Subgroups — #16828423

Order 3712.

364 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14271717	C₂	1	1	yes	maximal, characteristic
2	#14271293	C₂	1	1	yes	maximal, characteristic, complemented
3	#14271291	C₂	1	2	yes	maximal, complemented
4	Q₈×#6597	C₂	1	1	yes	maximal, characteristic, complemented
5	#14271287	C₂	1	1	yes	maximal, characteristic, complemented
6	#14271287	C₂	1	2	yes	maximal, complemented
7	#14271188	C₂	1	2	yes	maximal
8	#14271185	C₂	1	1	yes	maximal, characteristic
9	#14271120	C₂	1	2	yes	maximal
10	C₄×C₈×D₂₉	C₂	1	1	yes	maximal, characteristic
11	C₂₉×#446	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×Q₈×C₂₉	C₂²	1	1	yes	characteristic
13	#11745308	C₄	1	1	yes	characteristic
14	#11745306	C₄	1	1	yes	characteristic
15	C₂×#90772	C₂²	1	2	yes	—
16	C₂×C₈×D₂₉	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₂₉	C₂²	1	1	yes	characteristic
18	#11745259	C₄	1	1	yes	characteristic
19	#11745258	C₄	1	2	yes	—
20	#11745251	C₄	1	2	yes	—
21	#11745250	C₄	1	1	yes	characteristic
22	#11745234	C₂²	1	2	yes	—
23	C₄²×D₂₉	C₂²	1	1	yes	characteristic
24	C₄×Dic₅₈	C₂²	1	2	yes	—
25	C₄×Dic₅₈	C₂²	1	1	yes	characteristic
26	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₂₉×C₄⋊C₈	C₂²	1	2	yes	—
28	C₂₉×#97	C₂²	1	2	yes	—
29	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
30	#11745163	C₂²	1	2	yes	—
31	#11745162	C₂²	1	2	yes	—
32	#11745162	C₂²	1	2	yes	—
33	#11745162	C₂²	1	2	yes	—
34	C₈×Dic₂₉	C₂²	1	1	yes	characteristic
35	C₈×Dic₂₉	C₂²	1	2	yes	—
36	C₈×Dic₂₉	C₂²	1	1	yes	characteristic
37	C₄⋊#6597	C₂²	1	2	yes	—
38	C₄⋊#6597	C₂²	1	1	yes	characteristic
39	C₄×#6597	C₂²	1	1	yes	characteristic
40	C₄×#6597	C₂²	1	2	yes	—
41	C₂×Q₈×C₂₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
45	C₂×Dic₅₈	C₂×C₄	1	1	yes	characteristic
46	C₂×Dic₅₈	C₂×C₄	1	2	yes	—
47	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
48	C₂×C₈×C₂₉	C₂³	1	2	yes	—
49	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
50	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
51	C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
52	C₄²×C₂₉	C₂³	1	1	yes	characteristic
53	C₄²×C₂₉	C₂³	1	2	yes	—
54	#90781	C₂×C₄	1	2	yes	—
55	#90781	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₂₉	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	#90779	C₂×C₄	1	2	yes	—
62	#90779	C₂×C₄	1	2	yes	—
63	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
66	C₄×Dic₂₉	C₂³	1	2	yes	—
67	C₂×#6597	C₂³	1	1	yes	characteristic
68	C₂×#6597	C₂³	1	2	yes	—
69	C₂×#6597	C₂³	1	1	yes	characteristic
