Subgroups — #16828664

Order 3712.

346 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14271697	C₂	1	1	yes	maximal, characteristic
2	#14271291	C₂	1	2	yes	maximal, complemented
3	#14271238	C₂	1	2	yes	maximal, complemented
4	#14271237	C₂	1	2	yes	maximal, complemented
5	#14271236	C₂	1	2	yes	maximal, complemented
6	#14271223	C₂	1	2	yes	maximal, complemented
7	#14271188	C₂	1	2	yes	maximal, complemented
8	#14271187	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₉×#432	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
11	#11745292	C₄	1	1	yes	characteristic
12	#11745291	C₄	1	2	yes	—
13	C₄×C₂²⋊D₂₉	C₂²	1	2	yes	—
14	#11745286	C₄	1	1	yes	characteristic
15	C₂×#90777	C₂²	1	1	yes	characteristic
16	C₂×#90777	C₂²	1	1	yes	characteristic
17	C₂×#90772	C₂²	1	2	yes	—
18	C₂×C₈×D₂₉	C₂²	1	2	yes	—
19	#11745238	C₄	1	2	yes	—
20	#11745237	C₄	1	1	yes	characteristic
21	C₄×D₁₁₆	C₂²	1	1	yes	characteristic
22	#11745234	C₂²	1	2	yes	—
23	#11745232	C₄	1	1	yes	characteristic
24	C₄×Dic₅₈	C₂²	1	1	yes	characteristic
25	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₂₉×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₂₉×#97	C₂²	1	2	yes	—
29	#11745168	C₂²	1	2	yes	—
30	#11745168	C₂²	1	2	yes	—
31	#11745163	C₂²	1	2	yes	—
32	#11745162	C₂²	1	2	yes	—
33	#11745162	C₂²	1	2	yes	—
34	C₈×Dic₂₉	C₂²	1	2	yes	—
35	C₄⋊#6597	C₂²	1	2	yes	—
36	C₄⋊#6597	C₂²	1	2	yes	—
37	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
40	C₂×C₂²⋊D₂₉	C₂×C₄	1	2	yes	—
41	C₂×D₁₁₆	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₂₉	C₂³	1	2	yes	—
43	C₂×Dic₅₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×C₂₉	C₂³	1	2	yes	—
45	C₂×C₈×C₂₉	C₂³	1	2	yes	—
46	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
47	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
48	C₄²×C₂₉	C₂³	1	1	yes	characteristic
49	C₄²×C₂₉	C₂³	1	1	yes	characteristic
50	C₂²⋊Dic₂₉	C₂×C₄	1	2	yes	—
51	#90781	C₂×C₄	1	2	yes	—
52	#90781	C₂×C₄	1	2	yes	—
53	#90781	C₂×C₄	1	2	yes	—
54	#90781	C₂×C₄	1	2	yes	—
55	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
56	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
57	#90779	C₂×C₄	1	2	yes	—
58	#90779	C₂×C₄	1	2	yes	—
59	#90779	C₂×C₄	1	2	yes	—
60	#90779	C₂×C₄	1	2	yes	—
61	C₄×Dic₂₉	C₂³	1	2	yes	—
62	#90777	—	4	1	no	—
63	#90777	—	4	1	no	—
64	C₂×#6597	C₂³	1	2	yes	—
65	C₂×#6597	C₂³	1	2	yes	—
66	#90772	—	4	2	no	—
67	C₈×D₂₉	—	4	2	no	—
68	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×D₂₉	C₂²×C₄	1	2	yes	—
70	C₂×C₄×C₂₉	—	2	1	no	—
71	C₂×C₄×C₂₉	—	2	1	no	—
72	C₂×C₄×C₂₉	—	4	1	no	—
73	C₂×C₄×C₂₉	—	4	1	no	—
74	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
80	C₂²⋊D₂₉	—	4	2	no	—
81	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
82	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
83	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
84	D₁₁₆	—	4	1	no	—
85	C₄×D₂₉	—	2	2	no	—
86	C₄×D₂₉	—	2	2	no	—
87	Dic₅₈	—	4	1	no	—
88	C₈×C₂₉	—	2	2	no	—
89	C₈×C₂₉	—	2	2	no	—
90	#6597	—	2	2	no	—
91	#6597	—	2	2	no	—
92	#2995	—	29	1	no	maximal, Sylow
93	C₂²×C₂₉	—	2	1	no	—
94	C₂²×C₂₉	—	2	1	no	—
95	C₂²×C₂₉	—	2	1	no	—
96	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
97	C₂×D₂₉	—	2	2	no	—
98	C₂×D₂₉	—	2	2	no	—
99	C₂×D₂₉	—	2	2	no	—
100	C₄×C₂₉	—	2	1	no	—
101	C₄×C₂₉	—	2	1	no	—
102	C₄×C₂₉	—	2	1	no	—
103	C₄×C₂₉	—	2	1	no	—
104	C₄×C₂₉	—	2	1	no	—
105	C₄×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
106	C₄×C₂₉	2₋¹⁺⁴	1	1	yes	characteristic
107	Dic₂₉	—	2	2	no	—
108	Dic₂₉	—	2	2	no	—
109	Dic₂₉	—	2	2	no	—
110	#529	—	29	1	no	—
111	#446	—	29	2	no	—
112	#436	—	29	2	no	—
113	#435	—	29	2	no	—
114	#435	—	29	2	no	—
115	#433	—	29	2	no	—
116	#432	D₂₉	1	1	yes	characteristic, complemented
117	#424	—	29	2	no	—
118	#423	—	29	1	no	—
