Subgroups — #13259442

Order 1792.

642 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×#89351	C₂	1	2	yes	maximal
3	C₂×#89347	C₂	1	1	yes	maximal, characteristic
4	#11729723	C₂	1	1	yes	maximal, characteristic
5	C₂×#89326	C₂	1	2	yes	maximal
6	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
7	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₇×#117	C₂²	1	1	yes	characteristic
9	C₂×#6489	C₂²	1	2	yes	—
10	C₂²×#1136	C₂²	1	2	yes	—
11	C₇×#405	—	2	4	no	—
12	C₂×C₇×#97	C₄	1	2	yes	complemented
13	C₂×C₄×C₇×C₈	C₄	1	2	yes	complemented
14	#89351	C₄	1	8	yes	complemented
15	#89347	—	2	2	no	—
16	#89346	—	2	2	no	—
17	#89326	—	2	4	no	—
18	#10804	—	7	1	no	maximal, Sylow
19	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
20	C₂×C₇×M₁₆	—	2	2	no	—
21	C₂×C₇×M₁₆	—	2	1	no	—
22	C₂×C₇×M₁₆	—	2	1	no	—
23	C₂×C₇×M₁₆	C₂×C₄	1	4	yes	—
24	C₂²×C₇×C₈	—	2	2	no	—
25	C₂²×C₇×C₈	—	2	2	no	—
26	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
27	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₇×C₈	D₄	1	2	yes	—
30	C₂²×C₇×C₈	Q₈	1	2	yes	—
31	C₇×#117	—	2	2	no	—
32	C₇×#117	D₄	1	4	yes	—
33	C₂×C₇×C₄⋊C₄	—	2	1	no	—
34	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
35	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
36	C₂×C₂²⋊Dic₇	—	2	2	no	—
37	#6489	—	2	2	no	—
38	#6489	—	2	2	no	—
39	#6489	—	2	2	no	—
40	#6489	C₂×C₄	1	4	yes	—
41	C₂×C₄⋊Dic₇	—	2	2	no	—
42	C₂×C₄×Dic₇	—	2	2	no	—
43	C₂×#1136	—	2	2	no	—
44	C₂×#1136	—	2	2	no	—
45	C₂×#1136	—	2	2	no	—
46	C₂×#1136	—	2	2	no	—
47	C₂×#1136	—	2	2	no	—
48	C₂×#1136	C₂×C₄	1	4	yes	—
49	C₂²×C₇⋊C₈	—	2	2	no	—
50	C₇×#97	—	2	4	no	—
51	C₄×C₇×C₈	—	2	4	no	—
52	C₂×#405	D₇	1	1	yes	characteristic
53	C₂×#344	—	7	2	no	—
54	C₂×#341	—	7	1	no	—
55	#1735	—	7	1	no	—
56	C₂×H(3,ℤ/4)	—	7	2	no	—
57	C₂⁴×C₇	#41	1	1	yes	characteristic
58	C₂²×C₄×C₇	—	2	2	no	—
59	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₇	#41	1	2	yes	—
65	C₂²×C₄×C₇	#41	1	1	yes	characteristic
66	C₂²×C₄×C₇	#41	1	1	yes	characteristic
67	C₂²×C₄×C₇	#41	1	1	yes	characteristic
68	C₂²×C₄×C₇	#41	1	2	yes	—
69	C₂²×Dic₇	—	4	2	no	—
70	C₇×M₁₆	—	2	4	no	—
71	C₇×M₁₆	—	2	4	no	—
72	C₂×C₇×C₈	—	2	1	no	—
73	C₂×C₇×C₈	—	2	2	no	—
74	C₂×C₇×C₈	—	2	1	no	—
75	C₂×C₇×C₈	—	2	1	no	—
76	C₂×C₇×C₈	—	2	1	no	—
77	C₂×C₇×C₈	—	2	2	no	—
78	C₂×C₇×C₈	—	2	4	no	—
79	C₂×C₇×C₈	—	2	4	no	—
80	C₂×C₇×C₈	C₄²	1	4	yes	—
81	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
82	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
83	C₇×C₄⋊C₄	—	2	4	no	—
84	C₇×C₂²⋊C₄	—	2	4	no	—
85	C₄²×C₇	—	2	2	no	—
86	C₄²×C₇	C₂²⋊C₄	1	4	yes	—
87	C₂²⋊Dic₇	—	2	4	no	—
88	C₂²⋊Dic₇	—	4	2	no	—
89	C₄⋊Dic₇	—	2	4	no	—
90	C₄⋊Dic₇	—	4	2	no	—
91	C₄×Dic₇	—	2	4	no	—
92	C₄×Dic₇	—	4	2	no	—
93	#1136	—	2	4	no	—
94	#1136	—	2	4	no	—
95	#1136	—	4	2	no	—
96	#1136	—	4	2	no	—
97	C₂×C₇⋊C₈	—	2	4	no	—
98	C₂×C₇⋊C₈	—	4	2	no	—
99	C₂×C₇⋊C₈	—	4	2	no	—
100	C₂²×M₁₆	—	7	2	no	—
101	C₂²×M₁₆	C₂×D₇	1	1	yes	characteristic
102	C₂³×C₈	C₂×D₇	1	1	yes	characteristic
103	C₂×#117	—	7	2	no	—
104	C₂×#117	C₂×D₇	1	1	yes	characteristic
105	#405	—	2	4	no	—
106	C₂×#97	Dic₇	1	2	yes	complemented
