Subgroups — C₂×#10620964 #13059476

Order 1344.

660 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₂₁⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#66631	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×C₄×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Dic₃⋊Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#66575	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₃×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#10620964	C₂	1	8	yes	maximal, direct product factor
9	C₂×#6455	—	3	1	no	maximal, Hall
10	C₂³×D₂₁	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
12	C₂²×Dic₂₁	C₂²	1	1	yes	characteristic
13	C₂²×C₇×Dic₃	C₂²	1	1	yes	characteristic
14	C₂²×C₇×Dic₃	C₂²	1	1	yes	characteristic
15	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
16	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
17	C₂×D₂₁⋊C₄	C₄	1	8	yes	complemented
18	C₂×D₂₁⋊C₄	C₂²	1	4	yes	—
19	#66631	C₂²	1	4	yes	—
20	C₇×Dic₃⋊C₄	C₂²	1	4	yes	complemented
21	C₃×C₄×Dic₇	C₂²	1	4	yes	complemented
22	Dic₃⋊Dic₇	C₂²	1	4	yes	complemented
23	#66575	C₂²	1	4	yes	—
24	Dic₃×Dic₇	C₂²	1	4	yes	complemented
25	C₂²×C₄×D₇	—	3	1	no	—
26	C₂×C₇×C₄⋊C₄	—	3	1	no	—
27	C₂×(C₂×D₇)⋊C₄	—	3	1	no	—
28	C₂×(C₂×D₇)⋊C₄	—	3	1	no	—
29	C₂×C₄⋊Dic₇	—	3	1	no	—
30	C₂×C₄×Dic₇	—	3	1	no	—
31	C₂×C₄×Dic₇	D₃	1	1	yes	characteristic, complemented
32	#6455	—	3	8	no	—
33	C₂×#872	—	7	1	no	maximal, Hall
34	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
35	C₂²×D₂₁	—	2	2	no	—
36	C₂²×D₂₁	—	2	2	no	—
37	C₂²×D₂₁	C₂×C₄	1	4	yes	—
38	C₂²×D₂₁	C₂³	1	2	yes	—
39	C₂×C₃×C₄×C₇	—	2	2	no	—
40	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
41	C₂×Dic₂₁	—	2	2	no	—
42	C₂×Dic₂₁	C₂³	1	2	yes	—
43	C₂×C₇×Dic₃	—	2	2	no	—
44	C₂×C₇×Dic₃	C₂×C₄	1	4	yes	—
45	C₂×C₇×Dic₃	C₂³	1	2	yes	—
46	C₂×C₇×Dic₃	C₂³	1	2	yes	—
47	C₂×C₃×Dic₇	—	2	2	no	—
48	C₂×C₃×Dic₇	C₂×C₄	1	4	yes	—
49	C₂×C₃×Dic₇	C₂³	1	2	yes	—
50	C₂×C₃×Dic₇	C₂³	1	2	yes	—
51	D₂₁⋊C₄	C₂×C₄	1	16	yes	complemented
52	C₂³×D₇	—	3	1	no	—
53	C₂²×C₄×C₇	—	3	1	no	—
54	C₂²×C₄×C₇	—	3	1	no	—
55	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
56	C₂²×Dic₇	—	3	1	no	—
57	C₂²×Dic₇	C₂×D₃	1	1	yes	characteristic
58	C₂²×Dic₇	C₂×D₃	1	1	yes	characteristic
