Subgroups — C₂×C₄×#88110 #16796720

Order 3328.

1790 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88110	C₂	1	4	yes	maximal
2	C₂²×C₄×#1057	C₂	1	2	yes	maximal
3	C₄×#88110	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄²×Dic₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×#1057	C₂²	1	4	yes	—
6	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
7	C₂²×C₄×Dic₁₃	C₄	1	1	yes	characteristic
8	C₂²×C₄×Dic₁₃	C₄	1	2	yes	—
9	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄×Dic₁₃	C₂²	1	2	yes	—
11	C₂×#88110	C₄	1	32	yes	direct product factor
12	C₂×#88110	C₂²	1	16	yes	—
13	C₂×C₄×#1057	C₄	1	16	yes	complemented
14	C₂×C₄×#1057	C₂²	1	8	yes	—
15	C₄²×Dic₁₃	C₄	1	4	yes	—
16	C₄²×Dic₁₃	C₂²	1	4	yes	—
17	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂³×Dic₁₃	C₂×C₄	1	2	yes	—
20	C₂³×Dic₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂³×Dic₁₃	C₂³	1	1	yes	characteristic
22	C₂²×#1057	C₂×C₄	1	16	yes	—
23	C₂²×#1057	C₂×C₄	1	32	yes	—
24	C₂²×#1057	C₂³	1	8	yes	—
25	C₂×C₄²×C₁₃	—	2	4	no	—
26	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
27	C₂×C₄×Dic₁₃	—	2	4	no	—
28	C₂×C₄×Dic₁₃	—	2	8	no	—
29	C₂×C₄×Dic₁₃	C₂×C₄	1	4	yes	—
30	C₂×C₄×Dic₁₃	C₂×C₄	1	8	yes	—
31	C₂×C₄×Dic₁₃	C₂×C₄	1	4	yes	—
32	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
33	C₂×C₄×Dic₁₃	C₂×C₄	1	8	yes	—
34	C₂×C₄×Dic₁₃	C₂×C₄	1	16	yes	—
35	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
36	C₂×C₄×Dic₁₃	C₂³	1	4	yes	—
37	#88110	C₂×C₄	1	64	yes	direct product factor
38	C₄×#1057	C₂×C₄	1	32	yes	complemented
39	C₂×C₄×#97	—	13	1	no	maximal, Sylow
40	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	—	2	4	no	—
42	C₂²×C₄×C₁₃	—	2	4	no	—
43	C₂²×C₄×C₁₃	C₄²	1	8	yes	—
44	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
45	C₂²×C₄×C₁₃	M₁₆	1	4	yes	—
46	C₂²×C₄×C₁₃	M₁₆	1	4	yes	—
47	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
48	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
49	C₂²×Dic₁₃	—	2	4	no	—
50	C₂²×Dic₁₃	—	2	4	no	—
51	C₂²×Dic₁₃	C₄²	1	8	yes	—
52	C₂²×Dic₁₃	C₄²	1	4	yes	—
53	C₂²×Dic₁₃	M₁₆	1	8	yes	—
54	C₂²×Dic₁₃	C₂²×C₄	1	4	yes	—
55	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×Dic₁₃	C₂²×C₄	1	4	yes	—
57	C₂²×Dic₁₃	C₂²×C₄	1	8	yes	—
58	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
59	C₂²×Dic₁₃	C₂⁴	1	1	yes	characteristic
60	C₂×#1057	C₄²	1	64	yes	complemented
61	C₂×#1057	C₂²×C₄	1	16	yes	—
62	C₂×#1057	C₂²×C₄	1	32	yes	—
63	C₄²×C₁₃	—	2	8	no	—
64	C₄×Dic₁₃	—	2	8	no	—
65	C₄×Dic₁₃	—	2	16	no	—
66	C₄×Dic₁₃	C₄²	1	16	yes	—
67	C₄×Dic₁₃	C₄²	1	8	yes	—
68	C₄×Dic₁₃	C₂²×C₄	1	8	yes	—
69	C₄×Dic₁₃	C₂²×C₄	1	16	yes	—
70	C₂²×#97	—	13	4	no	—
71	C₂²×C₄×C₈	—	13	2	no	—
72	C₂×C₄³	—	13	1	no	—
73	C₄×#97	—	13	8	no	—
74	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
75	C₂³×C₁₃	C₂×C₄²	1	4	yes	—
76	C₂³×C₁₃	C₂×M₁₆	1	2	yes	—
77	C₂³×C₁₃	C₂×M₁₆	1	4	yes	—
78	C₂³×C₁₃	C₂×M₁₆	1	2	yes	—
