Subgroups — #13054000

Order 1344.

468 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₂₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#66575	C₂	1	2	yes	maximal
5	#10620529	C₂	1	2	yes	maximal, complemented
6	#89759	—	3	1	no	maximal, Hall
7	C₂³×D₂₁	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂²×Dic₂₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₂₁	—	2	2	no	—
11	C₂×C₄×D₂₁	—	2	2	no	—
12	C₂×C₄×D₂₁	C₄	1	4	yes	complemented
13	C₂²×C₇×Dic₃	C₂²	1	2	yes	—
14	C₂²×C₃×Dic₇	C₂²	1	2	yes	—
15	C₇×C₄⋊Dic₃	—	2	2	no	—
16	C₃×C₄⋊Dic₇	—	2	2	no	—
17	#66575	—	2	4	no	—
18	C₂²×C₄×D₇	—	3	1	no	—
19	C₂×C₇×C₄⋊C₄	—	3	1	no	—
20	C₂×(C₂×D₇)⋊C₄	—	3	2	no	—
21	C₂×C₄⋊Dic₇	D₃	1	1	yes	characteristic, complemented
22	C₇⋊#95	—	3	2	no	—
23	#4987	—	7	1	no	maximal, Hall
24	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
25	C₂²×D₂₁	—	2	1	no	—
26	C₂²×D₂₁	—	2	1	no	—
27	C₂²×D₂₁	—	2	1	no	—
28	C₂²×D₂₁	—	2	1	no	—
29	C₂²×D₂₁	C₂×C₄	1	2	yes	—
30	C₂²×D₂₁	D₄	1	2	yes	—
31	C₂²×D₂₁	Q₈	1	2	yes	—
32	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
34	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
35	C₂×Dic₂₁	—	2	1	no	—
36	C₂×Dic₂₁	—	2	1	no	—
37	C₂×Dic₂₁	C₂×C₄	1	2	yes	—
38	C₄×D₂₁	—	2	8	no	—
39	C₂×C₇×Dic₃	—	2	2	no	—
40	C₂×C₇×Dic₃	—	2	2	no	—
41	C₂×C₇×Dic₃	—	2	2	no	—
42	C₂×C₃×Dic₇	—	2	2	no	—
43	C₂×C₃×Dic₇	—	2	2	no	—
44	C₂×C₃×Dic₇	—	2	2	no	—
45	C₂³×D₇	—	3	1	no	—
46	C₂²×C₄×C₇	—	3	2	no	—
47	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
48	C₂²×Dic₇	—	3	1	no	—
49	C₂²×Dic₇	C₂×D₃	1	2	yes	—
50	C₂×C₄×D₇	—	3	4	no	—
51	C₂×C₄×D₇	—	6	2	no	—
52	C₂×C₄×D₇	—	6	2	no	—
53	C₇×C₄⋊C₄	—	6	2	no	—
54	(C₂×D₇)⋊C₄	—	6	4	no	—
55	C₄⋊Dic₇	—	2	2	no	—
56	C₂²×C₄×D₃	—	7	1	no	—
57	C₂×C₃×C₄⋊C₄	—	7	1	no	—
58	C₂×(C₂×D₃)⋊C₄	—	7	2	no	—
59	C₂×C₄⋊Dic₃	D₇	1	1	yes	characteristic, complemented
60	C₃⋊#95	—	7	2	no	—
61	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₃×C₇	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₃×C₇	#41	1	1	yes	characteristic
67	C₂²×C₃×C₇	#41	1	1	yes	characteristic
68	C₂×D₂₁	—	2	2	no	—
69	C₂×D₂₁	—	2	2	no	—
70	C₂×D₂₁	—	2	2	no	—
71	C₂×D₂₁	—	2	2	no	—
72	C₂×D₂₁	—	2	2	no	—
