Subgroups — #16828133

Order 3712.

400 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14271631	C₂	1	1	yes	maximal, characteristic
2	#14271225	C₂	1	1	yes	maximal, characteristic, complemented
3	#14271224	C₂	1	1	yes	maximal, characteristic, complemented
4	#14271222	C₂	1	1	yes	maximal, characteristic, complemented
5	#14271215	C₂	1	2	yes	maximal, complemented
6	#14271213	C₂	1	2	yes	maximal, complemented
7	#14271188	C₂	1	2	yes	maximal, complemented
8	#14271128	C₂	1	2	yes	maximal
9	#14271115	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₈×D₂₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₉×#433	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#90805	C₂²	1	1	yes	characteristic
13	C₂₉×#117	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
15	C₂×#90772	C₂²	1	2	yes	—
16	C₂×C₈×D₂₉	C₂²	1	2	yes	—
17	#11745255	C₄	1	2	yes	—
18	#11745249	C₄	1	2	yes	—
19	#11745243	C₄	1	2	yes	—
20	#11745243	C₄	1	2	yes	—
21	#11745234	C₂²	1	1	yes	characteristic
22	#11745234	C₂²	1	1	yes	characteristic
23	C₄²×D₂₉	C₂²	1	1	yes	characteristic
24	C₄²×D₂₉	C₂²	1	1	yes	characteristic
25	C₂₉×C₄⋊C₈	C₂²	1	2	yes	—
26	C₂₉×C₂²⋊C₈	C₂²	1	2	yes	—
27	C₂₉×#97	C₂²	1	1	yes	characteristic
28	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
29	C₂²⋊#6597	C₂²	1	2	yes	—
30	#11745168	C₂²	1	2	yes	—
31	#11745168	C₂²	1	2	yes	—
32	#11745162	C₂²	1	2	yes	—
33	#11745162	C₂²	1	2	yes	—
34	C₈×Dic₂₉	C₂²	1	2	yes	—
35	C₈×Dic₂₉	C₂²	1	2	yes	—
36	C₄⋊#6597	C₂²	1	2	yes	—
37	#11745151	C₂²	1	1	yes	characteristic
38	C₄×#6597	C₂²	1	1	yes	characteristic
39	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
40	C₂×C₂²⋊D₂₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₂²⋊D₂₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×Dic₂₉	C₂×C₄	1	2	yes	—
43	#90805	—	2	4	no	—
44	C₂×D₁₁₆	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₂₉	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₂₉	C₂×C₄	1	2	yes	—
47	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
48	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
49	C₂×Dic₅₈	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×C₂₉	C₂³	1	2	yes	—
51	C₂×C₈×C₂₉	C₂³	1	2	yes	—
52	C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
53	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
54	C₄²×C₂₉	C₂³	1	1	yes	characteristic
55	C₄²×C₂₉	C₂³	1	1	yes	characteristic
56	#90781	C₂×C₄	1	2	yes	—
57	#90781	C₂×C₄	1	2	yes	—
58	#90779	C₂×C₄	1	2	yes	—
59	#90779	C₂×C₄	1	2	yes	—
60	C₄×Dic₂₉	C₂×C₄	1	2	yes	—
61	C₄×Dic₂₉	C₂×C₄	1	2	yes	—
62	C₄×Dic₂₉	C₂×C₄	1	2	yes	—
63	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
