Subgroups — #16828652

Order 3712.

364 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14271294	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#6597	C₂	1	1	yes	maximal, characteristic, complemented
3	#14271241	C₂	1	1	yes	maximal, characteristic, complemented
4	#14271235	C₂	1	2	yes	maximal, complemented
5	D₄×#6597	C₂	1	2	yes	maximal, complemented
6	D₄×#6597	C₂	1	1	yes	maximal, characteristic, complemented
7	#14271224	C₂	1	2	yes	maximal, complemented
8	#14271216	C₂	1	1	yes	maximal, characteristic, complemented
9	#14271190	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊#6597	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₉×#529	C₂	1	1	yes	maximal, characteristic
12	C₂₉×#126	C₄	1	2	yes	—
13	C₂₉×#125	C₄	1	1	yes	characteristic
14	C₂₉×#124	C₄	1	1	yes	characteristic
15	C₂₉×#123	C₄	1	2	yes	—
16	C₂₉×C₂²⋊Q₈	C₄	1	1	yes	characteristic
17	C₂₉×#121	C₄	1	1	yes	characteristic
18	C₄×Q₈×C₂₉	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₂₉	C₂²	1	2	yes	—
20	C₄×D₄×C₂₉	C₂²	1	1	yes	characteristic
21	C₂₉×#117	C₂²	1	2	yes	—
22	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
23	C₂×#90777	C₂²	1	2	yes	—
24	C₂²×#6597	C₂²	1	2	yes	—
25	C₂²×#6597	C₂²	1	2	yes	—
26	C₂²⋊#6597	C₂²	1	1	yes	characteristic
27	C₂²⋊#6597	C₂²	1	2	yes	—
28	C₂²⋊#6597	C₂²	1	2	yes	—
29	C₂²⋊#6597	C₂²	1	1	yes	characteristic
30	C₄⋊#6597	C₂²	1	2	yes	—
31	C₄⋊#6597	C₂²	1	1	yes	characteristic
32	C₄⋊#6597	C₂²	1	1	yes	characteristic
33	C₄⋊#6597	C₂²	1	2	yes	—
34	C₄⋊#6597	C₂²	1	2	yes	—
35	C₄⋊#6597	C₂²	1	1	yes	characteristic
36	C₄⋊#6597	C₂²	1	1	yes	characteristic
37	#11745151	C₂²	1	2	yes	—
38	C₄×#6597	C₂²	1	1	yes	characteristic
39	C₄×#6597	C₂²	1	2	yes	—
40	C₄×#6597	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₂₉	C₂×C₄	1	1	yes	characteristic
42	C₂×D₄×C₂₉	C₂×C₄	1	2	yes	—
43	C₂×D₄×C₂₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
48	C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
49	C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
50	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
51	C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
52	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
53	C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
54	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
55	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
56	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
57	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
58	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
59	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
60	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
61	C₄²×C₂₉	C₂³	1	1	yes	characteristic
62	C₄²×C₂₉	C₂³	1	1	yes	characteristic
63	C₄²×C₂₉	C₂³	1	2	yes	—
64	#90777	—	4	2	no	—
65	C₂×#6597	—	4	2	no	—
