Subgroups — #16828219

Order 3712.

502 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11745310	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#90777	C₂	1	1	yes	maximal, characteristic, complemented
3	#14271241	C₂	1	2	yes	maximal, complemented
4	#14271235	C₂	1	2	yes	maximal, complemented
5	#14271235	C₂	1	2	yes	maximal, complemented
6	D₄×#6597	C₂	1	2	yes	maximal, complemented
7	#14271190	C₂	1	1	yes	maximal, characteristic, complemented
8	#14271130	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂²⋊#6597	C₂	1	1	yes	maximal, characteristic, complemented
10	#14271119	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₉×#525	C₂	1	1	yes	maximal, characteristic
12	C₂×#41×C₂₉	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₂₉	C₂²	1	1	yes	characteristic
14	C₂₉×#123	C₄	1	2	yes	—
15	C₂₉×C₂²⋊Q₈	C₄	1	1	yes	characteristic
16	C₂₉×C₂²⋊Q₈	C₄	1	1	yes	characteristic
17	C₂₉×#121	C₄	1	1	yes	characteristic
18	C₂₉×#121	C₄	1	1	yes	characteristic
19	C₂₉×#120	C₄	1	2	yes	—
20	C₄×D₄×C₂₉	C₂²	1	2	yes	—
21	C₄×D₄×C₂₉	C₂²	1	2	yes	—
22	C₂₉×#117	C₂²	1	1	yes	characteristic
23	#11745310	—	2	4	no	—
24	C₂×#90777	—	2	4	no	—
25	C₂×#90777	C₂²	1	2	yes	—
26	C₂×#90777	C₂²	1	1	yes	characteristic
27	C₂×#90777	C₂²	1	1	yes	characteristic
28	C₂×#90777	C₂²	1	2	yes	complemented
29	C₂²×#6597	C₂²	1	2	yes	—
30	C₂²×#6597	C₂²	1	1	yes	characteristic
31	C₂²×#6597	C₂²	1	1	yes	characteristic
32	C₂²×#6597	C₂²	1	2	yes	complemented
33	C₂²⋊#6597	C₂²	1	1	yes	characteristic
34	C₂²⋊#6597	C₂²	1	1	yes	characteristic
35	C₂²⋊#6597	C₂²	1	1	yes	characteristic
36	C₂²⋊#6597	C₂²	1	1	yes	characteristic
37	C₂²⋊#6597	C₂²	1	2	yes	complemented
38	C₂²⋊#6597	C₂²	1	2	yes	complemented
39	C₄⋊#6597	C₂²	1	2	yes	complemented
40	C₄⋊#6597	C₂²	1	2	yes	complemented
41	#11745151	C₂²	1	2	yes	complemented
42	C₄×#6597	C₂²	1	2	yes	complemented
43	C₂⁴×C₂₉	C₂×C₄	1	1	yes	characteristic
44	#41×C₂₉	—	2	2	no	—
45	#41×C₂₉	—	2	2	no	—
46	C₂×Q₈×C₂₉	C₂×C₄	1	1	yes	characteristic
47	C₂×D₄×C₂₉	C₂×C₄	1	2	yes	—
48	C₂×D₄×C₂₉	C₂×C₄	1	2	yes	—
49	C₂×D₄×C₂₉	C₂×C₄	1	1	yes	characteristic
50	C₂×D₄×C₂₉	C₂×C₄	1	2	yes	—
51	C₂²×C₄×C₂₉	—	2	2	no	—
52	C₂²×C₄×C₂₉	—	2	2	no	—
53	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
58	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
61	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
62	C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
63	C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
64	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
