Subgroups — C₂×#10621361 #13061571

Order 1344.

644 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₂₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×Dic₄₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₇×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₂×#66636	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Dic₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#66629	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#66629	C₂	1	1	yes	maximal, characteristic, complemented
8	#10621361	C₂	1	8	yes	maximal, direct product factor
9	C₂×#6441	—	3	1	no	maximal, Hall
10	C₂⁴×C₃×C₇	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
13	C₂²×Dic₂₁	—	2	4	no	—
14	C₂²×Dic₂₁	C₂²	1	4	yes	—
15	C₂²×Dic₂₁	C₂²	1	1	yes	characteristic
16	C₂²×Dic₂₁	C₂²	1	1	yes	characteristic
17	C₂²×Dic₂₁	C₂²	1	1	yes	characteristic
18	C₂²×Dic₂₁	C₂²	1	1	yes	characteristic
19	C₂×Dic₄₂	—	2	4	no	—
20	C₂×Dic₄₂	C₂²	1	4	yes	complemented
21	C₃×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
22	#66636	C₂²	1	4	yes	—
23	C₄⋊Dic₂₁	C₂²	1	4	yes	complemented
24	#66629	C₂²	1	4	yes	complemented
25	#66629	C₂²	1	4	yes	complemented
26	C₂³×Dic₇	—	3	1	no	—
27	C₂²×Dic₁₄	—	3	1	no	—
28	C₂×C₇×C₂²⋊C₄	D₃	1	1	yes	characteristic
29	C₂×C₂²⋊Dic₇	—	3	1	no	—
30	C₂×C₄⋊Dic₇	—	3	1	no	—
31	C₂×Dic₇⋊C₄	—	3	1	no	—
32	C₂×Dic₇⋊C₄	—	3	1	no	—
33	#6441	—	3	8	no	—
34	C₂×#878	—	7	1	no	maximal, Hall
35	C₂³×C₃×C₇	—	2	2	no	—
36	C₂³×C₃×C₇	—	2	2	no	—
37	C₂³×C₃×C₇	Q₈	1	4	yes	complemented
38	C₂³×C₃×C₇	C₂³	1	2	yes	—
39	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
40	C₂×C₃×C₄×C₇	—	2	2	no	—
41	C₂×C₃×C₄×C₇	—	2	2	no	—
42	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
43	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
44	C₂×Dic₂₁	—	2	2	no	—
45	C₂×Dic₂₁	—	2	2	no	—
46	C₂×Dic₂₁	—	2	2	no	—
47	C₂×Dic₂₁	—	2	8	no	—
48	C₂×Dic₂₁	D₄	1	4	yes	—
49	C₂×Dic₂₁	C₂³	1	2	yes	—
50	C₂×Dic₂₁	C₂³	1	2	yes	—
51	C₂×Dic₂₁	C₂³	1	2	yes	—
52	C₂×Dic₂₁	C₂³	1	2	yes	—
53	Dic₄₂	—	2	8	no	—
54	C₂⁴×C₇	C₂×D₃	1	1	yes	characteristic
55	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
56	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
57	C₂²×Dic₇	—	3	1	no	—
58	C₂²×Dic₇	—	3	4	no	—
59	C₂²×Dic₇	—	3	1	no	—
60	C₂²×Dic₇	—	3	1	no	—
61	C₂²×Dic₇	—	3	1	no	—
