Subgroups — #16828540

Order 3712.

412 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14271241	C₂	1	2	yes	maximal, complemented
2	#14271235	C₂	1	2	yes	maximal, complemented
3	#14271235	C₂	1	2	yes	maximal, complemented
4	D₄×#6597	C₂	1	2	yes	maximal, complemented
5	#14271224	C₂	1	2	yes	maximal, complemented
6	#14271130	C₂	1	2	yes	maximal, complemented
7	#14271127	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂²⋊#6597	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₉×#525	C₂	1	1	yes	maximal, characteristic
10	C₂×#41×C₂₉	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₉	C₂²	1	1	yes	characteristic
12	C₂₉×#123	C₄	1	2	yes	—
13	C₂₉×C₂²⋊Q₈	C₄	1	1	yes	characteristic
14	C₂₉×C₂²⋊Q₈	C₄	1	1	yes	characteristic
15	C₂₉×#121	C₄	1	1	yes	characteristic
16	C₂₉×#121	C₄	1	1	yes	characteristic
17	C₂₉×#120	C₄	1	2	yes	—
18	C₄×D₄×C₂₉	C₂²	1	2	yes	—
19	C₄×D₄×C₂₉	C₂²	1	2	yes	—
20	C₂₉×#117	C₂²	1	1	yes	characteristic
21	C₂×#90777	C₂²	1	1	yes	characteristic
22	C₂×#90777	C₂²	1	1	yes	characteristic
23	C₂×#90777	C₂²	1	2	yes	complemented
24	C₂²×#6597	C₂²	1	1	yes	characteristic
25	C₂²×#6597	C₂²	1	1	yes	characteristic
26	C₂²×#6597	C₂²	1	2	yes	complemented
27	C₂²⋊#6597	C₂²	1	2	yes	—
28	C₂²⋊#6597	C₂²	1	2	yes	—
29	C₂²⋊#6597	C₂²	1	2	yes	—
30	C₂²⋊#6597	C₂²	1	2	yes	—
31	C₂²⋊#6597	C₂²	1	2	yes	complemented
32	C₂²⋊#6597	C₂²	1	2	yes	complemented
33	C₄⋊#6597	C₂²	1	2	yes	complemented
34	C₄⋊#6597	C₂²	1	2	yes	complemented
35	#11745151	C₂²	1	2	yes	complemented
36	C₄×#6597	C₂²	1	2	yes	complemented
37	C₂⁴×C₂₉	C₂×C₄	1	1	yes	characteristic
38	#41×C₂₉	—	4	1	no	—
39	#41×C₂₉	—	4	1	no	—
40	C₂×Q₈×C₂₉	C₂×C₄	1	1	yes	characteristic
41	C₂×D₄×C₂₉	C₂×C₄	1	2	yes	—
42	C₂×D₄×C₂₉	C₂×C₄	1	2	yes	—
43	C₂×D₄×C₂₉	C₂×C₄	1	2	yes	—
44	C₂×D₄×C₂₉	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₉	—	4	1	no	—
46	C₂²×C₄×C₂₉	—	4	1	no	—
47	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
52	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
55	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
56	C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
57	C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
58	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
59	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
60	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
61	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
62	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
63	C₄²×C₂₉	C₂³	1	2	yes	—
64	#90777	—	4	1	no	—
65	#90777	—	4	1	no	—
66	#90777	—	4	2	no	—
67	C₂×#6597	—	4	1	no	—
68	C₂×#6597	—	4	1	no	—
69	C₂×#6597	—	4	2	no	—
70	C₂×#6597	C₂³	1	2	yes	—
71	C₂×#6597	C₂³	1	2	yes	—
72	C₂×#6597	C₂³	1	2	yes	—
73	C₂×#6597	C₂³	1	2	yes	—
74	C₂³×C₂₉	—	2	2	no	—
75	C₂³×C₂₉	—	4	1	no	—
76	C₂³×C₂₉	—	4	1	no	—
77	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
78	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
