Subgroups — #16828622

Order 3712.

394 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#90805	C₂	1	1	yes	maximal, characteristic
2	#14271287	C₂	1	2	yes	maximal, complemented
3	#14271240	C₂	1	2	yes	maximal, complemented
4	#14271239	C₂	1	4	yes	maximal, complemented
5	#14271216	C₂	1	1	yes	maximal, characteristic, complemented
6	#14271215	C₂	1	2	yes	maximal, complemented
7	#14271185	C₂	1	2	yes	maximal, complemented
8	C₂₉×#423	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#90805	C₄	1	1	yes	characteristic
10	C₂×M₁₆×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	C₂²	1	2	yes	—
14	#11745288	C₄	1	1	yes	characteristic
15	C₂×#90772	C₂²	1	4	yes	—
16	C₄×D₁₁₆	C₄	1	1	yes	characteristic
17	C₄×D₁₁₆	C₂²	1	1	yes	characteristic
18	#11745234	C₄	1	2	yes	—
19	C₄²×D₂₉	C₂²	1	2	yes	—
20	C₄×Dic₅₈	C₄	1	1	yes	characteristic
21	C₄×Dic₅₈	C₂²	1	1	yes	characteristic
22	C₂₉×C₄⋊C₈	C₂²	1	2	yes	—
23	C₂₉×C₄⋊C₈	C₂²	1	2	yes	—
24	C₂²⋊#6597	C₂²	1	2	yes	—
25	#11745168	C₂²	1	4	yes	—
26	#11745162	C₂²	1	4	yes	—
27	C₈×Dic₂₉	C₂²	1	4	yes	—
28	C₄⋊#6597	C₂²	1	1	yes	characteristic
29	C₄⋊#6597	C₂²	1	1	yes	characteristic
30	C₄×#6597	C₂²	1	2	yes	—
31	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊D₂₉	C₂×C₄	1	2	yes	—
35	#90805	—	2	4	no	—
36	C₂×D₁₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₂₉	C₂×C₄	1	2	yes	—
38	C₂×C₄×D₂₉	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₂₉	C₂³	1	2	yes	—
40	C₂×Dic₅₈	C₂×C₄	1	1	yes	characteristic
41	M₁₆×C₂₉	—	4	2	no	—
42	C₂×C₈×C₂₉	C₂³	1	4	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	C₂²⋊Dic₂₉	C₂×C₄	1	2	yes	—
48	#90781	C₂×C₄	1	2	yes	—
49	#90781	C₂×C₄	1	2	yes	—
50	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
51	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
52	#90779	C₂×C₄	1	2	yes	—
53	#90779	C₂×C₄	1	2	yes	—
54	C₄×Dic₂₉	C₂×C₄	1	2	yes	—
55	C₄×Dic₂₉	C₂×C₄	1	2	yes	—
56	C₄×Dic₂₉	C₂³	1	2	yes	—
57	C₂×#6597	C₂³	1	2	yes	—
58	C₂×#6597	C₂³	1	2	yes	—
59	#90772	—	4	4	no	—
60	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×D₂₉	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₂₉	—	2	2	no	—
63	C₂×C₄×C₂₉	—	2	2	no	—
64	C₂×C₄×C₂₉	—	2	1	no	—
65	C₂×C₄×C₂₉	—	2	1	no	—
66	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
72	C₂²⋊D₂₉	—	2	4	no	—
73	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
74	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
75	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
76	D₁₁₆	—	2	2	no	—
77	C₄×D₂₉	—	2	4	no	—
78	C₄×D₂₉	—	2	2	no	—
79	C₄×D₂₉	—	2	2	no	—
80	C₄×D₂₉	—	2	4	no	—
81	Dic₅₈	M₁₆	1	4	yes	complemented
82	C₈×C₂₉	—	2	4	no	—
83	#6597	—	2	2	no	—
84	#6597	—	2	2	no	—
85	#2991	—	29	1	no	maximal, Sylow
86	C₂²×C₂₉	—	2	1	no	—
87	C₂²×C₂₉	—	2	1	no	—
88	C₂²×C₂₉	—	2	1	no	—
89	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
90	C₂×D₂₉	—	2	2	no	—
91	C₂×D₂₉	—	2	2	no	—
92	C₂×D₂₉	—	2	2	no	—
93	C₄×C₂₉	—	2	1	no	—
94	C₄×C₂₉	—	2	1	no	—
95	C₄×C₂₉	—	2	1	no	—
96	C₄×C₂₉	C₂×M₁₆	1	2	yes	—
97	C₄×C₂₉	#131	1	2	yes	—
98	C₄×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
99	C₄×C₂₉	2₋¹⁺⁴	1	1	yes	characteristic
100	Dic₂₉	—	2	2	no	—
101	Dic₂₉	—	2	2	no	—
102	Dic₂₉	C₂×M₁₆	1	4	yes	—
103	C₄×#41	—	29	1	no	—
104	C₈×Q₈	—	29	2	no	—
105	#436	—	29	4	no	—
106	#436	—	29	2	no	—
107	#431	—	29	1	no	—
108	#431	—	29	2	no	—
109	#423	—	29	2	no	—
110	#423	D₂₉	1	1	yes	characteristic, complemented
111	C₂×C₂₉	—	4	1	no	—
112	C₂×C₂₉	#520	1	1	yes	characteristic
113	C₂×C₂₉	C₂²×M₁₆	1	1	yes	characteristic
