Subgroups — #16836190

Order 3712.

392 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	#14272188	C₂	1	2	yes	maximal, complemented
2	#14272187	C₂	1	2	yes	maximal, complemented
3	#14272186	C₂	1	1	yes	maximal, characteristic, complemented
4	#14272183	C₂	1	1	yes	maximal, characteristic, complemented
5	#14272181	C₂	1	1	yes	maximal, characteristic, complemented
6	#14272180	C₂	1	1	yes	maximal, characteristic, complemented
7	#14272166	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11745220	C₂	1	1	yes	maximal, characteristic, complemented
9	#14272136	C₂	1	2	yes	maximal
10	#14272127	C₂	1	2	yes	maximal, complemented
11	C₄×#90807	C₂	1	1	yes	maximal, characteristic
12	C₂×#90807	C₂²	1	1	yes	characteristic
13	C₂×#90800	C₂²	1	2	yes	—
14	C₂²×#6599	C₂²	1	2	yes	—
15	C₄×D₄×C₂₉	C₄	1	1	yes	characteristic
16	D₄×Dic₂₉	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊D₂₉	C₄	1	1	yes	characteristic
18	C₄×C₂²⋊D₂₉	C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
21	#11745254	C₂²	1	1	yes	characteristic
22	#11745249	C₄	1	1	yes	characteristic
23	#11745243	C₄	1	1	yes	characteristic
24	#11745243	C₄	1	1	yes	characteristic
25	#11745239	C₂²	1	1	yes	characteristic
26	#11745239	C₂²	1	1	yes	characteristic
27	C₄²×D₂₉	C₂²	1	1	yes	characteristic
28	C₄×Dic₅₈	C₄	1	1	yes	characteristic
29	C₂²⋊#6599	C₂²	1	2	yes	—
30	C₂²⋊#6599	C₂²	1	2	yes	—
31	#11745220	C₂²	1	1	yes	characteristic
32	#11745220	C₂²	1	1	yes	characteristic
33	#11745220	C₂²	1	2	yes	—
34	#11745220	C₂²	1	1	yes	characteristic
35	#11745220	C₂²	1	1	yes	characteristic
36	#11745219	C₂²	1	2	yes	—
37	#11745219	C₂²	1	2	yes	—
38	#11745218	C₂²	1	2	yes	—
39	#11745218	C₂²	1	2	yes	—
40	C₄⋊#6599	C₂²	1	2	yes	—
41	C₄×#6599	C₂²	1	2	yes	—
42	C₄×#6599	C₂²	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	C₂×C₂²⋊D₂₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊D₂₉	C₂×C₄	1	1	yes	characteristic
48	C₂²×Dic₂₉	C₂×C₄	1	1	yes	characteristic
49	C₂²×Dic₂₉	C₂×C₄	1	1	yes	characteristic
50	C₂²×Dic₂₉	C₂³	1	1	yes	characteristic
51	C₂²×Dic₂₉	C₂³	1	1	yes	characteristic
52	#90807	—	2	2	no	—
53	#90807	—	2	2	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×Dic₅₈	C₂×C₄	1	1	yes	characteristic
58	#90800	—	2	4	no	—
59	C₂×#6599	—	2	4	no	—
60	C₂×#6599	C₂³	1	2	yes	—
61	C₂×#6599	C₂³	1	2	yes	—
62	C₂×#6599	C₂³	1	2	yes	—
63	C₂×#6599	C₂³	1	2	yes	—
64	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
65	C₂²⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
66	C₂²⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
67	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
68	C₂²⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
69	C₂²⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
70	#90781	C₂×C₄	1	1	yes	characteristic
71	#90781	C₂×C₄	1	1	yes	characteristic
72	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
73	#90779	C₂×C₄	1	1	yes	characteristic
74	#90779	C₂×C₄	1	1	yes	characteristic
75	#90779	C₂×C₄	1	1	yes	characteristic
