Subgroups — C₂×C₄×#88202 #16726323

Order 3328.

3366 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×#88202	C₂	1	2	yes	maximal
2	C₄×#88202	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11718658	C₂	1	6	yes	maximal, complemented
4	C₂×C₄×#6099	C₂	1	3	yes	maximal, complemented
5	C₂²×C₄×#1055	C₂	1	3	yes	maximal, complemented
6	C₂×C₄×C₁₃×#41	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×#41	C₂²	1	1	yes	characteristic
8	C₄×C₁₃×#41	C₄	1	8	yes	—
9	C₄×C₁₃×#41	C₂²	1	8	yes	—
10	C₂×C₄×Q₈×C₁₃	C₄	1	2	yes	—
11	C₂×C₄×D₄×C₁₃	C₄	1	6	yes	—
12	C₂×C₁₃×#117	C₂²	1	3	yes	—
13	C₂²×C₄²×C₁₃	C₂²	1	3	yes	—
14	C₂×#88202	C₄	1	32	yes	direct product factor
15	C₂×#88202	C₂²	1	16	yes	—
16	C₂²×#6099	C₂²	1	6	yes	—
17	C₂³×#1055	C₂²	1	6	yes	—
18	#11718658	C₂²	1	48	yes	complemented
19	C₄×#6099	C₂²	1	24	yes	complemented
20	C₂×#88043	C₂²	1	6	yes	—
21	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
22	C₂×C₄×#1055	C₂²	1	3	yes	—
23	C₂×C₄×#1055	C₂²	1	6	yes	—
24	C₂×C₄×#1055	C₂²	1	24	yes	complemented
25	C₂×C₁₃×#41	C₂×C₄	1	16	yes	—
26	C₂×C₁₃×#41	C₂×C₄	1	4	yes	—
27	C₂×C₁₃×#41	C₂³	1	4	yes	—
28	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×D₄×C₁₃	C₂×C₄	1	3	yes	—
30	C₂³×C₄×C₁₃	C₂×C₄	1	6	yes	—
31	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
32	C₄×Q₈×C₁₃	C₂×C₄	1	16	yes	—
33	C₄×D₄×C₁₃	C₂×C₄	1	48	yes	—
34	C₁₃×#117	C₂×C₄	1	12	yes	—
35	C₁₃×#117	C₂³	1	12	yes	—
36	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	6	yes	—
37	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	6	yes	—
38	C₂×C₄²×C₁₃	C₂×C₄	1	6	yes	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	12	yes	—
40	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
41	C₂×C₄²×C₁₃	C₂³	1	12	yes	—
42	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
43	#88202	C₂×C₄	1	128	yes	direct product factor
44	C₂×#6099	C₂×C₄	1	48	yes	complemented
45	C₂×#6099	C₂³	1	24	yes	—
46	C₂²×#1055	C₂×C₄	1	48	yes	complemented
47	C₂²×#1055	C₂³	1	2	yes	—
48	C₂²×#1055	C₂³	1	6	yes	—
49	C₂²×#1055	C₂³	1	24	yes	—
50	#88043	C₂³	1	24	yes	—
51	C₄×#1055	C₂³	1	24	yes	—
52	C₄×#1055	C₂³	1	4	yes	—
53	C₄×#1055	C₂³	1	12	yes	—
54	C₂×C₄×#131	—	13	1	no	maximal, Sylow
55	C₂⁴×C₁₃	C₂²×C₄	1	3	yes	—
56	C₁₃×#41	C₄²	1	32	yes	—
57	C₁₃×#41	C₂²×C₄	1	32	yes	—
58	C₂×Q₈×C₁₃	C₄²	1	8	yes	—
59	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
60	C₂×D₄×C₁₃	C₄²	1	24	yes	—
61	C₂×D₄×C₁₃	C₂²×C₄	1	12	yes	—
62	C₂²×C₄×C₁₃	—	2	24	no	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	24	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	24	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
71	C₁₃×C₄⋊C₄	C₂²×C₄	1	24	yes	—
72	C₁₃×C₂²⋊C₄	C₂²×C₄	1	24	yes	—
73	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
74	C₄²×C₁₃	C₂²×C₄	1	6	yes	—
75	C₄²×C₁₃	C₂²×C₄	1	24	yes	—
76	C₄²×C₁₃	C₂⁴	1	2	yes	—
