Subgroups — #16828568

Order 3712.

402 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14271241	C₂	1	2	yes	maximal, complemented
2	#14271235	C₂	1	4	yes	maximal, complemented
3	D₄×#6597	C₂	1	2	yes	maximal, complemented
4	#14271220	C₂	1	1	yes	maximal, characteristic, complemented
5	#14271216	C₂	1	1	yes	maximal, characteristic, complemented
6	#14271130	C₂	1	4	yes	maximal, complemented
7	C₂₉×#532	C₂	1	1	yes	maximal, characteristic
8	C₂×#41×C₂₉	C₂²	1	2	yes	—
9	C₂₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
10	C₂₉×#127	C₄	1	1	yes	characteristic
11	C₂₉×#124	C₄	1	2	yes	—
12	C₂₉×#121	C₄	1	2	yes	—
13	C₂₉×#121	C₄	1	2	yes	—
14	C₄×D₄×C₂₉	C₂²	1	2	yes	—
15	C₄×D₄×C₂₉	C₂²	1	2	yes	—
16	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
17	C₂×#90777	C₂²	1	4	yes	—
18	C₂²×#6597	C₂²	1	4	yes	—
19	C₂²⋊#6597	C₂²	1	4	yes	—
20	C₂²⋊#6597	C₂²	1	4	yes	—
21	C₂²⋊#6597	C₂²	1	2	yes	—
22	C₂²⋊#6597	C₂²	1	2	yes	—
23	C₄⋊#6597	C₂²	1	2	yes	—
24	C₄⋊#6597	C₂²	1	2	yes	—
25	#11745151	C₂²	1	2	yes	—
26	C₄×#6597	C₂²	1	1	yes	characteristic
27	C₄×#6597	C₂²	1	1	yes	characteristic
28	#41×C₂₉	—	4	4	no	—
29	C₂×Q₈×C₂₉	C₂×C₄	1	2	yes	—
30	C₂×D₄×C₂₉	C₂×C₄	1	2	yes	—
31	C₂×D₄×C₂₉	C₂×C₄	1	2	yes	—
32	C₂×D₄×C₂₉	C₂×C₄	1	4	yes	—
33	C₂×D₄×C₂₉	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₂₉	C₂³	1	4	yes	—
37	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
39	C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
40	C₂²⋊C₄×C₂₉	C₂×C₄	1	4	yes	—
41	C₂²⋊C₄×C₂₉	C₂×C₄	1	4	yes	—
42	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₂₉	C₂³	1	1	yes	characteristic
45	C₄²×C₂₉	C₂³	1	1	yes	characteristic
46	#90777	—	4	4	no	—
47	C₂×#6597	—	4	4	no	—
48	C₂×#6597	C₂³	1	2	yes	—
49	C₂×#6597	C₂³	1	4	yes	—
50	C₂×#6597	C₂³	1	2	yes	—
51	C₂³×C₂₉	C₂²×C₄	1	4	yes	—
52	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
53	Q₈×C₂₉	—	4	2	no	—
54	D₄×C₂₉	—	2	4	no	—
55	D₄×C₂₉	—	4	2	no	—
56	D₄×C₂₉	—	4	2	no	—
57	D₄×C₂₉	—	4	4	no	—
58	C₂×C₄×C₂₉	—	2	2	no	—
59	C₂×C₄×C₂₉	—	2	4	no	—
60	C₂×C₄×C₂₉	—	2	4	no	—
61	C₂×C₄×C₂₉	—	2	2	no	—
62	C₂×C₄×C₂₉	—	4	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₂₉	C₂²×C₄	1	4	yes	—
68	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
70	#6597	—	2	2	no	—
71	#6597	—	2	4	no	—
72	#6597	—	2	2	no	—
73	#2981	—	29	1	no	maximal, Sylow
74	C₂²×C₂₉	—	2	1	no	—
75	C₂²×C₂₉	—	2	4	no	—
76	C₂²×C₂₉	—	2	1	no	—
77	C₂²×C₂₉	—	2	4	no	—
78	C₂²×C₂₉	—	2	1	no	—
79	C₂²×C₂₉	—	2	4	no	—
80	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
81	C₄×C₂₉	—	2	1	no	—
82	C₄×C₂₉	—	2	4	no	—
83	C₄×C₂₉	—	2	1	no	—
84	C₄×C₂₉	—	2	1	no	—
85	C₄×C₂₉	—	2	1	no	—
86	C₄×C₂₉	#131	1	2	yes	—
87	C₄×C₂₉	H(5,𝔽₂)	1	2	yes	—
88	#532	D₂₉	1	1	yes	characteristic
89	#436	—	29	2	no	—
90	#435	—	29	4	no	—
91	C₈×D₄	—	29	2	no	—
92	#432	—	29	1	no	—
93	#431	—	29	1	no	—
94	#408	—	29	4	no	—
95	C₂×C₂₉	—	4	1	no	—
96	C₂×C₂₉	—	4	4	no	—
97	C₂×C₂₉	#518	1	1	yes	characteristic
98	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
99	C₂×C₂₉	#568	1	1	yes	characteristic
100	C₂×#41	C₂×D₂₉	1	2	yes	—
101	C₂×M₁₆	—	29	4	no	—
102	C₂²×C₈	—	29	4	no	—
103	C₄⋊Q₈	Dic₂₉	1	1	yes	characteristic
