Subgroups — #16018428

Order 3328.

474 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125375	C₂	1	2	yes	maximal
2	#13117122	C₂	1	4	yes	maximal, complemented
3	#13117115	C₂	1	2	yes	maximal, complemented
4	#13117111	C₂	1	2	yes	maximal, complemented
5	#13116624	C₂	1	2	yes	maximal
6	#13116620	C₂	1	1	yes	maximal, characteristic
7	C₂×#11718096	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#2108	C₂	1	1	yes	maximal, characteristic
9	#11719224	—	2	2	no	—
10	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₁₃×#405	C₂²	1	1	yes	characteristic
13	#11718782	—	2	2	no	—
14	#11718767	C₂²	1	2	yes	—
15	C₂×#88058	C₂²	1	2	yes	—
16	#11718363	—	2	4	no	—
17	#11718344	C₂²	1	4	yes	—
18	#11718343	C₂²	1	4	yes	—
19	C₁₃×#346	C₂²	1	2	yes	—
20	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
21	#11718097	C₂²	1	4	yes	—
22	#11718097	C₂²	1	2	yes	—
23	#11718096	C₂²	1	1	yes	characteristic
24	#11718096	C₂²	1	2	yes	—
25	#11718096	C₂²	1	4	yes	—
26	#11718096	C₂²	1	1	yes	characteristic
27	#11718096	C₂²	1	2	yes	—
28	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
30	C₁₃×#117	D₄	1	1	yes	characteristic
31	#88180	—	2	2	no	—
32	C₂×C₄⋊Dic₁₃	—	2	2	no	—
33	#88144	—	4	2	no	—
34	#88142	—	4	2	no	—
35	#88132	—	4	4	no	—
36	#88124	—	2	4	no	—
37	#88123	—	2	4	no	—
38	C₁₃×M₃₂	Q₈	1	4	yes	complemented
39	C₂×C₁₃×C₁₆	Q₈	1	4	yes	complemented
40	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
41	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
42	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
43	C₁₃×#97	D₄	1	2	yes	—
44	C₄×C₈×C₁₃	C₂³	1	2	yes	—
45	#88058	—	2	2	no	—
46	#88058	—	2	4	no	—
47	#88058	C₂³	1	4	yes	—
48	#88058	C₂³	1	2	yes	—
49	#88058	C₂³	1	2	yes	—
50	#88057	—	2	2	no	—
51	#88057	—	2	4	no	—
52	#33729	—	13	1	no	maximal, Sylow
53	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×Dic₁₃	—	4	2	no	—
55	C₂×Dic₂₆	—	4	4	no	—
56	C₁₃×M₁₆	Q₁₆	1	4	yes	—
57	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
61	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
62	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
63	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
64	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
65	C₄²×C₁₃	C₂×D₄	1	2	yes	—
66	C₂²⋊Dic₁₃	—	4	2	no	—
67	C₄⋊Dic₁₃	—	2	2	no	—
68	C₄⋊Dic₁₃	—	2	4	no	—
69	C₄⋊Dic₁₃	—	2	2	no	—
70	C₄⋊Dic₁₃	—	4	4	no	—
71	C₄⋊Dic₁₃	—	4	2	no	—
72	#6101	—	4	4	no	—
73	#6101	—	4	4	no	—
74	C₄×Dic₁₃	—	4	2	no	—
75	C₁₃×C₁₆	C₂×Q₈	1	2	yes	—
76	C₁₃×C₁₆	C₂×Q₈	1	2	yes	—
77	C₁₃×C₁₆	C₂×Q₈	1	4	yes	—
78	#3164	—	13	2	no	—
79	#2255	—	13	4	no	—
80	#2253	—	13	2	no	—
81	#2252	—	13	2	no	—
82	#2159	—	13	2	no	—
83	#2158	—	13	1	no	—
84	C₂×#366	—	13	1	no	—
85	#2108	D₁₃	1	1	yes	characteristic
86	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
88	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
89	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
90	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
92	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
93	C₂×Dic₁₃	—	4	2	no	—
94	C₂×Dic₁₃	—	4	2	no	—
95	C₂×Dic₁₃	—	4	4	no	—
96	C₂×Dic₁₃	—	16	2	no	—
97	Dic₂₆	—	16	4	no	—
98	C₈×C₁₃	C₄⋊Q₈	1	2	yes	—
99	C₈×C₁₃	C₂×Q₁₆	1	4	yes	—
100	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
101	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
102	#544	—	26	2	no	—
103	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
104	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
105	#501	—	26	4	no	—
106	#500	—	13	4	no	—
107	#499	—	13	4	no	—
108	#428	—	26	2	no	—
109	#427	—	13	2	no	—
110	C₂×#107	—	13	2	no	—
111	#405	C₂×D₁₃	1	1	yes	characteristic
112	#367	—	13	2	no	—
113	#367	—	13	4	no	—
114	#366	—	13	2	no	—
115	#366	—	13	2	no	—
116	#366	—	13	1	no	—
117	#366	—	13	1	no	—
118	#366	—	13	4	no	—
119	#346	C₂×D₁₃	1	2	yes	—
120	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