77	C₄²×C₁₃	C₂⁴	1	6	yes	—
78	#6099	C₂²×C₄	1	96	yes	complemented
79	C₂×#1055	C₂²×C₄	1	8	yes	—
80	C₂×#1055	C₂²×C₄	1	24	yes	—
81	C₂×#1055	C₂²×C₄	1	96	yes	complemented
82	C₂×#1055	C₂⁴	1	4	yes	—
83	C₂×#1055	C₂⁴	1	12	yes	—
84	C₂²×#131	—	13	2	no	—
85	C₂×C₄×#41	D₁₃	1	1	yes	characteristic
86	C₄×#131	—	13	16	no	—
87	C₂×#405	—	13	6	no	—
88	C₂×C₄×M₁₆	—	13	3	no	—
89	C₂²×C₄×C₈	—	13	3	no	—
90	C₂³×C₁₃	—	2	12	no	—
91	C₂³×C₁₃	C₂×C₄²	1	12	yes	—
92	C₂³×C₁₃	C₂³×C₄	1	6	yes	—
93	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
94	Q₈×C₁₃	C₂×C₄²	1	16	yes	—
95	D₄×C₁₃	C₂×C₄²	1	48	yes	—
96	C₂×C₄×C₁₃	—	2	12	no	—
97	C₂×C₄×C₁₃	—	2	48	no	—
98	C₂×C₄×C₁₃	C₂×C₄²	1	12	yes	—
99	C₂×C₄×C₁₃	C₂×C₄²	1	24	yes	—
100	C₂×C₄×C₁₃	#131	1	8	yes	complemented
101	C₂×C₄×C₁₃	C₂³×C₄	1	12	yes	—
102	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
103	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
104	C₂×C₄×C₁₃	C₂³×C₄	1	24	yes	—
105	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
107	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
108	#1055	C₂³×C₄	1	8	yes	—
109	#1055	C₂³×C₄	1	24	yes	—
110	C₂²×#41	C₂×D₁₃	1	1	yes	characteristic
111	C₂×#131	—	13	32	no	—
112	C₂×#131	—	13	16	no	—
113	C₂²×M₁₆	—	13	6	no	—
114	C₂³×C₈	—	13	6	no	—
115	C₄×#41	Dic₁₃	1	8	yes	—
116	C₄×#41	C₂×D₁₃	1	8	yes	—
117	C₂×C₄×Q₈	Dic₁₃	1	2	yes	—
118	C₂×C₄×D₄	Dic₁₃	1	6	yes	—
119	C₂×#117	C₂×D₁₃	1	3	yes	—
120	C₂²×C₄²	C₂×D₁₃	1	3	yes	—
121	#405	—	13	48	no	—
122	C₄×M₁₆	—	13	24	no	—
123	C₂×#97	—	13	6	no	—
124	C₂×C₄×C₈	—	13	6	no	—
125	C₂×C₄×C₈	—	13	24	no	—
126	C₂×C₄×C₈	—	13	1	no	—
127	C₂×C₄×C₈	—	13	3	no	—
128	C₂²×C₁₃	—	2	24	no	—
129	C₂²×C₁₃	—	2	12	no	—
130	C₂²×C₁₃	C₂²×C₄²	1	12	yes	—
131	C₂²×C₁₃	C₂²×C₄²	1	2	yes	—
132	C₂²×C₁₃	C₂×#131	1	4	yes	—
133	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
134	C₄×C₁₃	—	2	12	no	—
135	C₄×C₁₃	C₂²×C₄²	1	2	yes	—
136	C₄×C₁₃	C₂²×C₄²	1	12	yes	—
137	C₄×C₁₃	C₂×#131	1	8	yes	complemented
138	C₄×C₁₃	C₂⁴×C₄	1	2	yes	—
139	C₂×#41	C₄×D₁₃	1	16	yes	—
140	C₂×#41	C₂×Dic₁₃	1	4	yes	—
141	C₂×#41	C₂²×D₁₃	1	4	yes	—
142	C₂²×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
143	C₂²×D₄	C₂×Dic₁₃	1	3	yes	—
144	C₂³×C₄	C₂×Dic₁₃	1	6	yes	—
145	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
146	#131	—	13	128	no	—
147	C₂×M₁₆	—	13	48	no	—
148	C₂×M₁₆	—	13	24	no	—
149	C₂²×C₈	—	13	48	no	—
150	C₂²×C₈	—	13	2	no	—
151	C₂²×C₈	—	13	6	no	—
152	C₂²×C₈	—	13	24	no	—
153	C₄×Q₈	C₂×Dic₁₃	1	16	yes	—
154	C₄×D₄	C₂×Dic₁₃	1	48	yes	—
155	#117	C₂×Dic₁₃	1	12	yes	—
156	#117	C₂²×D₁₃	1	12	yes	—
157	C₂×C₄⋊C₄	C₂×Dic₁₃	1	6	yes	—
158	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	6	yes	—
159	C₂×C₄²	C₂×Dic₁₃	1	6	yes	—
160	C₂×C₄²	C₂×Dic₁₃	1	12	yes	—
161	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
162	C₂×C₄²	C₂²×D₁₃	1	12	yes	—
