Subgroups — #16828057

Order 3712.

484 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11745263	C₂	1	2	yes	maximal
2	C₄×#90805	C₂	1	1	yes	maximal, characteristic
3	#14271189	C₂	1	2	yes	maximal, complemented
4	#14271188	C₂	1	4	yes	maximal, complemented
5	#14271187	C₂	1	1	yes	maximal, characteristic, complemented
6	#14271185	C₂	1	2	yes	maximal, complemented
7	C₄⋊C₈×D₂₉	C₂	1	2	yes	maximal, complemented
8	C₂×C₂₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#90805	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₂₉	C₂²	1	2	yes	—
11	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
12	C₄×C₂²⋊D₂₉	C₄	1	2	yes	—
13	C₄×C₂²⋊D₂₉	C₄	1	2	yes	—
14	#11745288	C₂²	1	1	yes	characteristic
15	C₂×#90777	C₂²	1	2	yes	—
16	#11745263	—	2	8	no	—
17	C₂×#90772	C₂²	1	4	yes	—
18	C₂×C₈×D₂₉	C₂²	1	4	yes	—
19	C₄×D₁₁₆	C₄	1	1	yes	characteristic
20	C₄×D₁₁₆	C₄	1	1	yes	characteristic
21	#11745234	C₂²	1	2	yes	—
22	C₄²×D₂₉	C₂²	1	2	yes	—
23	C₄×Dic₅₈	C₄	1	1	yes	characteristic
24	C₄×Dic₅₈	C₄	1	1	yes	characteristic
25	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₂₉×C₄⋊C₈	C₂²	1	2	yes	—
28	#11745162	C₂²	1	4	yes	—
29	#11745162	C₂²	1	4	yes	—
30	C₄⋊#6597	C₂²	1	2	yes	—
31	C₄⋊#6597	C₂²	1	2	yes	—
32	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
35	C₂×C₂²⋊D₂₉	C₂×C₄	1	2	yes	—
36	#90805	D₄	1	4	yes	—
37	#90805	Q₈	1	4	yes	—
38	C₂×D₁₁₆	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×D₂₉	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₂₉	C₂×C₄	1	2	yes	—
41	C₂×C₄×D₂₉	C₂³	1	2	yes	—
42	C₂×Dic₅₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×C₂₉	—	2	4	no	—
44	C₂×C₈×C₂₉	C₂³	1	2	yes	—
45	C₂×C₈×C₂₉	C₂³	1	2	yes	—
46	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
47	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
48	C₄²×C₂₉	C₂³	1	1	yes	characteristic
49	C₄²×C₂₉	C₂³	1	1	yes	characteristic
50	C₂²⋊Dic₂₉	C₂×C₄	1	2	yes	—
51	#90781	C₂×C₄	1	2	yes	—
52	#90781	C₂×C₄	1	2	yes	—
53	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
54	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
55	#90779	C₂×C₄	1	2	yes	—
56	#90779	C₂×C₄	1	2	yes	—
57	C₄×Dic₂₉	C₂×C₄	1	2	yes	—
58	C₄×Dic₂₉	C₂×C₄	1	2	yes	—
59	C₄×Dic₂₉	C₂³	1	2	yes	—
60	#90777	—	2	4	no	—
61	C₂×#6597	C₂³	1	4	yes	—
62	#90772	—	2	8	no	—
63	C₈×D₂₉	—	2	8	no	—
64	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
65	C₂²×D₂₉	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₂₉	—	2	2	no	—
67	C₂×C₄×C₂₉	—	4	1	no	—
68	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₂₉	C₂×Q₈	1	2	yes	—
75	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
76	C₂²⋊D₂₉	C₄⋊C₄	1	8	yes	—
77	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
78	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
79	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
80	D₁₁₆	C₄⋊C₄	1	4	yes	—
81	C₄×D₂₉	—	2	4	no	—
82	C₄×D₂₉	—	4	2	no	—
83	C₄×D₂₉	C₂×D₄	1	4	yes	—
84	C₄×D₂₉	C₂×Q₈	1	4	yes	—
85	Dic₅₈	C₄⋊C₄	1	4	yes	—
86	C₈×C₂₉	—	2	2	no	—
87	C₈×C₂₉	—	2	2	no	—
88	#6597	—	2	4	no	—
89	#2906	—	29	1	no	maximal, Sylow
90	C₂²×C₂₉	—	2	1	no	—
91	C₂²×C₂₉	—	2	1	no	—
92	C₂²×C₂₉	C₂×C₄⋊C₄	1	2	yes	—
93	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
94	C₂×D₂₉	—	2	2	no	—
95	C₂×D₂₉	—	2	2	no	—
96	C₂×D₂₉	C₂×C₄⋊C₄	1	4	yes	—
97	C₄×C₂₉	—	2	1	no	—
98	C₄×C₂₉	—	2	1	no	—
99	C₄×C₂₉	—	2	1	no	—
100	C₄×C₂₉	C₂×C₄⋊C₄	1	2	yes	—
101	C₄×C₂₉	#131	1	2	yes	complemented
102	C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
103	C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
104	Dic₂₉	—	2	2	no	—
105	Dic₂₉	—	2	2	no	—
106	Dic₂₉	C₂×C₄⋊C₄	1	4	yes	—
107	C₂×#131	—	29	2	no	—
108	C₄×#41	—	29	1	no	—
109	#424	—	29	2	no	—
110	#424	—	29	4	no	—
111	#423	—	29	1	no	—
112	#423	—	29	2	no	—
113	C₂×C₄⋊C₈	—	29	2	no	—
114	C₂×C₄⋊C₈	D₂₉	1	1	yes	characteristic, complemented