79	C₂³×C₂₉	C₂²×C₄	1	2	yes	—
80	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
81	Q₈×C₂₉	—	4	1	no	—
82	D₄×C₂₉	—	2	4	no	—
83	D₄×C₂₉	—	4	2	no	—
84	D₄×C₂₉	—	4	1	no	—
85	D₄×C₂₉	—	4	2	no	—
86	C₂×C₄×C₂₉	—	2	1	no	—
87	C₂×C₄×C₂₉	—	2	1	no	—
88	C₂×C₄×C₂₉	—	2	2	no	—
89	C₂×C₄×C₂₉	—	2	1	no	—
90	C₂×C₄×C₂₉	—	2	1	no	—
91	C₂×C₄×C₂₉	—	2	2	no	—
92	C₂×C₄×C₂₉	—	2	1	no	—
93	C₂×C₄×C₂₉	—	2	1	no	—
94	C₂×C₄×C₂₉	—	4	1	no	—
95	C₂×C₄×C₂₉	—	4	1	no	—
96	C₂×C₄×C₂₉	—	4	1	no	—
97	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
98	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
104	#6597	—	2	2	no	—
105	#6597	—	2	2	no	—
106	#6597	—	2	2	no	—
107	#6597	—	2	2	no	—
108	#2976	—	29	1	no	maximal, Sylow
109	C₂²×C₂₉	—	2	1	no	—
110	C₂²×C₂₉	—	2	1	no	—
111	C₂²×C₂₉	—	2	1	no	—
112	C₂²×C₂₉	—	2	1	no	—
113	C₂²×C₂₉	—	2	2	no	—
114	C₂²×C₂₉	—	2	2	no	—
115	C₂²×C₂₉	—	2	1	no	—
116	C₂²×C₂₉	—	2	1	no	—
117	C₂²×C₂₉	—	2	1	no	—
118	C₂²×C₂₉	—	2	2	no	—
119	C₂²×C₂₉	—	2	1	no	—
120	C₂²×C₂₉	—	4	4	no	—
121	C₂²×C₂₉	#131	1	2	yes	—
122	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
123	C₄×C₂₉	—	2	1	no	—
124	C₄×C₂₉	—	2	1	no	—
125	C₄×C₂₉	—	2	1	no	—
126	C₄×C₂₉	—	2	2	no	—
127	C₄×C₂₉	—	2	2	no	—
128	C₄×C₂₉	—	2	2	no	—
129	C₄×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
130	C₄×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
131	#525	D₂₉	1	1	yes	characteristic
132	#436	—	29	2	no	—
133	#435	—	29	2	no	—
134	#435	—	29	2	no	—
135	C₈×D₄	—	29	2	no	—
136	#433	—	29	2	no	—
137	#408	—	29	2	no	—
138	#407	—	29	1	no	—
139	C₂×C₂²⋊C₈	—	29	1	no	—
140	C₂×C₂₉	—	2	2	no	—
141	C₂×C₂₉	—	4	2	no	—
142	C₂×C₂₉	—	4	1	no	—
143	C₂×C₂₉	—	4	1	no	—
144	C₂×C₂₉	#518	1	1	yes	characteristic
145	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
146	C₂×C₂₉	#568	1	1	yes	characteristic
147	C₂×#41	C₂×D₂₉	1	1	yes	characteristic
148	C₂³×C₄	C₂×D₂₉	1	1	yes	characteristic
149	C₂×M₁₆	—	29	1	no	—
150	C₂×M₁₆	—	29	1	no	—
151	C₂×M₁₆	—	29	2	no	—
152	C₂²×C₈	—	29	1	no	—
153	C₂²×C₈	—	29	1	no	—
154	C₂²×C₈	—	29	2	no	—
155	#123	Dic₂₉	1	2	yes	—
156	C₂²⋊Q₈	Dic₂₉	1	1	yes	characteristic
157	C₂²⋊Q₈	Dic₂₉	1	1	yes	characteristic
158	#121	Dic₂₉	1	1	yes	characteristic
159	#121	Dic₂₉	1	1	yes	characteristic
160	#120	Dic₂₉	1	2	yes	—
161	C₄×D₄	C₂×D₂₉	1	2	yes	—
162	C₄×D₄	C₂×D₂₉	1	2	yes	—
163	#117	C₂×D₂₉	1	1	yes	characteristic
164	C₄⋊C₈	—	29	2	no	—
165	C₄⋊C₈	—	29	2	no	—
166	C₂²⋊C₈	—	29	2	no	—
167	C₂²⋊C₈	—	29	2	no	—
168	C₂²⋊C₈	—	29	2	no	—
169	C₂²⋊C₈	—	29	2	no	—
170	C₂²⋊C₈	—	29	2	no	—
171	C₂²⋊C₈	—	29	2	no	—
172	#97	—	29	2	no	—
173	C₄×C₈	—	29	2	no	—
174	C₂₉	#2976	1	1	yes	characteristic, complemented, Sylow
175	C₂⁴	C₂×Dic₂₉	1	1	yes	characteristic
176	#41	—	4	1	no	—
177	#41	—	4	1	no	—
178	C₂×Q₈	C₂×Dic₂₉	1	1	yes	characteristic
179	C₂×D₄	C₂×Dic₂₉	1	2	yes	—
180	C₂×D₄	C₂×Dic₂₉	1	2	yes	—
181	C₂×D₄	C₂×Dic₂₉	1	2	yes	—
182	C₂×D₄	C₂×Dic₂₉	1	1	yes	characteristic
