Subgroups — #16828481

Order 3712.

372 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	#14271697	C₂	1	1	yes	maximal, characteristic
2	#14271292	C₂	1	2	yes	maximal, complemented
3	#14271233	C₂	1	2	yes	maximal, complemented
4	#14271231	C₂	1	2	yes	maximal, complemented
5	#14271226	C₂	1	2	yes	maximal, complemented
6	#14271222	C₂	1	2	yes	maximal, complemented
7	#14271117	C₂	1	2	yes	maximal, complemented
8	C₂×#11745151	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₉×#431	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
11	#11745292	C₄	1	1	yes	characteristic
12	#11745291	C₄	1	2	yes	—
13	C₄×C₂²⋊D₂₉	C₂²	1	2	yes	—
14	#11745286	C₄	1	1	yes	characteristic
15	C₂²×#6597	C₂²	1	2	yes	—
16	C₂×#90772	C₂²	1	2	yes	—
17	C₂×C₈×D₂₉	C₂²	1	2	yes	—
18	#11745238	C₄	1	2	yes	—
19	#11745237	C₄	1	1	yes	characteristic
20	C₄×D₁₁₆	C₂²	1	1	yes	characteristic
21	#11745234	C₂²	1	2	yes	—
22	#11745232	C₄	1	1	yes	characteristic
23	C₄×Dic₅₈	C₂²	1	1	yes	characteristic
24	C₂₉×C₄⋊C₈	C₂²	1	2	yes	—
25	C₂₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
26	C₂₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
27	C₄×C₈×C₂₉	C₂²	1	2	yes	—
28	#11745168	C₂²	1	2	yes	—
29	#11745168	C₂²	1	2	yes	—
30	#11745163	C₂²	1	2	yes	—
31	#11745162	C₂²	1	2	yes	—
32	#11745162	C₂²	1	2	yes	—
33	C₈×Dic₂₉	C₂²	1	2	yes	—
34	#11745151	C₂²	1	2	yes	—
35	#11745151	C₂²	1	1	yes	characteristic
36	#11745151	C₂²	1	1	yes	characteristic
37	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
40	C₂×C₂²⋊D₂₉	C₂×C₄	1	2	yes	—
41	C₂×D₁₁₆	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₂₉	C₂³	1	2	yes	—
43	C₂×Dic₅₈	C₂×C₄	1	1	yes	characteristic
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	#90781	C₂×C₄	1	2	yes	—
54	#90781	C₂×C₄	1	2	yes	—
55	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
56	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
57	#90779	C₂×C₄	1	2	yes	—
58	#90779	C₂×C₄	1	2	yes	—
59	#90779	C₂×C₄	1	2	yes	—
60	#90779	C₂×C₄	1	2	yes	—
61	C₄×Dic₂₉	C₂³	1	2	yes	—
62	C₂×#6597	—	2	4	no	—
63	C₂×#6597	C₂³	1	2	yes	—
64	C₂×#6597	C₂³	1	2	yes	—
65	#90772	—	2	4	no	—
66	C₈×D₂₉	—	2	4	no	—
67	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×D₂₉	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₂₉	—	2	2	no	—
70	C₂×C₄×C₂₉	—	2	2	no	—
71	C₂×C₄×C₂₉	—	4	1	no	—
72	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
78	C₂²⋊D₂₉	—	2	4	no	—
79	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
80	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
81	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
82	D₁₁₆	—	4	1	no	—
83	C₄×D₂₉	—	2	2	no	—
84	C₄×D₂₉	—	2	2	no	—
85	Dic₅₈	—	4	1	no	—
86	C₈×C₂₉	—	2	2	no	—
87	C₈×C₂₉	—	2	2	no	—
88	#6597	—	2	2	no	—
89	#6597	#41	1	4	yes	—
90	#2968	—	29	1	no	maximal, Sylow
91	C₂²×C₂₉	—	2	1	no	—
92	C₂²×C₂₉	—	2	1	no	—
93	C₂²×C₂₉	#131	1	2	yes	—
94	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
95	C₂×D₂₉	—	2	2	no	—
96	C₂×D₂₉	—	2	2	no	—
97	C₂×D₂₉	—	2	2	no	—
98	C₄×C₂₉	—	2	1	no	—
99	C₄×C₂₉	—	2	1	no	—
100	C₄×C₂₉	—	2	1	no	—
101	C₄×C₂₉	—	2	1	no	—
102	C₄×C₂₉	—	2	1	no	—
103	C₄×C₂₉	C₂×#41	1	2	yes	—
104	Dic₂₉	—	2	2	no	—
105	Dic₂₉	—	2	2	no	—
106	Dic₂₉	—	2	2	no	—
107	#529	—	29	1	no	—
108	#446	—	29	2	no	—
109	#435	—	29	2	no	—
110	#435	—	29	2	no	—
111	C₈×D₄	—	29	2	no	—
112	#433	—	29	2	no	—
113	#431	D₂₉	1	1	yes	characteristic, complemented
114	#405	—	29	2	no	—
115	C₂×#97	—	29	1	no	—
