Subgroups — #16828947

Order 3712.

496 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	C₂×#11745312	C₂	1	2	yes	maximal, complemented
2	#14271713	C₂	1	1	yes	maximal, characteristic, complemented
3	#14271385	C₂	1	2	yes	maximal, complemented
4	#14271378	C₂	1	2	yes	maximal, complemented
5	#14271373	C₂	1	1	yes	maximal, characteristic, complemented
6	#14271368	C₂	1	1	yes	maximal, characteristic, complemented
7	#14271363	C₂	1	1	yes	maximal, characteristic, complemented
8	#14271358	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄⋊#6597	C₂	1	1	yes	maximal, characteristic, complemented
10	#14271157	C₂	1	2	yes	maximal, complemented
11	C₄×#41×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₉	C₂²	1	1	yes	characteristic
13	C₂×#90805	C₂²	1	2	yes	—
14	C₄×Q₈×C₂₉	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₂₉	C₂²	1	1	yes	characteristic
16	C₄×D₄×C₂₉	—	2	1	no	—
17	C₄×D₄×C₂₉	—	2	1	no	—
18	C₄×D₄×C₂₉	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₂₉	C₂²	1	1	yes	characteristic
20	C₂₉×#117	—	2	1	no	—
21	C₂₉×#117	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₂₉	—	2	1	no	—
23	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
24	#11745312	—	2	8	no	—
25	C₂×#90785	C₂²	1	2	yes	—
26	C₂×#90784	C₂²	1	2	yes	—
27	C₂×#90783	C₂²	1	2	yes	—
28	C₂×#90782	C₂²	1	2	yes	—
29	#11745292	—	2	1	no	—
30	#11745292	—	2	1	no	—
31	C₂²×#6597	C₂²	1	2	yes	—
32	#11745237	C₂²	1	2	yes	—
33	#11745236	C₂²	1	1	yes	characteristic
34	C₄×D₁₁₆	—	2	1	no	—
35	C₄×D₁₁₆	—	2	1	no	—
36	#11745231	C₂²	1	1	yes	characteristic
37	#11745158	C₂²	1	2	yes	—
38	#11745158	C₂²	1	2	yes	—
39	#11745157	C₂²	1	2	yes	—
40	#11745157	C₂²	1	2	yes	—
41	C₄⋊#6597	C₂²	1	1	yes	characteristic
42	C₄⋊#6597	C₂²	1	1	yes	characteristic
43	C₄⋊#6597	C₂²	1	2	yes	—
44	#41×C₂₉	—	2	2	no	—
45	#41×C₂₉	D₄	1	4	yes	complemented
46	C₂×Q₈×C₂₉	C₂³	1	1	yes	characteristic
47	C₂×D₄×C₂₉	—	2	1	no	—
48	C₂×D₄×C₂₉	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₂₉	—	2	1	no	—
50	C₂²×C₄×C₂₉	—	2	1	no	—
51	C₂²×C₄×C₂₉	—	2	1	no	—
52	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
55	C₂×C₂²⋊D₂₉	—	2	2	no	—
56	#90805	—	2	4	no	—
57	#90805	—	2	4	no	—
58	C₂×D₁₁₆	—	2	1	no	—
59	C₂×D₁₁₆	—	2	1	no	—
60	C₂×D₁₁₆	C₂³	1	2	yes	—
61	C₂×C₄×D₂₉	—	2	2	no	—
62	C₂×Dic₅₈	C₂³	1	2	yes	—
63	C₄⋊C₄×C₂₉	—	2	1	no	—
64	C₄⋊C₄×C₂₉	—	2	1	no	—
65	C₄⋊C₄×C₂₉	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₂₉	C₂³	1	1	yes	characteristic
67	C₂²⋊C₄×C₂₉	—	2	1	no	—
68	C₂²⋊C₄×C₂₉	—	2	1	no	—
69	C₂²⋊C₄×C₂₉	—	2	1	no	—
70	C₄²×C₂₉	—	2	1	no	—
71	C₄²×C₂₉	—	2	1	no	—
72	C₄²×C₂₉	—	2	1	no	—
73	C₄²×C₂₉	D₄	1	1	yes	characteristic
74	C₄²×C₂₉	D₄	1	1	yes	characteristic
75	C₄²×C₂₉	C₂³	1	1	yes	characteristic
76	C₄²×C₂₉	C₂³	1	1	yes	characteristic
77	#90785	—	2	4	no	—
78	#90784	—	2	4	no	—
79	#90783	—	2	4	no	—
80	#90782	—	2	4	no	—
81	#90781	—	2	2	no	—
82	#90781	—	2	2	no	—
83	C₄⋊Dic₂₉	—	2	1	no	—
84	C₄⋊Dic₂₉	—	2	1	no	—
85	C₂×#6597	—	2	4	no	—
86	C₂×#6597	C₂³	1	2	yes	—
87	C₂×#6597	C₂³	1	2	yes	—
88	C₂³×C₂₉	—	2	1	no	—
89	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
90	C₂²×D₂₉	—	2	2	no	—
91	Q₈×C₂₉	—	2	1	no	—
92	Q₈×C₂₉	C₂×D₄	1	2	yes	—
93	D₄×C₂₉	—	2	4	no	—
94	D₄×C₂₉	—	2	1	no	—