183	C₂²×C₄	—	4	1	no	—
184	C₂²×C₄	—	4	1	no	—
185	C₂²×C₄	C₂×Dic₂₉	1	1	yes	characteristic
186	C₂²×C₄	C₂×Dic₂₉	1	1	yes	characteristic
187	C₂²×C₄	C₂×Dic₂₉	1	1	yes	characteristic
188	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
189	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
190	C₂²×C₄	C₂²×D₂₉	1	2	yes	—
191	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
192	M₁₆	—	116	1	no	—
193	M₁₆	—	116	2	no	—
194	M₁₆	—	116	1	no	—
195	C₂×C₈	—	29	2	no	—
196	C₂×C₈	—	29	2	no	—
197	C₂×C₈	—	29	2	no	—
198	C₂×C₈	—	29	2	no	—
199	C₂×C₈	—	116	1	no	—
200	C₂×C₈	—	116	2	no	—
201	C₂×C₈	—	116	1	no	—
202	C₄⋊C₄	C₂×Dic₂₉	1	1	yes	characteristic
203	C₄⋊C₄	C₂×Dic₂₉	1	1	yes	characteristic
204	C₄⋊C₄	C₂×Dic₂₉	1	2	yes	—
205	C₄⋊C₄	C₂×Dic₂₉	1	2	yes	—
206	C₂²⋊C₄	C₂×Dic₂₉	1	2	yes	—
207	C₂²⋊C₄	C₂×Dic₂₉	1	2	yes	—
208	C₂²⋊C₄	C₂×Dic₂₉	1	2	yes	—
209	C₂²⋊C₄	C₂×Dic₂₉	1	2	yes	—
210	C₂²⋊C₄	C₂×Dic₂₉	1	2	yes	—
211	C₄²	C₂²×D₂₉	1	2	yes	—
212	C₂³	—	2	2	no	—
213	C₂³	—	4	1	no	—
214	C₂³	—	4	1	no	—
215	C₂³	C₂²×Dic₂₉	1	1	yes	characteristic
216	C₂³	C₂²×Dic₂₉	1	1	yes	characteristic
217	C₂³	C₂²×Dic₂₉	1	2	yes	—
218	C₂³	C₂²×Dic₂₉	1	1	yes	characteristic
219	Q₈	—	4	1	no	—
220	D₄	—	2	4	no	—
221	D₄	—	4	1	no	—
222	D₄	—	4	2	no	—
223	D₄	—	4	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	C₂²×Dic₂₉	1	2	yes	—
236	C₂×C₄	C₂²×Dic₂₉	1	2	yes	—
237	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
238	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
239	C₂×C₄	C₂²×Dic₂₉	1	2	yes	—
240	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
241	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
242	C₈	—	58	2	no	—
243	C₈	—	58	2	no	—
244	C₈	—	58	2	no	—
245	C₈	—	58	2	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	4	4	no	—
258	C₂²	#11745310	1	2	yes	—
259	C₂²	C₂³×Dic₂₉	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	#11745359	1	1	yes	characteristic
267	C₄	#11745359	1	1	yes	characteristic
268	C₂	—	2	2	no	—
269	C₂	—	4	1	no	—
270	C₂	—	4	1	no	—
271	C₂	—	4	2	no	—
272	C₂	#14271638	1	1	yes	characteristic
273	C₂	C₂×#11745310	1	1	yes	characteristic
274	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	#2976
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. #14271241	C₂	semidirect
2. #14271235	C₂	semidirect
3. #14271235	C₂	semidirect
4. D₄×#6597	C₂	semidirect
5. #14271224	C₂	semidirect
6. #14271130	C₂	semidirect
7. #14271127	C₂	semidirect
8. C₂×C₂²⋊#6597	C₂	semidirect
23. C₂×#90777	C₂²	semidirect
26. C₂²×#6597	C₂²	semidirect
31. C₂²⋊#6597	C₂²	semidirect
32. C₂²⋊#6597	C₂²	semidirect
33. C₄⋊#6597	C₂²	semidirect
34. C₄⋊#6597	C₂²	semidirect
35. #11745151	C₂²	semidirect
36. C₄×#6597	C₂²	semidirect
174. C₂₉	#2976	semidirect

Maximal subgroups

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

subgroup	index
#14271241	2
#14271235	2
D₄×#6597	2
#14271224	2
#14271130	2
#14271127	2
C₂×C₂²⋊#6597	2
C₂₉×#525	2
#2976	29