Subgroups — #13272057

Order 1792.

574 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×#89358	C₂	1	4	yes	maximal, complemented
3	#11727546	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#6386	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₇×D₈	—	2	1	no	—
6	C₂²×C₇×D₈	C₂²	1	2	yes	—
7	C₂×C₇×#127	—	2	1	no	—
8	C₇×#493	—	2	1	no	—
9	C₇×#493	—	2	1	no	—
10	C₇×#493	C₄	1	2	yes	—
11	C₇×#493	C₄	1	2	yes	—
12	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
13	C₂×#6393	—	2	1	no	—
14	C₂×#6393	C₂²	1	2	yes	—
15	C₂×#6386	C₂²	1	2	yes	—
16	#89696	—	2	1	no	—
17	#89358	—	2	8	no	—
18	#89256	—	2	1	no	—
19	C₄×C₇⋊C₁₆	—	2	1	no	—
20	#13213	—	7	1	no	maximal, Sylow
21	C₂²×C₇×D₄	—	2	2	no	—
22	C₂²×C₇×D₄	—	4	2	no	—
23	C₂×C₇×D₈	—	2	2	no	—
24	C₂×C₇×D₈	—	2	2	no	—
25	C₂×C₇×D₈	—	2	4	no	—
26	C₂×C₇×D₈	—	2	2	no	—
27	C₂×C₇×D₈	—	2	4	no	—
28	C₂×C₇×D₈	—	4	2	no	—
29	C₂×C₇×D₈	—	4	2	no	—
30	C₂×C₇×D₈	C₂×C₄	1	4	yes	—
31	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
33	C₇×#127	—	2	2	no	—
34	C₇×#127	—	2	2	no	—
35	C₇×#127	—	4	2	no	—
36	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
37	C₂×#1136	—	2	2	no	—
38	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
39	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
40	C₄×C₇×C₈	D₄	1	1	yes	characteristic
41	C₄×C₇×C₈	D₄	1	1	yes	characteristic
42	#6393	—	2	4	no	—
43	#6393	—	4	2	no	—
44	#6386	—	2	2	no	—
45	#6386	D₄	1	4	yes	complemented
46	C₂×C₇⋊C₁₆	—	2	2	no	—
47	C₄⋊(C₇⋊C₈)	—	4	2	no	—
48	C₂×#493	D₇	1	1	yes	characteristic
49	C₂×#361	—	7	4	no	—
50	C₄×M₃₂	—	7	1	no	—
51	#1848	—	7	1	no	—
52	C₂⁴×C₇	—	4	2	no	—
53	C₂×C₇×D₄	—	2	4	no	—
54	C₂×C₇×D₄	—	4	4	no	—
55	C₂×C₇×D₄	—	4	4	no	—
56	C₂×C₇×D₄	—	4	2	no	—
57	C₂×C₇×D₄	—	4	2	no	—
58	C₂×C₇×D₄	—	4	2	no	—
59	C₂×C₇×D₄	—	4	2	no	—
60	C₂×C₇×D₄	—	4	2	no	—
61	C₂×C₇×D₄	—	8	8	no	—
62	C₂²×C₄×C₇	SD₁₆	1	2	yes	—
63	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
64	C₇×D₈	—	2	8	no	—
65	C₇×D₈	—	4	4	no	—
66	C₇×D₈	—	4	4	no	—
67	C₇×D₈	—	4	4	no	—
68	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₇×C₈	D₈	1	2	yes	—
71	C₂×C₇×C₈	SD₁₆	1	2	yes	—
72	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
74	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
75	C₂×C₇×C₈	#41	1	2	yes	—
76	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₄²×C₇	D₈	1	2	yes	—
79	#1136	—	4	2	no	—
80	#1136	—	4	2	no	—
81	C₂×C₇⋊C₈	—	4	2	no	—
82	C₇⋊C₁₆	—	2	4	no	—
83	C₂²×D₈	—	2	1	no	—
84	C₂²×D₈	C₂×D₇	1	2	yes	—
85	C₂×#127	—	2	1	no	—
86	C₂×M₃₂	—	7	2	no	—
87	#493	—	2	1	no	—
88	#493	—	2	1	no	—
89	#493	Dic₇	1	2	yes	—