76	#90779	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	C₂×C₄	1	1	yes	characteristic
80	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
81	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
82	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
83	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
84	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
85	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
86	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
87	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
88	C₂²×D₂₉	C₂²×C₄	1	1	yes	characteristic
89	D₄×C₂₉	—	2	2	no	—
90	C₂×C₄×C₂₉	—	4	1	no	—
91	C₂×C₄×C₂₉	—	4	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
97	C₂²⋊D₂₉	—	2	2	no	—
98	C₂²⋊D₂₉	—	2	2	no	—
99	C₂×Dic₂₉	—	2	2	no	—
100	C₂×Dic₂₉	—	2	2	no	—
101	C₂×Dic₂₉	—	2	1	no	—
102	C₂×Dic₂₉	—	2	1	no	—
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	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
108	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
109	C₂×Dic₂₉	#41	1	2	yes	—
110	C₂×Dic₂₉	#41	1	2	yes	—
111	C₂×Dic₂₉	C₂⁴	1	1	yes	characteristic
112	C₄×D₂₉	—	2	2	no	—
113	C₄×D₂₉	—	2	1	no	—
114	C₄×D₂₉	—	2	1	no	—
115	C₄×D₂₉	—	4	1	no	—
116	Dic₅₈	—	2	2	no	—
117	#6599	—	2	2	no	—
118	#6599	—	2	2	no	—
119	#6599	—	2	2	no	—
120	#6599	—	2	2	no	—
121	#2925	—	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	—	2	1	no	—
126	C₂²×C₂₉	—	2	1	no	—
127	C₂²×C₂₉	#117	1	2	yes	—
128	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
129	C₂×D₂₉	—	2	1	no	—
130	C₂×D₂₉	—	2	1	no	—
131	C₂×D₂₉	—	2	1	no	—
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	—	2	1	no	—
137	Dic₂₉	—	2	1	no	—
138	Dic₂₉	—	2	1	no	—
139	Dic₂₉	—	2	1	no	—
140	Dic₂₉	—	2	1	no	—
141	Dic₂₉	#117	1	2	yes	—
142	Dic₂₉	#131	1	2	yes	—
143	Dic₂₉	C₂×#41	1	1	yes	characteristic
144	Dic₂₉	C₂×#41	1	1	yes	characteristic
145	C₄×#41	—	29	1	no	—
146	#433	—	29	2	no	—
147	#433	—	29	2	no	—
148	#432	—	29	1	no	—
149	#432	—	29	1	no	—
150	#431	—	29	1	no	—
151	#431	—	29	1	no	—
152	#423	—	29	1	no	—
153	C₂×C₄⋊C₈	—	29	1	no	—
154	#408	—	29	2	no	—
155	#405	—	29	2	no	—
156	C₂×C₂₉	—	2	2	no	—
157	C₂×C₂₉	—	4	1	no	—
158	C₂×C₂₉	C₂×#117	1	1	yes	characteristic
159	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
160	C₂×C₂₉	#568	1	1	yes	characteristic
161	D₂₉	—	4	1	no	—
162	C₂×#41	—	29	1	no	—
163	C₂×M₁₆	—	29	2	no	—
164	C₂²×C₈	—	29	2	no	—
165	C₄×Q₈	—	29	1	no	—
166	C₄×Q₈	—	29	1	no	—
167	C₄×D₄	—	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	#1181	1	1	yes	characteristic
173	#117	—	29	1	no	—
174	#117	—	29	1	no	—
175	#117	—	29	1	no	—
176	C₂×C₄²	—	29	1	no	—
177	C₂×C₄²	—	29	1	no	—
178	C₂×C₄²	—	29	1	no	—
179	C₄⋊C₈	—	29	1	no	—
180	C₄⋊C₈	—	29	1	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	C₂²⋊C₈	—	29	2	no	—
186	C₂²⋊C₈	—	29	2	no	—
187	C₂²⋊C₈	—	29	2	no	—
188	C₂²⋊C₈	—	29	2	no	—
189	#97	—	29	2	no	—
190	#97	—	29	2	no	—
191	C₄×C₈	—	29	2	no	—
192	C₄×C₈	—	29	2	no	—
193	C₂₉	#2925	1	1	yes	characteristic, complemented, Sylow
194	#41	—	58	2	no	—