59	C₂×C₄×D₇	—	3	8	no	—
60	C₂×C₄×D₇	—	3	4	no	—
61	C₇×C₄⋊C₄	—	3	4	no	—
62	(C₂×D₇)⋊C₄	—	3	4	no	—
63	(C₂×D₇)⋊C₄	—	3	4	no	—
64	C₄⋊Dic₇	—	3	4	no	—
65	C₄×Dic₇	—	3	4	no	—
66	C₄×Dic₇	C₂×D₃	1	4	yes	complemented
67	C₂²×C₄×D₃	—	7	1	no	—
68	C₂×C₃×C₄²	—	7	1	no	—
69	C₂×(C₂×D₃)⋊C₄	—	7	1	no	—
70	C₂×(C₂×D₃)⋊C₄	—	7	1	no	—
71	C₂×Dic₃⋊C₄	—	7	1	no	—
72	C₂×Dic₃⋊C₄	D₇	1	1	yes	characteristic, complemented
73	C₂×C₄×Dic₃	—	7	1	no	—
74	#872	—	7	8	no	—
75	C₂²×C₃×C₇	C₂²×C₄	1	2	yes	—
76	C₂²×C₃×C₇	#41	1	2	yes	—
77	C₂²×C₃×C₇	#41	1	2	yes	—
78	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
79	C₂×D₂₁	—	2	8	no	—
80	C₂×D₂₁	—	2	2	no	—
81	C₂×D₂₁	—	2	2	no	—
82	C₂×D₂₁	C₂²×C₄	1	4	yes	—
83	C₃×C₄×C₇	—	2	2	no	—
84	Dic₂₁	—	2	2	no	—
85	C₃×Dic₇	—	2	2	no	—
86	C₃×Dic₇	C₂²×C₄	1	4	yes	—
87	C₇×Dic₃	—	2	2	no	—
88	C₇×Dic₃	C₂²×C₄	1	4	yes	—
89	C₂×#117	—	21	1	no	Sylow
90	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
91	C₂²×D₇	—	3	4	no	—
92	C₂²×D₇	—	3	2	no	—
93	C₂²×D₇	—	6	2	no	—
94	C₂²×D₇	—	6	2	no	—
95	C₂×C₄×C₇	—	2	2	no	—
96	C₂×C₄×C₇	—	3	4	no	—
97	C₂×C₄×C₇	—	3	2	no	—
98	C₂×C₄×C₇	—	3	2	no	—
99	C₂×C₄×C₇	—	6	2	no	—
100	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
101	C₂×Dic₇	—	2	2	no	—
102	C₂×Dic₇	—	3	2	no	—
103	C₂×Dic₇	—	6	2	no	—
104	C₂×Dic₇	C₄×D₃	1	4	yes	—
105	C₂×Dic₇	C₂²×D₃	1	2	yes	—
106	C₂×Dic₇	C₂²×D₃	1	2	yes	—
107	C₄×D₇	—	3	16	no	—
108	C₂³×D₃	—	7	1	no	—
109	C₂²×C₃×C₄	—	7	1	no	—
110	C₂²×C₃×C₄	—	7	1	no	—
111	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
112	C₂²×Dic₃	—	7	1	no	—
113	C₂²×Dic₃	C₂×D₇	1	1	yes	characteristic
114	C₂²×Dic₃	C₂×D₇	1	1	yes	characteristic
115	C₂×C₄×D₃	—	7	8	no	—
116	C₂×C₄×D₃	—	7	4	no	—
117	C₃×C₄²	—	7	4	no	—
118	(C₂×D₃)⋊C₄	—	7	4	no	—
119	(C₂×D₃)⋊C₄	—	7	4	no	—
120	Dic₃⋊C₄	—	7	4	no	—
121	Dic₃⋊C₄	C₂×D₇	1	4	yes	complemented
122	C₄×Dic₃	—	7	4	no	—
123	C₂×C₃×C₇	#117	1	4	yes	complemented
124	C₂×C₃×C₇	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
126	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
127	D₂₁	—	2	4	no	—
128	C₂³×C₄	—	21	1	no	—
129	#117	—	21	8	no	—
130	C₂×C₄⋊C₄	—	3	1	no	—
131	C₂×C₄⋊C₄	—	21	1	no	—
132	C₂×C₂²⋊C₄	—	21	1	no	—
133	C₂×C₂²⋊C₄	—	21	1	no	—
134	C₂×C₄²	—	7	1	no	—