62	C₂²×Dic₇	—	6	4	no	—
63	C₂×Dic₁₄	—	3	4	no	—
64	C₂×Dic₁₄	—	6	4	no	—
65	C₇×C₂²⋊C₄	C₂×D₃	1	4	yes	—
66	C₂²⋊Dic₇	—	3	4	no	—
67	C₄⋊Dic₇	—	3	4	no	—
68	Dic₇⋊C₄	—	3	4	no	—
69	Dic₇⋊C₄	—	3	4	no	—
70	C₂³×Dic₃	—	7	1	no	—
71	C₂²×Dic₆	—	7	1	no	—
72	C₂×C₃×C₂²⋊C₄	D₇	1	1	yes	characteristic
73	C₂×#217	—	7	1	no	—
74	C₂×C₄⋊Dic₃	—	7	1	no	—
75	C₂×Dic₃⋊C₄	—	7	1	no	—
76	C₂×Dic₃⋊C₄	—	7	1	no	—
77	#878	—	7	8	no	—
78	C₂²×C₃×C₇	—	2	8	no	—
79	C₂²×C₃×C₇	—	2	2	no	—
80	C₂²×C₃×C₇	—	2	2	no	—
81	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
82	C₂²×C₃×C₇	C₂×Q₈	1	2	yes	—
83	C₂²×C₃×C₇	C₂×Q₈	1	4	yes	complemented
84	C₂²×C₃×C₇	#41	1	2	yes	—
85	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
86	C₃×C₄×C₇	—	2	2	no	—
87	C₃×C₄×C₇	—	2	2	no	—
88	Dic₂₁	—	2	2	no	—
89	Dic₂₁	—	2	2	no	—
90	Dic₂₁	—	2	2	no	—
91	Dic₂₁	C₂×D₄	1	4	yes	—
92	C₂×C₂²⋊Q₈	—	21	1	no	Sylow
93	C₂³×C₇	—	2	2	no	—
94	C₂³×C₇	—	2	2	no	—
95	C₂³×C₇	Dic₆	1	4	yes	complemented
96	C₂³×C₇	C₂²×D₃	1	2	yes	—
97	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
98	C₂×C₄×C₇	—	2	2	no	—
99	C₂×C₄×C₇	—	2	2	no	—
100	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
101	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
102	C₂×Dic₇	—	3	4	no	—
103	C₂×Dic₇	—	3	2	no	—
104	C₂×Dic₇	—	3	2	no	—
105	C₂×Dic₇	—	3	2	no	—
106	C₂×Dic₇	—	3	2	no	—
107	C₂×Dic₇	—	6	8	no	—
108	C₂×Dic₇	—	6	2	no	—
109	C₂×Dic₇	—	6	2	no	—
110	C₂×Dic₇	—	6	2	no	—
111	Dic₁₄	—	6	8	no	—
112	C₂⁴×C₃	C₂×D₇	1	1	yes	characteristic
113	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
114	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
115	C₂²×Dic₃	—	7	1	no	—
116	C₂²×Dic₃	—	7	4	no	—
117	C₂²×Dic₃	—	7	1	no	—
118	C₂²×Dic₃	—	7	1	no	—
119	C₂²×Dic₃	—	7	1	no	—
120	C₂²×Dic₃	—	14	4	no	—
121	C₂×Dic₆	—	7	4	no	—
122	C₂×Dic₆	—	14	4	no	—
123	C₃×C₂²⋊C₄	C₂×D₇	1	4	yes	—
124	#217	—	7	4	no	—
125	C₄⋊Dic₃	—	7	4	no	—
126	Dic₃⋊C₄	—	7	4	no	—
127	Dic₃⋊C₄	—	7	4	no	—
128	C₂×C₃×C₇	—	2	4	no	—
129	C₂×C₃×C₇	C₂²⋊Q₈	1	4	yes	complemented
130	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₃×C₇	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
133	C₂²×Q₈	—	21	1	no	—
134	C₂³×C₄	—	21	1	no	—
135	C₂²⋊Q₈	—	21	8	no	—
136	C₂×C₄⋊C₄	—	21	1	no	—
137	C₂×C₄⋊C₄	—	21	1	no	—
138	C₂×C₄⋊C₄	—	21	1	no	—
139	C₂×C₂²⋊C₄	—	21	1	no	—
140	C₂×C₂²⋊C₄	D₂₁	1	1	yes	characteristic