104	#127	Dic₂₉	1	1	yes	characteristic
105	#124	Dic₂₉	1	2	yes	—
106	#121	Dic₂₉	1	2	yes	—
107	#121	Dic₂₉	1	2	yes	—
108	C₄×D₄	C₂×D₂₉	1	2	yes	—
109	C₄×D₄	C₂×D₂₉	1	2	yes	—
110	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
111	C₄⋊C₈	—	29	2	no	—
112	C₄⋊C₈	—	29	2	no	—
113	C₂²⋊C₈	—	29	2	no	—
114	C₂²⋊C₈	—	29	2	no	—
115	C₂²⋊C₈	—	29	4	no	—
116	C₂²⋊C₈	—	29	4	no	—
117	#97	—	29	2	no	—
118	C₄×C₈	—	29	1	no	—
119	C₄×C₈	—	29	1	no	—
120	C₂₉	#2981	1	1	yes	characteristic, complemented, Sylow
121	#41	—	4	4	no	—
122	C₂×Q₈	C₂×Dic₂₉	1	2	yes	—
123	C₂×D₄	C₂×Dic₂₉	1	2	yes	—
124	C₂×D₄	C₂×Dic₂₉	1	2	yes	—
125	C₂×D₄	C₂×Dic₂₉	1	4	yes	—
126	C₂×D₄	C₂×Dic₂₉	1	2	yes	—
127	C₂²×C₄	C₂×Dic₂₉	1	1	yes	characteristic
128	C₂²×C₄	C₂×Dic₂₉	1	1	yes	characteristic
129	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
130	C₂²×C₄	C₂²×D₂₉	1	4	yes	—
131	M₁₆	—	116	4	no	—
132	C₂×C₈	—	29	2	no	—
133	C₂×C₈	—	29	2	no	—
134	C₂×C₈	—	29	4	no	—
135	C₂×C₈	—	116	4	no	—
136	C₄⋊C₄	C₂×Dic₂₉	1	2	yes	—
137	C₄⋊C₄	C₂×Dic₂₉	1	2	yes	—
138	C₂²⋊C₄	C₂×Dic₂₉	1	4	yes	—
139	C₂²⋊C₄	C₂×Dic₂₉	1	4	yes	—
140	C₄²	C₂×Dic₂₉	1	1	yes	characteristic
141	C₄²	C₂×Dic₂₉	1	1	yes	characteristic
142	C₄²	C₂²×D₂₉	1	1	yes	characteristic
143	C₄²	C₂²×D₂₉	1	1	yes	characteristic
144	C₂³	C₂²×Dic₂₉	1	4	yes	—
145	C₂³	C₂²×Dic₂₉	1	1	yes	characteristic
146	Q₈	—	4	2	no	—
147	D₄	—	2	4	no	—
148	D₄	—	4	2	no	—
149	D₄	—	4	4	no	—
150	D₄	—	4	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
157	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
158	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
159	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
160	C₂×C₄	C₂²×Dic₂₉	1	4	yes	—
161	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
162	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
163	C₈	—	58	2	no	—
164	C₈	—	58	4	no	—
165	C₈	—	58	2	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	4	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	4	no	—
170	C₂²	—	2	1	no	—
171	C₂²	—	2	4	no	—
172	C₂²	C₂³×Dic₂₉	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	4	no	—
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	#11745310	1	2	yes	—
179	C₄	#11745359	1	2	yes	—
180	C₂	—	4	1	no	—
181	C₂	—	4	4	no	—
182	C₂	#14271638	1	1	yes	characteristic
183	C₂	C₂×#11745310	1	1	yes	characteristic
184	C₂	#14271930	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₉
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting, prime radical	C₂₉×#532	C₂
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	#532	D₂₉
hypocenter	C₂₉	#2981
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ #532

Complements

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

N	H ≅ G/N	type
1. #14271241	C₂	semidirect
2. #14271235	C₂	semidirect
3. D₄×#6597	C₂	semidirect
4. #14271220	C₂	semidirect
5. #14271216	C₂	semidirect
6. #14271130	C₂	semidirect
120. C₂₉	#2981	semidirect

Maximal subgroups

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

subgroup	index
#14271241	2
#14271235	2
D₄×#6597	2
#14271220	2
#14271216	2
#14271130	2
C₂₉×#532	2
#2981	29