Subgroups — #16828409

Order 3712.

452 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14271930	C₂	1	2	yes	maximal, complemented
2	#14271285	C₂	1	4	yes	maximal, complemented
3	#14271280	C₂	1	4	yes	maximal, complemented
4	C₂×#11745167	C₂	1	1	yes	maximal, characteristic, complemented
5	#14271180	C₂	1	2	yes	maximal, complemented
6	#14271118	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₉×#135	C₂	1	1	yes	maximal, characteristic
8	C₂×#41×C₂₉	C₂²	1	2	yes	—
9	C₂₉×#135	C₄	1	2	yes	complemented
10	C₂₉×#135	C₄	1	2	yes	complemented
11	C₂₉×#135	C₂²	1	2	yes	—
12	C₂₉×#135	C₂²	1	2	yes	—
13	C₂×Q₁₆×C₂₉	C₄	1	1	yes	characteristic, complemented
14	C₂×SD₁₆×C₂₉	C₄	1	2	yes	complemented
15	C₂×D₈×C₂₉	C₄	1	1	yes	characteristic, complemented
16	C₂²×C₈×C₂₉	C₂²	1	1	yes	characteristic
17	#11745310	—	2	4	no	—
18	#11745310	C₂²	1	8	yes	complemented
19	#11745288	C₂²	1	1	yes	characteristic
20	C₂×#90777	—	2	4	no	—
21	C₂×#90777	C₂²	1	2	yes	—
22	C₂×#90777	C₂²	1	1	yes	characteristic
23	#11745185	C₂²	1	8	yes	complemented
24	#11745167	C₂²	1	2	yes	complemented
25	#11745167	C₂²	1	2	yes	complemented
26	#11745163	C₂²	1	2	yes	complemented
27	C₈×Dic₂₉	C₂²	1	2	yes	complemented
28	#41×C₂₉	—	2	2	no	—
29	#41×C₂₉	—	2	2	no	—
30	#41×C₂₉	C₂×C₄	1	4	yes	—
31	#41×C₂₉	C₂³	1	4	yes	—
32	C₂×Q₈×C₂₉	C₂×C₄	1	2	yes	—
33	C₂×D₄×C₂₉	—	2	2	no	—
34	C₂×D₄×C₂₉	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₂₉	—	2	2	no	—
36	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
37	Q₁₆×C₂₉	C₂×C₄	1	2	yes	—
38	Q₁₆×C₂₉	C₂×C₄	1	2	yes	—
39	SD₁₆×C₂₉	C₂×C₄	1	4	yes	—
40	SD₁₆×C₂₉	C₂×C₄	1	4	yes	—
41	D₈×C₂₉	C₂×C₄	1	2	yes	—
42	D₈×C₂₉	C₂×C₄	1	2	yes	—
43	C₂×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
48	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
49	C₂²⋊Dic₂₉	—	2	1	no	—
50	C₄⋊Dic₂₉	—	2	1	no	—
51	C₄×Dic₂₉	C₂³	1	2	yes	—
52	#90777	—	2	2	no	—
53	#90777	—	2	8	no	—
54	#90777	D₄	1	2	yes	—
55	#90777	D₄	1	2	yes	—
56	#90777	C₂³	1	4	yes	—
57	C₂×#6597	—	2	2	no	—
58	C₂×#6597	—	2	8	no	—
59	C₂×#6597	C₂³	1	2	yes	—
60	C₂³×C₂₉	—	2	2	no	—
61	C₂³×C₂₉	#41	1	1	yes	characteristic
62	Q₈×C₂₉	—	2	2	no	—
63	Q₈×C₂₉	C₂²×C₄	1	4	yes	—
64	D₄×C₂₉	—	2	4	no	—
65	D₄×C₂₉	—	2	4	no	—
66	D₄×C₂₉	—	2	2	no	—
67	D₄×C₂₉	C₂²×C₄	1	4	yes	—
68	C₂×C₄×C₂₉	—	2	2	no	—
69	C₂×C₄×C₂₉	—	2	4	no	—
70	C₂×C₄×C₂₉	—	2	4	no	—
71	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₂₉	#41	1	1	yes	characteristic
76	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
77	C₂×Dic₂₉	—	2	2	no	—
78	C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
