Subgroups — #13247643

Order 1792.

520 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#89696	C₂	1	1	yes	maximal, characteristic
2	C₄×C₇×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#11729284	C₂	1	1	yes	maximal, characteristic
4	C₂×#89253	C₂	1	2	yes	maximal, complemented
5	#11725460	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
7	C₂²×#1136	—	2	1	no	—
8	C₂×C₇×C₂²⋊C₈	C₄	1	2	yes	—
9	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
10	C₂×C₂²⋊(C₇⋊C₈)	—	2	1	no	—
11	#89696	—	2	4	no	—
12	C₂×C₄⋊(C₇⋊C₈)	—	2	1	no	—
13	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
16	C₇×#336	C₄	1	1	yes	characteristic
17	C₇×#336	C₄	1	1	yes	characteristic
18	#89319	—	2	1	no	—
19	#89253	—	2	4	no	—
20	#8832	—	7	1	no	maximal, Sylow
21	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₇	Q₈	1	1	yes	characteristic
24	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₇	—	2	2	no	—
27	C₂×C₄²×C₇	—	2	2	no	—
28	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₇	Q₈	1	1	yes	characteristic
31	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
32	C₂×#1136	—	2	4	no	—
33	C₂×#1136	—	4	2	no	—
34	C₂×#1136	—	4	2	no	—
35	C₂²×C₇⋊C₈	—	2	2	no	—
36	C₂²×C₇⋊C₈	—	2	1	no	—
37	C₂²×C₇⋊C₈	—	2	1	no	—
38	C₇×C₂²⋊C₈	—	2	4	no	—
39	C₄×C₇×C₈	—	2	2	no	—
40	C₄×C₇×C₈	—	2	2	no	—
41	C₂²⋊(C₇⋊C₈)	—	4	2	no	—
42	C₄⋊(C₇⋊C₈)	—	2	4	no	—
43	C₄⋊(C₇⋊C₈)	—	2	2	no	—
44	C₄⋊(C₇⋊C₈)	—	2	2	no	—
45	C₄⋊(C₇⋊C₈)	—	2	2	no	—
46	C₄⋊(C₇⋊C₈)	D₄	1	4	yes	—
47	C₂×#423	—	7	1	no	—
48	#1797	—	7	1	no	—
49	C₄×C₂²⋊C₈	D₇	1	1	yes	characteristic
50	C₂×#335	—	7	2	no	—
51	#1277	—	7	2	no	—
52	C₂⁴×C₇	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇	—	2	1	no	—
54	C₂²×C₄×C₇	—	2	1	no	—
55	C₂²×C₄×C₇	—	2	1	no	—
56	C₂²×C₄×C₇	—	2	1	no	—
57	C₂²×C₄×C₇	—	2	2	no	—
58	C₂²×C₄×C₇	—	2	2	no	—
59	C₂²×C₄×C₇	—	2	2	no	—
60	C₂²×C₄×C₇	—	2	2	no	—
61	C₂²×C₄×C₇	—	2	1	no	—
62	C₂²×C₄×C₇	—	2	1	no	—
63	C₂²×C₄×C₇	—	4	1	no	—
64	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇	D₈	1	2	yes	—
68	C₂²×C₄×C₇	Q₁₆	1	2	yes	—
69	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₇×C₈	—	2	2	no	—
73	C₂×C₇×C₈	—	2	2	no	—
74	C₂×C₇×C₈	—	2	2	no	—
75	C₂×C₇×C₈	—	2	2	no	—
76	C₂×C₇×C₈	—	2	2	no	—
77	C₂×C₇×C₈	M₁₆	1	4	yes	complemented
78	C₄²×C₇	—	2	2	no	—
79	C₄²×C₇	—	2	1	no	—
80	C₄²×C₇	—	2	2	no	—
81	C₄²×C₇	—	2	1	no	—
82	C₄²×C₇	C₂×D₄	1	2	yes	—
83	C₄²×C₇	#41	1	2	yes	—
84	#1136	—	4	8	no	—
85	C₂×C₇⋊C₈	—	2	4	no	—
86	C₂×C₇⋊C₈	—	4	2	no	—
87	C₂×C₇⋊C₈	—	4	2	no	—
88	C₂×C₇⋊C₈	—	4	1	no	—
89	C₂×C₇⋊C₈	—	4	1	no	—
90	C₂×C₇⋊C₈	—	4	1	no	—
91	C₂×C₇⋊C₈	—	4	1	no	—
92	C₂×C₇⋊C₈	—	4	1	no	—
93	C₂×C₇⋊C₈	—	4	1	no	—
94	C₂²×M₁₆	—	14	1	no	—
95	C₂²×C₄²	C₂×D₇	1	1	yes	characteristic
96	#423	—	14	4	no	—
97	C₂×C₄⋊C₈	—	7	2	no	—
98	C₂×C₄⋊C₈	—	7	1	no	—
99	C₂×C₄⋊C₈	—	7	1	no	—
100	C₂×C₄⋊C₈	—	14	1	no	—
101	C₂×C₂²⋊C₈	—	14	1	no	—
102	C₂×C₂²⋊C₈	Dic₇	1	2	yes	—
103	C₂×C₄×C₈	C₂×D₇	1	2	yes	—
104	#336	—	14	1	no	—
105	#336	Dic₇	1	1	yes	characteristic
106	#336	Dic₇	1	1	yes	characteristic
107	#335	—	14	4	no	—
108	C₂³×C₇	—	2	1	no	—
109	C₂³×C₇	—	2	1	no	—
110	C₂³×C₇	—	2	1	no	—
111	C₂³×C₇	—	2	1	no	—
112	C₂³×C₇	—	2	1	no	—
113	C₂³×C₇	—	2	1	no	—
114	C₂³×C₇	#107	1	2	yes	—