73	C₂×D₂₁	—	2	2	no	—
74	C₂×D₂₁	C₄⋊C₄	1	4	yes	complemented
75	C₃×C₄×C₇	C₂²⋊C₄	1	4	yes	complemented
76	Dic₂₁	—	2	2	no	—
77	C₃×Dic₇	—	4	2	no	—
78	C₇×Dic₃	—	4	2	no	—
79	#380	—	21	1	no	Sylow
80	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
81	C₂²×D₇	—	3	2	no	—
82	C₂²×D₇	—	3	2	no	—
83	C₂²×D₇	—	3	2	no	—
84	C₂²×D₇	—	6	1	no	—
85	C₂²×D₇	—	6	1	no	—
86	C₂²×D₇	—	6	1	no	—
87	C₂²×D₇	—	6	1	no	—
88	C₂×C₄×C₇	—	6	2	no	—
89	C₂×C₄×C₇	—	6	2	no	—
90	C₂×C₄×C₇	—	6	2	no	—
91	C₂×C₄×C₇	C₄×D₃	1	2	yes	—
92	C₂×C₄×C₇	D₁₂	1	2	yes	—
93	C₂×C₄×C₇	#67	1	2	yes	—
94	C₂×Dic₇	—	2	2	no	—
95	C₂×Dic₇	—	2	2	no	—
96	C₂×Dic₇	—	2	2	no	—
97	C₂×Dic₇	—	3	2	no	—
98	C₂×Dic₇	—	6	1	no	—
99	C₂×Dic₇	—	6	1	no	—
100	C₄×D₇	—	6	8	no	—
101	C₂³×D₃	—	7	1	no	—
102	C₂²×C₃×C₄	—	7	2	no	—
103	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
104	C₂²×Dic₃	—	7	1	no	—
105	C₂²×Dic₃	C₂×D₇	1	2	yes	—
106	C₂×C₄×D₃	—	7	4	no	—
107	C₂×C₄×D₃	—	14	2	no	—
108	C₂×C₄×D₃	—	14	2	no	—
109	C₃×C₄⋊C₄	—	14	2	no	—
110	(C₂×D₃)⋊C₄	—	14	4	no	—
111	C₄⋊Dic₃	—	2	2	no	—
112	C₂×C₃×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₃×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₃×C₇	#117	1	1	yes	characteristic
115	C₂×C₃×C₇	#121	1	1	yes	characteristic
116	C₂×C₃×C₇	#121	1	1	yes	characteristic
117	C₂×C₃×C₇	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₃×C₇	C₂²⋊Q₈	1	1	yes	characteristic
119	D₂₁	—	2	4	no	—
120	C₂³×C₄	—	21	1	no	—
121	C₂×C₄⋊C₄	—	3	1	no	—
122	C₂×C₄⋊C₄	—	7	1	no	—
123	C₂×C₂²⋊C₄	—	21	2	no	—
124	#95	—	21	2	no	—
125	C₂²×C₇	C₂×C₄×D₃	1	1	yes	characteristic
126	C₂²×C₇	C₂×D₁₂	1	1	yes	characteristic
127	C₂²×C₇	D₃×D₄	1	1	yes	characteristic
128	C₂²×C₇	#237	1	1	yes	characteristic
129	C₂²×C₇	D₃×Q₈	1	1	yes	characteristic
130	C₂²×C₇	#239	1	1	yes	characteristic
131	C₂²×C₇	C₂×#67	1	1	yes	characteristic
132	C₂×D₇	—	3	4	no	—
133	C₂×D₇	—	6	2	no	—
134	C₂×D₇	—	6	2	no	—
135	C₂×D₇	—	6	2	no	—
136	C₂×D₇	—	6	2	no	—
137	C₂×D₇	—	6	2	no	—
138	C₂×D₇	—	6	2	no	—
139	C₄×C₇	—	12	2	no	—
140	C₄×C₇	(C₂×D₃)⋊C₄	1	4	yes	complemented
141	Dic₇	—	4	2	no	—
142	Dic₇	—	6	2	no	—
143	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