141	C₂²×C₇	—	2	8	no	—
142	C₂²×C₇	—	2	2	no	—
143	C₂²×C₇	—	2	2	no	—
144	C₂²×C₇	C₂×Dic₆	1	2	yes	—
145	C₂²×C₇	C₂×Dic₆	1	4	yes	complemented
146	C₂²×C₇	D₃×D₄	1	2	yes	—
147	C₂²×C₇	#237	1	2	yes	—
148	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
149	C₄×C₇	—	2	2	no	—
150	C₄×C₇	—	2	2	no	—
151	Dic₇	—	3	4	no	—
152	Dic₇	—	6	2	no	—
153	Dic₇	—	6	2	no	—
154	Dic₇	—	6	2	no	—
155	C₂³×C₃	—	2	2	no	—
156	C₂³×C₃	—	2	2	no	—
157	C₂³×C₃	Dic₁₄	1	4	yes	complemented
158	C₂³×C₃	C₂²×D₇	1	2	yes	—
159	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
160	C₂×C₃×C₄	—	2	2	no	—
161	C₂×C₃×C₄	—	2	2	no	—
162	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
163	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
164	C₂×Dic₃	—	7	4	no	—
165	C₂×Dic₃	—	7	2	no	—
166	C₂×Dic₃	—	7	2	no	—
167	C₂×Dic₃	—	7	2	no	—
168	C₂×Dic₃	—	7	2	no	—
169	C₂×Dic₃	—	14	8	no	—
170	C₂×Dic₃	—	14	2	no	—
171	C₂×Dic₃	—	14	2	no	—
172	C₂×Dic₃	—	14	2	no	—
173	Dic₆	—	14	8	no	—
174	C₃×C₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Hall
175	C₂⁴	C₂×D₂₁	1	1	yes	characteristic
176	C₂×Q₈	—	21	4	no	—
177	C₂×Q₈	—	42	4	no	—
178	C₂²×C₄	—	21	1	no	—
179	C₂²×C₄	—	21	4	no	—
180	C₂²×C₄	—	21	1	no	—
181	C₂²×C₄	—	21	1	no	—
182	C₂²×C₄	—	21	1	no	—
183	C₂²×C₄	—	42	4	no	—
184	C₂²×C₄	C₂×D₂₁	1	1	yes	characteristic
185	C₂²×C₄	C₂×D₂₁	1	1	yes	characteristic
186	C₄⋊C₄	—	21	4	no	—
187	C₄⋊C₄	—	21	4	no	—
188	C₄⋊C₄	—	21	4	no	—
189	C₂²⋊C₄	—	21	4	no	—
190	C₂²⋊C₄	C₂×D₂₁	1	4	yes	—
191	C₂×C₇	—	2	4	no	—
192	C₂×C₇	#878	1	4	yes	complemented
193	C₂×C₇	C₂²×Dic₆	1	1	yes	characteristic
194	C₂×C₇	C₂×D₃×D₄	1	1	yes	characteristic
195	C₂×C₇	C₂×#237	1	1	yes	characteristic
196	C₂²×C₃	—	2	8	no	—
197	C₂²×C₃	—	2	2	no	—
198	C₂²×C₃	—	2	2	no	—
199	C₂²×C₃	C₂×Dic₁₄	1	2	yes	—
200	C₂²×C₃	C₂×Dic₁₄	1	4	yes	complemented
201	C₂²×C₃	D₄×D₇	1	2	yes	—
202	C₂²×C₃	#1159	1	2	yes	—
203	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
204	C₃×C₄	—	2	2	no	—
205	C₃×C₄	—	2	2	no	—
206	Dic₃	—	7	4	no	—
207	Dic₃	—	14	2	no	—
208	Dic₃	—	14	2	no	—
209	Dic₃	—	14	2	no	—
210	C₂³	—	2	2	no	—
211	C₂³	—	2	2	no	—
212	C₂³	Dic₄₂	1	4	yes	complemented
213	C₂³	C₂²×D₂₁	1	2	yes	—
214	C₂³	C₂²×D₂₁	1	1	yes	characteristic
215	Q₈	—	42	8	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	21	4	no	—
219	C₂×C₄	—	21	2	no	—