65	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
66	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
67	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
68	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
69	C₄²×C₂₉	C₂³	1	2	yes	—
70	#90777	—	2	2	no	—
71	#90777	—	2	4	no	—
72	#90777	—	2	4	no	—
73	#90777	D₄	1	4	yes	complemented
74	C₂×#6597	—	2	2	no	—
75	C₂×#6597	—	2	2	no	—
76	C₂×#6597	—	2	4	no	—
77	C₂×#6597	—	2	4	no	—
78	C₂×#6597	C₂³	1	1	yes	characteristic
79	C₂×#6597	C₂³	1	2	yes	—
80	C₂×#6597	C₂³	1	1	yes	characteristic
81	C₂×#6597	C₂³	1	1	yes	characteristic
82	C₂×#6597	C₂³	1	1	yes	characteristic
83	C₂×#6597	C₂³	1	2	yes	—
84	C₂³×C₂₉	—	2	2	no	—
85	C₂³×C₂₉	—	2	2	no	—
86	C₂³×C₂₉	—	4	1	no	—
87	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
88	C₂³×C₂₉	C₂²×C₄	1	2	yes	—
89	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
90	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
91	Q₈×C₂₉	—	2	2	no	—
92	D₄×C₂₉	—	2	4	no	—
93	D₄×C₂₉	—	2	4	no	—
94	D₄×C₂₉	—	2	2	no	—
95	D₄×C₂₉	—	4	2	no	—
96	C₂×C₄×C₂₉	—	2	2	no	—
97	C₂×C₄×C₂₉	—	2	2	no	—
98	C₂×C₄×C₂₉	—	2	2	no	—
99	C₂×C₄×C₂₉	—	2	1	no	—
100	C₂×C₄×C₂₉	—	2	2	no	—
101	C₂×C₄×C₂₉	—	2	1	no	—
102	C₂×C₄×C₂₉	—	2	1	no	—
103	C₂×C₄×C₂₉	—	2	1	no	—
104	C₂×C₄×C₂₉	—	2	2	no	—
105	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
106	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
110	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
112	C₂×C₄×C₂₉	#41	1	2	yes	—
113	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
114	#6597	—	2	1	no	—
115	#6597	—	2	1	no	—
116	#6597	—	2	2	no	—
117	#6597	—	2	1	no	—
118	#6597	—	2	1	no	—
119	#6597	C₂×D₄	1	4	yes	—
120	#2933	—	29	1	no	maximal, Sylow
121	C₂²×C₂₉	—	2	1	no	—
122	C₂²×C₂₉	—	2	1	no	—
123	C₂²×C₂₉	—	2	1	no	—
124	C₂²×C₂₉	—	2	1	no	—
125	C₂²×C₂₉	—	2	1	no	—
126	C₂²×C₂₉	—	2	2	no	—
127	C₂²×C₂₉	—	2	2	no	—
128	C₂²×C₂₉	—	2	1	no	—
129	C₂²×C₂₉	—	2	1	no	—
130	C₂²×C₂₉	—	2	2	no	—
131	C₂²×C₂₉	—	4	4	no	—
132	C₂²×C₂₉	C₄×D₄	1	2	yes	—
133	C₂²×C₂₉	#131	1	2	yes	complemented
134	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
135	C₄×C₂₉	—	2	1	no	—
136	C₄×C₂₉	—	2	1	no	—
137	C₄×C₂₉	—	2	2	no	—
138	C₄×C₂₉	—	2	2	no	—
139	C₄×C₂₉	—	2	2	no	—
140	C₄×C₂₉	C₄×D₄	1	2	yes	—
141	C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
142	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
143	C₂×#131	—	29	1	no	—
144	C₂²×M₁₆	—	29	1	no	—
145	#525	D₂₉	1	1	yes	characteristic
146	#436	—	29	2	no	—
147	#435	—	29	2	no	—
148	#435	—	29	2	no	—