90	#493	Dic₇	1	2	yes	—
91	C₂×#108	—	7	2	no	—
92	C₂×#108	—	14	1	no	—
93	#423	—	14	1	no	—
94	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
95	#361	—	14	8	no	—
96	#346	—	14	1	no	—
97	C₄×C₁₆	—	14	1	no	—
98	C₂³×C₇	—	4	4	no	—
99	C₂³×C₇	—	8	8	no	—
100	C₂³×C₇	—	8	2	no	—
101	C₂³×C₇	—	8	2	no	—
102	C₂³×C₇	C₂×SD₁₆	1	1	yes	characteristic
103	C₇×D₄	—	4	4	no	—
104	C₇×D₄	—	4	4	no	—
105	C₇×D₄	—	8	8	no	—
106	C₇×D₄	—	8	8	no	—
107	C₇×D₄	—	8	2	no	—
108	C₇×D₄	—	8	2	no	—
109	C₂×C₄×C₇	—	2	4	no	—
110	C₂×C₄×C₇	#102	1	2	yes	—
111	C₂×C₄×C₇	#102	1	2	yes	—
112	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₇	#124	1	1	yes	characteristic
114	C₂×C₄×C₇	#127	1	1	yes	characteristic
115	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
118	C₇×C₈	#102	1	2	yes	—
119	C₇×C₈	#102	1	2	yes	—
120	C₇×C₈	C₄×D₄	1	2	yes	—
121	C₇×C₈	#121	1	2	yes	—
122	C₇⋊C₈	—	8	2	no	—
123	C₂²×D₄	—	2	2	no	—
124	C₂²×D₄	—	4	2	no	—
125	C₂×D₈	—	2	2	no	—
126	C₂×D₈	—	2	2	no	—
127	C₂×D₈	—	2	4	no	—
128	C₂×D₈	—	2	4	no	—
129	C₂×D₈	—	2	2	no	—
130	C₂×D₈	—	4	2	no	—
131	C₂×D₈	—	4	2	no	—
132	C₂×D₈	C₂×Dic₇	1	4	yes	—
133	C₂×M₁₆	—	14	2	no	—
134	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
136	#127	—	2	2	no	—
137	#127	—	2	2	no	—
138	#127	—	4	2	no	—
139	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
140	M₃₂	—	7	4	no	—
141	M₃₂	—	14	2	no	—
142	C₂×C₁₆	—	14	2	no	—
143	#108	—	14	4	no	—
144	#108	—	28	2	no	—
145	C₄⋊C₈	—	28	2	no	—
146	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
147	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
148	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
149	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
150	C₂²×C₇	—	2	2	no	—
151	C₂²×C₇	—	8	8	no	—
152	C₂²×C₇	—	8	4	no	—
153	C₂²×C₇	—	16	4	no	—
154	C₂²×C₇	—	16	4	no	—
155	C₂²×C₇	C₂×#102	1	1	yes	characteristic
156	C₂²×C₇	#486	1	1	yes	characteristic
157	C₂²×C₇	#492	1	1	yes	characteristic
158	C₄×C₇	—	2	2	no	—
159	C₄×C₇	#361	1	4	yes	complemented
160	C₄×C₇	#390	1	1	yes	characteristic
161	C₄×C₇	C₂×#102	1	1	yes	characteristic
162	C₄×C₇	#486	1	1	yes	characteristic
163	C₄×C₇	#493	1	1	yes	characteristic
164	C₂⁴	—	4	2	no	—
165	C₂×D₄	—	2	4	no	—
166	C₂×D₄	—	4	4	no	—
167	C₂×D₄	—	4	4	no	—
168	C₂×D₄	—	4	2	no	—
169	C₂×D₄	—	4	2	no	—
170	C₂×D₄	—	4	2	no	—
171	C₂×D₄	—	4	2	no	—
172	C₂×D₄	—	4	2	no	—
173	C₂×D₄	—	8	8	no	—
174	C₂²×C₄	#1142	1	2	yes	—
175	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
176	D₈	—	2	8	no	—
177	D₈	—	4	4	no	—
178	D₈	—	4	4	no	—
179	D₈	—	4	4	no	—
180	M₁₆	—	28	2	no	—
181	M₁₆	—	28	2	no	—
182	C₂×C₈	—	28	2	no	—
