Subgroups — #16828655

Order 3712.

364 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14271700	C₂	1	1	yes	maximal, characteristic
2	Q₈×#6597	C₂	1	1	yes	maximal, characteristic, complemented
3	#14271240	C₂	1	1	yes	maximal, characteristic, complemented
4	#14271236	C₂	1	2	yes	maximal, complemented
5	C₈×D₁₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#14271228	C₂	1	2	yes	maximal, complemented
7	#14271225	C₂	1	2	yes	maximal, complemented
8	#14271214	C₂	1	1	yes	maximal, characteristic, complemented
9	#14271188	C₂	1	2	yes	maximal, complemented
10	C₄⋊C₈×D₂₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₉×#446	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×Q₈×C₂₉	C₂²	1	1	yes	characteristic
13	#11745309	C₄	1	1	yes	characteristic
14	#11745308	C₄	1	1	yes	characteristic
15	C₂×#90772	C₂²	1	2	yes	—
16	C₂×C₈×D₂₉	C₂²	1	2	yes	—
17	C₂×C₈×D₂₉	C₂²	1	2	yes	—
18	#11745260	C₄	1	2	yes	—
19	#11745257	C₄	1	1	yes	characteristic
20	#11745256	C₄	1	2	yes	—
21	#11745252	C₄	1	1	yes	characteristic
22	C₄×D₁₁₆	C₂²	1	2	yes	—
23	C₄×D₁₁₆	C₂²	1	1	yes	characteristic
24	#11745234	C₂²	1	2	yes	—
25	C₄²×D₂₉	C₂²	1	1	yes	characteristic
26	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₂₉×C₄⋊C₈	C₂²	1	2	yes	—
28	C₂₉×#97	C₂²	1	2	yes	—
29	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
30	#11745168	C₂²	1	1	yes	characteristic
31	#11745168	C₂²	1	2	yes	—
32	#11745168	C₂²	1	2	yes	—
33	#11745168	C₂²	1	1	yes	characteristic
34	#11745162	C₂²	1	2	yes	—
35	#11745162	C₂²	1	1	yes	characteristic
36	#11745162	C₂²	1	1	yes	characteristic
37	C₄⋊#6597	C₂²	1	2	yes	—
38	C₄⋊#6597	C₂²	1	1	yes	characteristic
39	C₄×#6597	C₂²	1	1	yes	characteristic
40	C₄×#6597	C₂²	1	2	yes	—
41	C₂×Q₈×C₂₉	C₂×C₄	1	1	yes	characteristic
42	C₂×D₁₁₆	C₂×C₄	1	1	yes	characteristic
43	C₂×D₁₁₆	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
47	C₂×C₄×D₂₉	C₂³	1	2	yes	—
48	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₂₉	C₂³	1	2	yes	—
51	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
52	C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
53	C₄²×C₂₉	C₂³	1	1	yes	characteristic
54	C₄²×C₂₉	C₂³	1	2	yes	—
55	#90781	C₂×C₄	1	2	yes	—
56	#90781	C₂×C₄	1	2	yes	—
57	#90781	C₂×C₄	1	2	yes	—
58	#90781	C₂×C₄	1	2	yes	—
59	#90781	C₂×C₄	1	2	yes	—
60	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
61	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
62	#90779	C₂×C₄	1	2	yes	—
63	#90779	C₂×C₄	1	2	yes	—
64	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
67	C₂×#6597	C₂³	1	1	yes	characteristic
68	C₂×#6597	C₂³	1	2	yes	—
69	C₂×#6597	C₂³	1	1	yes	characteristic