114	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
115	D₂₉	—	4	2	no	—
116	C₂×#41	—	29	1	no	—
117	C₂×M₁₆	—	29	4	no	—
118	C₂×M₁₆	C₂×D₂₉	1	2	yes	—
119	C₄×Q₈	—	29	1	no	—
120	C₄×Q₈	—	29	1	no	—
121	C₄×D₄	—	29	2	no	—
122	C₄×D₄	—	29	2	no	—
123	C₄×D₄	—	29	1	no	—
124	C₄×D₄	—	29	1	no	—
125	#117	—	29	1	no	—
126	#117	—	29	2	no	—
127	C₂×C₄²	—	29	2	no	—
128	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
129	C₄⋊C₈	—	29	4	no	—
130	C₄⋊C₈	—	29	1	no	—
131	C₄⋊C₈	—	29	1	no	—
132	C₄⋊C₈	C₂×D₂₉	1	2	yes	—
133	C₄⋊C₈	C₂×D₂₉	1	2	yes	—
134	C₂²⋊C₈	—	29	2	no	—
135	C₂²⋊C₈	—	29	4	no	—
136	C₄×C₈	—	29	4	no	—
137	C₄×C₈	—	29	2	no	—
138	C₂₉	#2991	1	1	yes	characteristic, complemented, Sylow
139	#41	—	58	4	no	—
140	C₂×Q₈	—	29	1	no	—
141	C₂×D₄	—	29	2	no	—
142	C₂×D₄	—	29	1	no	—
143	C₂²×C₄	—	29	2	no	—
144	C₂²×C₄	—	29	2	no	—
145	C₂²×C₄	—	29	2	no	—
146	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
147	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
148	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
149	M₁₆	—	4	2	no	—
150	M₁₆	—	116	4	no	—
151	C₂×C₈	—	29	2	no	—
152	C₂×C₈	—	29	2	no	—
153	C₂×C₈	C₂²×D₂₉	1	4	yes	—
154	C₄⋊C₄	—	29	2	no	—
155	C₄⋊C₄	—	29	2	no	—
156	C₄⋊C₄	—	29	1	no	—
157	C₄⋊C₄	—	29	1	no	—
158	C₂²⋊C₄	—	29	2	no	—
159	C₂²⋊C₄	—	29	2	no	—
160	C₂²⋊C₄	—	29	2	no	—
161	C₄²	—	29	2	no	—
162	C₄²	—	29	2	no	—
163	C₄²	—	29	2	no	—
164	C₄²	C₄×D₂₉	1	1	yes	characteristic
165	C₄²	C₄×D₂₉	1	1	yes	characteristic
166	C₄²	C₂²×D₂₉	1	1	yes	characteristic
167	C₄²	C₂²×D₂₉	1	1	yes	characteristic
168	C₂³	—	29	2	no	—
169	C₂³	C₂×C₄×D₂₉	1	1	yes	characteristic
170	Q₈	—	29	4	no	—
171	D₄	—	58	4	no	—
172	D₄	—	58	2	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	29	2	no	—
178	C₂×C₄	—	29	2	no	—
179	C₂×C₄	—	29	2	no	—
180	C₂×C₄	—	58	4	no	—
181	C₂×C₄	—	58	2	no	—
182	C₂×C₄	—	58	2	no	—
183	C₂×C₄	—	58	4	no	—
184	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
185	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
187	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
188	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
189	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
190	C₈	—	2	4	no	—
191	C₈	—	58	2	no	—
192	C₈	—	58	2	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	58	2	no	—
197	C₂²	—	58	2	no	—
198	C₂²	—	58	2	no	—
199	C₂²	C₂²×C₄×D₂₉	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	29	4	no	—
204	C₄	—	58	2	no	—
205	C₄	—	58	2	no	—
206	C₄	M₁₆×D₂₉	1	2	yes	—
207	C₄	#11745269	1	2	yes	—
208	C₄	#11745359	1	1	yes	characteristic
209	C₄	#11745362	1	1	yes	characteristic
210	C₂	—	4	1	no	—
211	C₂	—	116	2	no	—
212	C₂	#14271644	1	1	yes	characteristic
213	C₂	C₂×M₁₆×D₂₉	1	1	yes	characteristic
214	C₂	C₂×#11745269	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₉
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting	C₂₉×#423	C₂
prime radical	C₂×#90805	C₄
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	#423	D₂₉
hypocenter	C₂₉	#2991
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
2. #14271287	C₂	semidirect
3. #14271240	C₂	semidirect
4. #14271239	C₂	semidirect
5. #14271216	C₂	semidirect
6. #14271215	C₂	semidirect
7. #14271185	C₂	semidirect
8. C₂₉×#423	C₂	semidirect
81. Dic₅₈	M₁₆	semidirect
110. #423	D₂₉	semidirect
138. C₂₉	#2991	semidirect

Maximal subgroups

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

subgroup	index
C₄×#90805	2
#14271287	2
#14271240	2
#14271239	2
#14271216	2
#14271215	2
#14271185	2
C₂₉×#423	2
#2991	29