Subgroups — #13509358

Order 1792.

558 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11735920	C₂	1	1	yes	maximal, characteristic, complemented
2	#11735915	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#11729168	C₂	1	4	yes	maximal, complemented
5	#11726819	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#6374	C₂	1	4	yes	maximal, complemented
7	#11726815	C₂	1	1	yes	maximal, characteristic, complemented
8	#11726812	C₂	1	1	yes	maximal, characteristic, complemented
9	#11726800	C₂	1	1	yes	maximal, characteristic, complemented
10	#90168	—	2	1	no	—
11	#90161	—	2	1	no	—
12	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	#89888	—	2	1	no	—
15	#89888	C₂²	1	2	yes	—
16	#89887	—	2	1	no	—
17	#89887	C₂²	1	2	yes	—
18	C₂×#6374	C₂²	1	4	yes	—
19	#89471	C₂²	1	1	yes	characteristic
20	C₄×Dic₂₈	—	2	1	no	—
21	C₄×Dic₂₈	C₂²	1	2	yes	—
22	#89467	C₂²	1	1	yes	characteristic
23	#89466	—	2	1	no	—
24	C₄×D₅₆	—	2	1	no	—
25	C₄×D₅₆	C₂²	1	2	yes	—
26	#89454	C₂²	1	1	yes	characteristic
27	#89454	C₂²	1	1	yes	characteristic
28	#89453	—	2	1	no	—
29	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
30	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
31	C₄×C₇×C₁₆	C₂²	1	2	yes	—
32	#89304	C₂²	1	4	yes	—
33	#89300	C₂²	1	4	yes	—
34	#89298	C₂²	1	4	yes	—
35	#33736	—	7	1	no	maximal, Sylow
36	C₂×#1157	—	4	1	no	—
37	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
38	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
40	#6493	—	2	2	no	—
41	#6493	—	4	1	no	—
42	#6489	—	4	1	no	—
43	#6487	—	2	2	no	—
44	#6487	—	4	1	no	—
45	C₂×Dic₂₈	—	2	1	no	—
46	C₂×Dic₂₈	C₂³	1	2	yes	—
47	C₂×D₅₆	—	2	1	no	—
48	C₂×D₅₆	C₂³	1	2	yes	—
49	#6438	—	4	1	no	—
50	#6437	—	2	1	no	—
51	C₄×D₂₈	—	2	2	no	—
52	C₄×D₂₈	—	4	1	no	—
53	#6433	—	4	1	no	—
54	#6432	—	2	1	no	—
55	#6432	—	2	1	no	—
56	#6432	—	2	1	no	—
57	C₄×Dic₁₄	—	2	2	no	—
58	C₄×Dic₁₄	—	4	1	no	—
59	C₂×C₇×C₁₆	—	2	4	no	—
60	C₂×C₇×C₁₆	C₂³	1	2	yes	—
61	C₂×C₇×C₁₆	C₂³	1	2	yes	—
62	C₄×C₇×C₈	D₄	1	1	yes	characteristic
63	C₄×C₇×C₈	D₄	1	1	yes	characteristic
64	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
65	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
66	#6395	—	2	2	no	—
67	#6395	—	2	2	no	—
68	#6392	—	2	1	no	—
69	#6392	—	2	1	no	—
70	#6392	C₂³	1	2	yes	—
71	#6392	C₂³	1	2	yes	—
72	#6390	—	2	2	no	—
73	#6390	—	2	2	no	—
74	#6374	—	2	8	no	—
75	#3170	—	7	1	no	—
76	#3169	—	7	1	no	—
77	#2254	—	7	1	no	—
78	#2248	—	7	1	no	—
79	#2241	—	7	1	no	—
80	#2240	—	7	1	no	—
81	#2217	—	7	4	no	—
82	C₄×SD₃₂	—	7	4	no	—
83	C₂×C₄×C₁₆	D₇	1	1	yes	characteristic, complemented
84	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₂²⋊D₇	—	4	2	no	—