195	#41	—	58	2	no	—
196	C₂×Q₈	—	29	1	no	—
197	C₂×D₄	—	29	1	no	—
198	C₂×D₄	—	29	1	no	—
199	C₂×D₄	C₂×#1181	1	1	yes	characteristic
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	—	29	1	no	—
206	C₂²×C₄	—	29	1	no	—
207	C₂²×C₄	C₂×#1181	1	1	yes	characteristic
208	C₂²×C₄	C₂×#1181	1	1	yes	characteristic
209	M₁₆	—	58	4	no	—
210	C₂×C₈	—	29	2	no	—
211	C₂×C₈	—	29	2	no	—
212	C₂×C₈	—	29	2	no	—
213	C₂×C₈	—	29	2	no	—
214	C₂×C₈	—	58	4	no	—
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	C₂×#1181	1	1	yes	characteristic
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	C₂×#1181	1	1	yes	characteristic
226	C₂²⋊C₄	C₂×#1181	1	1	yes	characteristic
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	—	29	1	no	—
234	C₄²	—	29	1	no	—
235	C₄²	—	29	1	no	—
236	C₄²	C₂×#1181	1	1	yes	characteristic
237	C₂³	—	29	1	no	—
238	C₂³	C₂²×#1181	1	1	yes	characteristic
239	C₂³	C₂²×#1181	1	1	yes	characteristic
240	Q₈	—	58	2	no	—
241	D₄	—	2	2	no	—
242	D₄	—	58	2	no	—
243	D₄	—	58	2	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	29	2	no	—
247	C₂×C₄	—	29	2	no	—
248	C₂×C₄	—	29	1	no	—
249	C₂×C₄	—	29	1	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	2	no	—
258	C₂×C₄	—	58	1	no	—
259	C₂×C₄	—	58	1	no	—
260	C₂×C₄	—	58	2	no	—
261	C₂×C₄	—	58	2	no	—
262	C₂×C₄	—	116	1	no	—
263	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
264	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
265	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
266	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
267	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
268	C₈	—	58	2	no	—
269	C₈	—	58	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	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	58	1	no	—
278	C₂²	—	58	1	no	—
279	C₂²	—	58	1	no	—
280	C₂²	#11745345	1	2	yes	—
281	C₂²	C₂³×#1181	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	—	2	1	no	—
287	C₄	—	29	1	no	—
288	C₄	—	29	1	no	—
289	C₄	—	29	2	no	—
290	C₄	—	29	2	no	—
291	C₄	—	58	1	no	—
292	C₄	—	58	1	no	—
293	C₄	—	58	1	no	—
294	C₄	—	58	1	no	—
295	C₂	—	2	2	no	—
296	C₂	—	4	1	no	—
297	C₂	—	116	1	no	—
298	C₂	C₂×#11745345	1	1	yes	characteristic
299	C₂	D₂₉⋊#131	1	1	yes	characteristic
300	C₂	#14272241	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#1181
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting	C₄×D₄×C₂₉	C₄
prime radical	C₂×#90807	C₂²
cosocle	C₂×Dic₂₉	C₂⁴
hypercenter	C₄×D₄	#1181
hypocenter	C₂₉	#2925
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #14272188	C₂	semidirect
2. #14272187	C₂	semidirect
3. #14272186	C₂	semidirect
4. #14272183	C₂	semidirect
5. #14272181	C₂	semidirect
6. #14272180	C₂	semidirect
7. #14272166	C₂	semidirect
8. C₂×#11745220	C₂	semidirect
10. #14272127	C₂	semidirect
193. C₂₉	#2925	semidirect

Maximal subgroups

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

subgroup	index
#14272188	2
#14272187	2
#14272186	2
#14272183	2
#14272181	2
#14272180	2
#14272166	2
C₂×#11745220	2
#14272136	2
#14272127	2
C₄×#90807	2
#2925	29