Subgroups — #16835999

Order 3712.

532 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14272112	C₂	1	2	yes	maximal, complemented
2	#14272112	C₂	1	2	yes	maximal, complemented
3	#14272110	C₂	1	2	yes	maximal
4	#14272108	C₂	1	1	yes	maximal, characteristic, complemented
5	#14272107	C₂	1	1	yes	maximal, characteristic, complemented
6	#14272106	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊#1181	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²⋊C₄×#1181	C₂	1	2	yes	maximal
9	#14272099	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×(C₂×#1181)⋊C₄	C₂	1	1	yes	maximal, characteristic
11	#117×D₂₉	C₂	1	1	yes	maximal, characteristic
12	C₂³×#1181	C₂²	1	2	yes	—
13	C₂²×C₄×D₂₉	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊#1181	C₂²	1	2	yes	—
15	C₂×C₄⋊#1181	C₂²	1	2	yes	—
16	C₂×C₄⋊#1181	C₂²	1	2	yes	—
17	C₂×C₄×#1181	C₂²	1	2	yes	—
18	C₂×C₄×#1181	C₂²	1	2	yes	—
19	C₂₉×#117	C₄	1	1	yes	characteristic, complemented
20	#11745288	C₄	1	1	yes	characteristic, complemented
21	#11745254	C₄	1	1	yes	characteristic, complemented
22	#11745254	C₄	1	1	yes	characteristic, complemented
23	C₄⋊C₄×D₂₉	C₂²	1	1	yes	characteristic
24	C₄⋊C₄×D₂₉	C₂²	1	1	yes	characteristic
25	C₂²⋊C₄×D₂₉	C₂²	1	1	yes	characteristic
26	C₂²⋊C₄×D₂₉	C₂²	1	1	yes	characteristic
27	#11745239	C₄	1	1	yes	characteristic, complemented
28	#11745239	C₄	1	1	yes	characteristic, complemented
29	#11745234	C₄	1	1	yes	characteristic, complemented
30	#11745234	C₄	1	1	yes	characteristic, complemented
31	C₄²×D₂₉	C₂²	1	1	yes	characteristic
32	C₄²×D₂₉	C₂²	1	1	yes	characteristic
33	(C₂×#1181)⋊C₄	C₂²	1	1	yes	characteristic
34	(C₂×#1181)⋊C₄	C₂²	1	1	yes	characteristic
35	(C₂×#1181)⋊C₄	C₂²	1	2	yes	—
36	(C₂×#1181)⋊C₄	C₂²	1	1	yes	characteristic
37	(C₂×#1181)⋊C₄	C₂²	1	2	yes	—
38	(C₂×#1181)⋊C₄	C₂²	1	2	yes	—
39	(C₂×#1181)⋊C₄	C₂²	1	2	yes	—
40	(C₂×#1181)⋊C₄	C₂²	1	1	yes	characteristic
41	(C₂×#1181)⋊C₄	C₂²	1	2	yes	—
42	(C₂×#1181)⋊C₄	C₂²	1	2	yes	—
43	C₂³×D₂₉	C₂³	1	1	yes	characteristic
44	C₂²×#1181	—	2	4	no	—
45	C₂²×#1181	—	2	4	no	—
46	C₂²×#1181	—	4	2	no	—
47	C₂²×#1181	C₂³	1	2	yes	—
48	C₂²×#1181	C₂³	1	2	yes	—
49	C₂²×#1181	C₂³	1	2	yes	—
50	C₂²×#1181	C₂³	1	2	yes	—
51	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
52	C₂²×Dic₂₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₂₉	—	2	4	no	—
54	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
58	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
59	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
60	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
61	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
62	C₂²⋊#1181	—	2	4	no	—
63	C₄⋊#1181	—	2	4	no	—
64	C₄⋊#1181	—	4	2	no	—
65	C₄×#1181	—	2	4	no	—
66	C₄×#1181	—	4	2	no	—
67	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic, complemented
68	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic, complemented
69	C₂²⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
70	C₂²⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
71	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic, complemented