70	#90772	—	2	4	no	—
71	C₈×D₂₉	—	2	2	no	—
72	C₈×D₂₉	Q₈	1	4	yes	complemented
73	C₂²×D₂₉	C₂²×C₄	1	1	yes	characteristic
74	Q₈×C₂₉	—	2	2	no	—
75	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
80	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
82	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
83	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
85	C₄×D₂₉	—	2	2	no	—
86	C₄×D₂₉	—	2	2	no	—
87	C₄×D₂₉	C₂×Q₈	1	2	yes	—
88	C₄×D₂₉	#41	1	2	yes	—
89	Dic₅₈	—	2	2	no	—
90	Dic₅₈	—	2	4	no	—
91	C₈×C₂₉	—	2	1	no	—
92	C₈×C₂₉	—	2	2	no	—
93	C₈×C₂₉	C₂×Q₈	1	2	yes	—
94	#6597	—	2	1	no	—
95	#6597	—	2	2	no	—
96	#6597	C₂×Q₈	1	2	yes	—
97	#2960	—	29	1	no	maximal, Sylow
98	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
99	C₂×D₂₉	—	2	1	no	—
100	C₂×D₂₉	—	2	1	no	—
101	C₂×D₂₉	C₄×Q₈	1	2	yes	—
102	C₄×C₂₉	—	2	2	no	—
103	C₄×C₂₉	—	2	1	no	—
104	C₄×C₂₉	—	2	2	no	—
105	C₄×C₂₉	#131	1	2	yes	—
106	C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
107	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
108	Dic₂₉	—	2	2	no	—
109	Dic₂₉	—	2	2	no	—
110	Dic₂₉	—	2	1	no	—
111	Dic₂₉	C₄×Q₈	1	2	yes	—
112	Dic₂₉	#131	1	2	yes	—
113	#533	—	29	1	no	—
114	#446	—	29	2	no	—
115	#446	—	29	1	no	—
116	#446	D₂₉	1	1	yes	characteristic, complemented
117	C₈×Q₈	—	29	1	no	—
118	C₈×Q₈	—	29	2	no	—
119	C₈×Q₈	—	29	1	no	—
120	#424	—	29	2	no	—
121	#423	—	29	1	no	—
122	#405	—	29	2	no	—
123	C₂×C₄×C₈	—	29	1	no	—
124	C₂×C₂₉	C₂×C₄×Q₈	1	1	yes	characteristic
125	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
126	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
127	D₂₉	—	2	2	no	—
128	C₂×M₁₆	—	29	2	no	—
129	C₂²×C₈	—	29	1	no	—
130	C₂²×C₈	—	29	1	no	—
131	C₄⋊Q₈	—	29	1	no	—
132	C₄⋊Q₈	—	29	1	no	—
133	#125	—	29	2	no	—
134	C₂²⋊Q₈	—	29	1	no	—
135	C₂²⋊Q₈	—	29	2	no	—
136	C₂²⋊Q₈	—	29	1	no	—
137	C₄×Q₈	—	29	2	no	—
138	C₄×Q₈	—	29	1	no	—
139	C₄×Q₈	C₂×D₂₉	1	1	yes	characteristic
140	#117	—	29	2	no	—
141	C₂×C₄²	—	29	1	no	—
142	C₄⋊C₈	—	29	2	no	—
143	C₄⋊C₈	—	29	2	no	—
144	C₄⋊C₈	—	29	2	no	—
145	C₄⋊C₈	—	29	1	no	—
146	C₄⋊C₈	—	29	2	no	—
147	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic
148	C₄⋊C₈	C₂×D₂₉	1	2	yes	—
149	#97	—	29	2	no	—
150	#97	C₂×D₂₉	1	2	yes	—
151	C₄×C₈	—	29	1	no	—
152	C₄×C₈	—	29	1	no	—
153	C₄×C₈	—	29	2	no	—
154	C₄×C₈	—	29	1	no	—
155	C₄×C₈	—	29	2	no	—
156	C₄×C₈	C₂×D₂₉	1	1	yes	characteristic
157	C₂₉	#2960	1	1	yes	characteristic, complemented, Sylow
158	C₂×Q₈	—	29	2	no	—
159	C₂×Q₈	—	29	1	no	—
160	C₂×Q₈	C₄×D₂₉	1	1	yes	characteristic
161	C₂²×C₄	—	29	1	no	—
