C₂	semidirect
6. #14271238	C₂	semidirect
7. #14271219	C₂	semidirect
8. #14271217	C₂	semidirect
9. M₁₆×Dic₂₉	C₂	semidirect
10. C₄×#90772	C₂	semidirect
11. C₂₉×#436	C₂	semidirect
96. Dic₅₈	M₁₆	semidirect
99. C₈×C₂₉	#41	semidirect
102. #6597	#41	semidirect
127. #436	D₂₉	semidirect
166. C₂₉	#2971	semidirect
228. C₈	#90807	semidirect

Maximal subgroups

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

subgroup	index
C₄×#90807	2
C₈×Dic₅₈	2
#14271287	2
#14271241	2
#14271239	2
#14271238	2
#14271219	2
#14271217	2
M₁₆×Dic₂₉	2
C₄×#90772	2
C₂₉×#436	2
#2971	29