72	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic, complemented
73	C₂²⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
74	C₂²⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
75	#90781	C₂×C₄	1	1	yes	characteristic
76	#90781	C₂×C₄	1	1	yes	characteristic
77	#90781	C₂×C₄	1	1	yes	characteristic
78	#90781	C₂×C₄	1	1	yes	characteristic
79	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic, complemented
80	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic, complemented
81	#90779	C₂×C₄	1	1	yes	characteristic, complemented
82	#90779	C₂×C₄	1	1	yes	characteristic, complemented
83	#90779	C₂×C₄	1	1	yes	characteristic, complemented
84	#90779	C₂×C₄	1	1	yes	characteristic, complemented
85	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic, complemented
86	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic, complemented
87	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic, complemented
88	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic, complemented
89	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic, complemented
90	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic, complemented
91	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
92	C₂²×D₂₉	—	2	2	no	—
93	C₂²×D₂₉	—	2	1	no	—
94	C₂²×D₂₉	—	2	1	no	—
95	C₂²×D₂₉	C₂²×C₄	1	1	yes	characteristic
96	C₂²×D₂₉	#41	1	4	yes	—
97	C₂²×D₂₉	C₂⁴	1	1	yes	characteristic
98	C₂×#1181	—	2	2	no	—
99	C₂×#1181	—	2	2	no	—
100	C₂×#1181	—	2	2	no	—
101	C₂×#1181	—	2	2	no	—
102	C₂×#1181	—	2	2	no	—
103	C₂×#1181	—	2	2	no	—
104	C₂×#1181	—	2	2	no	—
105	C₂×#1181	—	2	2	no	—
106	C₂×#1181	—	2	2	no	—
107	C₂×#1181	—	2	2	no	—
108	C₂×#1181	—	2	2	no	—
109	C₂×#1181	—	4	8	no	—
110	C₂×#1181	#41	1	4	yes	—
111	C₂×C₄×C₂₉	—	2	2	no	—
112	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
118	C₂×Dic₂₉	—	2	2	no	—
119	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
120	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
121	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
122	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
123	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
124	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
125	C₄×D₂₉	—	2	2	no	—
126	C₄×D₂₉	—	2	2	no	—
127	C₄×D₂₉	—	2	2	no	—
128	C₄×D₂₉	—	2	2	no	—
129	C₄×D₂₉	—	2	2	no	—
130	C₄×D₂₉	—	2	2	no	—
131	C₄×D₂₉	—	2	1	no	—
132	C₄×D₂₉	—	2	1	no	—
133	C₄×D₂₉	—	2	1	no	—
134	C₄×D₂₉	—	2	1	no	—
135	#2343	—	29	1	no	maximal, Sylow
136	C₂²×C₂₉	—	2	1	no	—
137	C₂²×C₂₉	—	2	1	no	—
138	C₂²×C₂₉	#117	1	2	yes	—
139	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
140	C₂×D₂₉	—	2	1	no	—
141	C₂×D₂₉	—	2	1	no	—
142	C₂×D₂₉	—	2	4	no	—
143	C₂×D₂₉	—	2	4	no	—
144	C₂×D₂₉	#117	1	2	yes	—
145	C₂×D₂₉	C₂×#41	1	2	yes	—
146	C₂×D₂₉	C₂×#41	1	2	yes	—
147	C₂×D₂₉	H(5,𝔽₂)	1	1	yes	characteristic
148	C₂×D₂₉	2₋¹⁺⁴	1	1	yes	characteristic
149	#1181	—	2	4	no	—
150	#1181	—	4	2	no	—
151	#1181	—	4	2	no	—
152	#1181	—	4	2	no	—
153	C₄×C₂₉	—	2	1	no	—
154	C₄×C₂₉	—	2	1	no	—
155	C₄×C₂₉	—	2	1	no	—