70	#90772	—	4	2	no	—
71	C₈×D₂₉	—	4	2	no	—
72	C₈×D₂₉	—	4	2	no	—
73	C₂²×D₂₉	C₂²×C₄	1	2	yes	—
74	C₂²×D₂₉	C₂²×C₄	1	1	yes	characteristic
75	Q₈×C₂₉	—	2	2	no	—
76	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
81	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
85	D₁₁₆	—	2	4	no	—
86	D₁₁₆	—	2	2	no	—
87	C₄×D₂₉	—	2	2	no	—
88	C₄×D₂₉	—	2	2	no	—
89	C₄×D₂₉	—	2	1	no	—
90	C₄×D₂₉	—	2	2	no	—
91	C₄×D₂₉	—	2	1	no	—
92	C₄×D₂₉	—	2	2	no	—
93	C₈×C₂₉	—	2	1	no	—
94	C₈×C₂₉	—	2	2	no	—
95	C₈×C₂₉	—	2	1	no	—
96	#6597	—	2	2	no	—
97	#6597	—	2	1	no	—
98	#6597	—	2	1	no	—
99	#2994	—	29	1	no	maximal, Sylow
100	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
101	C₂×D₂₉	—	2	1	no	—
102	C₂×D₂₉	—	2	1	no	—
103	C₂×D₂₉	—	2	2	no	—
104	C₂×D₂₉	—	2	2	no	—
105	C₂×D₂₉	—	2	2	no	—
106	C₂×D₂₉	#131	1	2	yes	—
107	C₄×C₂₉	—	2	1	no	—
108	C₄×C₂₉	—	2	2	no	—
109	C₄×C₂₉	—	2	2	no	—
110	C₄×C₂₉	#131	1	2	yes	—
111	C₄×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
112	C₄×C₂₉	2₋¹⁺⁴	1	1	yes	characteristic
113	Dic₂₉	—	2	1	no	—
114	Dic₂₉	—	2	2	no	—
115	Dic₂₉	—	2	1	no	—
116	Dic₂₉	—	2	1	no	—
117	#529	—	29	1	no	—
118	#446	D₂₉	1	1	yes	characteristic, complemented
119	C₈×Q₈	—	29	1	no	—
120	#436	—	29	1	no	—
121	#435	—	29	2	no	—
122	C₈×D₄	—	29	2	no	—
123	C₈×D₄	—	29	1	no	—
124	#433	—	29	2	no	—
125	#431	—	29	1	no	—
126	#424	—	29	2	no	—
127	C₂×C₄⋊C₈	—	29	1	no	—
128	C₂×C₂₉	#520	1	1	yes	characteristic
129	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
130	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
131	D₂₉	—	2	2	no	—
132	D₂₉	—	4	2	no	—
133	C₂×M₁₆	—	29	2	no	—
134	C₂²×C₈	—	29	2	no	—
135	C₂²×C₈	—	29	2	no	—
136	#126	—	29	2	no	—
137	#125	—	29	1	no	—
138	#124	—	29	1	no	—
139	#123	—	29	2	no	—
140	C₂²⋊Q₈	—	29	1	no	—
141	#121	—	29	1	no	—
142	C₄×Q₈	C₂×D₂₉	1	1	yes	characteristic
143	C₄×D₄	—	29	2	no	—
144	C₄×D₄	—	29	1	no	—
145	#117	—	29	2	no	—
146	C₂×C₄²	—	29	1	no	—
147	C₄⋊C₈	—	29	2	no	—
148	C₄⋊C₈	—	29	1	no	—
149	C₄⋊C₈	—	29	1	no	—
150	C₄⋊C₈	—	29	1	no	—
151	C₄⋊C₈	—	29	2	no	—
152	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic
153	C₄⋊C₈	C₂×D₂₉	1	2	yes	—
154	C₂²⋊C₈	—	29	1	no	—
155	C₂²⋊C₈	—	29	1	no	—
156	C₂²⋊C₈	—	29	2	no	—
157	C₂²⋊C₈	—	29	2	no	—
158	#97	C₂×D₂₉	1	2	yes	—
159	C₄×C₈	—	29	1	no	—
160	C₄×C₈	—	29	2	no	—
161	C₄×C₈	C₂×D₂₉	1	1	yes	characteristic
162	C₂₉	#2994	1	1	yes	characteristic, complemented, Sylow
163	C₂×Q₈	C₄×D₂₉	1	1	yes	characteristic
164	C₂×D₄	—	29	2	no	—
165	C₂×D₄	—	29	1	no	—
166	C₂²×C₄	—	29	1	no	—
167	C₂²×C₄	—	29	1	no	—
168	C₂²×C₄	—	29	1	no	—
169	C₂²×C₄	—	29	2	no	—
170	M₁₆	—	116	2	no	—
171	C₂×C₈	—	29	1	no	—
172	C₂×C₈	—	29	2	no	—
173	C₂×C₈	—	29	1	no	—