79	C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
80	C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
81	C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
82	#6597	—	2	4	no	—
83	#6597	C₂×D₄	1	4	yes	—
84	#2957	—	29	1	no	maximal, Sylow
85	C₂²×C₂₉	—	2	4	no	—
86	C₂²×C₂₉	—	4	1	no	—
87	C₂²×C₂₉	—	8	2	no	—
88	C₂²×C₂₉	C₄×D₄	1	1	yes	characteristic
89	C₂²×C₂₉	C₄×D₄	1	1	yes	characteristic
90	C₂²×C₂₉	C₂×#41	1	1	yes	characteristic
91	C₄×C₂₉	—	2	4	no	—
92	C₄×C₂₉	C₄×D₄	1	1	yes	characteristic
93	C₄×C₂₉	C₄×D₄	1	1	yes	characteristic
94	C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
95	C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
96	Dic₂₉	—	4	2	no	—
97	C₂×#135	D₂₉	1	1	yes	characteristic
98	#568	—	29	2	no	—
99	#444	—	29	4	no	—
100	#443	—	29	4	no	—
101	C₂×#108	—	29	1	no	—
102	#421	—	29	2	no	—
103	#405	—	29	1	no	—
104	C₂×C₂₉	—	2	1	no	—
105	C₂×C₂₉	—	2	1	no	—
106	C₂×C₂₉	—	2	1	no	—
107	C₂×C₂₉	—	4	4	no	—
108	C₂×C₂₉	C₂×C₄×D₄	1	1	yes	characteristic
109	C₂×#41	C₂×D₂₉	1	2	yes	—
110	#135	Dic₂₉	1	2	yes	complemented
111	#135	Dic₂₉	1	2	yes	complemented
112	#135	C₂×D₂₉	1	2	yes	—
113	#135	C₂×D₂₉	1	2	yes	—
114	C₂×Q₁₆	Dic₂₉	1	1	yes	characteristic, complemented
115	C₂×SD₁₆	Dic₂₉	1	2	yes	complemented
116	C₂×D₈	Dic₂₉	1	1	yes	characteristic, complemented
117	#131	—	29	8	no	—
118	#131	—	58	4	no	—
119	C₂×M₁₆	—	29	1	no	—
120	C₂×M₁₆	—	29	2	no	—
121	C₂×M₁₆	—	58	4	no	—
122	C₂²×C₈	C₂×D₂₉	1	1	yes	characteristic
123	#117	—	29	1	no	—
124	#108	—	29	2	no	—
125	#108	—	29	2	no	—
126	#104	—	29	8	no	—
127	#97	—	29	2	no	—
128	C₄×C₈	—	29	2	no	—
129	C₂₉	#2957	1	1	yes	characteristic, complemented, Sylow
130	#41	—	2	2	no	—
131	#41	—	2	2	no	—
132	#41	C₂×Dic₂₉	1	4	yes	—
133	#41	C₂²×D₂₉	1	4	yes	—
134	C₂×Q₈	C₂×Dic₂₉	1	2	yes	—
135	C₂×D₄	—	2	2	no	—
136	C₂×D₄	C₂×Dic₂₉	1	2	yes	—
137	C₂²×C₄	—	2	2	no	—
138	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
139	Q₁₆	C₂×Dic₂₉	1	2	yes	—
140	Q₁₆	C₂×Dic₂₉	1	2	yes	—
141	SD₁₆	C₂×Dic₂₉	1	4	yes	—
142	SD₁₆	C₂×Dic₂₉	1	4	yes	—
143	D₈	C₂×Dic₂₉	1	2	yes	—
144	D₈	C₂×Dic₂₉	1	2	yes	—
145	M₁₆	—	29	2	no	—
146	M₁₆	—	29	2	no	—
147	M₁₆	—	29	4	no	—
148	M₁₆	—	58	8	no	—
149	M₁₆	—	58	2	no	—
150	C₂×C₈	—	29	2	no	—
151	C₂×C₈	—	58	8	no	—
152	C₂×C₈	—	58	2	no	—
153	C₂×C₈	C₂×Dic₂₉	1	1	yes	characteristic
154	C₂×C₈	C₂×Dic₂₉	1	1	yes	characteristic
155	C₂×C₈	C₂×Dic₂₉	1	1	yes	characteristic
156	C₂×C₈	C₂×Dic₂₉	1	1	yes	characteristic
157	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
158	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
