Subgroups — #13325486

Order 1792.

626 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11736710	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₈	C₂	1	1	yes	maximal, characteristic
3	C₇×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
4	#11727611	C₂	1	2	yes	maximal
5	#11727486	C₂	1	1	yes	maximal, characteristic
6	#11725841	C₂	1	4	yes	maximal, complemented
7	#11725810	C₂	1	2	yes	maximal, complemented
8	C₈×Dic₂₈	C₂	1	2	yes	maximal, complemented
9	C₂²×Dic₂₈	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₁₄	C₂²	1	2	yes	—
11	C₇×#431	C₂²	1	1	yes	characteristic
12	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₂×#6392	C₄	1	1	yes	characteristic
15	C₂×#6390	C₄	1	2	yes	—
16	#89698	C₂²	1	2	yes	—
17	#89600	—	2	4	no	—
18	#89491	—	2	2	no	—
19	#89490	—	2	2	no	—
20	C₄×Dic₂₈	C₄	1	2	yes	—
21	C₄×Dic₂₈	C₄	1	1	yes	characteristic
22	C₄×Dic₂₈	C₄	1	1	yes	characteristic
23	C₄×Dic₂₈	C₂²	1	2	yes	—
24	C₄×Dic₂₈	C₂²	1	1	yes	characteristic
25	C₄×Dic₂₈	C₂²	1	1	yes	characteristic
26	#89451	C₂²	1	4	yes	—
27	C₈×Dic₁₄	C₂²	1	4	yes	—
28	C₇×#322	C₂²	1	2	yes	—
29	C₇×C₈²	C₂²	1	2	yes	—
30	#89253	C₂²	1	2	yes	—
31	#89252	C₂²	1	2	yes	—
32	#89251	C₂²	1	4	yes	—
33	#89251	C₂²	1	4	yes	—
34	#18058	—	7	1	no	maximal, Sylow
35	C₂²×Dic₁₄	C₂×C₄	1	2	yes	—
36	C₂×C₇×M₁₆	—	2	1	no	—
37	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
40	C₂×C₄⋊Dic₇	C₂×C₄	1	2	yes	—
41	C₂×Dic₇⋊C₄	—	2	2	no	—
42	C₂×C₄×Dic₇	—	2	2	no	—
43	C₂×Dic₂₈	—	2	4	no	—
44	C₂×Dic₂₈	C₂×C₄	1	2	yes	—
45	C₂×Dic₂₈	C₂×C₄	1	1	yes	characteristic
46	C₂×Dic₂₈	C₂×C₄	1	1	yes	characteristic
47	C₄×Dic₁₄	—	2	2	no	—
48	C₄×Dic₁₄	—	2	2	no	—
49	C₄×Dic₁₄	C₂×C₄	1	2	yes	—
50	C₄×Dic₁₄	C₂×C₄	1	2	yes	—
51	C₄×Dic₁₄	C₂³	1	2	yes	—
52	C₄×Dic₁₄	C₂³	1	2	yes	—
53	C₇×C₄⋊C₈	D₄	1	2	yes	—
54	C₇×C₂²⋊C₈	D₄	1	1	yes	characteristic
55	C₇×C₂²⋊C₈	D₄	1	1	yes	characteristic
56	C₇×#97	—	2	1	no	—
57	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
58	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
59	C₄×C₇×C₈	C₂³	1	2	yes	—
60	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
61	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
62	C₄×C₇×C₈	C₂³	1	2	yes	—
63	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
64	#6392	C₂×C₄	1	2	yes	—
65	#6392	C₂×C₄	1	1	yes	characteristic
66	#6392	C₂×C₄	1	1	yes	characteristic
67	#6390	C₂×C₄	1	2	yes	—
68	#6390	C₂×C₄	1	2	yes	—
69	#6390	C₂×C₄	1	2	yes	—
70	#6390	C₂×C₄	1	2	yes	—
71	#6389	—	2	4	no	—
72	Dic₇⋊C₈	—	2	4	no	—
73	Dic₇⋊C₈	—	2	4	no	—