115	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₇	—	2	1	no	—
117	C₂×C₄×C₇	—	2	1	no	—
118	C₂×C₄×C₇	—	2	1	no	—
119	C₂×C₄×C₇	—	2	1	no	—
120	C₂×C₄×C₇	—	2	1	no	—
121	C₂×C₄×C₇	—	2	1	no	—
122	C₂×C₄×C₇	—	2	2	no	—
123	C₂×C₄×C₇	—	2	2	no	—
124	C₂×C₄×C₇	—	2	2	no	—
125	C₂×C₄×C₇	—	2	2	no	—
126	C₂×C₄×C₇	—	2	1	no	—
127	C₂×C₄×C₇	—	2	1	no	—
128	C₂×C₄×C₇	—	2	1	no	—
129	C₂×C₄×C₇	—	4	4	no	—
130	C₂×C₄×C₇	—	4	2	no	—
131	C₂×C₄×C₇	—	4	2	no	—
132	C₂×C₄×C₇	#107	1	2	yes	—
133	C₂×C₄×C₇	#108	1	2	yes	—
134	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
135	C₂×C₄×C₇	#120	1	1	yes	characteristic
136	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
137	C₂×C₄×C₇	#123	1	1	yes	characteristic
138	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
139	C₂×C₄×C₇	#131	1	1	yes	characteristic
140	C₂×C₄×C₇	#131	1	1	yes	characteristic
141	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
143	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₂×C₄×C₇	#137	1	1	yes	characteristic
145	C₇×C₈	—	2	4	no	—
146	C₇×C₈	—	4	2	no	—
147	C₇⋊C₈	—	4	4	no	—
148	C₇⋊C₈	—	8	1	no	—
149	C₇⋊C₈	—	8	1	no	—
150	C₂³×C₄	Dic₁₄	1	1	yes	characteristic
151	C₂³×C₄	D₂₈	1	1	yes	characteristic
152	C₂³×C₄	C₂×Dic₇	1	1	yes	characteristic
153	C₂×M₁₆	—	14	4	no	—
154	C₂×M₁₆	—	28	2	no	—
155	C₂×M₁₆	—	28	2	no	—
156	C₂²×C₈	—	14	2	no	—
157	C₂²×C₈	—	14	1	no	—
158	C₂²×C₈	—	14	1	no	—
159	C₂²×C₈	C₂×Dic₇	1	2	yes	—
160	C₂²×C₈	C₂×Dic₇	1	2	yes	—
161	C₂×C₄²	—	2	2	no	—
162	C₂×C₄²	—	2	2	no	—
163	C₂×C₄²	Dic₁₄	1	1	yes	characteristic
164	C₂×C₄²	D₂₈	1	1	yes	characteristic
165	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
166	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
167	C₄⋊C₈	—	7	4	no	—
168	C₄⋊C₈	—	14	4	no	—
169	C₄⋊C₈	—	14	2	no	—
170	C₄⋊C₈	—	14	2	no	—
171	C₄⋊C₈	—	14	2	no	—
172	C₂²⋊C₈	—	2	4	no	—
173	C₂²⋊C₈	—	28	2	no	—
174	C₄×C₈	—	2	2	no	—
175	C₄×C₈	—	2	2	no	—
176	C₂²×C₇	—	2	2	no	—
177	C₂²×C₇	—	2	2	no	—
178	C₂²×C₇	—	2	2	no	—
179	C₂²×C₇	—	4	1	no	—
180	C₂²×C₇	—	4	1	no	—
181	C₂²×C₇	—	4	1	no	—
182	C₂²×C₇	—	4	1	no	—
183	C₂²×C₇	#385	1	1	yes	characteristic
184	C₂²×C₇	#423	1	1	yes	characteristic
185	C₂²×C₇	#424	1	1	yes	characteristic
186	C₂²×C₇	C₂×#107	1	1	yes	characteristic
187	C₂²×C₇	#428	1	1	yes	characteristic
188	C₂²×C₇	C₂×#108	1	1	yes	characteristic
189	C₂²×C₇	#430	1	1	yes	characteristic
190	C₄×C₇	—	2	2	no	—
191	C₄×C₇	—	2	2	no	—
192	C₄×C₇	—	2	2	no	—
193	C₄×C₇	—	4	1	no	—
194	C₄×C₇	#435	1	2	yes	—
195	C₄×C₇	#436	1	2	yes	—
196	C₄×C₇	#447	1	1	yes	characteristic
197	C₄×C₇	#451	1	1	yes	characteristic
198	C₄×C₇	#480	1	1	yes	characteristic
199	C₄×C₇	#484	1	1	yes	characteristic
200	C₂⁴	C₄⋊Dic₇	1	1	yes	characteristic
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	4	1	no	—
212	C₂²×C₄	D₅₆	1	2	yes	—
213	C₂²×C₄	Dic₂₈	1	2	yes	—
214	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
215	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
216	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
217	C₂²×C₄	C₂×Dic₁₄	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
219	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
220	M₁₆	—	28	8	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	14	4	no	—
