Subgroups — C₄×#66728 #13061564

Order 1344.

620 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#66728	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄²×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#4268	C₂	1	4	yes	maximal, complemented
4	#10621409	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₈₄	C₂	1	2	yes	maximal, complemented
6	#10621355	C₂	1	2	yes	maximal, complemented
7	C₄²×D₂₁	C₂	1	2	yes	maximal, complemented
8	C₄×Dic₄₂	C₂	1	2	yes	maximal, complemented
9	C₄×#1157	—	3	1	no	maximal, Hall
10	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
11	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
12	C₂×#4268	C₂²	1	2	yes	—
13	#66728	C₄	1	8	yes	direct product factor
14	C₂×D₈₄	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₂₁	C₂²	1	4	yes	—
16	C₂×C₄×D₂₁	C₂²	1	2	yes	—
17	C₂×Dic₄₂	C₂²	1	1	yes	characteristic
18	C₃×C₄²×C₇	C₂²	1	2	yes	—
19	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
20	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
21	#66636	C₂²	1	2	yes	—
22	#66631	C₂²	1	4	yes	—
23	C₄⋊Dic₂₁	C₂²	1	2	yes	—
24	#66629	C₂²	1	4	yes	—
25	C₄×Dic₂₁	C₂²	1	4	yes	—
26	C₄×Dic₂₁	C₂²	1	2	yes	—
27	C₂×#1157	—	3	1	no	—
28	C₂×C₄²×C₇	D₃	1	1	yes	characteristic, complemented
29	C₄×C₂²⋊D₇	—	3	4	no	—
30	#6489	—	3	1	no	—
31	C₄×D₂₈	—	3	2	no	—
32	#6435	—	3	2	no	—
33	C₄²×D₇	—	3	2	no	—
34	C₄×Dic₁₄	—	3	2	no	—
35	C₄×#235	—	7	1	no	maximal, Hall
36	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
37	C₂²×D₂₁	C₂³	1	2	yes	—
38	C₂×C₃×C₄×C₇	—	2	4	no	—
39	C₂×C₃×C₄×C₇	C₂×C₄	1	4	yes	complemented
40	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
41	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
42	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
43	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
44	#4268	C₂×C₄	1	8	yes	complemented
45	C₂×Dic₂₁	C₂³	1	4	yes	—
46	C₂×Dic₂₁	C₂³	1	2	yes	—
47	D₈₄	C₂×C₄	1	4	yes	complemented
48	C₄×D₂₁	—	2	8	no	—
49	C₄×D₂₁	C₂×C₄	1	8	yes	complemented
50	Dic₄₂	C₂×C₄	1	4	yes	complemented
51	C₂²×C₄×C₇	C₂×D₃	1	2	yes	—
52	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
53	C₂×C₂²⋊D₇	—	3	2	no	—
54	#1157	—	3	8	no	—
55	C₂×D₂₈	—	3	1	no	—
56	C₂×C₄×D₇	—	3	2	no	—
57	C₂×C₄×D₇	—	3	4	no	—
58	C₂×Dic₁₄	—	3	1	no	—
59	C₄²×C₇	C₂×D₃	1	2	yes	—
60	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
61	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
62	C₂²⋊Dic₇	—	3	2	no	—
63	(C₂×D₇)⋊C₄	—	3	4	no	—
64	C₄⋊Dic₇	—	3	2	no	—
65	Dic₇⋊C₄	—	3	4	no	—
66	C₄×Dic₇	—	3	4	no	—
67	C₄×Dic₇	—	3	2	no	—
68	C₂×#235	—	7	1	no	—
69	C₂×C₃×C₄²	D₇	1	1	yes	characteristic, complemented
70	C₄×#67	—	7	4	no	—
71	#926	—	7	1	no	—
72	C₄×D₁₂	—	7	2	no	—
73	#872	—	7	2	no	—
74	C₄²×D₃	—	7	2	no	—
75	C₄×Dic₆	—	7	2	no	—
76	C₂²×C₃×C₇	—	2	2	no	—
77	C₂²×C₃×C₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
79	C₂×D₂₁	—	2	4	no	—