149	C₈×D₄	—	29	2	no	—
150	#424	—	29	1	no	—
151	#408	—	29	1	no	—
152	C₂×C₂²⋊C₈	—	29	1	no	—
153	#405	—	29	1	no	—
154	C₂×C₂₉	—	2	2	no	—
155	C₂×C₂₉	—	2	2	no	—
156	C₂×C₂₉	—	4	2	no	—
157	C₂×C₂₉	—	4	1	no	—
158	C₂×C₂₉	C₂×C₄×D₄	1	1	yes	characteristic
159	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
160	C₂×C₂₉	#568	1	1	yes	characteristic
161	C₂×#41	C₂×D₂₉	1	1	yes	characteristic
162	C₂³×C₄	C₂×D₂₉	1	1	yes	characteristic
163	#131	—	58	4	no	—
164	C₂×M₁₆	—	29	1	no	—
165	C₂×M₁₆	—	29	1	no	—
166	C₂×M₁₆	—	29	2	no	—
167	C₂×M₁₆	—	29	2	no	—
168	C₂×M₁₆	—	58	4	no	—
169	C₂²×C₈	—	29	2	no	—
170	C₂²×C₈	—	29	1	no	—
171	C₂²×C₈	—	29	1	no	—
172	C₂²×C₈	—	29	2	no	—
173	#123	Dic₂₉	1	2	yes	—
174	C₂²⋊Q₈	Dic₂₉	1	1	yes	characteristic
175	C₂²⋊Q₈	Dic₂₉	1	1	yes	characteristic
176	#121	Dic₂₉	1	1	yes	characteristic
177	#121	Dic₂₉	1	1	yes	characteristic
178	#120	Dic₂₉	1	2	yes	—
179	C₄×D₄	C₂×D₂₉	1	2	yes	—
180	C₄×D₄	C₂×D₂₉	1	2	yes	—
181	#117	C₂×D₂₉	1	1	yes	characteristic
182	C₄⋊C₈	—	29	2	no	—
183	C₄⋊C₈	—	29	2	no	—
184	C₂²⋊C₈	—	29	1	no	—
185	C₂²⋊C₈	—	29	1	no	—
186	C₂²⋊C₈	—	29	1	no	—
187	C₂²⋊C₈	—	29	1	no	—
188	C₂²⋊C₈	—	29	2	no	—
189	C₂²⋊C₈	—	29	2	no	—
190	#97	—	29	2	no	—
191	C₄×C₈	—	29	2	no	—
192	C₂₉	#2933	1	1	yes	characteristic, complemented, Sylow
193	C₂⁴	C₂×Dic₂₉	1	1	yes	characteristic
194	#41	—	2	2	no	—
195	#41	—	2	2	no	—
196	C₂×Q₈	C₂×Dic₂₉	1	1	yes	characteristic
197	C₂×D₄	C₂×Dic₂₉	1	2	yes	—
198	C₂×D₄	C₂×Dic₂₉	1	2	yes	—
199	C₂×D₄	C₂×Dic₂₉	1	1	yes	characteristic
200	C₂×D₄	C₂×Dic₂₉	1	2	yes	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	C₂×Dic₂₉	1	1	yes	characteristic
204	C₂²×C₄	C₂×Dic₂₉	1	1	yes	characteristic
205	C₂²×C₄	C₂×Dic₂₉	1	1	yes	characteristic
206	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₂₉	1	2	yes	—
208	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
209	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
210	M₁₆	—	29	4	no	—
211	M₁₆	—	58	4	no	—
212	M₁₆	—	58	4	no	—
213	M₁₆	—	58	2	no	—
214	C₂×C₈	—	29	2	no	—
215	C₂×C₈	—	29	1	no	—
216	C₂×C₈	—	29	1	no	—
217	C₂×C₈	—	29	1	no	—
218	C₂×C₈	—	29	1	no	—
219	C₂×C₈	—	29	2	no	—
220	C₂×C₈	—	58	4	no	—
221	C₂×C₈	—	58	4	no	—
222	C₂×C₈	—	58	2	no	—
223	C₂×C₈	—	58	2	no	—
224	C₄⋊C₄	C₂×Dic₂₉	1	1	yes	characteristic
225	C₄⋊C₄	C₂×Dic₂₉	1	1	yes	characteristic
226	C₄⋊C₄	C₂×Dic₂₉	1	2	yes	—
227	C₄⋊C₄	C₂×Dic₂₉	1	2	yes	—
228	C₂²⋊C₄	C₂×Dic₂₉	1	2	yes	—
229	C₂²⋊C₄	C₂×Dic₂₉	1	2	yes	—
230	C₂²⋊C₄	C₂×Dic₂₉	1	2	yes	—
231	C₂²⋊C₄	C₂×Dic₂₉	1	2	yes	—
232	C₂²⋊C₄	C₂×Dic₂₉	1	2	yes	—
