Subgroups — #13271972

Order 1792.

466 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#436	C₂	1	1	yes	maximal, characteristic
2	C₂×#89348	C₂	1	4	yes	maximal, complemented
3	#11727497	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#6386	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
6	C₂×C₄×C₇×D₄	C₄	1	1	yes	characteristic
7	C₇×#436	—	2	1	no	—
8	C₇×#436	—	2	1	no	—
9	C₇×#436	—	2	1	no	—
10	C₇×#436	—	2	1	no	—
11	C₂×C₇×C₄⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₇×C₂²⋊C₈	C₄	1	2	yes	—
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₂×#6386	C₂²	1	2	yes	—
15	C₂×#6386	C₂²	1	2	yes	—
16	#89348	—	2	8	no	—
17	C₄⋊(C₇⋊C₁₆)	—	2	1	no	—
18	C₄⋊(C₇⋊C₁₆)	—	2	1	no	—
19	#89256	—	2	1	no	—
20	C₄×C₇⋊C₁₆	—	2	1	no	—
21	#13194	—	7	1	no	maximal, Sylow
22	C₂²×C₇×D₄	C₈	1	1	yes	characteristic
23	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
24	C₂×C₇×M₁₆	—	2	4	no	—
25	C₂×C₇×M₁₆	—	2	2	no	—
26	C₂×C₇×M₁₆	—	2	2	no	—
27	C₂×C₇×M₁₆	—	2	4	no	—
28	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
29	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
31	C₄×C₇×D₄	—	2	1	no	—
32	C₄×C₇×D₄	—	2	1	no	—
33	C₄×C₇×D₄	—	2	1	no	—
34	C₄×C₇×D₄	—	2	1	no	—
35	C₂×C₇×C₄⋊C₄	C₈	1	1	yes	characteristic
36	C₂×C₇×C₂²⋊C₄	C₈	1	2	yes	—
37	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
38	C₇×C₄⋊C₈	—	2	1	no	—
39	C₇×C₄⋊C₈	—	2	1	no	—
40	C₇×C₂²⋊C₈	—	2	2	no	—
41	C₇×C₂²⋊C₈	—	2	2	no	—
42	C₄×C₇×C₈	D₄	1	1	yes	characteristic
43	C₄×C₇×C₈	D₄	1	1	yes	characteristic
44	C₄×C₇×C₈	D₄	1	1	yes	characteristic
45	C₄×C₇×C₈	D₄	1	1	yes	characteristic
46	#6386	—	2	2	no	—
47	#6386	—	2	4	no	—
48	#6386	D₄	1	4	yes	complemented
49	C₂×C₇⋊C₁₆	—	2	2	no	—
50	C₂×C₇⋊C₁₆	—	2	2	no	—
51	C₂×#436	D₇	1	1	yes	characteristic
52	#2150	—	7	1	no	—
53	C₂×#349	—	7	4	no	—
54	C₄×M₃₂	—	7	1	no	—
55	C₂⁴×C₇	C₂×C₈	1	2	yes	—
56	C₂×C₇×D₄	—	2	1	no	—
57	C₂×C₇×D₄	—	2	1	no	—
58	C₂×C₇×D₄	—	4	2	no	—
59	C₂²×C₄×C₇	—	2	2	no	—
60	C₂²×C₄×C₇	—	2	4	no	—
61	C₂²×C₄×C₇	—	2	2	no	—
62	C₂²×C₄×C₇	—	2	4	no	—
63	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
64	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
67	C₇×M₁₆	—	4	4	no	—
68	C₇×M₁₆	—	4	2	no	—
69	C₇×M₁₆	—	4	2	no	—
70	C₂×C₇×C₈	—	2	2	no	—
71	C₂×C₇×C₈	—	2	2	no	—
72	C₂×C₇×C₈	—	2	4	no	—
73	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
77	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
78	C₂×C₇×C₈	#41	1	2	yes	—
79	C₇×C₄⋊C₄	—	2	1	no	—
80	C₇×C₄⋊C₄	—	2	1	no	—
81	C₇×C₂²⋊C₄	—	2	2	no	—
82	C₇×C₂²⋊C₄	—	2	2	no	—
83	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
84	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
85	C₄²×C₇	M₁₆	1	1	yes	characteristic
86	C₄²×C₇	M₁₆	1	1	yes	characteristic
87	C₇⋊C₁₆	—	2	4	no	—
88	C₇⋊C₁₆	—	4	2	no	—
89	C₂²×M₁₆	C₂×D₇	1	2	yes	—
90	C₂×C₄×D₄	Dic₇	1	1	yes	characteristic