64	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
65	C₂×#6597	C₂³	1	2	yes	—
66	C₂×#6597	C₂³	1	2	yes	—
67	#90772	—	2	4	no	—
68	C₈×D₂₉	—	2	4	no	—
69	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×D₂₉	C₂²×C₄	1	1	yes	characteristic
71	C₂²×D₂₉	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₂₉	—	2	2	no	—
73	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
77	C₂²⋊D₂₉	—	2	2	no	—
78	C₂²⋊D₂₉	—	2	2	no	—
79	C₂×Dic₂₉	—	2	4	no	—
80	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
82	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
84	D₁₁₆	—	2	2	no	—
85	C₄×D₂₉	—	2	2	no	—
86	C₄×D₂₉	—	2	4	no	—
87	C₄×D₂₉	—	2	4	no	—
88	C₄×D₂₉	#41	1	4	yes	—
89	Dic₅₈	—	2	2	no	—
90	C₈×C₂₉	—	2	2	no	—
91	C₈×C₂₉	—	2	2	no	—
92	#6597	—	2	2	no	—
93	#6597	—	2	2	no	—
94	#2922	—	29	1	no	maximal, Sylow
95	C₂²×C₂₉	—	2	1	no	—
96	C₂²×C₂₉	—	2	2	no	—
97	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₂₉	—	2	2	no	—
99	C₂×D₂₉	—	2	1	no	—
100	C₂×D₂₉	—	2	2	no	—
101	C₂×D₂₉	#117	1	2	yes	—
102	C₄×C₂₉	—	2	1	no	—
103	C₄×C₂₉	—	2	2	no	—
104	C₄×C₂₉	—	2	2	no	—
105	C₄×C₂₉	C₂×#41	1	2	yes	—
106	Dic₂₉	—	2	1	no	—
107	Dic₂₉	—	2	2	no	—
108	Dic₂₉	#117	1	2	yes	—
109	Dic₂₉	#131	1	4	yes	—
110	C₄×#41	—	29	1	no	—
111	#433	—	29	1	no	—
112	#433	—	29	1	no	—
113	#433	—	29	1	no	—
114	#433	D₂₉	1	1	yes	characteristic, complemented
115	#431	—	29	2	no	—
116	#431	—	29	2	no	—
117	#424	—	29	2	no	—
118	#408	—	29	2	no	—
119	C₄×M₁₆	—	29	1	no	—
120	C₂×C₄×C₈	—	29	1	no	—
121	C₂×C₂₉	—	4	1	no	—
122	C₂×C₂₉	C₂×#117	1	1	yes	characteristic
123	C₂×C₂₉	C₂×#131	1	2	yes	—
124	D₂₉	—	2	2	no	—
125	D₂₉	—	4	1	no	—
126	C₂×#41	—	29	1	no	—
127	C₂×M₁₆	—	29	2	no	—
128	C₂²×C₈	—	29	2	no	—
129	C₄×Q₈	—	29	2	no	—
130	C₄×D₄	—	29	2	no	—
131	C₄×D₄	—	29	2	no	—
132	C₄×D₄	—	29	2	no	—
133	#117	—	29	1	no	—
134	#117	—	29	1	no	—
135	#117	C₂×D₂₉	1	1	yes	characteristic
136	C₂×C₄²	—	29	1	no	—
137	C₂×C₄²	—	29	1	no	—
138	C₂×C₄²	—	29	1	no	—
139	C₄⋊C₈	—	29	2	no	—
140	C₄⋊C₈	—	29	2	no	—
141	C₄⋊C₈	—	29	2	no	—
142	C₄⋊C₈	C₂×D₂₉	1	2	yes	—
143	C₂²⋊C₈	—	29	2	no	—
144	C₂²⋊C₈	—	29	2	no	—
145	C₂²⋊C₈	—	29	2	no	—
146	C₂²⋊C₈	C₂×D₂₉	1	2	yes	—
147	#97	—	29	1	no	—
148	#97	C₂×D₂₉	1	1	yes	characteristic
149	C₄×C₈	—	29	2	no	—
150	C₄×C₈	—	29	2	no	—
151	C₄×C₈	—	29	1	no	—
152	C₄×C₈	C₂×D₂₉	1	1	yes	characteristic
153	C₂₉	#2922	1	1	yes	characteristic, complemented, Sylow
154	#41	—	58	4	no	—
155	C₂×Q₈	—	29	1	no	—
156	C₂×D₄	—	29	1	no	—
157	C₂×D₄	—	29	1	no	—
158	C₂×D₄	—	29	1	no	—
159	C₂²×C₄	—	29	1	no	—
160	C₂²×C₄	—	29	2	no	—
161	C₂²×C₄	—	29	2	no	—
162	C₂²×C₄	—	29	2	no	—
163	C₂²×C₄	—	29	1	no	—