79	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	—	2	4	no	—
81	C₂×C₄×C₁₃	—	2	4	no	—
82	C₂×C₄×C₁₃	—	2	16	no	—
83	C₂×C₄×C₁₃	#97	1	16	yes	complemented
84	C₂×C₄×C₁₃	C₂×C₄²	1	8	yes	—
85	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
86	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
87	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
88	C₂×Dic₁₃	—	2	4	no	—
89	C₂×Dic₁₃	—	2	4	no	—
90	C₂×Dic₁₃	—	2	16	no	—
91	C₂×Dic₁₃	#97	1	16	yes	complemented
92	C₂×Dic₁₃	C₂×C₄²	1	8	yes	—
93	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
94	C₂×Dic₁₃	C₂×C₄²	1	4	yes	—
95	C₂×Dic₁₃	C₂×C₄²	1	16	yes	—
96	C₂×Dic₁₃	C₂×C₄²	1	4	yes	—
97	C₂×Dic₁₃	C₂×M₁₆	1	8	yes	—
98	C₂×Dic₁₃	C₂³×C₄	1	2	yes	—
99	C₂×Dic₁₃	C₂³×C₄	1	4	yes	—
100	#1057	C₂×C₄²	1	32	yes	complemented
101	C₂³×C₈	—	13	4	no	—
102	C₂²×C₄²	—	13	2	no	—
103	C₂²×C₄²	—	13	1	no	—
104	C₂²×C₄²	—	13	2	no	—
105	C₂²×C₄²	—	13	1	no	—
106	C₂²×C₄²	#261	1	1	yes	characteristic
107	C₂×#97	—	13	16	no	—
108	C₂×#97	—	13	32	no	—
109	C₂×C₄×C₈	—	13	8	no	—
110	C₂×C₄×C₈	—	13	16	no	—
111	C₄³	—	13	4	no	—
112	C₄³	—	13	4	no	—
113	C₂²×C₁₃	C₄³	1	4	yes	—
114	C₂²×C₁₃	C₂×#97	1	4	yes	—
115	C₂²×C₁₃	C₂×#97	1	4	yes	—
116	C₂²×C₁₃	C₄×M₁₆	1	8	yes	—
117	C₂²×C₁₃	C₄×M₁₆	1	8	yes	—
118	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
119	C₂²×C₁₃	C₂²×C₄²	1	2	yes	—
120	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
121	C₂²×C₁₃	C₂²×M₁₆	1	2	yes	—
122	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
123	C₄×C₁₃	—	2	8	no	—
124	C₄×C₁₃	C₂×#97	1	8	yes	complemented
125	Dic₁₃	—	2	8	no	—
126	Dic₁₃	C₄³	1	4	yes	—
127	Dic₁₃	C₂×#97	1	8	yes	complemented
128	Dic₁₃	C₂²×C₄²	1	4	yes	—
129	C₂³×C₄	—	13	1	no	—
130	C₂³×C₄	—	13	1	no	—
131	C₂³×C₄	—	13	2	no	—
132	C₂³×C₄	C₂×#261	1	2	yes	—
133	C₂³×C₄	C₂×#261	1	1	yes	characteristic
134	C₂²×C₈	—	13	8	no	—
135	C₂²×C₈	—	13	16	no	—
136	C₂²×C₈	—	13	32	no	—
137	C₂×C₄²	—	2	4	no	—
138	C₂×C₄²	—	13	8	no	—
139	C₂×C₄²	—	13	16	no	—
140	C₂×C₄²	—	13	2	no	—
141	C₂×C₄²	—	13	4	no	—
142	C₂×C₄²	—	13	2	no	—
143	C₂×C₄²	—	13	4	no	—
144	C₂×C₄²	—	13	8	no	—
145	C₂×C₄²	—	13	4	no	—
146	C₂×C₄²	—	26	8	no	—
147	C₂×C₄²	—	26	4	no	—
148	C₂×C₄²	C₂×#261	1	4	yes	—
149	#97	—	13	64	no	—
150	C₄×C₈	—	13	32	no	—
151	C₂×C₁₃	C₄×#97	1	8	yes	complemented
152	C₂×C₁₃	C₂×C₄³	1	1	yes	characteristic
153	C₂×C₁₃	C₂²×#97	1	1	yes	characteristic
154	C₂×C₁₃	C₂²×#97	1	1	yes	characteristic
155	C₂×C₁₃	C₂×C₄×M₁₆	1	2	yes	—
156	C₂×C₁₃	C₂×C₄×M₁₆	1	2	yes	—
157	C₂⁴	C₂²×#261	1	1	yes	characteristic
158	C₂²×C₄	—	2	4	no	—
159	C₂²×C₄	—	2	4	no	—
160	C₂²×C₄	—	13	8	no	—
161	C₂²×C₄	—	13	4	no	—
162	C₂²×C₄	—	13	8	no	—
163	C₂²×C₄	—	13	4	no	—
164	C₂²×C₄	—	13	1	no	—
165	C₂²×C₄	—	13	2	no	—
166	C₂²×C₄	—	13	1	no	—
167	C₂²×C₄	—	13	8	no	—
168	C₂²×C₄	—	13	4	no	—
169	C₂²×C₄	—	26	4	no	—
170	C₂²×C₄	—	26	4	no	—
171	C₂²×C₄	#6118	1	4	yes	—
172	C₂²×C₄	C₄×#261	1	8	yes	—
173	C₂²×C₄	C₄×#261	1	4	yes	—
