Subgroups — #16828488

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	#14271294	C₂	1	2	yes	maximal, complemented
2	#14271235	C₂	1	2	yes	maximal, complemented
3	#14271235	C₂	1	2	yes	maximal, complemented
4	D₄×#6597	C₂	1	2	yes	maximal, complemented
5	#14271224	C₂	1	2	yes	maximal, complemented
6	#14271216	C₂	1	1	yes	maximal, characteristic, complemented
7	#14271119	C₂	1	2	yes	maximal, complemented
8	C₂×#11745151	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₉×#529	C₂	1	1	yes	maximal, characteristic
10	C₂₉×#126	C₄	1	2	yes	—
11	C₂₉×#125	C₄	1	1	yes	characteristic
12	C₂₉×#124	C₄	1	1	yes	characteristic
13	C₂₉×#123	C₄	1	2	yes	—
14	C₂₉×C₂²⋊Q₈	C₄	1	1	yes	characteristic
15	C₂₉×#121	C₄	1	1	yes	characteristic
16	C₄×Q₈×C₂₉	C₂²	1	1	yes	characteristic
17	C₄×D₄×C₂₉	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₂₉	C₂²	1	2	yes	—
19	C₂₉×#117	C₂²	1	2	yes	—
20	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
21	C₂×#90777	C₂²	1	2	yes	—
22	C₂²×#6597	C₂²	1	2	yes	—
23	C₂²×#6597	C₂²	1	2	yes	—
24	C₂²⋊#6597	C₂²	1	1	yes	characteristic
25	C₂²⋊#6597	C₂²	1	2	yes	—
26	C₂²⋊#6597	C₂²	1	2	yes	—
27	C₂²⋊#6597	C₂²	1	1	yes	characteristic
28	C₄⋊#6597	C₂²	1	2	yes	—
29	C₄⋊#6597	C₂²	1	2	yes	—
30	C₄⋊#6597	C₂²	1	2	yes	—
31	#11745151	C₂²	1	2	yes	—
32	#11745151	C₂²	1	2	yes	—
33	#11745151	C₂²	1	1	yes	characteristic
34	#11745151	C₂²	1	1	yes	characteristic
35	C₄×#6597	C₂²	1	2	yes	—
36	C₄×#6597	C₂²	1	2	yes	—
37	C₂×Q₈×C₂₉	C₂×C₄	1	1	yes	characteristic
38	C₂×D₄×C₂₉	C₂×C₄	1	2	yes	—
39	C₂×D₄×C₂₉	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
44	C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
45	C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
46	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
47	C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
48	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
49	C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
50	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
51	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
52	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
53	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
54	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
55	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₂₉	C₂³	1	1	yes	characteristic
58	C₄²×C₂₉	C₂³	1	1	yes	characteristic
59	C₄²×C₂₉	C₂³	1	2	yes	—
60	#90777	—	2	4	no	—
61	C₂×#6597	—	2	4	no	—
62	C₂×#6597	—	2	4	no	—
63	C₂×#6597	C₂³	1	2	yes	—
64	C₂×#6597	C₂³	1	2	yes	—
65	C₂×#6597	C₂³	1	2	yes	—
66	C₂×#6597	C₂³	1	2	yes	—
67	C₂³×C₂₉	C₂²×C₄	1	2	yes	—
68	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
69	Q₈×C₂₉	—	4	1	no	—
70	D₄×C₂₉	—	2	4	no	—
71	D₄×C₂₉	—	4	1	no	—
72	C₂×C₄×C₂₉	—	2	2	no	—
73	C₂×C₄×C₂₉	—	2	2	no	—
74	C₂×C₄×C₂₉	—	2	2	no	—
75	C₂×C₄×C₂₉	—	2	2	no	—
76	C₂×C₄×C₂₉	—	4	1	no	—
77	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
86	#6597	—	2	2	no	—
87	#6597	—	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	—	2	2	no	—
94	C₂²×C₂₉	—	2	2	no	—
95	C₂²×C₂₉	—	2	2	no	—
96	C₂²×C₂₉	#131	1	2	yes	—
97	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
98	C₄×C₂₉	—	2	1	no	—
99	C₄×C₂₉	—	2	2	no	—
100	C₄×C₂₉	—	2	1	no	—
101	C₄×C₂₉	—	2	1	no	—
102	C₄×C₂₉	—	2	2	no	—
103	C₄×C₂₉	—	2	1	no	—
104	C₄×C₂₉	—	2	1	no	—
105	C₄×C₂₉	—	2	2	no	—
106	C₄×C₂₉	C₂×#41	1	2	yes	—
107	#529	D₂₉	1	1	yes	characteristic
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	—	29	1	no	—
114	#405	—	29	2	no	—
115	C₂×#97	—	29	1	no	—
116	C₂×C₂₉	—	2	2	no	—
117	C₂×C₂₉	—	4	2	no	—
118	C₂×C₂₉	C₄×#41	1	1	yes	characteristic
119	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
120	C₂×C₂₉	#568	1	1	yes	characteristic
121	C₂×M₁₆	—	29	2	no	—
