Subgroups — #13253789

Order 1792.

460 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	C₄×#6433	C₂	1	1	yes	maximal, characteristic
2	C₄×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#11727145	C₂	1	1	yes	maximal, characteristic
4	#11725461	C₂	1	2	yes	maximal
5	#11725461	C₂	1	2	yes	maximal
6	C₂×#6433	C₄	1	1	yes	characteristic
7	C₂×C₇×C₄⋊C₈	—	2	1	no	—
8	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
9	C₄³×C₇	C₂²	1	1	yes	characteristic
10	#89708	C₄	1	1	yes	characteristic
11	C₄×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
13	#89705	—	2	1	no	—
14	#89705	—	2	1	no	—
15	C₄×Dic₇⋊C₄	—	2	1	no	—
16	C₄×Dic₇⋊C₄	—	2	1	no	—
17	#89700	C₄	1	1	yes	characteristic
18	#89700	C₄	1	1	yes	characteristic
19	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
20	C₂×#6377	—	2	1	no	—
21	C₇×#336	—	2	1	no	—
22	#89319	—	2	1	no	—
23	#9867	—	7	1	no	maximal, Sylow
24	C₂²×C₇×C₈	—	2	2	no	—
25	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊Dic₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊Dic₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊Dic₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₇	C₂×C₄	1	1	yes	characteristic
36	C₂×Dic₇⋊C₄	—	2	1	no	—
37	C₂×Dic₇⋊C₄	—	2	1	no	—
38	C₂×Dic₇⋊C₄	—	2	1	no	—
39	C₂×Dic₇⋊C₄	—	2	1	no	—
40	C₂×C₄×Dic₇	—	2	1	no	—
41	C₂×C₄×Dic₇	—	2	1	no	—
42	C₂²×C₇⋊C₈	—	2	2	no	—
43	#6433	—	2	2	no	—
44	#6433	—	2	2	no	—
45	C₇×C₄⋊C₈	—	2	4	no	—
46	C₄×C₇×C₈	—	2	4	no	—
47	C₇⋊#95	—	2	1	no	—
48	C₇⋊#95	—	2	1	no	—
49	C₇⋊#95	—	2	1	no	—
50	C₇⋊#95	—	2	1	no	—
51	C₇⋊#95	—	2	1	no	—
52	C₇⋊#95	—	2	1	no	—
53	C₄⋊(C₇⋊C₈)	—	2	2	no	—
54	C₄⋊(C₇⋊C₈)	Q₈	1	4	yes	—
55	#6377	—	4	2	no	—
56	C₄×#125	—	7	1	no	—
57	#1767	—	7	1	no	—
58	C₄×C₄⋊C₈	D₇	1	1	yes	characteristic
59	#1279	—	7	2	no	—
60	#1279	—	7	2	no	—
61	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×Dic₇	—	2	1	no	—
69	C₂²×Dic₇	—	2	1	no	—
70	C₂²×Dic₇	—	2	1	no	—
71	C₂²×Dic₇	—	2	1	no	—
72	C₂×C₇×C₈	—	2	4	no	—
73	C₂×C₇×C₈	—	4	2	no	—
74	C₂×C₇×C₈	—	4	2	no	—
75	C₄²×C₇	—	2	2	no	—
76	C₄²×C₇	—	2	1	no	—
77	C₄²×C₇	—	2	1	no	—
78	C₄²×C₇	—	2	2	no	—
79	C₄²×C₇	—	2	2	no	—
80	C₄²×C₇	—	2	1	no	—
81	C₄²×C₇	—	2	1	no	—
82	C₄²×C₇	—	4	1	no	—
83	C₄²×C₇	C₂×Q₈	1	2	yes	—
84	C₄²×C₇	#41	1	2	yes	—
85	C₄⋊Dic₇	—	2	1	no	—
86	C₄⋊Dic₇	—	2	1	no	—
87	C₄⋊Dic₇	—	2	2	no	—
88	C₄⋊Dic₇	—	2	2	no	—
89	C₄⋊Dic₇	M₁₆	1	2	yes	—
90	C₄⋊Dic₇	M₁₆	1	2	yes	—
91	Dic₇⋊C₄	—	2	4	no	—
92	Dic₇⋊C₄	—	2	4	no	—
93	Dic₇⋊C₄	—	4	2	no	—
94	Dic₇⋊C₄	—	4	2	no	—
95	C₄×Dic₇	—	4	2	no	—
96	C₄×Dic₇	—	4	1	no	—
97	C₄×Dic₇	—	4	1	no	—
98	C₂×C₇⋊C₈	—	2	4	no	—
99	C₂×C₇⋊C₈	—	4	2	no	—
100	C₂×C₇⋊C₈	—	4	2	no	—
101	C₂×#125	—	7	1	no	—
102	C₂×C₄⋊C₈	—	2	1	no	—
103	C₂×C₄⋊C₈	—	7	2	no	—
104	C₂×#97	—	14	1	no	—
105	C₂×C₄×C₈	C₂×D₇	1	2	yes	—
106	#389	—	7	1	no	—
107	#389	—	7	1	no	—
108	#387	—	14	1	no	—
109	#387	—	14	1	no	—
110	#382	—	7	1	no	—
111	C₄×C₄⋊C₄	—	7	1	no	—
112	C₄×C₄⋊C₄	—	7	1	no	—
113	C₄×C₄⋊C₄	—	14	1	no	—
114	C₄×C₄⋊C₄	—	14	1	no	—