66	C₂×#6597	—	4	2	no	—
67	C₂×#6597	C₂³	1	2	yes	—
68	C₂×#6597	C₂³	1	1	yes	characteristic
69	C₂×#6597	C₂³	1	2	yes	—
70	C₂×#6597	C₂³	1	1	yes	characteristic
71	C₂×#6597	C₂³	1	1	yes	characteristic
72	C₂×#6597	C₂³	1	1	yes	characteristic
73	C₂³×C₂₉	C₂²×C₄	1	2	yes	—
74	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
75	Q₈×C₂₉	—	2	2	no	—
76	D₄×C₂₉	—	2	4	no	—
77	D₄×C₂₉	—	2	2	no	—
78	C₂×C₄×C₂₉	—	2	2	no	—
79	C₂×C₄×C₂₉	—	2	2	no	—
80	C₂×C₄×C₂₉	—	2	1	no	—
81	C₂×C₄×C₂₉	—	2	2	no	—
82	C₂×C₄×C₂₉	—	2	1	no	—
83	C₂×C₄×C₂₉	—	2	2	no	—
84	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
93	#6597	—	2	1	no	—
94	#6597	—	2	2	no	—
95	#6597	—	2	2	no	—
96	#6597	—	2	1	no	—
97	#6597	—	2	1	no	—
98	#6597	—	2	1	no	—
99	#2994	—	29	1	no	maximal, Sylow
100	C₂²×C₂₉	—	2	1	no	—
101	C₂²×C₂₉	—	2	1	no	—
102	C₂²×C₂₉	—	2	2	no	—
103	C₂²×C₂₉	—	2	2	no	—
104	C₂²×C₂₉	—	2	2	no	—
105	C₂²×C₂₉	#131	1	2	yes	—
106	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
107	C₄×C₂₉	—	2	1	no	—
108	C₄×C₂₉	—	2	2	no	—
109	C₄×C₂₉	—	2	1	no	—
110	C₄×C₂₉	—	2	1	no	—
111	C₄×C₂₉	—	2	1	no	—
112	C₄×C₂₉	—	2	2	no	—
113	C₄×C₂₉	—	2	2	no	—
114	C₄×C₂₉	#131	1	2	yes	—
115	C₄×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
116	C₄×C₂₉	2₋¹⁺⁴	1	1	yes	characteristic
117	#529	D₂₉	1	1	yes	characteristic
118	#446	—	29	1	no	—
119	C₈×Q₈	—	29	1	no	—
120	#436	—	29	1	no	—
121	#435	—	29	2	no	—
122	C₈×D₄	—	29	1	no	—
123	C₈×D₄	—	29	2	no	—
124	#433	—	29	2	no	—
125	#431	—	29	1	no	—
126	#424	—	29	2	no	—
127	C₂×C₄⋊C₈	—	29	1	no	—
128	C₂×C₂₉	—	2	2	no	—
129	C₂×C₂₉	—	4	2	no	—
130	C₂×C₂₉	#520	1	1	yes	characteristic
131	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
132	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
133	C₂×M₁₆	—	29	2	no	—
134	C₂²×C₈	—	29	2	no	—
135	C₂²×C₈	—	29	2	no	—
136	#126	Dic₂₉	1	2	yes	—
137	#125	Dic₂₉	1	1	yes	characteristic
138	#124	Dic₂₉	1	1	yes	characteristic
139	#123	Dic₂₉	1	2	yes	—
140	C₂²⋊Q₈	Dic₂₉	1	1	yes	characteristic
141	#121	Dic₂₉	1	1	yes	characteristic
142	C₄×Q₈	C₂×D₂₉	1	1	yes	characteristic
143	C₄×D₄	C₂×D₂₉	1	2	yes	—
144	C₄×D₄	C₂×D₂₉	1	1	yes	characteristic
145	#117	C₂×D₂₉	1	2	yes	—
146	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
147	C₄⋊C₈	—	29	2	no	—
148	C₄⋊C₈	—	29	1	no	—
149	C₄⋊C₈	—	29	1	no	—
150	C₄⋊C₈	—	29	1	no	—
151	C₄⋊C₈	—	29	2	no	—
152	C₄⋊C₈	—	29	2	no	—
153	C₄⋊C₈	—	29	1	no	—
154	C₂²⋊C₈	—	29	1	no	—
155	C₂²⋊C₈	—	29	1	no	—
156	C₂²⋊C₈	—	29	2	no	—
157	C₂²⋊C₈	—	29	2	no	—
158	#97	—	29	2	no	—
159	C₄×C₈	—	29	1	no	—
160	C₄×C₈	—	29	1	no	—
161	C₄×C₈	—	29	2	no	—
162	C₂₉	#2994	1	1	yes	characteristic, complemented, Sylow
163	C₂×Q₈	C₂×Dic₂₉	1	1	yes	characteristic
164	C₂×D₄	C₂×Dic₂₉	1	2	yes	—
165	C₂×D₄	C₂×Dic₂₉	1	1	yes	characteristic
166	C₂²×C₄	C₂×Dic₂₉	1	1	yes	characteristic
167	C₂²×C₄	C₂×Dic₂₉	1	1	yes	characteristic