183	C₂×C₈	#1141	1	2	yes	—
184	C₂×C₈	#1142	1	2	yes	—
185	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
186	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
187	C₂×C₈	D₄×D₇	1	2	yes	—
188	C₂×C₈	#1159	1	2	yes	—
189	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
190	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
191	C₄²	#1141	1	2	yes	—
192	C₄²	C₂²⋊Dic₇	1	1	yes	characteristic
193	C₄²	C₂²⋊Dic₇	1	1	yes	characteristic
194	C₁₆	—	14	4	no	—
195	C₂×C₇	—	2	1	no	—
196	C₂×C₇	—	2	1	no	—
197	C₂×C₇	—	16	4	no	—
198	C₂×C₇	#1957	1	1	yes	characteristic
199	C₂×C₇	C₂×#361	1	2	yes	—
200	C₂³	—	4	4	no	—
201	C₂³	—	8	8	no	—
202	C₂³	—	8	2	no	—
203	C₂³	—	8	2	no	—
204	C₂³	C₂×#1142	1	1	yes	characteristic
205	D₄	—	4	4	no	—
206	D₄	—	4	4	no	—
207	D₄	—	8	8	no	—
208	D₄	—	8	8	no	—
209	D₄	—	8	2	no	—
210	D₄	—	8	2	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	#6406	1	2	yes	—
213	C₂×C₄	#6406	1	2	yes	—
214	C₂×C₄	C₂×#1141	1	1	yes	characteristic
215	C₂×C₄	C₂×#1141	1	1	yes	characteristic
216	C₂×C₄	C₂×#1142	1	1	yes	characteristic
217	C₂×C₄	#6499	1	1	yes	characteristic
218	C₂×C₄	#6503	1	1	yes	characteristic
219	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
220	C₈	—	56	2	no	—
221	C₈	#6406	1	2	yes	—
222	C₈	#6406	1	2	yes	—
223	C₈	D₄×Dic₇	1	2	yes	—
224	C₈	#6501	1	2	yes	—
225	C₇	#13213	1	1	yes	characteristic, complemented, Sylow
226	C₂²	—	2	2	no	—
227	C₂²	—	8	4	no	—
228	C₂²	—	8	8	no	—
229	C₂²	—	16	4	no	—
230	C₂²	—	16	4	no	—
231	C₂²	#89842	1	1	yes	characteristic
232	C₂²	#89848	1	1	yes	characteristic
233	C₂²	C₂×#6406	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	#89358	1	4	yes	complemented
236	C₄	#89842	1	1	yes	characteristic
237	C₄	#89845	1	1	yes	characteristic
238	C₄	C₂×#6406	1	1	yes	characteristic
239	C₄	#89993	1	1	yes	characteristic
240	C₂	—	2	1	no	—
241	C₂	—	2	1	no	—
242	C₂	—	16	4	no	—
243	C₂	#11729518	1	1	yes	characteristic
244	C₂	C₂×#89358	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#6406
commutator	C₂×C₇×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇
Fitting, prime radical	C₂×C₇×#493	C₂
socle	C₂²×C₇	C₂×#102
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂×#493	D₇
hypocenter	C₇	#13213
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
2. C₂×#89358	C₂	semidirect
3. #11727546	C₂	semidirect
4. C₄×#6386	C₂	semidirect
45. #6386	D₄	semidirect
159. C₄×C₇	#361	semidirect
225. C₇	#13213	semidirect
235. C₄	#89358	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#493	2
C₂×#89358	2
#11727546	2
C₄×#6386	2
#13213	7