Subgroups — #16828611

Order 3712.

388 conjugacy classes of subgroups.

The subgroup lattice has 302 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	#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	#14271219	C₂	1	1	yes	maximal, characteristic, complemented
7	#14271215	C₂	1	1	yes	maximal, characteristic, complemented
8	#14271187	C₂	1	1	yes	maximal, characteristic, complemented
9	#14271185	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄⋊C₈×D₂₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₉×#432	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#90805	C₄	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	C₂²	1	1	yes	characteristic
17	C₄×C₂²⋊D₂₉	C₂²	1	1	yes	characteristic
18	#11745288	C₄	1	1	yes	characteristic
19	C₂×#90777	C₂²	1	2	yes	—
20	C₂×#90772	C₂²	1	2	yes	—
21	C₂×C₈×D₂₉	C₂²	1	2	yes	—
22	C₄×D₁₁₆	C₄	1	1	yes	characteristic
23	C₄×D₁₁₆	C₂²	1	1	yes	characteristic
24	#11745234	C₄	1	1	yes	characteristic
25	#11745234	C₄	1	1	yes	characteristic
26	C₄²×D₂₉	C₂²	1	1	yes	characteristic
27	C₄²×D₂₉	C₂²	1	1	yes	characteristic
28	C₄×Dic₅₈	C₄	1	1	yes	characteristic
29	C₄×Dic₅₈	C₂²	1	1	yes	characteristic
30	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₂₉×C₂²⋊C₈	C₂²	1	2	yes	—
33	C₂₉×#97	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	#90777	—	4	2	no	—
82	C₂×#6597	C₂³	1	2	yes	—
83	C₂×#6597	C₂³	1	2	yes	—
84	#90772	—	4	2	no	—
85	C₈×D₂₉	—	4	2	no	—
86	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
87	C₂²×D₂₉	C₂²×C₄	1	1	yes	characteristic
88	C₂²×D₂₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₂₉	—	2	2	no	—
90	C₂×C₄×C₂₉	—	2	1	no	—
91	C₂×C₄×C₂₉	—	2	1	no	—
92	C₂×C₄×C₂₉	—	4	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
99	C₂²⋊D₂₉	—	2	2	no	—
100	C₂²⋊D₂₉	—	2	2	no	—
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	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
106	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
107	D₁₁₆	—	2	2	no	—
108	C₄×D₂₉	—	2	2	no	—
109	C₄×D₂₉	—	2	1	no	—
110	C₄×D₂₉	—	2	1	no	—
111	C₄×D₂₉	—	2	1	no	—
112	C₄×D₂₉	—	2	1	no	—
113	C₄×D₂₉	—	4	1	no	—
114	C₄×D₂₉	—	4	1	no	—
115	C₄×D₂₉	M₁₆	1	4	yes	complemented
116	Dic₅₈	—	2	2	no	—
117	C₈×C₂₉	—	2	2	no	—
118	C₈×C₂₉	—	2	2	no	—
119	#6597	—	2	2	no	—
120	#6597	—	2	2	no	—
121	#2989	—	29	1	no	maximal, Sylow
122	C₂²×C₂₉	—	2	1	no	—
123	C₂²×C₂₉	—	2	1	no	—
124	C₂²×C₂₉	—	2	1	no	—
125	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
126	C₂×D₂₉	—	2	1	no	—
127	C₂×D₂₉	—	2	1	no	—