80	C₂×D₂₁	C₂²×C₄	1	4	yes	—
81	C₃×C₄×C₇	—	2	2	no	—
82	C₃×C₄×C₇	C₂²×C₄	1	2	yes	—
83	C₃×C₄×C₇	C₂²×C₄	1	2	yes	—
84	C₃×C₄×C₇	#41	1	4	yes	complemented
85	Dic₂₁	—	2	4	no	—
86	Dic₂₁	C₂²×C₄	1	4	yes	—
87	C₄×#41	—	21	1	no	Sylow
88	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
89	C₂²×D₇	—	3	2	no	—
90	C₂×C₄×C₇	—	2	4	no	—
91	C₂×C₄×C₇	C₄×D₃	1	4	yes	complemented
92	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
93	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
94	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
95	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
96	C₂²⋊D₇	—	3	8	no	—
97	C₂×Dic₇	—	3	2	no	—
98	C₂×Dic₇	—	3	4	no	—
99	D₂₈	—	3	4	no	—
100	C₄×D₇	—	3	8	no	—
101	C₄×D₇	—	6	8	no	—
102	Dic₁₄	—	3	4	no	—
103	C₂²×C₃×C₄	C₂×D₇	1	2	yes	—
104	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
105	C₂×#67	—	7	2	no	—
106	#235	—	7	8	no	—
107	C₂×D₁₂	—	7	1	no	—
108	C₂×C₄×D₃	—	7	2	no	—
109	C₂×C₄×D₃	—	7	4	no	—
110	C₂×Dic₆	—	7	1	no	—
111	C₃×C₄²	C₂×D₇	1	2	yes	—
112	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
113	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
114	#217	—	7	2	no	—
115	(C₂×D₃)⋊C₄	—	7	4	no	—
116	C₄⋊Dic₃	—	7	2	no	—
117	Dic₃⋊C₄	—	7	4	no	—
118	C₄×Dic₃	—	7	4	no	—
119	C₄×Dic₃	—	7	2	no	—
120	C₂×C₃×C₇	—	2	2	no	—
121	C₂×C₃×C₇	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₃×C₇	C₂×#41	1	2	yes	—
123	D₂₁	—	2	4	no	—
124	C₂×#41	—	21	1	no	—
125	C₄×Q₈	—	21	2	no	—
126	C₄×D₄	—	21	4	no	—
127	C₄×D₄	—	21	2	no	—
128	#117	—	21	1	no	—
129	#117	—	21	2	no	—
130	C₂×C₄²	—	21	2	no	—
131	C₂×C₄²	D₂₁	1	1	yes	characteristic, complemented
132	C₂²×C₇	—	2	2	no	—
133	C₂²×C₇	C₂×C₄×D₃	1	2	yes	—
134	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
135	C₂×D₇	—	3	4	no	—
136	C₂×D₇	—	6	4	no	—
137	C₄×C₇	—	2	2	no	—
138	C₄×C₇	C₂×C₄×D₃	1	2	yes	—
139	C₄×C₇	C₂×C₄×D₃	1	2	yes	—
140	C₄×C₇	#235	1	4	yes	complemented
141	Dic₇	—	3	4	no	—
142	Dic₇	—	6	4	no	—
143	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
144	C₂²×D₃	—	7	2	no	—
145	C₂×C₃×C₄	—	2	4	no	—
146	C₂×C₃×C₄	C₄×D₇	1	4	yes	complemented
147	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
148	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
151	#67	—	7	8	no	—
152	C₂×Dic₃	—	7	2	no	—
153	C₂×Dic₃	—	7	4	no	—
154	D₁₂	—	7	4	no	—
155	C₄×D₃	—	7	8	no	—
156	C₄×D₃	—	14	8	no	—
157	Dic₆	—	7	4	no	—
158	C₃×C₇	C₄×#41	1	1	yes	characteristic, complemented, Hall
159	#41	—	21	8	no	—
160	C₂×Q₈	—	21	1	no	—
161	C₂×D₄	—	21	2	no	—
162	C₂×D₄	—	21	1	no	—
163	C₂²×C₄	—	21	2	no	—
164	C₂²×C₄	—	21	4	no	—
165	C₂²×C₄	C₂×D₂₁	1	2	yes	—
166	C₂²×C₄	C₂×D₂₁	1	1	yes	characteristic
167	C₄⋊C₄	—	21	4	no	—
168	C₄⋊C₄	—	21	2	no	—
169	C₂²⋊C₄	—	21	2	no	—
170	C₂²⋊C₄	—	21	4	no	—
171	C₄²	—	21	2	no	—
172	C₄²	—	21	4	no	—
173	C₄²	C₂×D₂₁	1	1	yes	characteristic
174	C₄²	C₂×D₂₁	1	2	yes	—
175	C₄²	C₂×D₂₁	1	1	yes	characteristic
176	C₂×C₇	—	2	2	no	—