74	C₈×Dic₇	—	2	4	no	—
75	C₄⋊(C₇⋊C₈)	C₂³	1	4	yes	—
76	#6377	—	2	2	no	—
77	#2984	—	7	2	no	—
78	C₂×C₄×Q₁₆	—	7	1	no	—
79	#1591	—	7	2	no	—
80	#1587	—	7	4	no	—
81	C₈×Q₁₆	—	7	2	no	—
82	#1568	—	7	1	no	—
83	#1491	—	7	2	no	—
84	C₈×M₁₆	D₇	1	1	yes	characteristic
85	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇	#41	1	1	yes	characteristic
88	C₂²×Dic₇	—	2	2	no	—
89	C₂²×Dic₇	—	2	2	no	—
90	C₂×Dic₁₄	—	2	4	no	—
91	C₂×Dic₁₄	—	2	2	no	—
92	C₂×Dic₁₄	—	4	2	no	—
93	C₂×Dic₁₄	C₂²×C₄	1	2	yes	—
94	C₂×Dic₁₄	C₂²×C₄	1	2	yes	—
95	C₇×M₁₆	—	2	4	no	—
96	C₂×C₇×C₈	—	2	2	no	—
97	C₂×C₇×C₈	—	2	2	no	—
98	C₂×C₇×C₈	—	4	1	no	—
99	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
104	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
105	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₇	#41	1	1	yes	characteristic
108	C₄²×C₇	C₂⁴	1	1	yes	characteristic
109	C₄⋊Dic₇	—	2	2	no	—
110	C₄⋊Dic₇	C₂²×C₄	1	2	yes	—
111	C₄⋊Dic₇	C₂²×C₄	1	2	yes	—
112	Dic₇⋊C₄	—	2	2	no	—
113	Dic₇⋊C₄	—	2	2	no	—
114	Dic₇⋊C₄	—	2	2	no	—
115	Dic₇⋊C₄	—	2	2	no	—
116	C₄×Dic₇	—	2	2	no	—
117	C₄×Dic₇	—	2	2	no	—
118	C₄×Dic₇	—	2	2	no	—
119	C₄×Dic₇	—	2	2	no	—
120	C₂×C₇⋊C₈	—	2	4	no	—
121	Dic₂₈	—	2	4	no	—
122	Dic₂₈	—	2	2	no	—
123	Dic₂₈	M₁₆	1	4	yes	complemented
124	C₂²×Q₁₆	—	7	1	no	—
125	C₂×C₄×Q₈	—	7	2	no	—
126	#446	—	7	4	no	—
127	#446	—	14	4	no	—
128	C₈×Q₈	—	7	4	no	—
129	C₄×Q₁₆	—	7	1	no	—
130	C₄×Q₁₆	—	7	2	no	—
131	C₄×Q₁₆	—	7	1	no	—
132	C₄×Q₁₆	—	7	1	no	—
133	C₄×Q₁₆	—	7	2	no	—
134	C₄×Q₁₆	—	7	1	no	—
135	#432	—	7	2	no	—
136	#432	—	14	2	no	—
137	#431	—	14	2	no	—
138	#431	C₂×D₇	1	1	yes	characteristic
139	C₂×#107	—	7	1	no	—
140	C₂×#103	—	7	2	no	—
141	C₄×M₁₆	C₂×D₇	1	1	yes	characteristic
142	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
143	#335	—	7	2	no	—
144	#334	—	7	2	no	—
145	Q₈⋊C₈	—	7	4	no	—
146	Q₈⋊C₈	—	7	4	no	—
147	#322	C₂×D₇	1	2	yes	—
148	C₈²	C₂×D₇	1	2	yes	—
149	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₇	—	2	2	no	—
151	C₂×C₄×C₇	—	2	1	no	—
152	C₂×C₄×C₇	—	2	1	no	—
153	C₂×C₄×C₇	—	2	2	no	—
154	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
155	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
156	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
157	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
158	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
160	C₂×Dic₇	—	2	2	no	—
161	C₂×Dic₇	—	2	2	no	—
162	C₂×Dic₇	—	2	2	no	—
163	C₂×Dic₇	—	2	2	no	—
164	C₂×Dic₇	—	4	4	no	—
165	C₂×Dic₇	—	4	4	no	—
166	Dic₁₄	—	2	2	no	—
167	Dic₁₄	—	2	2	no	—