128	C₂×D₂₉	—	2	1	no	—
129	C₂×D₂₉	—	2	1	no	—
130	C₂×D₂₉	—	2	1	no	—
131	C₂×D₂₉	C₂×M₁₆	1	2	yes	—
132	C₄×C₂₉	—	2	1	no	—
133	C₄×C₂₉	—	2	1	no	—
134	C₄×C₂₉	—	2	1	no	—
135	C₄×C₂₉	—	2	1	no	—
136	C₄×C₂₉	C₂×M₁₆	1	2	yes	—
137	C₄×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
138	C₄×C₂₉	2₋¹⁺⁴	1	1	yes	characteristic
139	Dic₂₉	—	2	1	no	—
140	Dic₂₉	—	2	1	no	—
141	Dic₂₉	—	2	1	no	—
142	Dic₂₉	—	2	1	no	—
143	Dic₂₉	—	2	1	no	—
144	Dic₂₉	C₂×M₁₆	1	2	yes	—
145	C₄×#41	—	29	1	no	—
146	#446	—	29	2	no	—
147	#436	—	29	2	no	—
148	#435	—	29	2	no	—
149	#435	—	29	2	no	—
150	#432	—	29	1	no	—
151	#432	D₂₉	1	1	yes	characteristic, complemented
152	#431	—	29	1	no	—
153	#423	—	29	1	no	—
154	#423	—	29	1	no	—
155	C₂×C₄⋊C₈	—	29	1	no	—
156	C₂×C₂₉	—	4	1	no	—
157	C₂×C₂₉	#520	1	1	yes	characteristic
158	C₂×C₂₉	C₂²×M₁₆	1	1	yes	characteristic
159	C₂×C₂₉	#568	1	1	yes	characteristic
160	D₂₉	—	2	2	no	—
161	D₂₉	—	4	1	no	—
162	C₂×#41	—	29	1	no	—
163	C₂×M₁₆	—	29	2	no	—
164	C₂×M₁₆	—	29	2	no	—
165	C₂²×C₈	—	29	2	no	—
166	C₄×Q₈	—	29	1	no	—
167	C₄×Q₈	—	29	1	no	—
168	C₄×D₄	—	29	1	no	—
169	C₄×D₄	—	29	1	no	—
170	C₄×D₄	—	29	1	no	—
171	C₄×D₄	—	29	1	no	—
172	C₄×D₄	—	29	1	no	—
173	C₄×D₄	—	29	1	no	—
174	#117	—	29	1	no	—
175	#117	—	29	1	no	—
176	#117	—	29	1	no	—
177	C₂×C₄²	—	29	1	no	—
178	C₂×C₄²	—	29	1	no	—
179	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
180	C₄⋊C₈	—	29	2	no	—
181	C₄⋊C₈	—	29	2	no	—
182	C₄⋊C₈	—	29	1	no	—
183	C₄⋊C₈	—	29	1	no	—
184	C₄⋊C₈	—	29	2	no	—
185	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic
186	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic
187	C₂²⋊C₈	—	29	2	no	—
188	C₂²⋊C₈	—	29	2	no	—
189	C₂²⋊C₈	C₂×D₂₉	1	2	yes	—
190	#97	—	29	2	no	—
191	#97	C₂×D₂₉	1	2	yes	—
192	C₄×C₈	—	29	2	no	—
193	C₂₉	#2989	1	1	yes	characteristic, complemented, Sylow
194	#41	—	58	4	no	—
195	C₂×Q₈	—	29	1	no	—
196	C₂×D₄	—	29	1	no	—
197	C₂×D₄	—	29	1	no	—
198	C₂×D₄	—	29	1	no	—
199	C₂²×C₄	—	29	1	no	—
200	C₂²×C₄	—	29	1	no	—
201	C₂²×C₄	—	29	1	no	—
202	C₂²×C₄	—	29	1	no	—
203	C₂²×C₄	—	29	1	no	—
204	C₂²×C₄	—	29	1	no	—
205	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
206	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
208	M₁₆	—	116	2	no	—
209	M₁₆	—	116	2	no	—
210	C₂×C₈	—	29	2	no	—
211	C₂×C₈	—	29	2	no	—
212	C₂×C₈	—	116	2	no	—
213	C₂×C₈	C₂²×D₂₉	1	2	yes	—
214	C₂×C₈	C₂²×D₂₉	1	2	yes	—
215	C₄⋊C₄	—	29	1	no	—
216	C₄⋊C₄	—	29	1	no	—
217	C₄⋊C₄	—	29	1	no	—
218	C₄⋊C₄	—	29	1	no	—
219	C₄⋊C₄	—	29	1	no	—
220	C₄⋊C₄	—	29	1	no	—
221	C₂²⋊C₄	—	29	1	no	—