107	C₂×C₄×C₈	Dic₇	1	2	yes	complemented
108	#344	—	7	8	no	—
109	#341	—	14	2	no	—
110	#340	—	14	2	no	—
111	H(3,ℤ/4)	—	14	4	no	—
112	C₂³×C₇	—	4	2	no	—
113	C₂³×C₇	#117	1	2	yes	—
114	C₂³×C₇	#117	1	1	yes	characteristic
115	C₂³×C₇	#123	1	2	yes	—
116	C₂³×C₇	#124	1	1	yes	characteristic
117	C₂³×C₇	#125	1	1	yes	characteristic
118	C₂×C₄×C₇	—	2	4	no	—
119	C₂×C₄×C₇	—	4	4	no	—
120	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
121	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₇	#117	1	2	yes	—
124	C₂×C₄×C₇	#117	1	1	yes	characteristic
125	C₂×C₄×C₇	#117	1	1	yes	characteristic
126	C₂×C₄×C₇	#117	1	1	yes	characteristic
127	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
130	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
131	C₂×C₄×C₇	#121	1	2	yes	—
132	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
133	C₂×C₄×C₇	#123	1	2	yes	—
134	C₂×C₄×C₇	#124	1	1	yes	characteristic
135	C₂×C₄×C₇	#125	1	1	yes	characteristic
136	C₂×C₄×C₇	#126	1	2	yes	—
137	C₂×C₄×C₇	#126	1	2	yes	—
138	C₂×Dic₇	—	4	4	no	—
139	C₂×Dic₇	—	8	4	no	—
140	C₇×C₈	—	2	4	no	—
141	C₇×C₈	—	2	2	no	—
142	C₇×C₈	—	2	2	no	—
143	C₇⋊C₈	—	4	4	no	—
144	C₂³×C₄	C₂²×D₇	1	1	yes	characteristic
145	C₂×M₁₆	—	2	1	no	—
146	C₂×M₁₆	—	2	1	no	—
147	C₂×M₁₆	—	2	2	no	—
148	C₂×M₁₆	—	7	4	no	—
149	C₂×M₁₆	—	14	2	no	—
150	C₂×M₁₆	—	14	2	no	—
151	C₂×M₁₆	—	14	2	no	—
152	C₂×M₁₆	—	14	2	no	—
153	C₂×M₁₆	—	14	2	no	—
154	C₂×M₁₆	C₄×D₇	1	4	yes	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	14	2	no	—
158	C₂²×C₈	Dic₁₄	1	2	yes	—
159	C₂²×C₈	D₂₈	1	2	yes	—
160	C₂²×C₈	C₂×Dic₇	1	2	yes	—
161	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
162	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
163	#117	—	2	2	no	—
164	#117	—	7	4	no	—
165	#117	—	14	2	no	—
166	#117	—	14	2	no	—
167	#117	—	14	2	no	—
168	#117	C₂²⋊D₇	1	4	yes	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄⋊C₄	—	14	2	no	—
171	C₂×C₂²⋊C₄	—	2	1	no	—
172	C₂×C₂²⋊C₄	—	14	2	no	—
173	C₂×C₄²	—	14	2	no	—
174	C₂×C₄²	C₂×Dic₇	1	2	yes	—
175	#97	—	2	4	no	—
176	C₄×C₈	—	2	4	no	—
177	C₂²×C₇	—	2	2	no	—
178	C₂²×C₇	—	2	2	no	—
179	C₂²×C₇	—	2	2	no	—
180	C₂²×C₇	—	4	2	no	—
181	C₂²×C₇	—	4	2	no	—
182	C₂²×C₇	#376	1	1	yes	characteristic
183	C₂²×C₇	#381	1	1	yes	characteristic
184	C₂²×C₇	#382	1	1	yes	characteristic
185	C₂²×C₇	#387	1	2	yes	—
186	C₂²×C₇	#388	1	2	yes	—
187	C₄×C₇	—	4	4	no	—
188	C₄×C₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
189	C₄×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic
190	C₄×C₇	#380	1	1	yes	characteristic
191	C₄×C₇	#383	1	1	yes	characteristic
192	C₄×C₇	#387	1	2	yes	—
193	C₄×C₇	#388	1	2	yes	—
194	Dic₇	—	8	4	no	—
195	C₂⁴	#1157	1	1	yes	characteristic
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	28	2	no	—
198	C₂²×C₄	C₂×Dic₁₄	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₄×D₇	1	2	yes	—
200	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
201	C₂²×C₄	#1157	1	2	yes	—
202	C₂²×C₄	#1157	1	1	yes	characteristic
203	C₂²×C₄	#1157	1	1	yes	characteristic
204	C₂²×C₄	#1157	1	2	yes	—
205	C₂²×C₄	#1157	1	1	yes	characteristic
