Subgroups — #16828558

Order 3712.

410 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14271713	C₂	1	1	yes	maximal, characteristic
2	#14271242	C₂	1	2	yes	maximal, complemented
3	#14271240	C₂	1	2	yes	maximal, complemented
4	C₈×D₁₁₆	C₂	1	2	yes	maximal, complemented
5	#14271228	C₂	1	2	yes	maximal, complemented
6	#14271216	C₂	1	1	yes	maximal, characteristic, complemented
7	#14271128	C₂	1	4	yes	maximal, complemented
8	C₂₉×#431	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#90805	C₂²	1	2	yes	—
10	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
11	#11745292	C₄	1	2	yes	—
12	#11745292	C₄	1	2	yes	—
13	C₂×#90772	C₂²	1	4	yes	—
14	C₂×C₈×D₂₉	C₂²	1	4	yes	—
15	#11745237	C₄	1	2	yes	—
16	#11745236	C₄	1	1	yes	characteristic
17	C₄×D₁₁₆	C₂²	1	2	yes	—
18	C₄×D₁₁₆	C₂²	1	2	yes	—
19	#11745231	C₄	1	1	yes	characteristic
20	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂₉×C₂²⋊C₈	C₂²	1	2	yes	—
23	C₄×C₈×C₂₉	C₂²	1	2	yes	—
24	C₂²⋊#6597	C₂²	1	2	yes	—
25	#11745168	C₂²	1	4	yes	—
26	#11745168	C₂²	1	4	yes	—
27	C₄⋊#6597	C₂²	1	1	yes	characteristic
28	C₄⋊#6597	C₂²	1	1	yes	characteristic
29	C₄×#6597	C₂²	1	2	yes	—
30	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
33	C₂×C₂²⋊D₂₉	C₂×C₄	1	4	yes	—
34	#90805	—	4	2	no	—
35	#90805	—	4	2	no	—
36	C₂×D₁₁₆	C₂×C₄	1	2	yes	—
37	C₂×D₁₁₆	C₂×C₄	1	2	yes	—
38	C₂×D₁₁₆	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₂₉	C₂³	1	4	yes	—
40	C₂×Dic₅₈	C₂×C₄	1	2	yes	—
41	C₂×C₈×C₂₉	C₂³	1	2	yes	—
42	C₂×C₈×C₂₉	C₂³	1	2	yes	—
43	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
45	C₄²×C₂₉	C₂³	1	1	yes	characteristic
46	C₄²×C₂₉	C₂³	1	1	yes	characteristic
47	#90781	C₂×C₄	1	4	yes	—
48	#90781	C₂×C₄	1	4	yes	—
49	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
50	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
51	C₂×#6597	C₂³	1	2	yes	—
52	C₂×#6597	C₂³	1	2	yes	—
53	#90772	—	4	4	no	—
54	C₈×D₂₉	—	4	4	no	—
55	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
56	C₂²×D₂₉	C₂²×C₄	1	4	yes	—
57	C₂×C₄×C₂₉	—	2	2	no	—
58	C₂×C₄×C₂₉	—	2	2	no	—
59	C₂×C₄×C₂₉	—	2	1	no	—
60	C₂×C₄×C₂₉	—	2	1	no	—
61	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
67	C₂²⋊D₂₉	—	4	4	no	—
68	C₂×Dic₂₉	C₂²×C₄	1	4	yes	—
69	D₁₁₆	—	2	4	no	—
70	D₁₁₆	—	2	4	no	—
71	D₁₁₆	—	4	2	no	—
72	C₄×D₂₉	—	2	4	no	—
73	C₄×D₂₉	—	2	4	no	—
74	Dic₅₈	—	4	2	no	—
75	C₈×C₂₉	—	2	2	no	—
76	C₈×C₂₉	—	2	2	no	—
77	#6597	—	2	2	no	—
78	#6597	—	2	2	no	—
79	#2979	—	29	1	no	maximal, Sylow
80	C₂²×C₂₉	—	2	1	no	—
81	C₂²×C₂₉	—	2	1	no	—
82	C₂²×C₂₉	—	2	1	no	—
83	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
84	C₂×D₂₉	—	2	4	no	—
85	C₂×D₂₉	—	2	4	no	—
86	C₂×D₂₉	—	2	4	no	—
87	C₄×C₂₉	—	2	1	no	—
88	C₄×C₂₉	—	2	1	no	—
89	C₄×C₂₉	—	2	1	no	—
90	C₄×C₂₉	#131	1	2	yes	—
91	C₄×C₂₉	#131	1	2	yes	—
92	C₄×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
93	C₄×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
94	Dic₂₉	—	2	4	no	—
95	#532	—	29	1	no	—
96	#436	—	29	2	no	—
97	#436	—	29	2	no	—
98	C₈×D₄	—	29	2	no	—
99	C₈×D₄	—	29	2	no	—
100	#431	—	29	1	no	—
101	#431	D₂₉	1	1	yes	characteristic, complemented
102	#408	—	29	4	no	—
103	C₂×C₂₉	—	4	1	no	—
104	C₂×C₂₉	#518	1	1	yes	characteristic
105	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
106	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