168	Dic₁₄	—	2	2	no	—
169	Dic₁₄	—	4	4	no	—
170	Dic₁₄	C₂×M₁₆	1	4	yes	—
171	C₇×C₈	—	2	4	no	—
172	C₇×C₈	—	2	1	no	—
173	C₇×C₈	—	2	1	no	—
174	C₇×C₈	—	2	1	no	—
175	C₇×C₈	—	2	2	no	—
176	C₇×C₈	—	2	2	no	—
177	C₇×C₈	C₂×M₁₆	1	2	yes	—
178	C₇⋊C₈	—	4	4	no	—
179	C₂²×Q₈	—	7	2	no	—
180	C₂×Q₁₆	—	7	1	no	—
181	C₂×Q₁₆	—	7	2	no	—
182	C₂×Q₁₆	—	7	1	no	—
183	C₂×Q₁₆	—	14	4	no	—
184	C₂×M₁₆	—	2	1	no	—
185	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
186	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
187	C₄×Q₈	—	7	2	no	—
188	C₄×Q₈	—	7	2	no	—
189	C₄×Q₈	—	7	2	no	—
190	C₄×Q₈	—	7	2	no	—
191	C₄×Q₈	—	14	2	no	—
192	C₄×Q₈	—	14	2	no	—
193	C₂×C₄⋊C₄	—	7	2	no	—
194	C₂×C₄⋊C₄	—	14	2	no	—
195	C₂×C₄²	—	14	2	no	—
196	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
197	#107	—	7	2	no	—
198	#107	—	7	1	no	—
199	#107	—	7	1	no	—
200	C₄⋊C₈	—	7	4	no	—
201	C₄⋊C₈	—	14	4	no	—
202	C₄⋊C₈	—	14	4	no	—
203	C₄⋊C₈	D₂₈	1	2	yes	—
204	#103	—	7	2	no	—
205	#103	—	7	2	no	—
206	#103	—	7	2	no	—
207	#103	—	7	2	no	—
208	C₂²⋊C₈	—	14	2	no	—
209	C₂²⋊C₈	D₂₈	1	1	yes	characteristic
210	C₂²⋊C₈	D₂₈	1	1	yes	characteristic
211	#97	—	2	1	no	—
212	#97	—	14	4	no	—
213	#97	—	14	2	no	—
214	C₄×C₈	—	14	4	no	—
215	C₄×C₈	C₄×D₇	1	1	yes	characteristic
216	C₄×C₈	C₄×D₇	1	1	yes	characteristic
217	C₄×C₈	C₂²×D₇	1	2	yes	—
218	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
219	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
220	C₄×C₈	C₂²×D₇	1	2	yes	—
221	C₂²×C₇	—	2	1	no	—
222	C₂²×C₇	—	2	1	no	—
223	C₂²×C₇	#443	1	2	yes	—
224	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
225	C₄×C₇	—	2	1	no	—
226	C₄×C₇	—	2	1	no	—
227	C₄×C₇	—	2	1	no	—
228	C₄×C₇	—	2	1	no	—
229	C₄×C₇	C₂²×M₁₆	1	1	yes	characteristic
230	C₄×C₇	#568	1	1	yes	characteristic
231	C₄×C₇	#578	1	2	yes	—
232	Dic₇	—	2	4	no	—
233	Dic₇	—	4	2	no	—
234	Dic₇	—	4	2	no	—
235	Dic₇	—	4	2	no	—
236	C₂×Q₈	—	7	2	no	—
237	C₂×Q₈	—	7	2	no	—
238	C₂×Q₈	—	14	4	no	—
239	C₂×Q₈	—	14	2	no	—
240	C₂×Q₈	—	28	2	no	—
241	C₂²×C₄	—	14	2	no	—
242	C₂²×C₄	—	14	2	no	—
243	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
245	C₂²×C₄	#1157	1	1	yes	characteristic
246	Q₁₆	—	7	4	no	—
247	Q₁₆	—	14	4	no	—
248	Q₁₆	—	14	2	no	—
249	M₁₆	—	2	4	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	14	4	no	—
254	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
255	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
256	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
257	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