174	C₂×C₈	—	116	2	no	—
175	C₂×C₈	—	116	2	no	—
176	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
177	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₂₉	1	2	yes	—
179	C₄⋊C₄	—	29	2	no	—
180	C₄⋊C₄	—	29	2	no	—
181	C₄⋊C₄	—	29	1	no	—
182	C₄⋊C₄	—	29	2	no	—
183	C₄⋊C₄	C₄×D₂₉	1	1	yes	characteristic
184	C₄⋊C₄	C₄×D₂₉	1	2	yes	—
185	C₂²⋊C₄	—	29	2	no	—
186	C₂²⋊C₄	—	29	2	no	—
187	C₂²⋊C₄	—	29	2	no	—
188	C₂²⋊C₄	—	29	2	no	—
189	C₂²⋊C₄	—	29	2	no	—
190	C₄²	—	29	1	no	—
191	C₄²	—	29	1	no	—
192	C₄²	—	29	1	no	—
193	C₄²	C₂²×D₂₉	1	1	yes	characteristic
194	C₄²	C₂²×D₂₉	1	2	yes	—
195	C₂³	—	29	1	no	—
196	C₂³	—	29	2	no	—
197	Q₈	—	2	2	no	—
198	D₄	—	58	4	no	—
199	D₄	—	58	2	no	—
200	C₂×C₄	—	29	1	no	—
201	C₂×C₄	—	29	1	no	—
202	C₂×C₄	—	29	1	no	—
203	C₂×C₄	—	29	2	no	—
204	C₂×C₄	—	58	2	no	—
205	C₂×C₄	—	58	1	no	—
206	C₂×C₄	—	58	2	no	—
207	C₂×C₄	—	58	1	no	—
208	C₂×C₄	—	58	2	no	—
209	C₂×C₄	—	58	2	no	—
210	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
212	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
213	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
214	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	1	no	—
217	C₈	—	2	2	no	—
218	C₈	—	58	1	no	—
219	C₈	—	58	2	no	—
220	C₈	—	58	1	no	—
221	C₂²	—	29	2	no	—
222	C₂²	—	58	1	no	—
223	C₂²	—	58	1	no	—
224	C₂²	—	58	2	no	—
225	C₂²	—	58	2	no	—
226	C₂²	—	58	2	no	—
227	C₂²	C₂²×C₄×D₂₉	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	2	no	—
231	C₄	—	58	1	no	—
232	C₄	—	58	2	no	—
233	C₄	—	58	1	no	—
234	C₄	—	58	1	no	—
235	C₄	#11745269	1	2	yes	—
236	C₄	#11745362	1	1	yes	characteristic
237	C₄	#11745364	1	1	yes	characteristic
238	C₂	—	58	2	no	—
239	C₂	—	116	2	no	—
240	C₂	#14271645	1	1	yes	characteristic
241	C₂	#131×D₂₉	1	1	yes	characteristic
242	C₂	C₂×#11745269	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₉
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting	C₂₉×#446	C₂
prime radical	#11745257	C₄
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	#446	D₂₉
hypocenter	C₂₉	#2994
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. Q₈×#6597	C₂	semidirect
3. #14271240	C₂	semidirect
4. #14271236	C₂	semidirect
5. C₈×D₁₁₆	C₂	semidirect
6. #14271228	C₂	semidirect
7. #14271225	C₂	semidirect
8. #14271214	C₂	semidirect
9. #14271188	C₂	semidirect
10. C₄⋊C₈×D₂₉	C₂	semidirect
11. C₂₉×#446	C₂	semidirect
118. #446	D₂₉	semidirect
162. C₂₉	#2994	semidirect

Maximal subgroups

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

subgroup	index
#14271700	2
Q₈×#6597	2
#14271240	2
#14271236	2
C₈×D₁₁₆	2
#14271228	2
#14271225	2
#14271214	2
#14271188	2
C₄⋊C₈×D₂₉	2
C₂₉×#446	2
#2994	29