95	D₄×C₂₉	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₂₉	—	2	1	no	—
97	C₂×C₄×C₂₉	—	2	4	no	—
98	C₂×C₄×C₂₉	—	2	2	no	—
99	C₂×C₄×C₂₉	—	2	1	no	—
100	C₂×C₄×C₂₉	—	2	1	no	—
101	C₂×C₄×C₂₉	—	4	1	no	—
102	C₂×C₄×C₂₉	—	4	2	no	—
103	C₂×C₄×C₂₉	—	4	2	no	—
104	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₂₉	#41	1	2	yes	—
111	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
112	C₂²⋊D₂₉	—	4	4	no	—
113	C₂×Dic₂₉	—	2	2	no	—
114	D₁₁₆	—	2	2	no	—
115	D₁₁₆	—	2	2	no	—
116	D₁₁₆	—	4	2	no	—
117	D₁₁₆	—	4	2	no	—
118	C₄×D₂₉	—	4	4	no	—
119	Dic₅₈	—	2	2	no	—
120	Dic₅₈	—	2	2	no	—
121	#6597	—	2	2	no	—
122	#6597	—	2	2	no	—
123	#3035	—	29	1	no	maximal, Sylow
124	C₂²×C₂₉	—	2	1	no	—
125	C₂²×C₂₉	—	2	1	no	—
126	C₂²×C₂₉	—	2	2	no	—
127	C₂²×C₂₉	—	4	1	no	—
128	C₂²×C₂₉	—	4	1	no	—
129	C₂²×C₂₉	#121	1	2	yes	—
130	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
131	C₂×D₂₉	—	4	2	no	—
132	C₂×D₂₉	—	4	2	no	—
133	C₂×D₂₉	—	4	2	no	—
134	C₄×C₂₉	—	2	2	no	—
135	C₄×C₂₉	—	2	1	no	—
136	C₄×C₂₉	—	2	1	no	—
137	C₄×C₂₉	—	2	1	no	—
138	C₄×C₂₉	—	4	1	no	—
139	C₄×C₂₉	—	4	1	no	—
140	C₄×C₂₉	#121	1	2	yes	—
141	C₄×C₂₉	#135	1	2	yes	complemented
142	C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
143	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
144	Dic₂₉	—	4	2	no	—
145	C₂×#135	—	29	2	no	—
146	#532	—	29	1	no	—
147	C₄×#41	D₂₉	1	1	yes	characteristic, complemented
148	#464	—	29	2	no	—
149	#463	—	29	2	no	—
150	#462	—	29	1	no	—
151	#461	—	29	1	no	—
152	#460	—	29	1	no	—
153	#459	—	29	1	no	—
154	C₂×C₄⋊C₈	—	29	1	no	—
155	#416	—	29	2	no	—
156	C₂×C₂₉	—	2	2	no	—
157	C₂×C₂₉	—	4	2	no	—
158	C₂×C₂₉	C₂×#121	1	1	yes	characteristic
159	C₂×C₂₉	C₂×#135	1	1	yes	characteristic
160	C₂×C₂₉	#575	1	1	yes	characteristic
161	D₂₉	—	8	2	no	—
162	C₂×#41	—	29	2	no	—
163	C₂×#41	C₂×D₂₉	1	1	yes	characteristic
164	#135	—	58	8	no	—
165	C₂×Q₁₆	—	29	2	no	—
166	C₂×SD₁₆	—	29	2	no	—
167	C₂×SD₁₆	—	29	2	no	—
168	C₂×D₈	—	29	2	no	—
169	C₂²×C₈	—	29	2	no	—
170	C₄⋊Q₈	—	29	1	no	—
171	#127	—	29	1	no	—
172	#124	—	29	2	no	—
173	#121	—	58	1	no	—
174	#121	—	58	1	no	—
175	C₄×Q₈	C₂×D₂₉	1	1	yes	characteristic
176	C₄×Q₈	C₂×D₂₉	1	1	yes	characteristic
177	C₄×D₄	—	2	1	no	—
178	C₄×D₄	—	2	1	no	—
179	C₄×D₄	—	58	1	no	—
180	C₄×D₄	—	58	1	no	—
181	C₄×D₄	C₂×D₂₉	1	1	yes	characteristic
182	C₄×D₄	C₂×D₂₉	1	1	yes	characteristic
183	#117	—	2	1	no	—
184	#117	C₂×D₂₉	1	1	yes	characteristic
185	C₂×C₄²	—	2	1	no	—
186	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
187	C₄⋊C₈	—	29	1	no	—
188	C₄⋊C₈	—	29	1	no	—
189	C₄⋊C₈	—	29	2	no	—
190	#103	—	29	2	no	—
191	#103	—	29	2	no	—
192	#102	—	29	2	no	—
193	#102	—	29	2	no	—
194	C₂₉	#3035	1	1	yes	characteristic, complemented, Sylow
195	#41	—	2	2	no	—
196	#41	—	58	4	no	—
197	#41	—	58	4	no	—
198	#41	D₁₁₆	1	4	yes	complemented
199	C₂×Q₈	—	29	2	no	—
200	C₂×Q₈	C₂²×D₂₉	1	1	yes	characteristic
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	—	29	2	no	—
203	C₂×D₄	—	58	2	no	—
204	C₂×D₄	—	58	1	no	—
205	C₂×D₄	—	58	1	no	—
206	C₂×D₄	C₂²×D₂₉	1	1	yes	characteristic
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	58	2	no	—
211	C₂²×C₄	C₂²⋊D₂₉	1	1	yes	characteristic
212	C₂²×C₄	C₂²⋊D₂₉	1	1	yes	characteristic