164	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
165	M₁₆	—	58	4	no	—
166	C₂×C₈	—	29	2	no	—
167	C₂×C₈	—	29	2	no	—
168	C₂×C₈	—	58	4	no	—
169	C₂×C₈	C₂²×D₂₉	1	2	yes	—
170	C₂×C₈	C₂²×D₂₉	1	2	yes	—
171	C₄⋊C₄	—	29	2	no	—
172	C₄⋊C₄	—	29	2	no	—
173	C₄⋊C₄	C₄×D₂₉	1	2	yes	—
174	C₂²⋊C₄	—	29	2	no	—
175	C₂²⋊C₄	—	29	2	no	—
176	C₂²⋊C₄	C₄×D₂₉	1	2	yes	—
177	C₄²	—	29	2	no	—
178	C₄²	—	29	2	no	—
179	C₄²	—	29	2	no	—
180	C₄²	—	29	1	no	—
181	C₄²	—	29	1	no	—
182	C₄²	C₂²×D₂₉	1	1	yes	characteristic
183	C₄²	C₂²×D₂₉	1	1	yes	characteristic
184	C₂³	—	29	1	no	—
185	C₂³	—	29	1	no	—
186	C₂³	C₂×C₄×D₂₉	1	1	yes	characteristic
187	Q₈	—	58	2	no	—
188	D₄	—	58	2	no	—
189	D₄	—	58	2	no	—
190	D₄	—	58	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	29	2	no	—
193	C₂×C₄	—	29	1	no	—
194	C₂×C₄	—	29	4	no	—
195	C₂×C₄	—	29	2	no	—
196	C₂×C₄	—	29	1	no	—
197	C₂×C₄	—	58	4	no	—
198	C₂×C₄	—	58	4	no	—
199	C₂×C₄	—	58	4	no	—
200	C₂×C₄	—	58	2	no	—
201	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
202	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
203	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
204	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	—	2	2	no	—
207	C₈	—	58	2	no	—
208	C₈	—	58	2	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	29	2	no	—
212	C₂²	—	58	2	no	—
213	C₂²	—	58	1	no	—
214	C₂²	—	58	2	no	—
215	C₂²	C₂²×C₄×D₂₉	1	1	yes	characteristic
216	C₄	—	2	1	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	29	2	no	—
220	C₄	—	29	4	no	—
221	C₄	—	58	1	no	—
222	C₄	—	58	2	no	—
223	C₄	#41×D₂₉	1	2	yes	—
224	C₂	—	4	1	no	—
225	C₂	—	58	2	no	—
226	C₂	—	116	1	no	—
227	C₂	#117×D₂₉	1	1	yes	characteristic
228	C₂	#131×D₂₉	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₉
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting	C₂₉×#433	C₂
prime radical	C₂×#90805	C₂²
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	#433	D₂₉
hypocenter	C₂₉	#2922
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ #433

Complements

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

N	H ≅ G/N	type
2. #14271225	C₂	semidirect
3. #14271224	C₂	semidirect
4. #14271222	C₂	semidirect
5. #14271215	C₂	semidirect
6. #14271213	C₂	semidirect
7. #14271188	C₂	semidirect
9. #14271115	C₂	semidirect
10. C₄×C₈×D₂₉	C₂	semidirect
11. C₂₉×#433	C₂	semidirect
114. #433	D₂₉	semidirect
153. C₂₉	#2922	semidirect

Maximal subgroups

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

subgroup	index
#14271631	2
#14271225	2
#14271224	2
#14271222	2
#14271215	2
#14271213	2
#14271188	2
#14271128	2
#14271115	2
C₄×C₈×D₂₉	2
C₂₉×#433	2
#2922	29