115	C₂×C₂₉	—	2	2	no	—
116	C₂×C₂₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
118	C₂×C₂₉	#568	1	1	yes	characteristic
119	D₂₉	—	2	4	no	—
120	C₂×#41	—	29	1	no	—
121	#131	—	58	8	no	—
122	C₂×M₁₆	—	29	2	no	—
123	C₂×M₁₆	—	29	4	no	—
124	C₂²×C₈	—	29	4	no	—
125	C₂²×C₈	C₂×D₂₉	1	2	yes	—
126	C₄×Q₈	—	29	1	no	—
127	C₄×Q₈	—	29	1	no	—
128	C₄×D₄	—	29	2	no	—
129	C₄×D₄	—	29	2	no	—
130	C₄×D₄	—	29	1	no	—
131	C₄×D₄	—	29	1	no	—
132	#117	—	29	1	no	—
133	#117	—	29	2	no	—
134	C₂×C₄²	—	29	2	no	—
135	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
136	C₄⋊C₈	—	29	4	no	—
137	C₄⋊C₈	—	29	4	no	—
138	C₄⋊C₈	—	29	2	no	—
139	C₄⋊C₈	—	29	2	no	—
140	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic
141	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic
142	C₄⋊C₈	C₂×D₂₉	1	2	yes	—
143	C₂₉	#2906	1	1	yes	characteristic, complemented, Sylow
144	#41	—	29	4	no	—
145	#41	—	29	4	no	—
146	C₂×Q₈	—	29	1	no	—
147	C₂×D₄	—	29	2	no	—
148	C₂×D₄	—	29	1	no	—
149	C₂²×C₄	—	29	2	no	—
150	C₂²×C₄	—	29	2	no	—
151	C₂²×C₄	—	29	2	no	—
152	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
153	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
154	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
155	M₁₆	—	58	8	no	—
156	M₁₆	—	58	4	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	29	4	no	—
159	C₂×C₈	—	58	8	no	—
160	C₂×C₈	C₂²×D₂₉	1	2	yes	—
161	C₂×C₈	C₂²×D₂₉	1	2	yes	—
162	C₄⋊C₄	—	29	2	no	—
163	C₄⋊C₄	—	29	2	no	—
164	C₄⋊C₄	—	29	1	no	—
165	C₄⋊C₄	—	29	1	no	—
166	C₂²⋊C₄	—	29	2	no	—
167	C₂²⋊C₄	—	29	2	no	—
168	C₂²⋊C₄	—	29	2	no	—
169	C₄²	—	29	2	no	—
170	C₄²	—	29	2	no	—
171	C₄²	—	29	2	no	—
172	C₄²	C₄×D₂₉	1	1	yes	characteristic
173	C₄²	C₄×D₂₉	1	1	yes	characteristic
174	C₄²	C₂²×D₂₉	1	1	yes	characteristic
175	C₄²	C₂²×D₂₉	1	1	yes	characteristic
176	C₂³	—	29	2	no	—
177	C₂³	C₂×C₄×D₂₉	1	1	yes	characteristic
178	Q₈	—	29	4	no	—
179	D₄	—	29	8	no	—
180	D₄	—	29	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	4	1	no	—
183	C₂×C₄	—	29	2	no	—
184	C₂×C₄	—	29	4	no	—
185	C₂×C₄	—	29	4	no	—
186	C₂×C₄	—	29	2	no	—
187	C₂×C₄	—	29	2	no	—
188	C₂×C₄	—	58	4	no	—
189	C₂×C₄	—	116	2	no	—
190	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
191	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
192	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
195	C₂×C₄	D₄×D₂₉	1	2	yes	—
196	C₂×C₄	Q₈×D₂₉	1	2	yes	—
197	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	—	2	2	no	—
200	C₈	—	58	4	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	29	4	no	—
204	C₂²	—	58	2	no	—
205	C₂²	—	58	2	no	—
206	C₂²	C₄⋊C₄×D₂₉	1	2	yes	—
207	C₂²	C₂²×C₄×D₂₉	1	1	yes	characteristic
208	C₄	—	2	1	no	—
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	29	4	no	—
212	C₄	—	58	2	no	—
213	C₄	—	58	2	no	—
214	C₄	C₄⋊C₄×D₂₉	1	2	yes	—
215	C₄	#11745263	1	2	yes	complemented
216	C₄	C₂×D₄×D₂₉	1	1	yes	characteristic
217	C₄	C₂×Q₈×D₂₉	1	1	yes	characteristic
218	C₂	—	2	2	no	—
219	C₂	—	58	4	no	—
220	C₂	C₂×C₄⋊C₄×D₂₉	1	1	yes	characteristic
221	C₂	C₂×#11745263	1	1	yes	characteristic
222	C₂	#14271929	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₉
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting	C₂×C₂₉×C₄⋊C₈	C₂
prime radical	C₂×#90805	C₂²
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₂₉
hypocenter	C₂₉	#2906
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. #14271189	C₂	semidirect
4. #14271188	C₂	semidirect
5. #14271187	C₂	semidirect
6. #14271185	C₂	semidirect
7. C₄⋊C₈×D₂₉	C₂	semidirect
8. C₂×C₂₉×C₄⋊C₈	C₂	semidirect
101. C₄×C₂₉	#131	semidirect
114. C₂×C₄⋊C₈	D₂₉	semidirect
143. C₂₉	#2906	semidirect
215. C₄	#11745263	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11745263	2
C₄×#90805	2
#14271189	2
#14271188	2
#14271187	2
#14271185	2
C₄⋊C₈×D₂₉	2
C₂×C₂₉×C₄⋊C₈	2
#2906	29