Subgroups — #16828131

Order 3712.

400 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#90805	C₂	1	1	yes	maximal, characteristic
2	#14271223	C₂	1	2	yes	maximal, complemented
3	#14271221	C₂	1	2	yes	maximal, complemented
4	#14271216	C₂	1	1	yes	maximal, characteristic, complemented
5	#14271214	C₂	1	1	yes	maximal, characteristic, complemented
6	#14271214	C₂	1	1	yes	maximal, characteristic, complemented
7	#14271188	C₂	1	2	yes	maximal, complemented
8	#14271128	C₂	1	2	yes	maximal
9	C₄×#90772	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₈×D₂₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₉×#431	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#90805	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
14	C₄×C₂²⋊D₂₉	C₄	1	1	yes	characteristic
15	C₄×C₂²⋊D₂₉	C₄	1	1	yes	characteristic
16	C₄×C₂²⋊D₂₉	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊D₂₉	C₄	1	1	yes	characteristic
18	#11745288	C₂²	1	1	yes	characteristic
19	C₂×#90772	C₂²	1	2	yes	—
20	C₂×C₈×D₂₉	C₂²	1	2	yes	—
21	C₄×D₁₁₆	C₄	1	1	yes	characteristic
22	C₄×D₁₁₆	C₄	1	1	yes	characteristic
23	#11745234	C₂²	1	1	yes	characteristic
24	#11745234	C₂²	1	1	yes	characteristic
25	C₄²×D₂₉	C₂²	1	1	yes	characteristic
26	C₄²×D₂₉	C₂²	1	1	yes	characteristic
27	C₄×Dic₅₈	C₄	1	1	yes	characteristic
28	C₄×Dic₅₈	C₄	1	1	yes	characteristic
29	C₂₉×C₄⋊C₈	C₂²	1	2	yes	—
30	C₂₉×C₂²⋊C₈	C₂²	1	2	yes	—
31	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
32	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
33	C₂²⋊#6597	C₂²	1	2	yes	—
34	#11745168	C₂²	1	2	yes	—
35	#11745168	C₂²	1	2	yes	—
36	#11745163	C₂²	1	2	yes	—
37	#11745162	C₂²	1	2	yes	—
38	#11745162	C₂²	1	2	yes	—
39	C₈×Dic₂₉	C₂²	1	2	yes	—
40	C₄⋊#6597	C₂²	1	2	yes	—
41	C₄×#6597	C₂²	1	1	yes	characteristic
42	C₄×#6597	C₂²	1	1	yes	characteristic
43	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₂₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊D₂₉	C₂×C₄	1	1	yes	characteristic
48	#90805	—	2	4	no	—
49	C₂×D₁₁₆	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
55	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
56	C₂×Dic₅₈	C₂×C₄	1	1	yes	characteristic
57	C₂×C₈×C₂₉	C₂³	1	2	yes	—
58	C₂×C₈×C₂₉	C₂³	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	C₂²⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
64	C₂²⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
65	#90781	C₂×C₄	1	1	yes	characteristic
66	#90781	C₂×C₄	1	1	yes	characteristic
67	#90781	C₂×C₄	1	1	yes	characteristic
68	#90781	C₂×C₄	1	1	yes	characteristic
69	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
70	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
71	#90779	C₂×C₄	1	1	yes	characteristic
72	#90779	C₂×C₄	1	1	yes	characteristic
73	#90779	C₂×C₄	1	1	yes	characteristic
74	#90779	C₂×C₄	1	1	yes	characteristic
75	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
76	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
77	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
78	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
79	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
80	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
81	C₂×#6597	C₂³	1	2	yes	—
82	C₂×#6597	C₂³	1	2	yes	—
83	#90772	—	2	4	no	—