159	C₄⋊C₄	—	58	1	no	—
160	C₂²⋊C₄	—	58	1	no	—
161	C₄²	—	29	2	no	—
162	C₂³	—	2	2	no	—
163	C₂³	#90807	1	1	yes	characteristic
164	Q₈	—	2	2	no	—
165	Q₈	C₂²×Dic₂₉	1	4	yes	—
166	D₄	—	2	4	no	—
167	D₄	—	2	4	no	—
168	D₄	—	2	2	no	—
169	D₄	C₂²×Dic₂₉	1	4	yes	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	58	2	no	—
174	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
175	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
176	C₂×C₄	#90807	1	1	yes	characteristic
177	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
178	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
179	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
180	C₈	—	29	4	no	—
181	C₈	—	58	4	no	—
182	C₈	C₂²×Dic₂₉	1	1	yes	characteristic
183	C₈	C₂²×Dic₂₉	1	1	yes	characteristic
184	C₈	C₂²×Dic₂₉	1	1	yes	characteristic
185	C₈	C₂²×Dic₂₉	1	1	yes	characteristic
186	C₂²	—	2	4	no	—
187	C₂²	—	4	1	no	—
188	C₂²	—	8	2	no	—
189	C₂²	D₄×Dic₂₉	1	1	yes	characteristic
190	C₂²	D₄×Dic₂₉	1	1	yes	characteristic
191	C₂²	C₂×#90807	1	1	yes	characteristic
192	C₄	—	2	4	no	—
193	C₄	—	116	2	no	—
194	C₄	D₄×Dic₂₉	1	1	yes	characteristic
195	C₄	D₄×Dic₂₉	1	1	yes	characteristic
196	C₄	C₂×D₄×D₂₉	1	1	yes	characteristic
197	C₄	C₂³×Dic₂₉	1	1	yes	characteristic
198	C₂	—	2	1	no	—
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	—	4	4	no	—
202	C₂	C₂×D₄×Dic₂₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₂₉
commutator	C₄×C₂₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₉
Fitting, prime radical	C₂×C₂₉×#135	C₂
socle	C₂×C₂₉	C₂×C₄×D₄
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	C₂×#135	D₂₉
hypocenter	C₂₉	#2957
LCS	G ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. #14271930	C₂	semidirect
2. #14271285	C₂	semidirect
3. #14271280	C₂	semidirect
4. C₂×#11745167	C₂	semidirect
5. #14271180	C₂	semidirect
6. #14271118	C₂	semidirect
9. C₂₉×#135	C₄	semidirect
10. C₂₉×#135	C₄	semidirect
13. C₂×Q₁₆×C₂₉	C₄	semidirect
14. C₂×SD₁₆×C₂₉	C₄	semidirect
15. C₂×D₈×C₂₉	C₄	semidirect
18. #11745310	C₂²	semidirect
23. #11745185	C₂²	semidirect
24. #11745167	C₂²	semidirect
25. #11745167	C₂²	semidirect
26. #11745163	C₂²	semidirect
27. C₈×Dic₂₉	C₂²	semidirect
110. #135	Dic₂₉	semidirect
111. #135	Dic₂₉	semidirect
114. C₂×Q₁₆	Dic₂₉	semidirect
115. C₂×SD₁₆	Dic₂₉	semidirect
116. C₂×D₈	Dic₂₉	semidirect
129. C₂₉	#2957	semidirect

Maximal subgroups

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

subgroup	index
#14271930	2
#14271285	2
#14271280	2
C₂×#11745167	2
#14271180	2
#14271118	2
C₂×C₂₉×#135	2
#2957	29