163	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
164	#97	—	13	24	no	—
165	C₄×C₈	—	13	4	no	—
166	C₄×C₈	—	13	12	no	—
167	C₄×C₈	—	13	24	no	—
168	C₂×C₁₃	—	2	12	no	—
169	C₂×C₁₃	C₄×#131	1	4	yes	complemented
170	C₂×C₁₃	C₂³×C₄²	1	1	yes	characteristic
171	C₂×C₁₃	C₂²×#131	1	2	yes	—
172	C₂⁴	C₂²×Dic₁₃	1	3	yes	—
173	#41	C₄×Dic₁₃	1	32	yes	—
174	#41	C₂×C₄×D₁₃	1	32	yes	—
175	C₂×Q₈	C₄×Dic₁₃	1	8	yes	—
176	C₂×Q₈	C₂²×Dic₁₃	1	4	yes	—
177	C₂×D₄	C₄×Dic₁₃	1	24	yes	—
178	C₂×D₄	C₂²×Dic₁₃	1	12	yes	—
179	C₂²×C₄	—	2	24	no	—
180	C₂²×C₄	C₂×C₄×D₁₃	1	24	yes	—
181	C₂²×C₄	C₂²×Dic₁₃	1	6	yes	—
182	C₂²×C₄	C₂²×Dic₁₃	1	24	yes	—
183	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
184	C₂²×C₄	C₂²×Dic₁₃	1	6	yes	—
185	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
186	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
187	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
188	M₁₆	—	13	96	no	—
189	C₂×C₈	—	13	8	no	—
190	C₂×C₈	—	13	24	no	—
191	C₂×C₈	—	13	96	no	—
192	C₂×C₈	—	13	4	no	—
193	C₂×C₈	—	13	12	no	—
194	C₄⋊C₄	C₂²×Dic₁₃	1	24	yes	—
195	C₂²⋊C₄	C₂²×Dic₁₃	1	24	yes	—
196	C₄²	C₂²×Dic₁₃	1	2	yes	—
197	C₄²	C₂²×Dic₁₃	1	6	yes	—
198	C₄²	C₂²×Dic₁₃	1	24	yes	—
199	C₄²	C₂³×D₁₃	1	2	yes	—
200	C₄²	C₂³×D₁₃	1	6	yes	—
201	C₁₃	C₂×C₄×#131	1	1	yes	characteristic, complemented, Sylow
202	C₂³	—	2	12	no	—
203	C₂³	C₂×C₄×Dic₁₃	1	12	yes	—
204	C₂³	C₂³×Dic₁₃	1	6	yes	—
205	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
206	Q₈	C₂×C₄×Dic₁₃	1	16	yes	—
207	D₄	C₂×C₄×Dic₁₃	1	48	yes	—
208	C₂×C₄	—	2	12	no	—
209	C₂×C₄	—	2	48	no	—
210	C₂×C₄	C₂×C₄×Dic₁₃	1	12	yes	—
211	C₂×C₄	C₂×C₄×Dic₁₃	1	24	yes	—
212	C₂×C₄	#88202	1	8	yes	direct product factor
213	C₂×C₄	C₂²×C₄×D₁₃	1	4	yes	—
214	C₂×C₄	C₂²×C₄×D₁₃	1	24	yes	—
215	C₂×C₄	C₂³×Dic₁₃	1	12	yes	—
216	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
217	C₂×C₄	C₂³×Dic₁₃	1	6	yes	—
218	C₂×C₄	C₂³×Dic₁₃	1	4	yes	—
219	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
220	C₈	—	13	8	no	—
221	C₈	—	13	24	no	—
222	C₂²	—	2	12	no	—
223	C₂²	—	2	24	no	—
224	C₂²	C₂²×C₄×Dic₁₃	1	12	yes	—
225	C₂²	C₂²×C₄×Dic₁₃	1	2	yes	—
226	C₂²	C₂×#88202	1	4	yes	—
227	C₂²	C₂⁴×Dic₁₃	1	1	yes	characteristic
228	C₄	—	2	12	no	—
229	C₄	C₂²×C₄×Dic₁₃	1	2	yes	—
230	C₄	C₂²×C₄×Dic₁₃	1	12	yes	—
231	C₄	C₂×#88202	1	8	yes	direct product factor
232	C₄	C₂³×C₄×D₁₃	1	2	yes	—
233	C₂	—	2	12	no	—
234	C₂	C₄×#88202	1	4	yes	direct product factor
235	C₂	C₂³×C₄×Dic₁₃	1	1	yes	characteristic
236	C₂	C₂²×#88202	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₁₃
commutator	C₂×C₁₃	C₂³×C₄²
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂×C₄×C₁₃×#41	C₂
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂²×C₁₃×#41	C₂²
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂×C₄×#41	D₁₃
hypocenter	C₁₃	C₂×C₄×#131
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×#41