144	C₂²×D₃	—	7	2	no	—
145	C₂²×D₃	—	7	2	no	—
146	C₂²×D₃	—	7	2	no	—
147	C₂²×D₃	—	14	1	no	—
148	C₂²×D₃	—	14	1	no	—
149	C₂²×D₃	—	14	1	no	—
150	C₂²×D₃	—	14	1	no	—
151	C₂×C₃×C₄	—	14	2	no	—
152	C₂×C₃×C₄	—	14	2	no	—
153	C₂×C₃×C₄	—	14	2	no	—
154	C₂×C₃×C₄	C₄×D₇	1	2	yes	—
155	C₂×C₃×C₄	D₂₈	1	2	yes	—
156	C₂×C₃×C₄	C₂²⋊D₇	1	2	yes	—
157	C₂×Dic₃	—	2	2	no	—
158	C₂×Dic₃	—	2	2	no	—
159	C₂×Dic₃	—	2	2	no	—
160	C₂×Dic₃	—	7	2	no	—
161	C₂×Dic₃	—	14	1	no	—
162	C₂×Dic₃	—	14	1	no	—
163	C₄×D₃	—	14	8	no	—
164	C₃×C₇	#380	1	1	yes	characteristic, complemented, Hall
165	C₂⁴	—	21	1	no	—
166	C₂²×C₄	—	3	2	no	—
167	C₂²×C₄	—	7	2	no	—
168	C₂²×C₄	—	21	1	no	—
169	C₂²×C₄	—	21	4	no	—
170	C₂²×C₄	—	42	2	no	—
171	C₂²×C₄	—	42	2	no	—
172	C₂²×C₄	D₃×D₇	1	1	yes	characteristic
173	C₄⋊C₄	—	6	2	no	—
174	C₄⋊C₄	—	14	2	no	—
175	C₂²⋊C₄	—	42	4	no	—
176	C₂×C₇	D₃×C₄⋊C₄	1	1	yes	characteristic
177	C₂×C₇	#892	1	1	yes	characteristic
178	C₂×C₇	#895	1	1	yes	characteristic
179	C₂×C₇	#897	1	1	yes	characteristic
180	C₂×C₇	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
181	C₂×C₇	#938	1	1	yes	characteristic
182	C₂×C₇	#946	1	1	yes	characteristic
183	D₇	—	6	4	no	—
184	C₂²×C₃	C₂×C₄×D₇	1	1	yes	characteristic
185	C₂²×C₃	C₂×D₂₈	1	1	yes	characteristic
186	C₂²×C₃	D₄×D₇	1	1	yes	characteristic
187	C₂²×C₃	#1159	1	1	yes	characteristic
188	C₂²×C₃	Q₈×D₇	1	1	yes	characteristic
189	C₂²×C₃	#1161	1	1	yes	characteristic
190	C₂²×C₃	C₂×C₂²⋊D₇	1	1	yes	characteristic
191	C₂×D₃	—	7	4	no	—
192	C₂×D₃	—	14	2	no	—
193	C₂×D₃	—	14	2	no	—
194	C₂×D₃	—	14	2	no	—
195	C₂×D₃	—	14	2	no	—
196	C₂×D₃	—	14	2	no	—
197	C₂×D₃	—	14	2	no	—
198	C₃×C₄	—	28	2	no	—
199	C₃×C₄	(C₂×D₇)⋊C₄	1	4	yes	complemented
200	Dic₃	—	4	2	no	—
201	Dic₃	—	14	2	no	—
202	C₂³	—	21	2	no	—
203	C₂³	—	21	2	no	—
204	C₂³	—	21	2	no	—
205	C₂³	—	42	1	no	—
206	C₂³	—	42	1	no	—
207	C₂³	—	42	1	no	—
208	C₂³	—	42	1	no	—
209	C₂³	C₂×D₃×D₇	1	1	yes	characteristic
210	C₂×C₄	—	6	2	no	—
211	C₂×C₄	—	6	2	no	—
212	C₂×C₄	—	6	2	no	—
213	C₂×C₄	—	14	2	no	—
214	C₂×C₄	—	14	2	no	—
215	C₂×C₄	—	14	2	no	—
216	C₂×C₄	—	21	2	no	—
217	C₂×C₄	—	42	1	no	—
218	C₂×C₄	—	42	1	no	—
219	C₂×C₄	—	42	8	no	—
220	C₂×C₄	D₂₁⋊C₄	1	2	yes	—
