Subgroups — C₂×#13116985 #16017545

Order 3328.

574 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	C₂×C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic
2	#13116985	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718081	C₂	1	2	yes	maximal
4	C₂×C₄⋊#6090	C₂	1	1	yes	maximal, characteristic
5	C₂×#11718084	C₂	1	2	yes	maximal
6	C₂×C₁₃×#500	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
8	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
9	C₂×C₄×Dic₂₆	—	2	1	no	—
10	C₁₃×#500	—	2	2	no	—
11	C₁₃×#500	C₂²	1	4	yes	—
12	C₂×C₁₃×#107	—	2	1	no	—
13	C₂×C₁₃×#107	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×#88058	C₂²	1	1	yes	characteristic
16	C₂×#88056	—	2	1	no	—
17	C₂²×#6090	C₂²	1	2	yes	—
18	C₄×Dic₅₂	—	2	2	no	—
19	C₄×Dic₅₂	C₂²	1	4	yes	—
20	#11718084	C₂²	1	8	yes	—
21	#11718081	C₂²	1	8	yes	—
22	C₄⋊#6090	C₂²	1	4	yes	—
23	C₂²×Q₈×C₁₃	—	4	1	no	—
24	C₂²×Dic₂₆	—	2	1	no	—
25	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
27	C₁₃×C₄⋊Q₈	—	2	4	no	—
28	C₁₃×C₄⋊Q₈	—	4	2	no	—
29	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
30	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
31	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
32	C₂×C₄⋊Dic₁₃	—	2	1	no	—
33	C₂×#6101	—	4	1	no	—
34	C₂×C₄×Dic₁₃	—	4	1	no	—
35	C₂×Dic₅₂	—	2	1	no	—
36	C₂×Dic₅₂	—	2	2	no	—
37	C₂×Dic₅₂	Q₈	1	4	yes	complemented
38	C₂×Dic₅₂	C₂³	1	2	yes	—
39	C₄×Dic₂₆	—	2	4	no	—
40	C₄×Dic₂₆	—	4	2	no	—
41	C₁₃×#107	—	2	4	no	—
42	C₁₃×#107	—	2	2	no	—
43	C₁₃×#107	C₂³	1	4	yes	—
44	C₄×C₈×C₁₃	D₄	1	2	yes	—
45	C₄×C₈×C₁₃	C₂³	1	2	yes	—
46	#88058	—	2	1	no	—
47	#88058	C₂³	1	2	yes	—
48	#88056	—	2	4	no	—
49	C₂×#6090	—	2	4	no	—
50	C₂×#6090	C₂³	1	4	yes	—
51	C₂×#2227	—	13	1	no	maximal, Sylow
52	C₂×Q₈×C₁₃	—	4	4	no	—
53	C₂×Q₈×C₁₃	—	8	4	no	—
54	C₂²×C₄×C₁₃	—	4	2	no	—
55	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×Dic₁₃	—	4	1	no	—
59	C₂²×Dic₁₃	—	4	1	no	—
60	C₂×Dic₂₆	—	2	4	no	—
61	C₂×Dic₂₆	—	2	2	no	—
62	C₂×Dic₂₆	—	4	1	no	—
63	C₂×Dic₂₆	—	4	2	no	—
64	C₂×C₈×C₁₃	—	2	2	no	—
65	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
68	C₂×C₈×C₁₃	#41	1	2	yes	—
69	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
70	C₁₃×C₄⋊C₄	—	2	4	no	—
71	C₁₃×C₄⋊C₄	—	4	2	no	—
72	C₁₃×C₄⋊C₄	—	4	4	no	—
73	C₄²×C₁₃	D₈	1	2	yes	—
74	C₄²×C₁₃	C₂×D₄	1	2	yes	—
75	C₄⋊Dic₁₃	—	2	2	no	—
76	C₄⋊Dic₁₃	—	4	1	no	—
77	#6101	—	4	4	no	—
78	C₄×Dic₁₃	—	4	4	no	—
79	Dic₅₂	—	2	4	no	—
80	Dic₅₂	—	2	2	no	—
81	Dic₅₂	C₂×Q₈	1	4	yes	complemented
82	#6090	—	2	4	no	—
83	C₂×#500	D₁₃	1	1	yes	characteristic
84	C₂×C₄×Q₁₆	—	13	1	no	—
85	#2227	—	13	8	no	—
86	C₂×#366	—	13	2	no	—
87	C₂×C₄⋊C₁₆	—	13	1	no	—
88	C₂×#358	—	13	2	no	—
