Subgroups — #16836269

Order 3712.

452 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14272242	C₂	1	2	yes	maximal, complemented
2	#14272208	C₂	1	4	yes	maximal, complemented
3	#14272207	C₂	1	2	yes	maximal, complemented
4	#14272205	C₂	1	2	yes	maximal, complemented
5	C₂×#11745212	C₂	1	1	yes	maximal, characteristic, complemented
6	#14272160	C₂	1	2	yes	maximal, complemented
7	#14272126	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₂₉	C₂	1	1	yes	maximal, characteristic
9	#41×D₂₉	C₂²	1	2	yes	—
10	#11745348	C₂²	1	4	yes	complemented
11	#11745346	—	2	4	no	—
12	#11745346	C₂²	1	4	yes	complemented
13	#11745345	C₂²	1	1	yes	characteristic
14	D₂₉⋊M₁₆	—	2	2	no	—
15	D₂₉⋊M₁₆	C₂²	1	1	yes	characteristic
16	C₂×#90796	—	2	2	no	—
17	C₂×#90796	C₂²	1	2	yes	—
18	C₂₉×#135	C₄	1	1	yes	characteristic, complemented
19	#11745312	C₄	1	2	yes	complemented
20	#11745281	C₂²	1	1	yes	characteristic
21	Q₁₆×D₂₉	C₄	1	1	yes	characteristic, complemented
22	#11745278	C₂²	1	2	yes	—
23	SD₁₆×D₂₉	C₄	1	2	yes	complemented
24	#11745274	C₂²	1	1	yes	characteristic
25	D₈×D₂₉	C₄	1	1	yes	characteristic, complemented
26	#11745266	C₄	1	1	yes	characteristic, complemented
27	C₂×C₈×D₂₉	C₂²	1	1	yes	characteristic
28	#11745226	C₂²	1	4	yes	complemented
29	#11745224	C₂²	1	4	yes	complemented
30	#11745212	C₂²	1	2	yes	complemented
31	#11745212	C₂²	1	2	yes	complemented
32	#11745208	C₂²	1	2	yes	complemented
33	C₈×#1181	C₂²	1	2	yes	complemented
34	#41×C₂₉	C₂×C₄	1	2	yes	—
35	#90809	C₂³	1	2	yes	—
36	Q₈×D₂₉	C₂×C₄	1	2	yes	—
37	#90807	—	2	2	no	—
38	#90807	C₂³	1	2	yes	—
39	D₄×D₂₉	—	2	2	no	—
40	D₄×D₂₉	C₂×C₄	1	2	yes	—
41	#90805	—	2	2	no	—
42	#90805	C₂×C₄	1	2	yes	—
43	C₂×C₄×D₂₉	—	2	2	no	—
44	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
45	C₂²⋊#1181	—	2	1	no	—
46	#90800	—	2	4	no	—
47	#90800	D₄	1	2	yes	—
48	C₂×#6599	—	2	2	no	—
49	C₂×#6599	—	2	4	no	—
50	C₄⋊#1181	—	2	1	no	—
51	C₄×#1181	C₂³	1	2	yes	—
52	#90796	—	2	2	no	—
53	#90796	—	2	4	no	—
54	#90796	D₄	1	1	yes	characteristic
55	#90796	D₄	1	1	yes	characteristic
56	#90796	C₂³	1	4	yes	—
57	D₂₉⋊C₈	—	2	4	no	—
58	D₂₉⋊C₈	C₂³	1	2	yes	—
59	Q₁₆×C₂₉	C₂×C₄	1	1	yes	characteristic
60	SD₁₆×C₂₉	C₂×C₄	1	2	yes	—
61	D₈×C₂₉	C₂×C₄	1	1	yes	characteristic
62	C₂×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
63	#90785	C₂×C₄	1	2	yes	—
64	#90784	C₂×C₄	1	2	yes	—
65	#90783	C₂×C₄	1	2	yes	—
66	#90782	C₂×C₄	1	2	yes	—
67	C₂×#6597	C₂×C₄	1	1	yes	characteristic
68	Dic₁₁₆	C₂×C₄	1	1	yes	characteristic
69	D₂₃₂	C₂×C₄	1	1	yes	characteristic
70	#90773	C₂×C₄	1	2	yes	—
71	C₈×D₂₉	C₂×C₄	1	1	yes	characteristic
72	C₈×D₂₉	C₂×C₄	1	1	yes	characteristic
73	C₈×D₂₉	C₂³	1	1	yes	characteristic
74	C₈×D₂₉	C₂³	1	1	yes	characteristic
75	C₂²×D₂₉	—	2	2	no	—
76	C₂²×D₂₉	#41	1	1	yes	characteristic
77	C₂×#1181	—	2	2	no	—
78	Q₈×C₂₉	C₂²×C₄	1	2	yes	—
79	D₄×C₂₉	—	2	2	no	—