116	C₂×C₂₉	—	2	2	no	—
117	C₂×C₂₉	C₄×#41	1	1	yes	characteristic
118	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
119	C₂×C₂₉	#568	1	1	yes	characteristic
120	D₂₉	—	4	2	no	—
121	C₂×M₁₆	—	29	2	no	—
122	C₂²×C₈	—	29	2	no	—
123	C₂²×C₈	—	29	2	no	—
124	#126	—	29	2	no	—
125	#125	—	29	1	no	—
126	#124	—	29	1	no	—
127	#123	—	29	2	no	—
128	C₂²⋊Q₈	—	29	1	no	—
129	#121	—	29	1	no	—
130	C₄×Q₈	—	29	1	no	—
131	C₄×D₄	—	29	2	no	—
132	C₄×D₄	—	29	1	no	—
133	#117	—	29	2	no	—
134	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
135	C₄⋊C₈	—	29	2	no	—
136	C₄⋊C₈	—	29	2	no	—
137	C₄⋊C₈	C₂×D₂₉	1	2	yes	—
138	C₂²⋊C₈	—	29	2	no	—
139	C₂²⋊C₈	—	29	2	no	—
140	C₂²⋊C₈	C₂×D₂₉	1	1	yes	characteristic
141	C₂²⋊C₈	C₂×D₂₉	1	1	yes	characteristic
142	#97	—	29	2	no	—
143	#97	—	29	1	no	—
144	#97	—	29	1	no	—
145	#97	—	29	2	no	—
146	C₄×C₈	—	29	2	no	—
147	C₄×C₈	C₂×D₂₉	1	2	yes	—
148	C₂₉	#2968	1	1	yes	characteristic, complemented, Sylow
149	C₂×Q₈	—	29	1	no	—
150	C₂×D₄	—	29	2	no	—
151	C₂×D₄	—	29	1	no	—
152	C₂²×C₄	—	29	2	no	—
153	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
154	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
155	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
156	M₁₆	—	58	4	no	—
157	C₂×C₈	—	29	2	no	—
158	C₂×C₈	—	29	2	no	—
159	C₂×C₈	—	58	4	no	—
160	C₂×C₈	—	58	4	no	—
161	C₂×C₈	C₂²×D₂₉	1	2	yes	—
162	C₂×C₈	C₂²×D₂₉	1	2	yes	—
163	C₄⋊C₄	—	29	2	no	—
164	C₄⋊C₄	—	29	2	no	—
165	C₄⋊C₄	—	29	1	no	—
166	C₄⋊C₄	—	29	2	no	—
167	C₄⋊C₄	—	29	1	no	—
168	C₄⋊C₄	—	29	2	no	—
169	C₂²⋊C₄	—	29	2	no	—
170	C₂²⋊C₄	—	29	2	no	—
171	C₂²⋊C₄	—	29	2	no	—
172	C₂²⋊C₄	—	29	2	no	—
173	C₂²⋊C₄	—	29	2	no	—
174	C₄²	—	29	2	no	—
175	C₄²	C₄×D₂₉	1	1	yes	characteristic
176	C₄²	C₄×D₂₉	1	1	yes	characteristic
177	C₄²	C₂²×D₂₉	1	1	yes	characteristic
178	C₄²	C₂²×D₂₉	1	1	yes	characteristic
179	C₂³	—	29	2	no	—
180	C₂³	C₂×C₄×D₂₉	1	1	yes	characteristic
181	Q₈	—	116	1	no	—
182	D₄	—	58	4	no	—
183	D₄	—	116	1	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	4	1	no	—
187	C₂×C₄	—	29	2	no	—
188	C₂×C₄	—	29	2	no	—
189	C₂×C₄	—	29	2	no	—
190	C₂×C₄	—	58	2	no	—
191	C₂×C₄	—	58	2	no	—
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	C₂×C₄×D₂₉	1	1	yes	characteristic
196	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
197	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	—	2	2	no	—
200	C₈	—	29	4	no	—
201	C₈	—	58	2	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	58	2	no	—
205	C₂²	—	58	2	no	—
206	C₂²	—	58	2	no	—
207	C₂²	#11745263	1	2	yes	—
208	C₂²	C₂²×C₄×D₂₉	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	58	2	no	—
215	C₄	—	58	2	no	—
216	C₄	—	58	2	no	—
217	C₄	#41×D₂₉	1	2	yes	—
218	C₂	—	2	2	no	—
219	C₂	—	116	2	no	—
220	C₂	#14271631	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	C₂₉×#431	C₂
prime radical	#11745292	C₄
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	#431	D₂₉
hypocenter	C₂₉	#2968
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
2. #14271292	C₂	semidirect
3. #14271233	C₂	semidirect
4. #14271231	C₂	semidirect
5. #14271226	C₂	semidirect
6. #14271222	C₂	semidirect
7. #14271117	C₂	semidirect
8. C₂×#11745151	C₂	semidirect
9. C₂₉×#431	C₂	semidirect
113. #431	D₂₉	semidirect
148. C₂₉	#2968	semidirect

Maximal subgroups

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

subgroup	index
#14271697	2
#14271292	2
#14271233	2
#14271231	2
#14271226	2
#14271222	2
#14271117	2
C₂×#11745151	2
C₂₉×#431	2
#2968	29