122	C₂²×C₈	—	29	2	no	—
123	C₂²×C₈	—	29	2	no	—
124	#126	Dic₂₉	1	2	yes	—
125	#125	Dic₂₉	1	1	yes	characteristic
126	#124	Dic₂₉	1	1	yes	characteristic
127	#123	Dic₂₉	1	2	yes	—
128	C₂²⋊Q₈	Dic₂₉	1	1	yes	characteristic
129	#121	Dic₂₉	1	1	yes	characteristic
130	C₄×Q₈	C₂×D₂₉	1	1	yes	characteristic
131	C₄×D₄	C₂×D₂₉	1	2	yes	—
132	C₄×D₄	C₂×D₂₉	1	1	yes	characteristic
133	#117	C₂×D₂₉	1	2	yes	—
134	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
135	C₄⋊C₈	—	29	2	no	—
136	C₄⋊C₈	—	29	2	no	—
137	C₄⋊C₈	—	29	2	no	—
138	C₂²⋊C₈	—	29	1	no	—
139	C₂²⋊C₈	—	29	1	no	—
140	C₂²⋊C₈	—	29	2	no	—
141	C₂²⋊C₈	—	29	2	no	—
142	#97	—	29	1	no	—
143	#97	—	29	1	no	—
144	#97	—	29	2	no	—
145	#97	—	29	2	no	—
146	C₄×C₈	—	29	2	no	—
147	C₄×C₈	—	29	2	no	—
148	C₂₉	#2968	1	1	yes	characteristic, complemented, Sylow
149	C₂×Q₈	C₂×Dic₂₉	1	1	yes	characteristic
150	C₂×D₄	C₂×Dic₂₉	1	2	yes	—
151	C₂×D₄	C₂×Dic₂₉	1	1	yes	characteristic
152	C₂²×C₄	C₂×Dic₂₉	1	1	yes	characteristic
153	C₂²×C₄	C₂×Dic₂₉	1	1	yes	characteristic
154	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
155	C₂²×C₄	C₂²×D₂₉	1	2	yes	—
156	M₁₆	—	58	4	no	—
157	C₂×C₈	—	29	2	no	—
158	C₂×C₈	—	29	2	no	—
159	C₂×C₈	—	29	2	no	—
160	C₂×C₈	—	29	2	no	—
161	C₂×C₈	—	58	4	no	—
162	C₂×C₈	—	58	4	no	—
163	C₄⋊C₄	C₂×Dic₂₉	1	2	yes	—
164	C₄⋊C₄	C₂×Dic₂₉	1	2	yes	—
165	C₄⋊C₄	C₂×Dic₂₉	1	1	yes	characteristic
166	C₄⋊C₄	C₂×Dic₂₉	1	2	yes	—
167	C₄⋊C₄	C₂×Dic₂₉	1	1	yes	characteristic
168	C₄⋊C₄	C₂×Dic₂₉	1	2	yes	—
169	C₂²⋊C₄	C₂×Dic₂₉	1	2	yes	—
170	C₂²⋊C₄	C₂×Dic₂₉	1	2	yes	—
171	C₂²⋊C₄	C₂×Dic₂₉	1	2	yes	—
172	C₂²⋊C₄	C₂×Dic₂₉	1	2	yes	—
173	C₂²⋊C₄	C₂×Dic₂₉	1	2	yes	—
174	C₄²	C₂×Dic₂₉	1	1	yes	characteristic
175	C₄²	C₂×Dic₂₉	1	1	yes	characteristic
176	C₄²	C₂²×D₂₉	1	1	yes	characteristic
177	C₄²	C₂²×D₂₉	1	2	yes	—
178	C₄²	C₂²×D₂₉	1	1	yes	characteristic
179	C₂³	C₂²×Dic₂₉	1	2	yes	—
180	C₂³	C₂²×Dic₂₉	1	1	yes	characteristic
181	Q₈	—	4	1	no	—
182	D₄	—	2	4	no	—
183	D₄	—	4	1	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	1	no	—
189	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
190	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
191	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
192	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
193	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
194	C₂×C₄	C₂²×Dic₂₉	1	2	yes	—
195	C₂×C₄	C₂²×Dic₂₉	1	2	yes	—
196	C₂×C₄	C₂²×Dic₂₉	1	2	yes	—
197	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
198	C₈	—	29	4	no	—
199	C₈	—	58	2	no	—
200	C₈	—	58	2	no	—
201	C₈	—	58	2	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	2	2	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	#11745310	1	2	yes	—
208	C₂²	C₂³×Dic₂₉	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	2	2	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	2	no	—
217	C₄	#41×D₂₉	1	2	yes	—
218	C₂	—	2	2	no	—
219	C₂	—	4	2	no	—
220	C₂	#41×Dic₂₉	1	1	yes	characteristic
221	C₂	C₂×#11745310	1	1	yes	characteristic
222	C₂	#14271930	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₉
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting	C₂₉×#529	C₂
prime radical	C₂₉×#121	C₄
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	#529	D₂₉
hypocenter	C₂₉	#2968
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ #529

Complements

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

N	H ≅ G/N	type
1. #14271294	C₂	semidirect
2. #14271235	C₂	semidirect
3. #14271235	C₂	semidirect
4. D₄×#6597	C₂	semidirect
5. #14271224	C₂	semidirect
6. #14271216	C₂	semidirect
7. #14271119	C₂	semidirect
8. C₂×#11745151	C₂	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
#14271294	2
#14271235	2
D₄×#6597	2
#14271224	2
#14271216	2
#14271119	2
C₂×#11745151	2
C₂₉×#529	2
#2968	29