80	D₄×C₂₉	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₂₉	—	2	2	no	—
82	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
83	C₂²⋊D₂₉	—	2	2	no	—
84	C₂²⋊D₂₉	—	2	2	no	—
85	C₂²⋊D₂₉	—	2	2	no	—
86	C₂×Dic₂₉	—	2	2	no	—
87	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
88	D₁₁₆	—	2	2	no	—
89	D₁₁₆	C₂²×C₄	1	2	yes	—
90	C₄×D₂₉	—	2	2	no	—
91	C₄×D₂₉	—	2	2	no	—
92	C₄×D₂₉	—	2	2	no	—
93	C₄×D₂₉	C₂²×C₄	1	1	yes	characteristic
94	C₄×D₂₉	C₂×D₄	1	1	yes	characteristic
95	C₄×D₂₉	#41	1	1	yes	characteristic
96	C₄×D₂₉	C₂⁴	1	1	yes	characteristic
97	Dic₅₈	—	2	2	no	—
98	Dic₅₈	C₂²×C₄	1	2	yes	—
99	#6599	—	2	4	no	—
100	#6599	C₂×D₄	1	2	yes	—
101	#6599	C₂×D₄	1	2	yes	—
102	C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
103	C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
104	#6597	C₂²×C₄	1	1	yes	characteristic
105	#6597	C₂²×C₄	1	1	yes	characteristic
106	#2957	—	29	1	no	maximal, Sylow
107	C₂²×C₂₉	—	2	2	no	—
108	C₂²×C₂₉	C₄×D₄	1	1	yes	characteristic
109	C₂×D₂₉	—	2	2	no	—
110	C₂×D₂₉	—	4	1	no	—
111	C₂×D₂₉	—	8	2	no	—
112	C₂×D₂₉	C₄×D₄	1	1	yes	characteristic
113	C₂×D₂₉	C₂×#41	1	1	yes	characteristic
114	#1181	—	4	2	no	—
115	C₄×C₂₉	—	2	2	no	—
116	C₄×C₂₉	C₄×D₄	1	1	yes	characteristic
117	C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
118	Dic₂₉	—	2	2	no	—
119	Dic₂₉	C₄×D₄	1	1	yes	characteristic
120	Dic₂₉	C₂²×D₄	1	1	yes	characteristic
121	C₂×#135	—	29	1	no	—
122	#568	—	29	2	no	—
123	#444	—	29	4	no	—
124	#443	—	29	2	no	—
125	#443	—	29	2	no	—
126	C₂×#108	—	29	1	no	—
127	#421	—	29	2	no	—
128	#405	—	29	1	no	—
129	C₂×C₂₉	—	2	1	no	—
130	C₂×C₂₉	—	4	2	no	—
131	C₂×C₂₉	C₂×C₄×D₄	1	1	yes	characteristic
132	D₂₉	—	2	1	no	—
133	D₂₉	—	2	1	no	—
134	D₂₉	—	4	2	no	—
135	C₂×#41	—	29	2	no	—
136	#135	—	29	2	no	—
137	#135	—	29	1	no	—
138	#135	—	29	2	no	—
139	#135	—	29	1	no	—
140	#135	—	29	1	no	—
141	#135	#1181	1	1	yes	characteristic, complemented
142	C₂×Q₁₆	—	29	1	no	—
143	C₂×SD₁₆	—	29	2	no	—
144	C₂×D₈	—	29	1	no	—
145	#131	—	29	4	no	—
146	#131	—	29	4	no	—
147	#131	—	58	4	no	—
148	C₂×M₁₆	—	29	1	no	—
149	C₂×M₁₆	—	29	2	no	—
150	C₂×M₁₆	—	58	2	no	—
151	C₂×M₁₆	—	58	2	no	—
152	C₂²×C₈	—	29	1	no	—
153	#117	—	29	1	no	—
154	#108	—	29	2	no	—
155	#108	—	29	2	no	—
156	#104	—	29	4	no	—
157	#104	—	29	4	no	—
158	#97	—	29	2	no	—
159	C₄×C₈	—	29	2	no	—
160	C₂₉	#2957	1	1	yes	characteristic, complemented, Sylow
161	#41	—	29	2	no	—
162	#41	—	29	2	no	—
163	#41	—	29	2	no	—
164	#41	—	58	2	no	—
165	#41	—	58	2	no	—
166	#41	C₂×#1181	1	2	yes	—
167	C₂×Q₈	—	29	2	no	—
168	C₂×D₄	—	29	2	no	—
169	C₂×D₄	—	58	2	no	—
170	C₂²×C₄	—	29	1	no	—
171	C₂²×C₄	—	58	2	no	—
172	Q₁₆	—	29	2	no	—
173	Q₁₆	—	29	1	no	—
174	Q₁₆	C₂×#1181	1	1	yes	characteristic
175	SD₁₆	—	29	2	no	—
176	SD₁₆	—	29	2	no	—
177	SD₁₆	—	29	2	no	—
178	SD₁₆	C₂×#1181	1	2	yes	—
179	D₈	—	29	2	no	—
180	D₈	—	29	1	no	—
181	D₈	C₂×#1181	1	1	yes	characteristic