156	C₄×C₂₉	—	2	1	no	—
157	C₄×C₂₉	—	2	1	no	—
158	C₄×C₂₉	—	2	1	no	—
159	Dic₂₉	—	2	1	no	—
160	Dic₂₉	—	2	1	no	—
161	Dic₂₉	—	2	1	no	—
162	Dic₂₉	—	2	1	no	—
163	Dic₂₉	—	2	1	no	—
164	Dic₂₉	—	2	1	no	—
165	C₂×#117	—	29	1	no	—
166	#387	—	29	2	no	—
167	#387	—	29	2	no	—
168	#385	—	29	2	no	—
169	#383	—	29	1	no	—
170	#382	—	29	1	no	—
171	#381	—	29	1	no	—
172	C₄×C₄⋊C₄	—	29	1	no	—
173	C₄×C₂²⋊C₄	—	29	2	no	—
174	#376	—	29	1	no	—
175	C₂×#95	—	29	1	no	—
176	C₂×C₂₉	—	2	2	no	—
177	C₂×C₂₉	C₂×#117	1	1	yes	characteristic
178	C₂×C₂₉	C₄×#41	1	1	yes	characteristic
179	C₂×C₂₉	#520	1	1	yes	characteristic
180	D₂₉	—	2	2	no	—
181	D₂₉	#551	1	2	yes	—
182	D₂₉	#552	1	2	yes	—
183	C₂³×C₄	—	29	1	no	—
184	C₂³×C₄	—	29	2	no	—
185	#117	—	29	1	no	—
186	#117	—	29	1	no	—
187	#117	—	29	1	no	—
188	#117	—	29	1	no	—
189	#117	—	29	1	no	—
190	#117	—	29	1	no	—
191	#117	—	29	1	no	—
192	#117	#1181	1	1	yes	characteristic, complemented
193	C₂×C₄⋊C₄	—	29	1	no	—
194	C₂×C₄⋊C₄	—	29	1	no	—
195	C₂×C₄⋊C₄	—	29	2	no	—
196	C₂×C₄⋊C₄	—	29	2	no	—
197	C₂×C₂²⋊C₄	—	29	1	no	—
198	C₂×C₂²⋊C₄	—	29	1	no	—
199	C₂×C₂²⋊C₄	—	29	2	no	—
200	C₂×C₄²	—	29	1	no	—
201	C₂×C₄²	—	29	1	no	—
202	C₂×C₄²	—	29	2	no	—
203	C₂×C₄²	—	29	2	no	—
204	#95	—	29	1	no	—
205	#95	—	29	1	no	—
206	#95	—	29	2	no	—
207	#95	—	29	1	no	—
208	#95	—	29	2	no	—
209	#95	—	29	2	no	—
210	#95	—	29	2	no	—
211	#95	—	29	1	no	—
212	#95	—	29	2	no	—
213	#95	—	29	2	no	—
214	C₂₉	#2343	1	1	yes	characteristic, complemented, Sylow
215	C₂⁴	—	29	1	no	—
216	C₂²×C₄	—	29	2	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	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	2	no	—
227	C₂²×C₄	—	29	2	no	—
228	C₂²×C₄	—	29	2	no	—
229	C₂²×C₄	—	58	4	no	—
230	C₂²×C₄	—	58	4	no	—
231	C₂²×C₄	—	58	4	no	—
232	C₂²×C₄	—	116	2	no	—
233	C₂²×C₄	C₂×#1181	1	1	yes	characteristic
234	C₄⋊C₄	—	29	1	no	—
235	C₄⋊C₄	—	29	1	no	—
236	C₄⋊C₄	—	29	1	no	—
237	C₄⋊C₄	—	29	1	no	—
238	C₄⋊C₄	—	29	1	no	—
239	C₄⋊C₄	—	29	1	no	—
240	C₄⋊C₄	—	58	4	no	—
241	C₄⋊C₄	—	116	2	no	—
242	C₄⋊C₄	C₂×#1181	1	1	yes	characteristic, complemented
243	C₄⋊C₄	C₂×#1181	1	1	yes	characteristic, complemented
244	C₂²⋊C₄	—	29	1	no	—
245	C₂²⋊C₄	—	29	1	no	—
246	C₂²⋊C₄	—	29	1	no	—
247	C₂²⋊C₄	—	29	1	no	—
248	C₂²⋊C₄	—	29	1	no	—
249	C₂²⋊C₄	—	29	1	no	—
250	C₂²⋊C₄	—	58	4	no	—
251	C₂²⋊C₄	C₂×#1181	1	1	yes	characteristic
252	C₂²⋊C₄	C₂×#1181	1	1	yes	characteristic
253	C₄²	—	29	1	no	—
254	C₄²	—	29	1	no	—
255	C₄²	—	29	1	no	—
256	C₄²	—	29	1	no	—
257	C₄²	—	29	1	no	—
258	C₄²	—	29	1	no	—
259	C₄²	—	58	4	no	—
260	C₄²	—	116	2	no	—
261	C₄²	C₂×#1181	1	1	yes	characteristic, complemented
262	C₄²	C₂×#1181	1	1	yes	characteristic, complemented
263	C₂³	—	29	1	no	—
264	C₂³	—	29	4	no	—
265	C₂³	—	29	1	no	—
266	C₂³	—	58	2	no	—
267	C₂³	—	58	1	no	—
268	C₂³	—	58	1	no	—
269	C₂³	C₂²×#1181	1	1	yes	characteristic
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	29	4	no	—
272	C₂×C₄	—	29	1	no	—
273	C₂×C₄	—	29	1	no	—
274	C₂×C₄	—	29	1	no	—
275	C₂×C₄	—	29	1	no	—
276	C₂×C₄	—	29	1	no	—
277	C₂×C₄	—	29	1	no	—
278	C₂×C₄	—	58	2	no	—
279	C₂×C₄	—	58	1	no	—