91	C₂×M₃₂	—	7	2	no	—
92	C₂×M₃₂	—	7	2	no	—
93	#436	—	2	1	no	—
94	#436	—	2	1	no	—
95	#436	—	2	1	no	—
96	#436	—	2	1	no	—
97	C₂×C₄⋊C₈	Dic₇	1	1	yes	characteristic
98	C₂×C₂²⋊C₈	Dic₇	1	2	yes	—
99	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
100	C₄⋊C₁₆	—	14	1	no	—
101	C₄⋊C₁₆	—	14	1	no	—
102	#349	—	14	8	no	—
103	#346	—	14	1	no	—
104	C₄×C₁₆	—	14	1	no	—
105	C₂³×C₇	—	2	2	no	—
106	C₂³×C₇	—	2	4	no	—
107	C₂³×C₇	—	4	2	no	—
108	C₂³×C₇	—	4	2	no	—
109	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
110	C₇×D₄	—	4	2	no	—
111	C₇×D₄	—	4	2	no	—
112	C₂×C₄×C₇	—	2	2	no	—
113	C₂×C₄×C₇	—	2	2	no	—
114	C₂×C₄×C₇	—	2	4	no	—
115	C₂×C₄×C₇	—	4	1	no	—
116	C₂×C₄×C₇	—	4	2	no	—
117	C₂×C₄×C₇	—	4	2	no	—
118	C₂×C₄×C₇	—	4	4	no	—
119	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
120	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
121	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
122	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
123	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
126	C₂×C₄×C₇	#131	1	2	yes	—
127	C₇×C₈	—	2	1	no	—
128	C₇×C₈	—	2	1	no	—
129	C₇×C₈	—	4	2	no	—
130	C₇×C₈	#121	1	2	yes	—
131	C₇×C₈	#124	1	1	yes	characteristic
132	C₇×C₈	#127	1	1	yes	characteristic
133	C₂²×D₄	C₇⋊C₈	1	1	yes	characteristic
134	C₂³×C₄	C₂×Dic₇	1	2	yes	—
135	C₂×M₁₆	—	2	4	no	—
136	C₂×M₁₆	—	2	4	no	—
137	C₂×M₁₆	—	2	2	no	—
138	C₂×M₁₆	—	2	2	no	—
139	C₂²×C₈	C₂×Dic₇	1	2	yes	—
140	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
141	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
142	C₄×D₄	—	2	1	no	—
143	C₄×D₄	—	2	1	no	—
144	C₄×D₄	—	2	1	no	—
145	C₄×D₄	—	2	1	no	—
146	C₂×C₄⋊C₄	C₇⋊C₈	1	1	yes	characteristic
147	C₂×C₂²⋊C₄	C₇⋊C₈	1	2	yes	—
148	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
149	M₃₂	—	7	4	no	—
150	M₃₂	—	14	2	no	—
151	M₃₂	—	14	4	no	—
152	C₂×C₁₆	—	14	2	no	—
153	C₂×C₁₆	—	14	2	no	—
154	C₄⋊C₈	—	2	1	no	—
155	C₄⋊C₈	—	2	1	no	—
156	C₂²⋊C₈	—	2	2	no	—
157	C₂²⋊C₈	—	2	2	no	—
158	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
159	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
160	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
162	C₂²×C₇	—	2	1	no	—
163	C₂²×C₇	—	2	1	no	—
164	C₂²×C₇	—	4	2	no	—
165	C₂²×C₇	—	4	2	no	—
166	C₂²×C₇	—	4	2	no	—
167	C₂²×C₇	—	8	4	no	—
168	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
169	C₂²×C₇	C₈×D₄	1	2	yes	—
170	C₄×C₇	—	2	1	no	—
171	C₄×C₇	—	2	1	no	—
172	C₄×C₇	—	2	1	no	—
173	C₄×C₇	—	4	2	no	—
174	C₄×C₇	#349	1	2	yes	complemented
175	C₄×C₇	#390	1	1	yes	characteristic
176	C₄×C₇	#408	1	1	yes	characteristic
177	C₄×C₇	#436	1	2	yes	—
178	C₂⁴	C₂×C₇⋊C₈	1	2	yes	—
179	C₂×D₄	—	2	1	no	—
180	C₂×D₄	—	2	1	no	—
181	C₂×D₄	—	4	2	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	C₂×C₇⋊C₈	1	2	yes	—
187	C₂²×C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