135	C₂×C₄²	—	21	1	no	—
136	C₂²×C₇	C₂×C₄×D₃	1	2	yes	—
137	C₂²×C₇	#235	1	2	yes	—
138	C₂²×C₇	#235	1	2	yes	—
139	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
140	C₂×D₇	—	3	4	no	—
141	C₂×D₇	—	6	8	no	—
142	C₂×D₇	—	6	2	no	—
143	C₂×D₇	—	6	2	no	—
144	C₄×C₇	—	2	2	no	—
145	C₄×C₇	—	3	4	no	—
146	C₄×C₇	—	6	2	no	—
147	Dic₇	—	2	2	no	—
148	Dic₇	—	6	2	no	—
149	Dic₇	C₂×C₄×D₃	1	4	yes	—
150	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
151	C₂²×D₃	—	7	4	no	—
152	C₂²×D₃	—	7	2	no	—
153	C₂²×D₃	—	14	2	no	—
154	C₂²×D₃	—	14	2	no	—
155	C₂×C₃×C₄	—	2	2	no	—
156	C₂×C₃×C₄	—	7	4	no	—
157	C₂×C₃×C₄	—	7	2	no	—
158	C₂×C₃×C₄	—	7	2	no	—
159	C₂×C₃×C₄	—	14	2	no	—
160	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
161	C₂×Dic₃	—	2	2	no	—
162	C₂×Dic₃	—	7	2	no	—
163	C₂×Dic₃	—	14	2	no	—
164	C₂×Dic₃	C₄×D₇	1	4	yes	—
165	C₂×Dic₃	C₂²×D₇	1	2	yes	—
166	C₂×Dic₃	C₂²×D₇	1	2	yes	—
167	C₄×D₃	—	7	16	no	—
168	C₃×C₇	C₂×#117	1	1	yes	characteristic, complemented, Hall
169	C₂⁴	—	21	1	no	—
170	C₂²×C₄	—	3	1	no	—
171	C₂²×C₄	—	3	1	no	—
172	C₂²×C₄	—	7	1	no	—
173	C₂²×C₄	—	7	1	no	—
174	C₂²×C₄	—	21	4	no	—
175	C₂²×C₄	—	21	8	no	—
176	C₂²×C₄	—	21	1	no	—
177	C₂²×C₄	D₃×D₇	1	1	yes	characteristic
178	C₄⋊C₄	—	3	4	no	—
179	C₄⋊C₄	—	21	4	no	—
180	C₂²⋊C₄	—	21	4	no	—
181	C₂²⋊C₄	—	21	4	no	—
182	C₄²	—	7	4	no	—
183	C₄²	—	21	4	no	—
184	C₂×C₇	#872	1	4	yes	complemented
185	C₂×C₇	C₂²×C₄×D₃	1	1	yes	characteristic
186	C₂×C₇	C₂×#235	1	1	yes	characteristic
187	C₂×C₇	C₂×#235	1	1	yes	characteristic
188	D₇	—	6	4	no	—
189	C₂²×C₃	C₂×C₄×D₇	1	2	yes	—
190	C₂²×C₃	#1159	1	2	yes	—
191	C₂²×C₃	#1161	1	2	yes	—
192	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
193	C₂×D₃	—	7	4	no	—
194	C₂×D₃	—	14	8	no	—
195	C₂×D₃	—	14	2	no	—
196	C₂×D₃	—	14	2	no	—
197	C₃×C₄	—	2	2	no	—
198	C₃×C₄	—	7	4	no	—
199	C₃×C₄	—	14	2	no	—
200	Dic₃	—	2	2	no	—
201	Dic₃	—	14	2	no	—
202	Dic₃	C₂×C₄×D₇	1	4	yes	—
203	C₂³	—	21	4	no	—
204	C₂³	—	21	2	no	—
205	C₂³	—	42	2	no	—
206	C₂³	—	42	2	no	—
207	C₂³	C₂×D₃×D₇	1	1	yes	characteristic
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	3	4	no	—
210	C₂×C₄	—	3	2	no	—
211	C₂×C₄	—	3	2	no	—
212	C₂×C₄	—	6	2	no	—
213	C₂×C₄	—	7	4	no	—
214	C₂×C₄	—	7	2	no	—