115	C₄³	C₂×D₇	1	1	yes	characteristic
116	#336	—	2	1	no	—
117	#336	—	14	1	no	—
118	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	1	no	—
123	C₂×C₄×C₇	—	2	1	no	—
124	C₂×C₄×C₇	—	2	1	no	—
125	C₂×C₄×C₇	—	2	1	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	—	2	1	no	—
130	C₂×C₄×C₇	—	2	1	no	—
131	C₂×C₄×C₇	—	2	1	no	—
132	C₂×C₄×C₇	#104	1	2	yes	—
133	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
134	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
135	C₂×C₄×C₇	#124	1	1	yes	characteristic
136	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
137	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
138	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
139	C₂×C₄×C₇	#131	1	2	yes	—
140	C₂×C₄×C₇	#135	1	2	yes	—
141	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₄×C₇	#137	1	1	yes	characteristic
143	C₂×Dic₇	—	2	2	no	—
144	C₂×Dic₇	—	2	2	no	—
145	C₂×Dic₇	—	4	1	no	—
146	C₂×Dic₇	—	4	1	no	—
147	C₂×Dic₇	—	4	1	no	—
148	C₂×Dic₇	—	4	1	no	—
149	C₂×Dic₇	—	4	1	no	—
150	C₂×Dic₇	—	4	1	no	—
151	C₂×Dic₇	—	4	1	no	—
152	C₂×Dic₇	—	4	1	no	—
153	C₂×Dic₇	—	4	1	no	—
154	C₂×Dic₇	—	4	1	no	—
155	C₇×C₈	—	4	4	no	—
156	C₇⋊C₈	—	4	4	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	14	2	no	—
159	#125	—	14	2	no	—
160	#125	—	14	2	no	—
161	C₂×C₄⋊C₄	—	7	1	no	—
162	C₂×C₄⋊C₄	—	7	1	no	—
163	C₂×C₄⋊C₄	—	7	1	no	—
164	C₂×C₄⋊C₄	—	7	1	no	—
165	C₂×C₄⋊C₄	—	14	1	no	—
166	C₂×C₄⋊C₄	—	14	1	no	—
167	C₂×C₄⋊C₄	—	14	1	no	—
168	C₂×C₄⋊C₄	—	14	1	no	—
169	C₂×C₄²	—	14	1	no	—
170	C₂×C₄²	—	14	1	no	—
171	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
172	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
173	C₂×C₄²	D₂₈	1	1	yes	characteristic
174	C₂×C₄²	D₂₈	1	1	yes	characteristic
175	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
176	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
177	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
178	C₄⋊C₈	—	2	4	no	—
179	C₄⋊C₈	—	7	4	no	—
180	C₄⋊C₈	—	14	2	no	—
181	#97	—	28	2	no	—
182	C₄×C₈	—	2	4	no	—
183	#95	—	14	1	no	—
184	#95	—	14	1	no	—
185	#95	—	14	1	no	—
186	#95	—	14	1	no	—
187	#95	—	14	1	no	—
188	#95	—	14	1	no	—
189	C₂²×C₇	#391	1	1	yes	characteristic
190	C₂²×C₇	#407	1	1	yes	characteristic
191	C₂²×C₇	#408	1	1	yes	characteristic
192	C₂²×C₇	#416	1	1	yes	characteristic
193	C₂²×C₇	#417	1	1	yes	characteristic
194	C₂²×C₇	C₂×#104	1	1	yes	characteristic
195	C₂²×C₇	#421	1	1	yes	characteristic
196	C₄×C₇	—	2	1	no	—
197	C₄×C₇	—	2	1	no	—
198	C₄×C₇	—	2	2	no	—
199	C₄×C₇	—	4	1	no	—
200	C₄×C₇	—	4	1	no	—
201	C₄×C₇	—	4	1	no	—
202	C₄×C₇	#446	1	2	yes	—
203	C₄×C₇	#446	1	2	yes	—
204	C₄×C₇	#478	1	2	yes	—
205	C₄×C₇	#479	1	2	yes	—
206	Dic₇	—	4	2	no	—
207	Dic₇	—	4	2	no	—
208	Dic₇	—	8	1	no	—
209	Dic₇	—	8	1	no	—
210	C₂²×C₄	—	14	1	no	—
211	C₂²×C₄	—	14	1	no	—
212	C₂²×C₄	—	14	1	no	—
213	C₂²×C₄	—	14	1	no	—
214	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
215	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