177	C₂×C₇	C₂²×C₄×D₃	1	1	yes	characteristic
178	C₂×C₇	C₂×#235	1	2	yes	—
179	D₇	—	6	4	no	—
180	C₂²×C₃	—	2	2	no	—
181	C₂²×C₃	C₂×C₄×D₇	1	2	yes	—
182	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
183	C₂×D₃	—	7	4	no	—
184	C₂×D₃	—	14	4	no	—
185	C₃×C₄	—	2	2	no	—
186	C₃×C₄	C₂×C₄×D₇	1	2	yes	—
187	C₃×C₄	C₂×C₄×D₇	1	2	yes	—
188	C₃×C₄	#1157	1	4	yes	complemented
189	Dic₃	—	7	4	no	—
190	Dic₃	—	14	4	no	—
191	C₂³	—	21	2	no	—
192	C₂³	C₂²×D₂₁	1	1	yes	characteristic
193	Q₈	—	21	4	no	—
194	D₄	—	21	8	no	—
195	D₄	—	21	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	21	2	no	—
198	C₂×C₄	—	21	8	no	—
199	C₂×C₄	—	21	4	no	—
200	C₂×C₄	—	42	8	no	—
201	C₂×C₄	C₄×D₂₁	1	4	yes	complemented
202	C₂×C₄	C₂²×D₂₁	1	2	yes	—
203	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
204	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
205	C₂×C₄	C₂²×D₂₁	1	2	yes	—
206	C₇	C₄×#235	1	1	yes	characteristic, complemented, Sylow
207	C₂×C₃	—	2	2	no	—
208	C₂×C₃	C₂²×C₄×D₇	1	1	yes	characteristic
209	C₂×C₃	C₂×#1157	1	2	yes	—
210	D₃	—	14	4	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	21	4	no	—
213	C₂²	—	42	4	no	—
214	C₂²	C₂×C₄×D₂₁	1	2	yes	—
215	C₂²	C₂³×D₂₁	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	21	4	no	—
218	C₄	—	42	4	no	—
219	C₄	C₂×C₄×D₂₁	1	2	yes	—
220	C₄	C₂×C₄×D₂₁	1	2	yes	—
221	C₄	#66728	1	4	yes	direct product factor
222	C₃	C₄×#1157	1	1	yes	characteristic, complemented, Sylow
223	C₂	—	2	2	no	—
224	C₂	—	42	4	no	—
225	C₂	C₂²×C₄×D₂₁	1	1	yes	characteristic
226	C₂	C₂×#66728	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂×D₂₁
commutator	C₂×C₃×C₇	C₂³×C₄
Frattini	C₂²	C₂³×D₂₁
Fitting	C₂×C₃×C₄²×C₇	C₂
socle, cosocle	C₂²×C₃×C₇	C₂⁴
prime radical	C₂×#66728	C₂
hypercenter	C₂×C₄²	D₂₁
hypocenter	C₃×C₇	C₄×#41
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₄²×C₇	C₂	semidirect
3. C₄×#4268	C₂	semidirect
4. #10621409	C₂	semidirect
5. C₄×D₈₄	C₂	semidirect
6. #10621355	C₂	semidirect
7. C₄²×D₂₁	C₂	semidirect
8. C₄×Dic₄₂	C₂	semidirect
13. #66728	C₄	direct (N × H)
28. C₂×C₄²×C₇	D₃	semidirect
39. C₂×C₃×C₄×C₇	C₂×C₄	semidirect
44. #4268	C₂×C₄	semidirect
47. D₈₄	C₂×C₄	semidirect
49. C₄×D₂₁	C₂×C₄	semidirect
50. Dic₄₂	C₂×C₄	semidirect
69. C₂×C₃×C₄²	D₇	semidirect
84. C₃×C₄×C₇	#41	semidirect
91. C₂×C₄×C₇	C₄×D₃	semidirect
131. C₂×C₄²	D₂₁	semidirect
140. C₄×C₇	#235	semidirect
146. C₂×C₃×C₄	C₄×D₇	semidirect
158. C₃×C₇	C₄×#41	semidirect
188. C₃×C₄	#1157	semidirect
201. C₂×C₄	C₄×D₂₁	semidirect
206. C₇	C₄×#235	semidirect
221. C₄	#66728	direct (N × H)
222. C₃	C₄×#1157	semidirect

Maximal subgroups

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

subgroup	index
C₂×#66728	2
C₂×C₃×C₄²×C₇	2
C₄×#4268	2
#10621409	2
C₄×D₈₄	2
#10621355	2
C₄²×D₂₁	2
C₄×Dic₄₂	2
C₄×#1157	3
C₄×#235	7

Subgroups — C4×#66728 #13061564

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#66728 #13061564