213	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
214	Q₁₆	—	58	4	no	—
215	SD₁₆	—	58	4	no	—
216	SD₁₆	—	58	4	no	—
217	D₈	—	58	4	no	—
218	C₂×C₈	—	29	2	no	—
219	C₂×C₈	—	29	2	no	—
220	C₂×C₈	—	58	4	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	58	1	no	—
224	C₄⋊C₄	—	58	1	no	—
225	C₄⋊C₄	C₂²×D₂₉	1	1	yes	characteristic
226	C₄⋊C₄	C₂²×D₂₉	1	1	yes	characteristic
227	C₂²⋊C₄	—	2	1	no	—
228	C₂²⋊C₄	—	2	1	no	—
229	C₂²⋊C₄	—	2	1	no	—
230	C₂²⋊C₄	—	58	2	no	—
231	C₂²⋊C₄	—	58	2	no	—
232	C₄²	—	2	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	2	1	no	—
235	C₄²	C₂²⋊D₂₉	1	1	yes	characteristic
236	C₄²	C₂²⋊D₂₉	1	1	yes	characteristic
237	C₄²	C₂²×D₂₉	1	1	yes	characteristic
238	C₄²	C₂²×D₂₉	1	1	yes	characteristic
239	C₂³	—	2	1	no	—
240	C₂³	—	58	2	no	—
241	C₂³	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
242	Q₈	—	2	1	no	—
243	Q₈	—	58	2	no	—
244	Q₈	—	58	2	no	—
245	Q₈	C₂×D₁₁₆	1	2	yes	—
246	D₄	—	2	4	no	—
247	D₄	—	2	1	no	—
248	D₄	—	58	2	no	—
249	D₄	—	58	2	no	—
250	D₄	—	116	2	no	—
251	D₄	—	116	4	no	—
252	D₄	—	116	2	no	—
253	D₄	C₂×D₁₁₆	1	2	yes	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	58	2	no	—
263	C₂×C₄	—	116	4	no	—
264	C₂×C₄	C₂×D₁₁₆	1	2	yes	—
265	C₂×C₄	#90805	1	2	yes	—
266	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
267	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
268	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
269	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
270	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
271	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
272	C₈	—	58	2	no	—
273	C₈	—	58	2	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	4	1	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	116	2	no	—
280	C₂²	—	116	2	no	—
281	C₂²	—	116	2	no	—
282	C₂²	#11745292	1	2	yes	—
283	C₂²	C₂²×C₂²⋊D₂₉	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	4	1	no	—
289	C₄	—	4	1	no	—
290	C₄	—	116	2	no	—
291	C₄	#11745292	1	2	yes	—
292	C₄	#11745312	1	2	yes	complemented
293	C₄	C₂²×D₁₁₆	1	1	yes	characteristic
294	C₄	C₂×#90805	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	—	4	2	no	—
297	C₂	—	232	2	no	—
298	C₂	C₂×#11745292	1	1	yes	characteristic
299	C₂	C₂×#11745312	1	1	yes	characteristic
300	C₂	#14271961	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₉
commutator, cosocle	C₂×C₄×C₂₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₉
Fitting	C₄×#41×C₂₉	C₂
socle	C₂²×C₂₉	C₂²×D₄
hypercenter	C₄×#41	D₂₉
hypocenter	C₂₉	#3035
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#11745312	C₂	semidirect
2. #14271713	C₂	semidirect
3. #14271385	C₂	semidirect
4. #14271378	C₂	semidirect
5. #14271373	C₂	semidirect
6. #14271368	C₂	semidirect
7. #14271363	C₂	semidirect
8. #14271358	C₂	semidirect
9. C₂×C₄⋊#6597	C₂	semidirect
10. #14271157	C₂	semidirect
11. C₄×#41×C₂₉	C₂	semidirect
45. #41×C₂₉	D₄	semidirect
141. C₄×C₂₉	#135	semidirect
147. C₄×#41	D₂₉	semidirect
194. C₂₉	#3035	semidirect
198. #41	D₁₁₆	semidirect
292. C₄	#11745312	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11745312	2
#14271713	2
#14271385	2
#14271378	2
#14271373	2
#14271368	2
#14271363	2
#14271358	2
C₂×C₄⋊#6597	2
#14271157	2
C₄×#41×C₂₉	2
#3035	29