220	C₂×C₄	—	21	2	no	—
221	C₂×C₄	—	21	2	no	—
222	C₂×C₄	—	21	2	no	—
223	C₂×C₄	—	42	8	no	—
224	C₂×C₄	—	42	2	no	—
225	C₂×C₄	—	42	2	no	—
226	C₂×C₄	—	42	2	no	—
227	C₂×C₄	C₂²×D₂₁	1	2	yes	—
228	C₂×C₄	C₂²×D₂₁	1	2	yes	—
229	C₇	C₂×#878	1	1	yes	characteristic, complemented, Sylow
230	C₂×C₃	—	2	4	no	—
231	C₂×C₃	#6441	1	4	yes	complemented
232	C₂×C₃	C₂²×Dic₁₄	1	1	yes	characteristic
233	C₂×C₃	C₂×D₄×D₇	1	1	yes	characteristic
234	C₂×C₃	C₂×#1159	1	1	yes	characteristic
235	C₂²	—	2	8	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	C₂×Dic₄₂	1	2	yes	—
239	C₂²	C₂×Dic₄₂	1	4	yes	complemented
240	C₂²	D₄×D₂₁	1	2	yes	—
241	C₂²	#66730	1	2	yes	—
242	C₂²	C₂³×D₂₁	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	2	2	no	—
245	C₄	—	21	4	no	—
246	C₄	—	42	2	no	—
247	C₄	—	42	2	no	—
248	C₄	—	42	2	no	—
249	C₃	C₂×#6441	1	1	yes	characteristic, complemented, Sylow
250	C₂	—	2	4	no	—
251	C₂	#10621361	1	4	yes	direct product factor
252	C₂	C₂²×Dic₄₂	1	1	yes	characteristic
253	C₂	C₂×D₄×D₂₁	1	1	yes	characteristic
254	C₂	C₂×#66730	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₂₁
commutator, cosocle	C₂²×C₃×C₇	C₂⁴
Frattini	C₂²	C₂³×D₂₁
Fitting	C₂×C₃×C₇×C₂²⋊C₄	C₂
socle	C₂³×C₃×C₇	C₂³
prime radical	C₂⁴×C₃×C₇	C₂²
hypercenter	C₂×C₂²⋊C₄	D₂₁
hypocenter	C₃×C₇	C₂×C₂²⋊Q₈
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×Dic₄₂	C₂	semidirect
5. C₂×C₄⋊Dic₂₁	C₂	semidirect
6. C₂×#66629	C₂	semidirect
7. C₂×#66629	C₂	semidirect
8. #10621361	C₂	direct (N × H)
20. C₂×Dic₄₂	C₂²	semidirect
23. C₄⋊Dic₂₁	C₂²	semidirect
24. #66629	C₂²	semidirect
25. #66629	C₂²	semidirect
37. C₂³×C₃×C₇	Q₈	semidirect
83. C₂²×C₃×C₇	C₂×Q₈	semidirect
95. C₂³×C₇	Dic₆	semidirect
129. C₂×C₃×C₇	C₂²⋊Q₈	semidirect
145. C₂²×C₇	C₂×Dic₆	semidirect
157. C₂³×C₃	Dic₁₄	semidirect
174. C₃×C₇	C₂×C₂²⋊Q₈	semidirect
192. C₂×C₇	#878	semidirect
200. C₂²×C₃	C₂×Dic₁₄	semidirect
212. C₂³	Dic₄₂	semidirect
229. C₇	C₂×#878	semidirect
231. C₂×C₃	#6441	semidirect
239. C₂²	C₂×Dic₄₂	semidirect
249. C₃	C₂×#6441	semidirect
251. C₂	#10621361	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Dic₂₁	2
C₂²×Dic₄₂	2
C₂×C₃×C₇×C₂²⋊C₄	2
C₂×#66636	2
C₂×C₄⋊Dic₂₁	2
C₂×#66629	2
#10621361	2
C₂×#6441	3
C₂×#878	7

Subgroups — C2×#10621361 #13061571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10621361 #13061571