227	C₂×C₈	—	28	2	no	—
228	C₂×C₈	—	28	2	no	—
229	C₂×C₈	—	28	1	no	—
230	C₂×C₈	—	28	1	no	—
231	C₂×C₈	—	28	1	no	—
232	C₂×C₈	—	28	1	no	—
233	C₂×C₈	—	28	1	no	—
234	C₂×C₈	—	28	1	no	—
235	C₂×C₈	#1136	1	4	yes	complemented
236	C₄²	—	2	2	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	2	2	no	—
240	C₄²	D₄×D₇	1	2	yes	—
241	C₄²	#1159	1	2	yes	—
242	C₂×C₇	—	4	2	no	—
243	C₂×C₇	#1567	1	1	yes	characteristic
244	C₂×C₇	#1569	1	1	yes	characteristic
245	C₂×C₇	#1571	1	1	yes	characteristic
246	C₂×C₇	#1574	1	1	yes	characteristic
247	C₂×C₇	#1851	1	1	yes	characteristic
248	C₂×C₇	#1852	1	1	yes	characteristic
249	C₂×C₇	#1855	1	1	yes	characteristic
250	C₂³	—	2	1	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	#6392	1	2	yes	—
257	C₂³	C₂×C₄⋊Dic₇	1	1	yes	characteristic
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	#6392	1	2	yes	—
275	C₂×C₄	#6393	1	2	yes	—
276	C₂×C₄	#6441	1	2	yes	—
277	C₂×C₄	#6445	1	1	yes	characteristic
278	C₂×C₄	#6449	1	1	yes	characteristic
279	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
280	C₂×C₄	C₂×Dic₂₈	1	1	yes	characteristic
281	C₂×C₄	#6471	1	1	yes	characteristic
282	C₂×C₄	#6472	1	1	yes	characteristic
283	C₂×C₄	C₂×#1136	1	2	yes	—
284	C₂×C₄	D₄×Dic₇	1	2	yes	—
285	C₂×C₄	#6511	1	1	yes	characteristic
286	C₂×C₄	#6511	1	1	yes	characteristic
287	C₈	—	2	4	no	—
288	C₈	—	4	2	no	—
289	C₈	—	28	4	no	—
290	C₈	—	56	1	no	—
291	C₈	—	56	1	no	—
292	C₇	#8832	1	1	yes	characteristic, complemented, Sylow
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	1	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	#89696	1	1	yes	characteristic
301	C₂²	#89722	1	1	yes	characteristic
302	C₂²	#89771	1	1	yes	characteristic
303	C₂²	C₂×#6392	1	1	yes	characteristic
304	C₂²	C₂×#6393	1	1	yes	characteristic
305	C₂²	#89893	1	1	yes	characteristic
306	C₂²	#89897	1	1	yes	characteristic
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	4	1	no	—
311	C₄	#89494	1	1	yes	characteristic
312	C₄	#89504	1	1	yes	characteristic
313	C₄	#89508	1	1	yes	characteristic
314	C₄	#89511	1	1	yes	characteristic
315	C₄	#89783	1	2	yes	—
316	C₄	#89784	1	2	yes	—
317	C₂	—	4	2	no	—
318	C₂	#11727204	1	1	yes	characteristic
319	C₂	#11727485	1	1	yes	characteristic
320	C₂	#11727605	1	1	yes	characteristic
321	C₂	#11727681	1	1	yes	characteristic
322	C₂	#11727708	1	1	yes	characteristic
323	C₂	#11728504	1	1	yes	characteristic
324	C₂	#11728507	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₈
commutator	C₂²×C₄×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₄×C₇×C₂²⋊C₈	C₂
socle	C₂³×C₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₇	C₄⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	C₄×C₂²⋊C₈	D₇
hypocenter	C₇	#8832
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₄×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
4. C₂×#89253	C₂	semidirect
5. #11725460	C₂	semidirect
77. C₂×C₇×C₈	M₁₆	semidirect
235. C₂×C₈	#1136	semidirect
292. C₇	#8832	semidirect

Maximal subgroups

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

subgroup	index
C₂×#89696	2
C₄×C₇×C₂²⋊C₈	2
#11729284	2
C₂×#89253	2
#11725460	2
#8832	7