222	C₂²⋊C₄	—	29	1	no	—
223	C₂²⋊C₄	—	29	1	no	—
224	C₂²⋊C₄	—	29	1	no	—
225	C₂²⋊C₄	—	29	1	no	—
226	C₂²⋊C₄	—	29	1	no	—
227	C₄²	—	29	1	no	—
228	C₄²	—	29	1	no	—
229	C₄²	—	29	1	no	—
230	C₄²	—	29	1	no	—
231	C₄²	—	29	1	no	—
232	C₄²	—	29	1	no	—
233	C₄²	C₄×D₂₉	1	1	yes	characteristic
234	C₄²	C₄×D₂₉	1	1	yes	characteristic
235	C₄²	C₂²×D₂₉	1	1	yes	characteristic
236	C₄²	C₂²×D₂₉	1	1	yes	characteristic
237	C₂³	—	29	1	no	—
238	C₂³	—	29	1	no	—
239	C₂³	C₂×C₄×D₂₉	1	1	yes	characteristic
240	Q₈	—	58	2	no	—
241	D₄	—	58	2	no	—
242	D₄	—	58	2	no	—
243	D₄	—	58	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	29	1	no	—
249	C₂×C₄	—	29	4	no	—
250	C₂×C₄	—	29	1	no	—
251	C₂×C₄	—	29	1	no	—
252	C₂×C₄	—	29	1	no	—
253	C₂×C₄	—	29	1	no	—
254	C₂×C₄	—	29	1	no	—
255	C₂×C₄	—	58	1	no	—
256	C₂×C₄	—	58	1	no	—
257	C₂×C₄	—	58	1	no	—
258	C₂×C₄	—	58	1	no	—
259	C₂×C₄	—	58	2	no	—
260	C₂×C₄	—	116	1	no	—
261	C₂×C₄	—	116	1	no	—
262	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
263	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
264	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
266	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
267	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	2	2	no	—
270	C₈	—	58	2	no	—
271	C₈	—	58	2	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	29	2	no	—
276	C₂²	—	58	1	no	—
277	C₂²	—	58	1	no	—
278	C₂²	—	58	1	no	—
279	C₂²	—	58	1	no	—
280	C₂²	—	58	1	no	—
281	C₂²	C₂²×C₄×D₂₉	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	29	2	no	—
287	C₄	—	58	1	no	—
288	C₄	—	58	1	no	—
289	C₄	—	58	1	no	—
290	C₄	—	58	1	no	—
291	C₄	—	58	1	no	—
292	C₄	M₁₆×D₂₉	1	2	yes	—
293	C₄	#11745364	1	1	yes	characteristic
294	C₄	#11745365	1	1	yes	characteristic
295	C₂	—	4	1	no	—
296	C₂	—	58	2	no	—
297	C₂	—	116	1	no	—
298	C₂	#14271643	1	1	yes	characteristic
299	C₂	C₂×M₁₆×D₂₉	1	1	yes	characteristic
300	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	C₂×#90805	C₄
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	#432	D₂₉
hypocenter	C₂₉	#2989
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. #14271219	C₂	semidirect
7. #14271215	C₂	semidirect
8. #14271187	C₂	semidirect
9. #14271185	C₂	semidirect
10. C₄⋊C₈×D₂₉	C₂	semidirect
11. C₂₉×#432	C₂	semidirect
115. C₄×D₂₉	M₁₆	semidirect
151. #432	D₂₉	semidirect
193. C₂₉	#2989	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
#14271291	2
#14271238	2
#14271237	2
#14271236	2
#14271219	2
#14271215	2
#14271187	2
#14271185	2
C₄⋊C₈×D₂₉	2
C₂₉×#432	2
#2989	29