119	C₂×C₂₉	—	4	1	no	—
120	C₂×C₂₉	#520	1	1	yes	characteristic
121	C₂×C₂₉	#568	1	1	yes	characteristic
122	C₂×C₂₉	#568	1	1	yes	characteristic
123	D₂₉	—	4	2	no	—
124	C₂×M₁₆	—	29	1	no	—
125	C₂×M₁₆	—	29	1	no	—
126	C₂×M₁₆	—	29	2	no	—
127	C₂²×C₈	—	29	2	no	—
128	#126	—	29	2	no	—
129	#125	—	29	1	no	—
130	#124	—	29	1	no	—
131	#123	—	29	2	no	—
132	C₂²⋊Q₈	—	29	1	no	—
133	#121	—	29	1	no	—
134	C₄×Q₈	—	29	1	no	—
135	C₄×D₄	—	29	2	no	—
136	C₄×D₄	—	29	1	no	—
137	#117	—	29	2	no	—
138	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
139	C₄⋊C₈	—	29	2	no	—
140	C₄⋊C₈	—	29	2	no	—
141	C₄⋊C₈	—	29	2	no	—
142	C₄⋊C₈	—	29	2	no	—
143	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic
144	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic
145	C₂²⋊C₈	—	29	2	no	—
146	C₂²⋊C₈	—	29	2	no	—
147	C₂²⋊C₈	C₂×D₂₉	1	2	yes	—
148	#97	—	29	2	no	—
149	#97	C₂×D₂₉	1	2	yes	—
150	C₄×C₈	—	29	2	no	—
151	C₂₉	#2995	1	1	yes	characteristic, complemented, Sylow
152	C₂×Q₈	—	29	1	no	—
153	C₂×D₄	—	29	2	no	—
154	C₂×D₄	—	29	1	no	—
155	C₂²×C₄	—	29	2	no	—
156	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
157	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
159	M₁₆	—	116	2	no	—
160	M₁₆	—	116	1	no	—
161	M₁₆	—	116	1	no	—
162	C₂×C₈	—	29	2	no	—
163	C₂×C₈	—	29	2	no	—
164	C₂×C₈	—	116	2	no	—
165	C₂×C₈	C₂²×D₂₉	1	2	yes	—
166	C₂×C₈	C₂²×D₂₉	1	2	yes	—
167	C₄⋊C₄	—	29	2	no	—
168	C₄⋊C₄	—	29	2	no	—
169	C₄⋊C₄	—	29	1	no	—
170	C₄⋊C₄	—	29	2	no	—
171	C₄⋊C₄	—	29	1	no	—
172	C₄⋊C₄	—	29	2	no	—
173	C₂²⋊C₄	—	29	2	no	—
174	C₂²⋊C₄	—	29	2	no	—
175	C₂²⋊C₄	—	29	2	no	—
176	C₂²⋊C₄	—	29	2	no	—
177	C₂²⋊C₄	—	29	2	no	—
178	C₄²	—	29	2	no	—
179	C₄²	C₄×D₂₉	1	1	yes	characteristic
180	C₄²	C₄×D₂₉	1	1	yes	characteristic
181	C₄²	C₂²×D₂₉	1	1	yes	characteristic
182	C₄²	C₂²×D₂₉	1	1	yes	characteristic
183	C₂³	—	29	2	no	—
184	C₂³	C₂×C₄×D₂₉	1	1	yes	characteristic
185	Q₈	—	116	1	no	—
186	D₄	—	116	1	no	—
187	D₄	—	116	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	4	1	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	29	2	no	—
193	C₂×C₄	—	29	2	no	—
194	C₂×C₄	—	29	2	no	—
195	C₂×C₄	—	58	2	no	—
196	C₂×C₄	—	58	2	no	—
197	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
199	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
200	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
201	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
202	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
203	C₈	—	2	2	no	—
204	C₈	—	2	2	no	—
205	C₈	—	58	2	no	—
206	C₈	—	58	2	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	58	2	no	—
211	C₂²	—	58	2	no	—
212	C₂²	—	58	2	no	—
213	C₂²	C₂²×C₄×D₂₉	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	58	2	no	—
220	C₄	—	58	2	no	—
221	C₄	—	58	2	no	—
222	C₄	#11745364	1	1	yes	characteristic
223	C₄	#11745365	1	1	yes	characteristic
224	C₂	—	4	1	no	—
225	C₂	—	116	2	no	—
226	C₂	#14271643	1	1	yes	characteristic
227	C₂	#14271929	1	1	yes	characteristic
228	C₂	#14271929	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₉
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting	C₂₉×#432	C₂
prime radical	#11745292	C₄
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	#432	D₂₉
hypocenter	C₂₉	#2995
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
2. #14271291	C₂	semidirect
3. #14271238	C₂	semidirect
4. #14271237	C₂	semidirect
5. #14271236	C₂	semidirect
6. #14271223	C₂	semidirect
7. #14271188	C₂	semidirect
8. #14271187	C₂	semidirect
9. C₂₉×#432	C₂	semidirect
116. #432	D₂₉	semidirect
151. C₂₉	#2995	semidirect

Maximal subgroups

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

subgroup	index
#14271697	2
#14271291	2
#14271238	2
#14271237	2
#14271236	2
#14271223	2
#14271188	2
#14271187	2
C₂₉×#432	2
#2995	29