280	C₂×C₄	—	58	2	no	—
281	C₂×C₄	—	58	1	no	—
282	C₂×C₄	—	58	2	no	—
283	C₂×C₄	—	58	2	no	—
284	C₂×C₄	—	58	2	no	—
285	C₂×C₄	—	58	2	no	—
286	C₂×C₄	—	58	1	no	—
287	C₂×C₄	—	58	1	no	—
288	C₂×C₄	—	58	2	no	—
289	C₂×C₄	—	58	2	no	—
290	C₂×C₄	—	58	2	no	—
291	C₂×C₄	—	58	2	no	—
292	C₂×C₄	—	58	2	no	—
293	C₂×C₄	—	58	2	no	—
294	C₂×C₄	—	58	2	no	—
295	C₂×C₄	—	58	2	no	—
296	C₂×C₄	—	58	2	no	—
297	C₂×C₄	—	58	2	no	—
298	C₂×C₄	—	58	2	no	—
299	C₂×C₄	—	58	2	no	—
300	C₂×C₄	—	116	8	no	—
301	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
302	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
303	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
304	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
305	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
306	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	29	2	no	—
310	C₂²	—	29	2	no	—
311	C₂²	—	29	1	no	—
312	C₂²	—	29	1	no	—
313	C₂²	—	29	2	no	—
314	C₂²	—	58	4	no	—
315	C₂²	—	58	4	no	—
316	C₂²	—	58	1	no	—
317	C₂²	—	58	1	no	—
318	C₂²	#11745345	1	2	yes	—
319	C₂²	C₂³×#1181	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	58	4	no	—
327	C₄	—	58	1	no	—
328	C₄	—	58	1	no	—
329	C₄	—	58	1	no	—
330	C₄	—	58	1	no	—
331	C₄	—	58	1	no	—
332	C₄	—	58	1	no	—
333	C₄	—	116	2	no	—
334	C₄	—	116	2	no	—
335	C₄	—	116	2	no	—
336	C₂	—	2	2	no	—
337	C₂	—	29	2	no	—
338	C₂	—	29	2	no	—
339	C₂	—	58	2	no	—
340	C₂	C₂×#11745345	1	1	yes	characteristic
341	C₂	#41×#1181	1	1	yes	characteristic
342	C₂	#14272232	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#1181
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting	C₂₉×#117	C₄
prime radical	C₂³×D₂₉	C₂³
cosocle	C₂²×D₂₉	C₂⁴
hypercenter	#117	#1181
hypocenter	C₂₉	#2343
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
1. #14272112	C₂	semidirect
2. #14272112	C₂	semidirect
4. #14272108	C₂	semidirect
5. #14272107	C₂	semidirect
7. C₄×C₄⋊#1181	C₂	semidirect
9. #14272099	C₂	semidirect
19. C₂₉×#117	C₄	semidirect
20. #11745288	C₄	semidirect
21. #11745254	C₄	semidirect
22. #11745254	C₄	semidirect
27. #11745239	C₄	semidirect
28. #11745239	C₄	semidirect
29. #11745234	C₄	semidirect
30. #11745234	C₄	semidirect
67. C₄⋊C₄×C₂₉	C₂×C₄	semidirect
68. C₄⋊C₄×C₂₉	C₂×C₄	semidirect
71. C₄²×C₂₉	C₂×C₄	semidirect
72. C₄²×C₂₉	C₂×C₄	semidirect
79. C₄⋊Dic₂₉	C₂×C₄	semidirect
80. C₄⋊Dic₂₉	C₂×C₄	semidirect
81. #90779	C₂×C₄	semidirect
82. #90779	C₂×C₄	semidirect
83. #90779	C₂×C₄	semidirect
84. #90779	C₂×C₄	semidirect
85. C₄×Dic₂₉	C₂×C₄	semidirect
86. C₄×Dic₂₉	C₂×C₄	semidirect
87. C₄×Dic₂₉	C₂×C₄	semidirect
88. C₄×Dic₂₉	C₂×C₄	semidirect
89. C₄×Dic₂₉	C₂×C₄	semidirect
90. C₄×Dic₂₉	C₂×C₄	semidirect
192. #117	#1181	semidirect
214. C₂₉	#2343	semidirect
242. C₄⋊C₄	C₂×#1181	semidirect
243. C₄⋊C₄	C₂×#1181	semidirect
261. C₄²	C₂×#1181	semidirect
262. C₄²	C₂×#1181	semidirect

Maximal subgroups

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

subgroup	index
#14272112	2
#14272110	2
#14272108	2
#14272107	2
#14272106	2
C₄×C₄⋊#1181	2
C₂²⋊C₄×#1181	2
#14272099	2
C₂×(C₂×#1181)⋊C₄	2
#117×D₂₉	2
#2343	29