189	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
190	M₁₆	—	4	2	no	—
191	M₁₆	—	4	4	no	—
192	M₁₆	—	4	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
197	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
198	C₂×C₈	D₄×D₇	1	2	yes	—
199	C₂×C₈	#1159	1	2	yes	—
200	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
201	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
202	C₄⋊C₄	—	2	1	no	—
203	C₄⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	2	2	no	—
205	C₂²⋊C₄	—	2	2	no	—
206	C₄²	#1136	1	1	yes	characteristic
207	C₄²	#1136	1	1	yes	characteristic
208	C₄²	C₂²⋊Dic₇	1	1	yes	characteristic
209	C₄²	C₂²⋊Dic₇	1	1	yes	characteristic
210	C₁₆	—	14	4	no	—
211	C₁₆	—	28	2	no	—
212	C₂×C₇	—	2	2	no	—
213	C₂×C₇	—	8	2	no	—
214	C₂×C₇	#1954	1	1	yes	characteristic
215	C₂×C₇	C₂×#349	1	1	yes	characteristic
216	C₂×C₇	#2115	1	1	yes	characteristic
217	C₂³	—	2	2	no	—
218	C₂³	—	2	4	no	—
219	C₂³	—	4	2	no	—
220	C₂³	—	4	2	no	—
221	C₂³	C₂²×C₇⋊C₈	1	1	yes	characteristic
222	D₄	—	4	2	no	—
223	D₄	—	4	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
232	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
233	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
234	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
235	C₂×C₄	C₂×#1136	1	1	yes	characteristic
236	C₂×C₄	D₄×Dic₇	1	2	yes	—
237	C₂×C₄	#6511	1	2	yes	—
238	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	4	2	no	—
242	C₈	#6499	1	1	yes	characteristic
243	C₈	#6501	1	2	yes	—
244	C₈	#6503	1	1	yes	characteristic
245	C₇	#13194	1	1	yes	characteristic, complemented, Sylow
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	4	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	8	4	no	—
252	C₂²	D₄×C₇⋊C₈	1	2	yes	—
253	C₂²	C₂×C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	4	2	no	—
258	C₄	#89348	1	2	yes	complemented
259	C₄	#89784	1	2	yes	—
260	C₄	#89993	1	1	yes	characteristic
261	C₄	#90006	1	1	yes	characteristic
262	C₂	—	2	2	no	—
263	C₂	—	8	2	no	—
264	C₂	#11729368	1	1	yes	characteristic
265	C₂	C₂×#89348	1	1	yes	characteristic
266	C₂	#11729761	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₇
commutator	C₂³×C₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₂×C₇×#436	C₂
socle	C₂²×C₇	C₂×C₂²⋊C₈
prime radical	C₂²×C₇×D₄	C₈
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂×#436	D₇
hypocenter	C₇	#13194
LCS	G ▷ C₂³×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#436

Complements

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

N	H ≅ G/N	type
2. C₂×#89348	C₂	semidirect
3. #11727497	C₂	semidirect
4. C₄×#6386	C₂	semidirect
48. #6386	D₄	semidirect
174. C₄×C₇	#349	semidirect
245. C₇	#13194	semidirect
258. C₄	#89348	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#436	2
C₂×#89348	2
#11727497	2
C₄×#6386	2
#13194	7