84	C₈×D₂₉	—	2	4	no	—
85	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
86	C₂²×D₂₉	C₂²×C₄	1	1	yes	characteristic
87	C₂²×D₂₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₂₉	—	2	2	no	—
89	C₂×C₄×C₂₉	—	2	2	no	—
90	C₂×C₄×C₂₉	—	2	2	no	—
91	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₂₉	—	2	2	no	—
98	C₂²⋊D₂₉	—	2	2	no	—
99	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
100	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
101	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
102	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
103	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
104	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
105	D₁₁₆	—	2	2	no	—
106	C₄×D₂₉	—	2	2	no	—
107	C₄×D₂₉	—	2	2	no	—
108	C₄×D₂₉	—	2	2	no	—
109	C₄×D₂₉	—	2	2	no	—
110	C₄×D₂₉	—	2	2	no	—
111	C₄×D₂₉	#41	1	4	yes	—
112	Dic₅₈	—	2	2	no	—
113	C₈×C₂₉	—	2	2	no	—
114	C₈×C₂₉	—	2	2	no	—
115	#6597	—	2	2	no	—
116	#6597	—	2	2	no	—
117	#2922	—	29	1	no	maximal, Sylow
118	C₂²×C₂₉	—	2	1	no	—
119	C₂²×C₂₉	—	2	1	no	—
120	C₂²×C₂₉	—	2	1	no	—
121	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
122	C₂×D₂₉	—	2	1	no	—
123	C₂×D₂₉	—	2	1	no	—
124	C₂×D₂₉	—	2	1	no	—
125	C₂×D₂₉	—	2	1	no	—
126	C₂×D₂₉	—	2	1	no	—
127	C₂×D₂₉	#117	1	2	yes	—
128	C₄×C₂₉	—	2	1	no	—
129	C₄×C₂₉	—	2	1	no	—
130	C₄×C₂₉	—	2	1	no	—
131	C₄×C₂₉	#131	1	2	yes	—
132	C₄×C₂₉	#131	1	2	yes	—
133	C₄×C₂₉	C₂×#41	1	2	yes	—
134	Dic₂₉	—	2	1	no	—
135	Dic₂₉	—	2	1	no	—
136	Dic₂₉	—	2	1	no	—
137	Dic₂₉	—	2	1	no	—
138	Dic₂₉	—	2	1	no	—
139	Dic₂₉	#117	1	2	yes	—
140	C₄×#41	—	29	1	no	—
141	#433	—	29	2	no	—
142	#433	—	29	2	no	—
143	#431	—	29	1	no	—
144	#431	—	29	1	no	—
145	#431	—	29	1	no	—
146	#431	D₂₉	1	1	yes	characteristic, complemented
147	#424	—	29	2	no	—
148	#408	—	29	2	no	—
149	C₄×M₁₆	—	29	1	no	—
150	C₂×C₄×C₈	—	29	1	no	—
151	C₂×C₂₉	—	4	1	no	—
152	C₂×C₂₉	C₂×#117	1	1	yes	characteristic
153	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
154	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
155	D₂₉	—	2	2	no	—
156	D₂₉	—	4	1	no	—
157	C₂×#41	—	29	1	no	—
158	C₂×M₁₆	—	29	2	no	—
159	C₂²×C₈	—	29	2	no	—
160	C₄×Q₈	—	29	1	no	—
161	C₄×Q₈	—	29	1	no	—
162	C₄×D₄	—	29	1	no	—
163	C₄×D₄	—	29	1	no	—
164	C₄×D₄	—	29	1	no	—
165	C₄×D₄	—	29	1	no	—
166	C₄×D₄	—	29	1	no	—
167	C₄×D₄	—	29	1	no	—
168	#117	—	29	1	no	—
169	#117	—	29	1	no	—
170	#117	—	29	1	no	—
171	C₂×C₄²	—	29	1	no	—
172	C₂×C₄²	—	29	1	no	—
173	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
174	C₄⋊C₈	—	29	2	no	—
175	C₄⋊C₈	—	29	2	no	—
176	C₄⋊C₈	—	29	2	no	—
177	C₄⋊C₈	C₂×D₂₉	1	2	yes	—
178	C₂²⋊C₈	—	29	2	no	—
179	C₂²⋊C₈	—	29	2	no	—
180	C₂²⋊C₈	—	29	2	no	—
181	C₂²⋊C₈	C₂×D₂₉	1	2	yes	—
182	#97	—	29	2	no	—
183	C₄×C₈	—	29	2	no	—
184	C₄×C₈	—	29	1	no	—
185	C₄×C₈	—	29	1	no	—
186	C₄×C₈	C₂×D₂₉	1	1	yes	characteristic
187	C₄×C₈	C₂×D₂₉	1	1	yes	characteristic
188	C₂₉	#2922	1	1	yes	characteristic, complemented, Sylow
189	#41	—	58	4	no	—
190	C₂×Q₈	—	29	1	no	—
191	C₂×D₄	—	29	1	no	—
192	C₂×D₄	—	29	1	no	—
193	C₂×D₄	—	29	1	no	—
194	C₂²×C₄	—	29	1	no	—
195	C₂²×C₄	—	29	1	no	—
196	C₂²×C₄	—	29	1	no	—
197	C₂²×C₄	—	29	1	no	—
198	C₂²×C₄	—	29	1	no	—
199	C₂²×C₄	—	29	1	no	—
200	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