206	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₂²⋊D₇	1	2	yes	—
208	M₁₆	—	2	4	no	—
209	M₁₆	—	2	4	no	—
210	M₁₆	—	14	4	no	—
211	M₁₆	—	14	4	no	—
212	M₁₆	—	28	2	no	—
213	M₁₆	—	28	2	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	14	4	no	—
223	C₂×C₈	—	28	2	no	—
224	C₂×C₈	—	28	2	no	—
225	C₂×C₈	C₄×Dic₇	1	4	yes	—
226	C₂×C₈	C₄⋊Dic₇	1	2	yes	—
227	C₂×C₈	C₄⋊Dic₇	1	2	yes	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	14	4	no	—
230	C₄⋊C₄	—	28	2	no	—
231	C₂²⋊C₄	—	2	4	no	—
232	C₂²⋊C₄	—	14	4	no	—
233	C₂²⋊C₄	—	28	2	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	14	4	no	—
236	C₄²	—	28	2	no	—
237	C₄²	C₂²⋊Dic₇	1	4	yes	—
238	C₂×C₇	—	2	2	no	—
239	C₂×C₇	—	2	1	no	—
240	C₂×C₇	—	2	1	no	—
241	C₂×C₇	—	2	2	no	—
242	C₂×C₇	#1434	1	1	yes	characteristic
243	C₂×C₇	#1833	1	2	yes	—
244	C₂³	—	4	2	no	—
245	C₂³	#6433	1	1	yes	characteristic
246	C₂³	#6435	1	2	yes	—
247	C₂³	#6438	1	1	yes	characteristic
248	C₂³	#6489	1	1	yes	characteristic
249	C₂³	#6492	1	2	yes	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	28	4	no	—
253	C₂×C₄	—	56	4	no	—
254	C₂×C₄	C₄×Dic₁₄	1	2	yes	—
255	C₂×C₄	#6433	1	1	yes	characteristic
256	C₂×C₄	#6435	1	2	yes	—
257	C₂×C₄	C₄×D₂₈	1	2	yes	—
258	C₂×C₄	#6438	1	1	yes	characteristic
259	C₂×C₄	#6439	1	2	yes	—
260	C₂×C₄	#6439	1	2	yes	—
261	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
262	C₂×C₄	#6487	1	2	yes	—
263	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
264	C₂×C₄	#6489	1	1	yes	characteristic
265	C₂×C₄	#6489	1	1	yes	characteristic
266	C₂×C₄	#6489	1	1	yes	characteristic
267	C₂×C₄	C₄×C₂²⋊D₇	1	2	yes	—
268	C₂×C₄	C₄×C₂²⋊D₇	1	2	yes	—
269	C₂×C₄	#6492	1	2	yes	—
270	C₂×C₄	#6493	1	2	yes	—
271	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
272	C₈	—	2	4	no	—
273	C₈	—	2	2	no	—
274	C₈	—	2	2	no	—
275	C₈	—	28	4	no	—
276	C₇	#10804	1	1	yes	characteristic, complemented, Sylow
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	#89704	1	1	yes	characteristic
283	C₂²	#89705	1	2	yes	—
284	C₂²	#89708	1	1	yes	characteristic
285	C₂²	#89712	1	2	yes	—
286	C₂²	#89983	1	1	yes	characteristic
287	C₄	—	4	4	no	—
288	C₄	—	56	4	no	—
289	C₄	#89705	1	2	yes	—
290	C₄	C₄×C₄⋊Dic₇	1	1	yes	characteristic
291	C₄	#89709	1	1	yes	characteristic
292	C₄	#89712	1	2	yes	—
293	C₄	C₄×C₂²⋊Dic₇	1	1	yes	characteristic
294	C₄	#89984	1	1	yes	characteristic
295	C₂	—	2	1	no	—
296	C₂	—	2	2	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	2	no	—
299	C₂	#11727650	1	2	yes	—
300	C₂	#11729288	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6489
commutator	C₂×C₄×C₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₇
Fitting	C₂×C₇×#405	C₂
socle	C₂²×C₇	#382
prime radical	C₂⁴×C₇	#41
cosocle	C₂³×C₄×C₇	C₂³
hypercenter	C₂×#405	D₇
hypocenter	C₇	#10804
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
12. C₂×C₇×#97	C₄	semidirect
13. C₂×C₄×C₇×C₈	C₄	semidirect
14. #89351	C₄	semidirect
106. C₂×#97	Dic₇	semidirect
107. C₂×C₄×C₈	Dic₇	semidirect
276. C₇	#10804	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#405	2
C₂×#89351	2
C₂×#89347	2
#11729723	2
C₂×#89326	2
#10804	7