121	C₂²×C₁₃	—	2	1	no	—
122	C₂²×C₁₃	—	2	1	no	—
123	C₂²×C₁₃	#500	1	2	yes	—
124	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
125	C₄×C₁₃	—	2	2	no	—
126	C₄×C₁₃	—	2	2	no	—
127	C₄×C₁₃	#500	1	2	yes	—
128	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
129	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
130	Dic₁₃	—	8	2	no	—
131	Dic₁₃	—	8	4	no	—
132	Dic₁₃	—	8	2	no	—
133	C₂×M₁₆	D₅₂	1	1	yes	characteristic
134	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
135	C₄⋊Q₈	—	26	4	no	—
136	C₄⋊Q₈	—	52	2	no	—
137	#125	—	26	4	no	—
138	#125	—	52	2	no	—
139	C₂²⋊Q₈	—	52	4	no	—
140	#117	—	26	2	no	—
141	#117	D₅₂	1	1	yes	characteristic
142	C₂×C₄⋊C₄	—	26	2	no	—
143	M₃₂	Dic₂₆	1	4	yes	complemented
144	C₂×C₁₆	Dic₂₆	1	4	yes	complemented
145	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
146	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
147	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
148	#107	—	13	4	no	—
149	#107	—	13	2	no	—
150	#107	—	13	2	no	—
151	#107	—	26	2	no	—
152	#107	—	26	4	no	—
153	#106	—	26	2	no	—
154	#106	—	26	4	no	—
155	#97	D₅₂	1	2	yes	—
156	C₄×C₈	C₂²×D₁₃	1	2	yes	—
157	C₂×C₁₃	—	2	2	no	—
158	C₂×C₁₃	C₂×#500	1	1	yes	characteristic
159	C₂×C₁₃	#3415	1	1	yes	characteristic
160	C₂×C₁₃	#3417	1	1	yes	characteristic
161	C₂×Q₈	—	52	4	no	—
162	C₂²×C₄	—	52	2	no	—
163	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
164	M₁₆	Dic₅₂	1	4	yes	—
165	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
166	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
167	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
168	C₂×C₈	C₂×D₅₂	1	2	yes	—
169	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
170	C₄⋊C₄	—	26	2	no	—
171	C₄⋊C₄	—	26	4	no	—
172	C₄⋊C₄	—	26	2	no	—
173	C₄⋊C₄	—	52	2	no	—
174	C₄⋊C₄	—	52	4	no	—
175	C₄⋊C₄	—	52	4	no	—
176	C₄⋊C₄	—	52	4	no	—
177	C₄⋊C₄	D₁₀₄	1	1	yes	characteristic
178	C₄⋊C₄	D₁₀₄	1	1	yes	characteristic
179	C₂²⋊C₄	—	52	2	no	—
180	C₂²⋊C₄	D₁₀₄	1	2	yes	—
181	C₄²	—	52	2	no	—
182	C₄²	C₂×D₅₂	1	2	yes	—
183	C₁₆	C₂×Dic₂₆	1	2	yes	—
184	C₁₆	C₂×Dic₂₆	1	2	yes	—
185	C₁₆	C₂×Dic₂₆	1	4	yes	—
186	C₁₃	#33729	1	1	yes	characteristic, complemented, Sylow
187	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
188	Q₈	—	208	4	no	—
189	C₂×C₄	—	52	2	no	—
190	C₂×C₄	—	52	2	no	—
191	C₂×C₄	—	52	4	no	—
192	C₂×C₄	—	208	2	no	—
193	C₂×C₄	#88123	1	2	yes	—
194	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
195	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
196	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
197	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
198	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
199	C₈	#88123	1	2	yes	—
200	C₈	C₂×Dic₅₂	1	4	yes	—
201	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
202	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	#11718343	1	2	yes	—
206	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	104	4	no	—
210	C₄	—	104	2	no	—
211	C₄	—	104	2	no	—
212	C₄	#11718343	1	2	yes	—
213	C₄	C₂×#88123	1	1	yes	characteristic
214	C₄	C₂²×Dic₅₂	1	1	yes	characteristic
215	C₂	—	2	2	no	—
216	C₂	C₂×#11718343	1	1	yes	characteristic
217	C₂	#13127977	1	1	yes	characteristic
218	C₂	#13127989	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2108	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂³×C₁₃	C₂×D₈
hypercenter	#2108	D₁₃
hypocenter	C₁₃	#33729
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #13117122	C₂	semidirect
3. #13117115	C₂	semidirect
4. #13117111	C₂	semidirect
38. C₁₃×M₃₂	Q₈	semidirect
39. C₂×C₁₃×C₁₆	Q₈	semidirect
143. M₃₂	Dic₂₆	semidirect
144. C₂×C₁₆	Dic₂₆	semidirect
186. C₁₃	#33729	semidirect

Maximal subgroups

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

subgroup	index
#13125375	2
#13117122	2
#13117115	2
#13117111	2
#13116624	2
#13116620	2
C₂×#11718096	2
C₁₃×#2108	2
#33729	13