201	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
202	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
203	M₁₆	—	58	4	no	—
204	C₂×C₈	—	29	2	no	—
205	C₂×C₈	—	29	2	no	—
206	C₂×C₈	—	58	4	no	—
207	C₂×C₈	C₂²×D₂₉	1	2	yes	—
208	C₂×C₈	C₂²×D₂₉	1	2	yes	—
209	C₄⋊C₄	—	29	1	no	—
210	C₄⋊C₄	—	29	1	no	—
211	C₄⋊C₄	—	29	1	no	—
212	C₄⋊C₄	—	29	1	no	—
213	C₄⋊C₄	—	29	1	no	—
214	C₄⋊C₄	—	29	1	no	—
215	C₂²⋊C₄	—	29	1	no	—
216	C₂²⋊C₄	—	29	1	no	—
217	C₂²⋊C₄	—	29	1	no	—
218	C₂²⋊C₄	—	29	1	no	—
219	C₂²⋊C₄	—	29	1	no	—
220	C₂²⋊C₄	—	29	1	no	—
221	C₄²	—	29	1	no	—
222	C₄²	—	29	1	no	—
223	C₄²	—	29	1	no	—
224	C₄²	—	29	1	no	—
225	C₄²	—	29	1	no	—
226	C₄²	—	29	1	no	—
227	C₄²	C₄×D₂₉	1	1	yes	characteristic
228	C₄²	C₄×D₂₉	1	1	yes	characteristic
229	C₄²	C₂²×D₂₉	1	1	yes	characteristic
230	C₄²	C₂²×D₂₉	1	1	yes	characteristic
231	C₂³	—	29	1	no	—
232	C₂³	—	29	1	no	—
233	C₂³	C₂×C₄×D₂₉	1	1	yes	characteristic
234	Q₈	—	58	2	no	—
235	D₄	—	58	2	no	—
236	D₄	—	58	2	no	—
237	D₄	—	58	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	29	1	no	—
242	C₂×C₄	—	29	4	no	—
243	C₂×C₄	—	29	1	no	—
244	C₂×C₄	—	29	1	no	—
245	C₂×C₄	—	29	1	no	—
246	C₂×C₄	—	29	1	no	—
247	C₂×C₄	—	29	1	no	—
248	C₂×C₄	—	58	2	no	—
249	C₂×C₄	—	58	2	no	—
250	C₂×C₄	—	58	2	no	—
251	C₂×C₄	—	58	2	no	—
252	C₂×C₄	—	58	2	no	—
253	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
254	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
255	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
256	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
258	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
259	C₈	—	2	2	no	—
260	C₈	—	2	2	no	—
261	C₈	—	58	2	no	—
262	C₈	—	58	2	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	29	2	no	—
267	C₂²	—	58	1	no	—
268	C₂²	—	58	1	no	—
269	C₂²	—	58	1	no	—
270	C₂²	—	58	1	no	—
271	C₂²	—	58	1	no	—
272	C₂²	C₂²×C₄×D₂₉	1	1	yes	characteristic
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	29	2	no	—
277	C₄	—	58	1	no	—
278	C₄	—	58	1	no	—
279	C₄	—	58	1	no	—
280	C₄	—	58	1	no	—
281	C₄	—	58	1	no	—
282	C₄	#11745263	1	2	yes	—
283	C₄	#11745269	1	2	yes	—
284	C₄	#41×D₂₉	1	2	yes	—
285	C₂	—	4	1	no	—
286	C₂	—	58	2	no	—
287	C₂	—	116	1	no	—
288	C₂	#117×D₂₉	1	1	yes	characteristic
289	C₂	C₂×#11745263	1	1	yes	characteristic
290	C₂	C₂×#11745269	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂²×D₂₉
commutator, socle	C₂²×C₂₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₉
Fitting	C₂₉×#431	C₂
prime radical	C₂×#90805	C₂²
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	#431	D₂₉
hypocenter	C₂₉	#2922
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₄² ◁ #431

Complements

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

N	H ≅ G/N	type
2. #14271223	C₂	semidirect
3. #14271221	C₂	semidirect
4. #14271216	C₂	semidirect
5. #14271214	C₂	semidirect
6. #14271214	C₂	semidirect
7. #14271188	C₂	semidirect
9. C₄×#90772	C₂	semidirect
10. C₄×C₈×D₂₉	C₂	semidirect
11. C₂₉×#431	C₂	semidirect
146. #431	D₂₉	semidirect
188. C₂₉	#2922	semidirect

Maximal subgroups

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

subgroup	index
C₄×#90805	2
#14271223	2
#14271221	2
#14271216	2
#14271214	2
#14271188	2
#14271128	2
C₄×#90772	2
C₄×C₈×D₂₉	2
C₂₉×#431	2
#2922	29