221	C₂×C₄	#4246	1	2	yes	—
222	C₂×C₄	#4247	1	2	yes	—
223	C₇	#4987	1	1	yes	characteristic, complemented, Sylow
224	C₂×C₃	D₇×C₄⋊C₄	1	1	yes	characteristic
225	C₂×C₃	#6455	1	1	yes	characteristic
226	C₂×C₃	#6458	1	1	yes	characteristic
227	C₂×C₃	#6460	1	1	yes	characteristic
228	C₂×C₃	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
229	C₂×C₃	#6501	1	1	yes	characteristic
230	C₂×C₃	#6509	1	1	yes	characteristic
231	D₃	—	14	4	no	—
232	C₂²	—	21	4	no	—
233	C₂²	—	42	2	no	—
234	C₂²	—	42	2	no	—
235	C₂²	—	42	2	no	—
236	C₂²	—	42	2	no	—
237	C₂²	—	42	2	no	—
238	C₂²	—	42	2	no	—
239	C₂²	#66670	1	1	yes	characteristic
240	C₂²	#66672	1	1	yes	characteristic
241	C₂²	D₂₁⋊Q₈	1	1	yes	characteristic
242	C₂²	#66681	1	1	yes	characteristic
243	C₂²	C₂×D₂₁⋊C₄	1	1	yes	characteristic
244	C₂²	C₂×#4246	1	1	yes	characteristic
245	C₂²	C₂×#4247	1	1	yes	characteristic
246	C₄	—	12	2	no	—
247	C₄	—	28	2	no	—
248	C₄	—	42	2	no	—
249	C₄	#66575	1	4	yes	complemented
250	C₃	#89759	1	1	yes	characteristic, complemented, Sylow
251	C₂	—	42	4	no	—
252	C₂	#10620909	1	1	yes	characteristic
253	C₂	#10620945	1	1	yes	characteristic
254	C₂	#10620952	1	1	yes	characteristic
255	C₂	#10620999	1	1	yes	characteristic
256	C₂	#10621027	1	1	yes	characteristic
257	C₂	#10621028	1	1	yes	characteristic
258	C₂	C₂×#66575	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂×D₃×D₇
commutator	C₂²×C₃×C₇	C₂²×C₄
Fitting	C₂²×C₃×C₄×C₇	C₂²
socle, cosocle	C₂³×C₃×C₇	C₂³
prime radical	C₂³×D₂₁	C₂²
hypercenter	C₂²×C₄	D₃×D₇
hypocenter	C₃×C₇	#380
LCS	G ▷ C₂²×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
2. C₂×C₇×C₄⋊Dic₃	C₂	semidirect
3. C₂×C₃×C₄⋊Dic₇	C₂	semidirect
5. #10620529	C₂	semidirect
12. C₂×C₄×D₂₁	C₄	semidirect
21. C₂×C₄⋊Dic₇	D₃	semidirect
59. C₂×C₄⋊Dic₃	D₇	semidirect
74. C₂×D₂₁	C₄⋊C₄	semidirect
75. C₃×C₄×C₇	C₂²⋊C₄	semidirect
140. C₄×C₇	(C₂×D₃)⋊C₄	semidirect
164. C₃×C₇	#380	semidirect
199. C₃×C₄	(C₂×D₇)⋊C₄	semidirect
223. C₇	#4987	semidirect
249. C₄	#66575	semidirect
250. C₃	#89759	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₂₁	2
C₂×C₇×C₄⋊Dic₃	2
C₂×C₃×C₄⋊Dic₇	2
C₂×#66575	2
#10620529	2
#89759	3
#4987	7