89	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
90	Q₈×C₁₃	—	8	8	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	4	4	no	—
93	C₂×C₄×C₁₃	—	8	4	no	—
94	C₂×C₄×C₁₃	Q₃₂	1	4	yes	—
95	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
96	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
97	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
99	C₂×C₄×C₁₃	#137	1	2	yes	—
100	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
101	C₂×Dic₁₃	—	4	2	no	—
102	C₂×Dic₁₃	—	4	4	no	—
103	C₂×Dic₁₃	—	4	2	no	—
104	C₂×Dic₁₃	—	8	2	no	—
105	Dic₂₆	—	2	4	no	—
106	Dic₂₆	—	4	2	no	—
107	Dic₂₆	—	4	4	no	—
108	C₈×C₁₃	—	2	2	no	—
109	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
110	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
111	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
112	C₂²×Q₁₆	—	13	1	no	—
113	C₂×C₄⋊Q₈	—	2	1	no	—
114	C₂×C₄×Q₈	—	26	1	no	—
115	C₂²×C₁₆	—	13	2	no	—
116	#500	—	2	2	no	—
117	#500	C₂×D₁₃	1	4	yes	—
118	C₄×Q₁₆	—	13	4	no	—
119	C₄×Q₁₆	—	26	2	no	—
120	C₂×#107	—	2	1	no	—
121	C₂×#107	—	13	1	no	—
122	C₂×#107	C₂×D₁₃	1	2	yes	—
123	C₂×#103	—	26	1	no	—
124	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
125	#366	—	13	8	no	—
126	C₄⋊C₁₆	—	13	4	no	—
127	#358	—	13	8	no	—
128	C₂²×C₁₃	#474	1	2	yes	—
129	C₂²×C₁₃	C₂×Q₃₂	1	2	yes	—
130	C₂²×C₁₃	#509	1	2	yes	—
131	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
132	C₄×C₁₃	—	2	2	no	—
133	C₄×C₁₃	—	8	4	no	—
134	C₄×C₁₃	#474	1	2	yes	—
135	C₄×C₁₃	C₂×Q₃₂	1	4	yes	—
136	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
137	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
138	Dic₁₃	—	4	4	no	—
139	Dic₁₃	—	8	2	no	—
140	C₂²×Q₈	—	4	1	no	—
141	C₂²×Q₈	—	26	1	no	—
142	C₂×Q₁₆	—	13	4	no	—
143	C₂×Q₁₆	—	13	2	no	—
144	C₂×Q₁₆	—	26	2	no	—
145	C₂×Q₁₆	—	26	1	no	—
146	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
147	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
148	C₄⋊Q₈	—	2	4	no	—
149	C₄⋊Q₈	—	4	2	no	—
150	C₄×Q₈	—	26	4	no	—
151	C₄×Q₈	—	52	2	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	—	4	1	no	—
154	C₂×C₄⋊C₄	—	26	1	no	—
155	C₂×C₄⋊C₄	—	52	1	no	—
156	C₂×C₄²	—	52	1	no	—
157	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
158	C₂×C₁₆	—	13	4	no	—
159	C₂×C₁₆	—	26	4	no	—
160	#107	—	2	4	no	—
161	#107	—	2	2	no	—
162	#107	—	13	2	no	—
163	#107	—	26	1	no	—
164	#107	C₂²×D₁₃	1	4	yes	—
165	#103	—	26	4	no	—
166	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
167	C₄×C₈	C₂²×D₁₃	1	2	yes	—
168	C₂×C₁₃	#2227	1	4	yes	complemented
169	C₂×C₁₃	C₂×#474	1	1	yes	characteristic
170	C₂×C₁₃	C₂²×Q₃₂	1	1	yes	characteristic
171	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
172	C₂×Q₈	—	4	4	no	—
173	C₂×Q₈	—	8	4	no	—