162	C₂²×C₄	—	29	1	no	—
163	C₂²×C₄	—	29	1	no	—
164	M₁₆	—	58	4	no	—
165	C₂×C₈	—	29	1	no	—
166	C₂×C₈	—	29	4	no	—
167	C₂×C₈	—	29	1	no	—
168	C₂×C₈	—	29	2	no	—
169	C₂×C₈	—	58	2	no	—
170	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₂₉	1	2	yes	—
172	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
173	C₄⋊C₄	—	29	2	no	—
174	C₄⋊C₄	—	29	2	no	—
175	C₄⋊C₄	—	29	2	no	—
176	C₄⋊C₄	—	29	2	no	—
177	C₄⋊C₄	—	29	2	no	—
178	C₄⋊C₄	—	29	2	no	—
179	C₄⋊C₄	—	29	1	no	—
180	C₄⋊C₄	C₄×D₂₉	1	2	yes	—
181	C₄⋊C₄	C₄×D₂₉	1	1	yes	characteristic
182	C₂²⋊C₄	—	29	2	no	—
183	C₂²⋊C₄	—	29	2	no	—
184	C₄²	—	29	1	no	—
185	C₄²	—	29	1	no	—
186	C₄²	—	29	1	no	—
187	C₄²	—	29	2	no	—
188	C₄²	C₂²×D₂₉	1	1	yes	characteristic
189	C₄²	C₂²×D₂₉	1	2	yes	—
190	C₂³	—	29	1	no	—
191	Q₈	—	2	2	no	—
192	Q₈	—	58	4	no	—
193	Q₈	—	58	2	no	—
194	C₂×C₄	—	29	1	no	—
195	C₂×C₄	—	29	1	no	—
196	C₂×C₄	—	29	2	no	—
197	C₂×C₄	—	29	2	no	—
198	C₂×C₄	—	29	1	no	—
199	C₂×C₄	—	29	2	no	—
200	C₂×C₄	—	29	2	no	—
201	C₂×C₄	—	58	2	no	—
202	C₂×C₄	—	58	2	no	—
203	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
205	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
207	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
208	C₈	—	2	1	no	—
209	C₈	—	2	2	no	—
210	C₈	—	29	2	no	—
211	C₈	—	58	1	no	—
212	C₈	—	58	2	no	—
213	C₈	Q₈×D₂₉	1	2	yes	—
214	C₂²	—	29	2	no	—
215	C₂²	—	58	1	no	—
216	C₂²	—	58	1	no	—
217	C₂²	C₂²×C₄×D₂₉	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	29	2	no	—
222	C₄	—	29	2	no	—
223	C₄	—	58	2	no	—
224	C₄	—	58	1	no	—
225	C₄	—	58	2	no	—
226	C₄	#11745269	1	2	yes	—
227	C₄	C₂×Q₈×D₂₉	1	1	yes	characteristic
228	C₄	#41×D₂₉	1	1	yes	characteristic
229	C₂	—	58	2	no	—
230	C₂	C₄×Q₈×D₂₉	1	1	yes	characteristic
231	C₂	#131×D₂₉	1	1	yes	characteristic
232	C₂	C₂×#11745269	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₉
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting	C₂₉×#446	C₂
prime radical	C₂²×D₂₉	C₂²×C₄
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	#446	D₂₉
hypocenter	C₂₉	#2960
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #14271293	C₂	semidirect
3. #14271291	C₂	semidirect
4. Q₈×#6597	C₂	semidirect
5. #14271287	C₂	semidirect
6. #14271287	C₂	semidirect
11. C₂₉×#446	C₂	semidirect
72. C₈×D₂₉	Q₈	semidirect
116. #446	D₂₉	semidirect
157. C₂₉	#2960	semidirect

Maximal subgroups

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

subgroup	index
#14271717	2
#14271293	2
#14271291	2
Q₈×#6597	2
#14271287	2
#14271188	2
#14271185	2
#14271120	2
C₄×C₈×D₂₉	2
C₂₉×#446	2
#2960	29