107	D₂₉	—	4	4	no	—
108	C₂×#41	—	29	2	no	—
109	C₂×M₁₆	—	29	4	no	—
110	C₂²×C₈	—	29	4	no	—
111	C₄⋊Q₈	—	29	1	no	—
112	#127	—	29	1	no	—
113	#124	—	29	2	no	—
114	#121	—	29	2	no	—
115	#121	—	29	2	no	—
116	C₄×D₄	—	29	2	no	—
117	C₄×D₄	—	29	2	no	—
118	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
119	C₄⋊C₈	—	29	1	no	—
120	C₄⋊C₈	—	29	1	no	—
121	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic
122	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic
123	C₂²⋊C₈	—	29	2	no	—
124	C₂²⋊C₈	—	29	4	no	—
125	C₂²⋊C₈	—	29	4	no	—
126	C₂²⋊C₈	C₂×D₂₉	1	2	yes	—
127	C₄×C₈	—	29	2	no	—
128	C₄×C₈	C₂×D₂₉	1	2	yes	—
129	C₂₉	#2979	1	1	yes	characteristic, complemented, Sylow
130	#41	—	116	2	no	—
131	#41	—	116	2	no	—
132	C₂×Q₈	—	29	2	no	—
133	C₂×D₄	—	29	4	no	—
134	C₂×D₄	—	29	2	no	—
135	C₂×D₄	—	29	2	no	—
136	C₂×D₄	—	29	2	no	—
137	C₂²×C₄	—	29	4	no	—
138	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
139	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
140	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
141	M₁₆	—	116	4	no	—
142	C₂×C₈	—	29	2	no	—
143	C₂×C₈	—	29	2	no	—
144	C₂×C₈	—	116	4	no	—
145	C₂×C₈	C₂²×D₂₉	1	2	yes	—
146	C₂×C₈	C₂²×D₂₉	1	2	yes	—
147	C₄⋊C₄	—	29	2	no	—
148	C₄⋊C₄	—	29	2	no	—
149	C₂²⋊C₄	—	29	4	no	—
150	C₂²⋊C₄	—	29	4	no	—
151	C₄²	C₄×D₂₉	1	1	yes	characteristic
152	C₄²	C₄×D₂₉	1	1	yes	characteristic
153	C₄²	C₂²×D₂₉	1	1	yes	characteristic
154	C₄²	C₂²×D₂₉	1	1	yes	characteristic
155	C₂³	—	29	4	no	—
156	C₂³	C₂×C₄×D₂₉	1	1	yes	characteristic
157	Q₈	—	116	2	no	—
158	D₄	—	58	4	no	—
159	D₄	—	58	4	no	—
160	D₄	—	116	2	no	—
161	D₄	—	116	4	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	29	4	no	—
167	C₂×C₄	—	58	4	no	—
168	C₂×C₄	—	58	4	no	—
169	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
170	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
172	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
173	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
174	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	—	2	2	no	—
177	C₈	—	58	2	no	—
178	C₈	—	58	2	no	—
179	C₂²	—	2	1	no	—
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	—	58	4	no	—
183	C₂²	—	58	4	no	—
184	C₂²	—	58	4	no	—
185	C₂²	C₂²×C₄×D₂₉	1	1	yes	characteristic
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	2	1	no	—
189	C₄	—	58	4	no	—
190	C₄	#11745263	1	2	yes	—
191	C₄	#11745269	1	2	yes	—
192	C₄	#11745364	1	1	yes	characteristic
193	C₄	#11745364	1	1	yes	characteristic
194	C₂	—	4	1	no	—
195	C₂	—	116	4	no	—
196	C₂	#14271637	1	1	yes	characteristic
197	C₂	C₂×#11745263	1	1	yes	characteristic
198	C₂	C₂×#11745269	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	#14271713	C₂
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	#431	D₂₉
hypocenter	C₂₉	#2979
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. #14271242	C₂	semidirect
3. #14271240	C₂	semidirect
4. C₈×D₁₁₆	C₂	semidirect
5. #14271228	C₂	semidirect
6. #14271216	C₂	semidirect
7. #14271128	C₂	semidirect
8. C₂₉×#431	C₂	semidirect
101. #431	D₂₉	semidirect
129. C₂₉	#2979	semidirect

Maximal subgroups

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

subgroup	index
#14271713	2
#14271242	2
#14271240	2
C₈×D₁₁₆	2
#14271228	2
#14271216	2
#14271128	2
C₂₉×#431	2
#2979	29