215	C₂×C₄	—	7	2	no	—
216	C₂×C₄	—	14	2	no	—
217	C₂×C₄	—	21	16	no	—
218	C₂×C₄	—	21	2	no	—
219	C₂×C₄	—	42	2	no	—
220	C₂×C₄	C₂×D₃×D₇	1	2	yes	—
221	C₇	C₂×#872	1	1	yes	characteristic, complemented, Sylow
222	C₂×C₃	#6455	1	4	yes	complemented
223	C₂×C₃	C₂²×C₄×D₇	1	1	yes	characteristic
224	C₂×C₃	C₂×#1159	1	1	yes	characteristic
225	C₂×C₃	C₂×#1161	1	1	yes	characteristic
226	D₃	—	14	4	no	—
227	C₂²	—	21	4	no	—
228	C₂²	—	42	8	no	—
229	C₂²	—	42	2	no	—
230	C₂²	—	42	2	no	—
231	C₂²	#66677	1	2	yes	—
232	C₂²	C₄×D₃×D₇	1	2	yes	—
233	C₂²	#66686	1	2	yes	—
234	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	3	4	no	—
237	C₄	—	6	2	no	—
238	C₄	—	7	4	no	—
239	C₄	—	14	2	no	—
240	C₄	—	42	2	no	—
241	C₃	C₂×#6455	1	1	yes	characteristic, complemented, Sylow
242	C₂	—	42	4	no	—
243	C₂	#10620964	1	4	yes	direct product factor
244	C₂	C₂×#66677	1	1	yes	characteristic
245	C₂	C₂×C₄×D₃×D₇	1	1	yes	characteristic
246	C₂	C₂×#66686	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₃×D₇
commutator	C₂×C₃×C₇	C₂³×C₄
Frattini	C₂²	C₂²×D₃×D₇
Fitting	C₂²×C₃×C₄×C₇	C₂²
socle	C₂³×C₃×C₇	C₂³
prime radical	C₂³×D₂₁	C₂²
cosocle	C₂²×C₃×C₇	C₂⁴
hypercenter	C₂²×C₄	D₃×D₇
hypocenter	C₃×C₇	C₂×#117
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
3. C₂×C₇×Dic₃⋊C₄	C₂	semidirect
4. C₂×C₃×C₄×Dic₇	C₂	semidirect
5. C₂×Dic₃⋊Dic₇	C₂	semidirect
7. C₂×Dic₃×Dic₇	C₂	semidirect
8. #10620964	C₂	direct (N × H)
17. C₂×D₂₁⋊C₄	C₄	semidirect
20. C₇×Dic₃⋊C₄	C₂²	semidirect
21. C₃×C₄×Dic₇	C₂²	semidirect
22. Dic₃⋊Dic₇	C₂²	semidirect
24. Dic₃×Dic₇	C₂²	semidirect
31. C₂×C₄×Dic₇	D₃	semidirect
51. D₂₁⋊C₄	C₂×C₄	semidirect
66. C₄×Dic₇	C₂×D₃	semidirect
72. C₂×Dic₃⋊C₄	D₇	semidirect
121. Dic₃⋊C₄	C₂×D₇	semidirect
123. C₂×C₃×C₇	#117	semidirect
168. C₃×C₇	C₂×#117	semidirect
184. C₂×C₇	#872	semidirect
221. C₇	C₂×#872	semidirect
222. C₂×C₃	#6455	semidirect
241. C₃	C₂×#6455	semidirect
243. C₂	#10620964	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×D₂₁⋊C₄	2
C₂×#66631	2
C₂×C₇×Dic₃⋊C₄	2
C₂×C₃×C₄×Dic₇	2
C₂×Dic₃⋊Dic₇	2
C₂×#66575	2
C₂×Dic₃×Dic₇	2
#10620964	2
C₂×#6455	3
C₂×#872	7

Subgroups — C2×#10620964 #13059476

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10620964 #13059476