182	M₁₆	—	29	1	no	—
183	M₁₆	—	29	1	no	—
184	M₁₆	—	29	2	no	—
185	M₁₆	—	29	4	no	—
186	M₁₆	—	58	4	no	—
187	M₁₆	—	58	4	no	—
188	M₁₆	—	58	2	no	—
189	C₂×C₈	—	29	2	no	—
190	C₂×C₈	—	29	1	no	—
191	C₂×C₈	—	29	1	no	—
192	C₂×C₈	—	29	1	no	—
193	C₂×C₈	—	29	1	no	—
194	C₂×C₈	—	29	1	no	—
195	C₂×C₈	—	58	4	no	—
196	C₂×C₈	—	58	4	no	—
197	C₂×C₈	—	58	2	no	—
198	C₂×C₈	C₂×#1181	1	1	yes	characteristic
199	C₄⋊C₄	—	58	1	no	—
200	C₂²⋊C₄	—	58	1	no	—
201	C₄²	—	29	2	no	—
202	C₂³	—	29	1	no	—
203	C₂³	—	58	2	no	—
204	Q₈	—	29	2	no	—
205	Q₈	—	58	2	no	—
206	Q₈	C₂²×#1181	1	2	yes	—
207	D₄	—	2	2	no	—
208	D₄	—	29	2	no	—
209	D₄	—	58	2	no	—
210	D₄	—	58	2	no	—
211	D₄	—	58	2	no	—
212	D₄	—	58	2	no	—
213	D₄	C₂²×#1181	1	2	yes	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	29	1	no	—
216	C₂×C₄	—	29	1	no	—
217	C₂×C₄	—	29	1	no	—
218	C₂×C₄	—	29	1	no	—
219	C₂×C₄	—	29	1	no	—
220	C₂×C₄	—	58	2	no	—
221	C₂×C₄	—	58	2	no	—
222	C₂×C₄	—	58	2	no	—
223	C₂×C₄	—	58	2	no	—
224	C₂×C₄	—	58	2	no	—
225	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
226	C₈	—	29	2	no	—
227	C₈	—	29	2	no	—
228	C₈	—	29	1	no	—
229	C₈	—	29	1	no	—
230	C₈	—	58	4	no	—
231	C₈	C₂²×#1181	1	1	yes	characteristic
232	C₈	C₂²×#1181	1	1	yes	characteristic
233	C₂²	—	2	2	no	—
234	C₂²	—	29	1	no	—
235	C₂²	—	29	1	no	—
236	C₂²	—	58	2	no	—
237	C₂²	—	116	1	no	—
238	C₂²	—	232	2	no	—
239	C₂²	D₄×#1181	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	29	1	no	—
242	C₄	—	29	1	no	—
243	C₄	—	58	2	no	—
244	C₄	—	116	2	no	—
245	C₄	D₄×#1181	1	1	yes	characteristic
246	C₄	C₂³×#1181	1	1	yes	characteristic
247	C₂	—	2	1	no	—
248	C₂	—	4	2	no	—
249	C₂	—	58	1	no	—
250	C₂	—	58	1	no	—
251	C₂	—	116	2	no	—
252	C₂	C₂×D₄×#1181	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#1181
commutator	C₄×C₂₉	C₂³×C₄
Frattini	C₄	C₂³×#1181
Fitting	C₂₉×#135	C₄
socle	C₂×C₂₉	C₂×C₄×D₄
prime radical	#135×D₂₉	C₂
cosocle	C₄×D₂₉	C₂⁴
hypercenter	#135	#1181
hypocenter	C₂₉	#2957
LCS	G ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #14272242	C₂	semidirect
2. #14272208	C₂	semidirect
3. #14272207	C₂	semidirect
4. #14272205	C₂	semidirect
5. C₂×#11745212	C₂	semidirect
6. #14272160	C₂	semidirect
7. #14272126	C₂	semidirect
10. #11745348	C₂²	semidirect
12. #11745346	C₂²	semidirect
18. C₂₉×#135	C₄	semidirect
19. #11745312	C₄	semidirect
21. Q₁₆×D₂₉	C₄	semidirect
23. SD₁₆×D₂₉	C₄	semidirect
25. D₈×D₂₉	C₄	semidirect
26. #11745266	C₄	semidirect
28. #11745226	C₂²	semidirect
29. #11745224	C₂²	semidirect
30. #11745212	C₂²	semidirect
31. #11745212	C₂²	semidirect
32. #11745208	C₂²	semidirect
33. C₈×#1181	C₂²	semidirect
141. #135	#1181	semidirect
160. C₂₉	#2957	semidirect

Maximal subgroups

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

subgroup	index
#14272242	2
#14272208	2
#14272207	2
#14272205	2
C₂×#11745212	2
#14272160	2
#14272126	2
#135×D₂₉	2
#2957	29