233	C₄²	C₂²×D₂₉	1	2	yes	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	4	1	no	—
237	C₂³	C₂²×Dic₂₉	1	1	yes	characteristic
238	C₂³	C₂²×Dic₂₉	1	2	yes	—
239	C₂³	C₂²×Dic₂₉	1	1	yes	characteristic
240	C₂³	C₂²×Dic₂₉	1	1	yes	characteristic
241	Q₈	—	2	2	no	—
242	D₄	—	2	4	no	—
243	D₄	—	2	4	no	—
244	D₄	—	2	2	no	—
245	D₄	—	4	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	D₄×D₂₉	1	2	yes	—
256	C₂×C₄	#90807	1	2	yes	—
257	C₂×C₄	C₂²×Dic₂₉	1	2	yes	—
258	C₂×C₄	C₂²×Dic₂₉	1	2	yes	—
259	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
260	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
261	C₂×C₄	C₂²×Dic₂₉	1	2	yes	—
262	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
263	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
264	C₈	—	29	4	no	—
265	C₈	—	58	1	no	—
266	C₈	—	58	2	no	—
267	C₈	—	58	1	no	—
268	C₈	—	58	1	no	—
269	C₈	—	58	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	4	no	—
281	C₂²	D₄×Dic₂₉	1	2	yes	—
282	C₂²	#11745310	1	2	yes	complemented
283	C₂²	C₂³×Dic₂₉	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	D₄×Dic₂₉	1	2	yes	—
290	C₄	C₂×D₄×D₂₉	1	1	yes	characteristic
291	C₄	C₂×#90807	1	1	yes	characteristic
292	C₂	—	2	2	no	—
293	C₂	—	2	2	no	—
294	C₂	—	4	1	no	—
295	C₂	—	4	2	no	—
296	C₂	C₂×D₄×Dic₂₉	1	1	yes	characteristic
297	C₂	C₂×#11745310	1	1	yes	characteristic
298	C₂	#14271930	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₉
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting, prime radical	C₂₉×#525	C₂
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	#525	D₂₉
hypocenter	C₂₉	#2933
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ #525

Complements

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

N	H ≅ G/N	type
1. C₂×#11745310	C₂	semidirect
2. C₂²×#90777	C₂	semidirect
3. #14271241	C₂	semidirect
4. #14271235	C₂	semidirect
5. #14271235	C₂	semidirect
6. D₄×#6597	C₂	semidirect
7. #14271190	C₂	semidirect
8. #14271130	C₂	semidirect
9. C₂×C₂²⋊#6597	C₂	semidirect
10. #14271119	C₂	semidirect
28. C₂×#90777	C₂²	semidirect
32. C₂²×#6597	C₂²	semidirect
37. C₂²⋊#6597	C₂²	semidirect
38. C₂²⋊#6597	C₂²	semidirect
39. C₄⋊#6597	C₂²	semidirect
40. C₄⋊#6597	C₂²	semidirect
41. #11745151	C₂²	semidirect
42. C₄×#6597	C₂²	semidirect
73. #90777	D₄	semidirect
133. C₂²×C₂₉	#131	semidirect
192. C₂₉	#2933	semidirect
282. C₂²	#11745310	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11745310	2
C₂²×#90777	2
#14271241	2
#14271235	2
D₄×#6597	2
#14271190	2
#14271130	2
C₂×C₂²⋊#6597	2
#14271119	2
C₂₉×#525	2
#2933	29