174	C₂×Q₈	—	26	4	no	—
175	C₂×Q₈	—	26	2	no	—
176	C₂×Q₈	—	52	1	no	—
177	C₂×Q₈	—	52	2	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	—	52	1	no	—
180	C₂²×C₄	—	52	1	no	—
181	C₂²×C₄	#6104	1	1	yes	characteristic
182	C₂²×C₄	#6104	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
184	Q₁₆	—	13	4	no	—
185	Q₁₆	—	26	4	no	—
186	Q₁₆	—	26	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	Q₈×D₁₃	1	2	yes	—
189	C₂×C₈	#6131	1	2	yes	—
190	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
191	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
192	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	4	4	no	—
196	C₄⋊C₄	—	26	2	no	—
197	C₄⋊C₄	—	52	1	no	—
198	C₄⋊C₄	—	52	4	no	—
199	C₄²	—	52	4	no	—
200	C₄²	#6104	1	2	yes	—
201	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
202	C₁₆	—	26	4	no	—
203	C₁₃	C₂×#2227	1	1	yes	characteristic, complemented, Sylow
204	C₂³	C₂×#6104	1	1	yes	characteristic
205	Q₈	—	8	8	no	—
206	Q₈	—	26	4	no	—
207	Q₈	—	52	2	no	—
208	Q₈	—	52	4	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	8	4	no	—
212	C₂×C₄	—	52	2	no	—
213	C₂×C₄	—	52	4	no	—
214	C₂×C₄	—	52	2	no	—
215	C₂×C₄	—	104	2	no	—
216	C₂×C₄	#88069	1	4	yes	—
217	C₂×C₄	C₂×#6104	1	2	yes	—
218	C₂×C₄	C₂×#6104	1	1	yes	characteristic
219	C₂×C₄	C₂×#6104	1	2	yes	—
220	C₂×C₄	#88196	1	2	yes	—
221	C₂×C₄	#88200	1	2	yes	—
222	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	#88200	1	2	yes	—
225	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
226	C₈	C₂×#6131	1	1	yes	characteristic
227	C₂²	#11718748	1	2	yes	—
228	C₂²	#11718808	1	2	yes	—
229	C₂²	C₂×#88069	1	2	yes	—
230	C₂²	C₂²×#6104	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	8	4	no	—
233	C₄	—	52	4	no	—
234	C₄	—	104	2	no	—
235	C₄	#11718748	1	2	yes	—
236	C₄	C₂×#88069	1	4	yes	—
237	C₄	C₂×#88196	1	1	yes	characteristic
238	C₄	C₂×#88200	1	1	yes	characteristic
239	C₂	#13116985	1	4	yes	direct product factor
240	C₂	C₂×#11718748	1	1	yes	characteristic
241	C₂	C₂²×#88069	1	1	yes	characteristic
242	C₂	C₂×#11718808	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#500	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₈
hypercenter	C₂×#500	D₁₃
hypocenter	C₁₃	C₂×#2227
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
2. #13116985	C₂	direct (N × H)
37. C₂×Dic₅₂	Q₈	semidirect
81. Dic₅₂	C₂×Q₈	semidirect
168. C₂×C₁₃	#2227	semidirect
203. C₁₃	C₂×#2227	semidirect
239. C₂	#13116985	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₅₂	2
#13116985	2
C₂×#11718081	2
C₂×C₄⋊#6090	2
C₂×#11718084	2
C₂×C₁₃×#500	2
C₂×#2227	13

Subgroups — C2×#13116985 #16017545

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116985 #16017545