174	C₂²×C₄	#6122	1	4	yes	—
175	C₂²×C₄	C₂²×#261	1	4	yes	—
176	C₂²×C₄	C₂²×#261	1	2	yes	—
177	C₂×C₈	—	13	16	no	—
178	C₂×C₈	—	13	32	no	—
179	C₂×C₈	—	13	64	no	—
180	C₄²	—	2	8	no	—
181	C₄²	—	13	8	no	—
182	C₄²	—	13	16	no	—
183	C₄²	—	13	8	no	—
184	C₄²	—	13	16	no	—
185	C₄²	—	26	16	no	—
186	C₄²	—	26	8	no	—
187	C₁₃	C₂×C₄×#97	1	1	yes	characteristic, complemented, Sylow
188	C₂³	C₂×#6118	1	2	yes	—
189	C₂³	D₁₃⋊M₁₆	1	4	yes	—
190	C₂³	C₂×C₄×#261	1	2	yes	—
191	C₂³	C₂×C₄×#261	1	4	yes	—
192	C₂³	C₂×#6122	1	2	yes	—
193	C₂³	C₂³×#261	1	1	yes	characteristic
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	2	16	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	13	16	no	—
198	C₂×C₄	—	13	16	no	—
199	C₂×C₄	—	13	8	no	—
200	C₂×C₄	—	13	2	no	—
201	C₂×C₄	—	13	4	no	—
202	C₂×C₄	—	13	4	no	—
203	C₂×C₄	—	13	2	no	—
204	C₂×C₄	—	13	4	no	—
205	C₂×C₄	—	13	8	no	—
206	C₂×C₄	—	26	16	no	—
207	C₂×C₄	—	26	4	no	—
208	C₂×C₄	—	26	4	no	—
209	C₂×C₄	#88110	1	16	yes	direct product factor
210	C₂×C₄	C₂×#6118	1	4	yes	—
211	C₂×C₄	C₂×C₄×#261	1	8	yes	—
212	C₂×C₄	C₂×C₄×#261	1	4	yes	—
213	C₂×C₄	C₂×#6122	1	4	yes	—
214	C₈	—	13	32	no	—
215	C₂²	#11719047	1	4	yes	—
216	C₂²	C₄×#6118	1	8	yes	—
217	C₂²	C₄²×#261	1	4	yes	—
218	C₂²	C₂×#88110	1	4	yes	—
219	C₂²	C₄×#6122	1	8	yes	—
220	C₂²	C₂²×#6118	1	1	yes	characteristic
221	C₂²	C₂×D₁₃⋊M₁₆	1	2	yes	—
222	C₂²	C₂²×C₄×#261	1	1	yes	characteristic
223	C₂²	C₂²×C₄×#261	1	2	yes	—
224	C₂²	C₂²×#6122	1	1	yes	characteristic
225	C₄	—	2	8	no	—
226	C₄	—	13	8	no	—
227	C₄	—	13	4	no	—
228	C₄	—	13	4	no	—
229	C₄	—	26	8	no	—
230	C₄	C₂×#88110	1	8	yes	direct product factor
231	C₂	C₄×#88110	1	8	yes	direct product factor
232	C₂	C₂×C₄²×#261	1	1	yes	characteristic
233	C₂	C₂²×#88110	1	1	yes	characteristic
234	C₂	C₂×#11719047	1	1	yes	characteristic
235	C₂	C₂×C₄×#6122	1	2	yes	—
236	C₂	C₂×C₄×#6118	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂×#261
commutator	C₂×C₁₃	C₂×C₄³
Frattini	C₂³	C₂³×#261
Fitting	C₂²×C₄²×C₁₃	C₄
socle, prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂²×Dic₁₃	C₂⁴
hypercenter	C₂²×C₄²	#261
hypocenter	C₁₃	C₂×C₄×#97
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
3. C₄×#88110	C₂	direct (N × H)
11. C₂×#88110	C₄	direct (N × H)
13. C₂×C₄×#1057	C₄	semidirect
37. #88110	C₂×C₄	direct (N × H)
38. C₄×#1057	C₂×C₄	semidirect
60. C₂×#1057	C₄²	semidirect
83. C₂×C₄×C₁₃	#97	semidirect
91. C₂×Dic₁₃	#97	semidirect
100. #1057	C₂×C₄²	semidirect
124. C₄×C₁₃	C₂×#97	semidirect
127. Dic₁₃	C₂×#97	semidirect
151. C₂×C₁₃	C₄×#97	semidirect
187. C₁₃	C₂×C₄×#97	semidirect
209. C₂×C₄	#88110	direct (N × H)
230. C₄	C₂×#88110	direct (N × H)
231. C₂	C₄×#88110	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#88110	2
C₂²×C₄×#1057	2
C₄×#88110	2
C₂×C₄²×Dic₁₃	2
C₂×C₄×#97	13

Subgroups — C2×C4×#88110 #16796720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#88110 #16796720