258	C₂×C₈	C₂×D₂₈	1	2	yes	—
259	C₂×C₈	C₂×D₂₈	1	2	yes	—
260	C₄⋊C₄	—	7	2	no	—
261	C₄⋊C₄	—	7	2	no	—
262	C₄⋊C₄	—	14	2	no	—
263	C₄⋊C₄	—	14	2	no	—
264	C₄⋊C₄	—	14	2	no	—
265	C₄⋊C₄	—	14	2	no	—
266	C₄⋊C₄	—	14	2	no	—
267	C₄²	—	14	2	no	—
268	C₄²	—	14	2	no	—
269	C₄²	—	14	2	no	—
270	C₄²	—	14	2	no	—
271	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
272	C₄²	C₂×D₂₈	1	1	yes	characteristic
273	C₄²	#1157	1	1	yes	characteristic
274	C₄²	C₂³×D₇	1	1	yes	characteristic
275	C₂×C₇	—	2	2	no	—
276	C₂×C₇	D₄×M₁₆	1	1	yes	characteristic
277	C₂×C₇	#2950	1	1	yes	characteristic
278	C₂×C₇	C₂×#443	1	1	yes	characteristic
279	C₂³	C₄×D₂₈	1	1	yes	characteristic
280	Q₈	—	7	4	no	—
281	Q₈	—	14	2	no	—
282	Q₈	—	14	2	no	—
283	Q₈	—	14	2	no	—
284	Q₈	—	28	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	14	2	no	—
290	C₂×C₄	—	14	2	no	—
291	C₂×C₄	—	14	2	no	—
292	C₂×C₄	—	14	2	no	—
293	C₂×C₄	—	28	4	no	—
294	C₂×C₄	—	28	4	no	—
295	C₂×C₄	C₄×D₂₈	1	1	yes	characteristic
296	C₂×C₄	C₄×D₂₈	1	1	yes	characteristic
297	C₂×C₄	C₄×D₂₈	1	1	yes	characteristic
298	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
299	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
300	C₂×C₄	C₂×#1157	1	1	yes	characteristic
301	C₈	—	2	4	no	—
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	—	2	2	no	—
306	C₈	—	2	2	no	—
307	C₈	—	28	4	no	—
308	C₈	D₇×M₁₆	1	2	yes	—
309	C₇	#18058	1	1	yes	characteristic, complemented, Sylow
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	#89472	1	2	yes	—
313	C₂²	C₂×C₄×D₂₈	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	14	4	no	—
319	C₄	—	28	2	no	—
320	C₄	—	28	2	no	—
321	C₄	—	28	2	no	—
322	C₄	C₂×D₇×M₁₆	1	1	yes	characteristic
323	C₄	#90436	1	1	yes	characteristic
324	C₄	#90444	1	2	yes	—
325	C₂	—	2	2	no	—
326	C₂	C₂×#89472	1	1	yes	characteristic
327	C₂	M₁₆×D₂₈	1	1	yes	characteristic
328	C₂	#11736018	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₈
commutator	C₂×C₄×C₇	C₂³×C₄
Frattini	C₄²	C₂³×D₇
Fitting	C₇×C₈×M₁₆	C₂
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	C₂³×C₇	C₄×D₄
cosocle	C₄²×C₇	C₂⁴
hypercenter	C₈×M₁₆	D₇
hypocenter	C₇	#18058
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
6. #11725841	C₂	semidirect
7. #11725810	C₂	semidirect
8. C₈×Dic₂₈	C₂	semidirect
123. Dic₂₈	M₁₆	semidirect
309. C₇	#18058	semidirect

Maximal subgroups

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

subgroup	index
#11736710	2
C₂×C₄×Dic₂₈	2
C₇×C₈×M₁₆	2
#11727611	2
#11727486	2
#11725841	2
#11725810	2
C₈×Dic₂₈	2
#18058	7