216	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
217	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	14	4	no	—
225	C₂×C₈	—	28	2	no	—
226	C₂×C₈	—	28	2	no	—
227	C₄⋊C₄	—	7	2	no	—
228	C₄⋊C₄	—	7	2	no	—
229	C₄⋊C₄	—	14	4	no	—
230	C₄⋊C₄	—	14	4	no	—
231	C₄⋊C₄	—	14	1	no	—
232	C₄⋊C₄	—	14	1	no	—
233	C₄⋊C₄	—	14	2	no	—
234	C₄⋊C₄	—	14	2	no	—
235	C₄⋊C₄	—	28	2	no	—
236	C₄⋊C₄	—	28	2	no	—
237	C₄²	—	2	2	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	2	1	no	—
242	C₄²	—	2	1	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	28	2	no	—
246	C₄²	—	28	1	no	—
247	C₄²	—	28	1	no	—
248	C₄²	#1159	1	2	yes	—
249	C₄²	Q₈×D₇	1	2	yes	—
250	C₂×C₇	#1488	1	1	yes	characteristic
251	C₂×C₇	#1489	1	1	yes	characteristic
252	C₂×C₇	#1494	1	1	yes	characteristic
253	C₂×C₇	#1495	1	1	yes	characteristic
254	C₂×C₇	#1985	1	1	yes	characteristic
255	C₂×C₇	#1986	1	1	yes	characteristic
256	C₂×C₇	#1989	1	1	yes	characteristic
257	C₂³	C₂×(C₂×D₇)⋊C₄	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	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	14	2	no	—
272	C₂×C₄	—	14	2	no	—
273	C₂×C₄	—	28	1	no	—
274	C₂×C₄	—	28	1	no	—
275	C₂×C₄	—	28	1	no	—
276	C₂×C₄	—	28	1	no	—
277	C₂×C₄	—	28	1	no	—
278	C₂×C₄	—	28	1	no	—
279	C₂×C₄	—	28	1	no	—
280	C₂×C₄	—	28	1	no	—
281	C₂×C₄	—	28	1	no	—
282	C₂×C₄	—	28	1	no	—
283	C₂×C₄	#6379	1	2	yes	—
284	C₂×C₄	#6450	1	2	yes	—
285	C₂×C₄	#6460	1	2	yes	—
286	C₂×C₄	#6464	1	2	yes	—
287	C₂×C₄	#6467	1	2	yes	—
288	C₂×C₄	D₇×M₁₆	1	1	yes	characteristic
289	C₂×C₄	D₇×M₁₆	1	1	yes	characteristic
290	C₂×C₄	#6471	1	1	yes	characteristic
291	C₂×C₄	#6472	1	1	yes	characteristic
292	C₂×C₄	#6499	1	1	yes	characteristic
293	C₂×C₄	#6507	1	1	yes	characteristic
294	C₈	—	4	4	no	—
295	C₈	—	28	4	no	—
296	C₇	#9867	1	1	yes	characteristic, complemented, Sylow
297	C₂²	C₂×#6379	1	1	yes	characteristic
298	C₂²	#89746	1	1	yes	characteristic
299	C₂²	#89777	1	1	yes	characteristic
300	C₂²	#89882	1	1	yes	characteristic
301	C₂²	#89885	1	1	yes	characteristic
302	C₂²	#89898	1	1	yes	characteristic
303	C₂²	#89905	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	28	2	no	—
311	C₄	—	28	2	no	—
312	C₄	—	56	1	no	—
313	C₄	—	56	1	no	—
314	C₄	#89600	1	2	yes	—
315	C₄	#89600	1	2	yes	—
316	C₄	#89601	1	2	yes	—
317	C₄	#89619	1	2	yes	—
318	C₂	#11727247	1	1	yes	characteristic
319	C₂	#11727490	1	1	yes	characteristic
320	C₂	#11727608	1	1	yes	characteristic
321	C₂	#11728215	1	1	yes	characteristic
322	C₂	#11728227	1	1	yes	characteristic
323	C₂	#11728517	1	1	yes	characteristic
324	C₂	#11728518	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×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	C₄×C₄⋊C₈	D₇
hypocenter	C₇	#9867
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
296. C₇	#9867	semidirect

Maximal subgroups

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

subgroup	index
C₄×#6433	2
C₄×C₇×C₄⋊C₈	2
#11727145	2
#11725461	2
#9867	7