168	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₂₉	1	2	yes	—
170	M₁₆	—	116	2	no	—
171	C₂×C₈	—	29	1	no	—
172	C₂×C₈	—	29	1	no	—
173	C₂×C₈	—	29	1	no	—
174	C₂×C₈	—	29	2	no	—
175	C₂×C₈	—	29	1	no	—
176	C₂×C₈	—	29	2	no	—
177	C₂×C₈	—	116	2	no	—
178	C₂×C₈	—	116	2	no	—
179	C₄⋊C₄	C₂×Dic₂₉	1	2	yes	—
180	C₄⋊C₄	C₂×Dic₂₉	1	2	yes	—
181	C₄⋊C₄	C₂×Dic₂₉	1	1	yes	characteristic
182	C₄⋊C₄	C₂×Dic₂₉	1	2	yes	—
183	C₄⋊C₄	C₂×Dic₂₉	1	1	yes	characteristic
184	C₄⋊C₄	C₂×Dic₂₉	1	2	yes	—
185	C₂²⋊C₄	C₂×Dic₂₉	1	2	yes	—
186	C₂²⋊C₄	C₂×Dic₂₉	1	2	yes	—
187	C₂²⋊C₄	C₂×Dic₂₉	1	2	yes	—
188	C₂²⋊C₄	C₂×Dic₂₉	1	2	yes	—
189	C₂²⋊C₄	C₂×Dic₂₉	1	2	yes	—
190	C₄²	C₂×Dic₂₉	1	1	yes	characteristic
191	C₄²	C₂×Dic₂₉	1	1	yes	characteristic
192	C₄²	C₂²×D₂₉	1	1	yes	characteristic
193	C₄²	C₂²×D₂₉	1	2	yes	—
194	C₄²	C₂²×D₂₉	1	1	yes	characteristic
195	C₂³	C₂²×Dic₂₉	1	2	yes	—
196	C₂³	C₂²×Dic₂₉	1	1	yes	characteristic
197	Q₈	—	2	2	no	—
198	D₄	—	2	4	no	—
199	D₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
207	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
208	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
209	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
210	C₂×C₄	C₂²×Dic₂₉	1	2	yes	—
211	C₂×C₄	C₂²×Dic₂₉	1	2	yes	—
212	C₂×C₄	C₂²×Dic₂₉	1	2	yes	—
213	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
214	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
215	C₈	—	58	1	no	—
216	C₈	—	58	2	no	—
217	C₈	—	58	2	no	—
218	C₈	—	58	1	no	—
219	C₈	—	58	1	no	—
220	C₈	—	58	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	#11745310	1	2	yes	—
227	C₂²	C₂³×Dic₂₉	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	#11745310	1	2	yes	—
236	C₄	#11745364	1	1	yes	characteristic
237	C₄	#11745365	1	1	yes	characteristic
238	C₂	—	2	2	no	—
239	C₂	—	4	2	no	—
240	C₂	#14271646	1	1	yes	characteristic
241	C₂	C₂×#11745310	1	1	yes	characteristic
242	C₂	C₂×#11745310	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₉
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting	C₂₉×#529	C₂
prime radical	C₂₉×#121	C₄
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	#529	D₂₉
hypocenter	C₂₉	#2994
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ #529

Complements

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

N	H ≅ G/N	type
1. #14271294	C₂	semidirect
2. Q₈×#6597	C₂	semidirect
3. #14271241	C₂	semidirect
4. #14271235	C₂	semidirect
5. D₄×#6597	C₂	semidirect
6. D₄×#6597	C₂	semidirect
7. #14271224	C₂	semidirect
8. #14271216	C₂	semidirect
9. #14271190	C₂	semidirect
10. C₂×C₄⋊#6597	C₂	semidirect
162. C₂₉	#2994	semidirect

Maximal subgroups

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

subgroup	index
#14271294	2
Q₈×#6597	2
#14271241	2
#14271235	2
D₄×#6597	2
#14271224	2
#14271216	2
#14271190	2
C₂×C₄⋊#6597	2
C₂₉×#529	2
#2994	29