Complements

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

N	H ≅ G/N	type
2. C₄×#88202	C₂	direct (N × H)
3. C₂×#11718658	C₂	semidirect
4. C₂×C₄×#6099	C₂	semidirect
5. C₂²×C₄×#1055	C₂	semidirect
14. C₂×#88202	C₄	direct (N × H)
18. #11718658	C₂²	semidirect
19. C₄×#6099	C₂²	semidirect
24. C₂×C₄×#1055	C₂²	semidirect
43. #88202	C₂×C₄	direct (N × H)
44. C₂×#6099	C₂×C₄	semidirect
46. C₂²×#1055	C₂×C₄	semidirect
78. #6099	C₂²×C₄	semidirect
81. C₂×#1055	C₂²×C₄	semidirect
100. C₂×C₄×C₁₃	#131	semidirect
137. C₄×C₁₃	C₂×#131	semidirect
169. C₂×C₁₃	C₄×#131	semidirect
201. C₁₃	C₂×C₄×#131	semidirect
212. C₂×C₄	#88202	direct (N × H)
231. C₄	C₂×#88202	direct (N × H)
234. C₂	C₄×#88202	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×#88202	2
C₄×#88202	2
C₂×#11718658	2
C₂×C₄×#6099	2
C₂²×C₄×#1055	2
C₂×C₄×C₁₃×#41	2
C₂×C₄×#131	13

Subgroups — C2×C4×#88202 #16726323

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#88202 #16726323