88	#1157	—	8	4	no	—
89	C₂×D₂₈	—	2	2	no	—
90	C₂×D₂₈	—	4	1	no	—
91	C₂×D₂₈	—	4	1	no	—
92	C₂×C₄×D₇	—	4	2	no	—
93	C₂×Dic₁₄	—	2	2	no	—
94	C₂×Dic₁₄	—	4	1	no	—
95	C₂×Dic₁₄	—	4	1	no	—
96	C₂×C₇×C₈	—	2	2	no	—
97	C₂×C₇×C₈	—	2	2	no	—
98	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
100	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
102	C₄²×C₇	D₈	1	1	yes	characteristic
103	C₄²×C₇	D₈	1	1	yes	characteristic
104	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
106	C₂²⋊Dic₇	—	4	2	no	—
107	(C₂×D₇)⋊C₄	—	4	2	no	—
108	(C₂×D₇)⋊C₄	—	4	2	no	—
109	C₄⋊Dic₇	—	2	2	no	—
110	C₄⋊Dic₇	—	2	2	no	—
111	C₄⋊Dic₇	—	4	1	no	—
112	C₄⋊Dic₇	—	4	1	no	—
113	C₄⋊Dic₇	—	4	1	no	—
114	C₄⋊Dic₇	—	4	1	no	—
115	Dic₇⋊C₄	—	4	2	no	—
116	Dic₇⋊C₄	—	4	2	no	—
117	C₄×Dic₇	—	4	2	no	—
118	Dic₂₈	—	2	4	no	—
119	Dic₂₈	—	4	2	no	—
120	D₅₆	—	2	4	no	—
121	D₅₆	—	4	2	no	—
122	C₇×C₁₆	—	2	2	no	—
123	C₇×C₁₆	#41	1	4	yes	complemented
124	#533	—	14	1	no	—
125	#532	—	14	1	no	—
126	C₂×SD₃₂	—	7	4	no	—
127	C₂²×C₁₆	C₂×D₇	1	2	yes	—
128	#500	—	7	1	no	—
129	#500	—	7	1	no	—
130	#494	—	7	1	no	—
131	#493	—	7	1	no	—
132	#487	—	14	1	no	—
133	#486	—	14	1	no	—
134	#467	—	7	2	no	—
135	#467	—	14	1	no	—
136	#466	—	7	2	no	—
137	#466	—	14	1	no	—
138	C₄×Q₁₆	—	7	2	no	—
139	C₄×Q₁₆	—	14	1	no	—
140	C₄×D₈	—	7	2	no	—
141	C₄×D₈	—	14	1	no	—
142	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
143	#367	—	7	4	no	—
144	#358	—	7	4	no	—
145	#357	—	7	4	no	—
146	C₄×C₁₆	C₂×D₇	1	2	yes	—
147	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
148	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
149	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
150	C₂²×D₇	—	4	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	SD₃₂	1	4	yes	complemented
155	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
158	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
160	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
161	C₂²⋊D₇	—	8	4	no	—
162	C₂×Dic₇	—	4	2	no	—
163	C₂×Dic₇	—	4	2	no	—
164	C₂×Dic₇	—	4	2	no	—
165	D₂₈	—	4	2	no	—
166	D₂₈	—	4	2	no	—
167	D₂₈	—	8	2	no	—
168	D₂₈	—	8	2	no	—
169	C₄×D₇	—	8	4	no	—
170	Dic₁₄	—	4	2	no	—
171	Dic₁₄	—	4	2	no	—
172	Dic₁₄	—	8	2	no	—
173	Dic₁₄	—	8	2	no	—
174	C₇×C₈	—	2	1	no	—
175	C₇×C₈	—	2	1	no	—
176	C₇×C₈	#135	1	2	yes	—
177	C₇×C₈	C₂×#41	1	2	yes	—
178	C₂×#41	—	28	1	no	—
179	C₂×Q₁₆	—	7	2	no	—
180	C₂×Q₁₆	—	14	1	no	—
181	C₂×D₈	—	7	2	no	—
182	C₂×D₈	—	14	1	no	—
183	C₂²×C₈	D₂₈	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
185	C₄⋊Q₈	—	14	1	no	—
186	C₄⋊Q₈	—	14	1	no	—
187	C₄⋊Q₈	—	14	1	no	—
188	#127	—	14	1	no	—
189	#125	—	28	1	no	—
190	#124	—	28	1	no	—
191	C₂²⋊Q₈	—	14	2	no	—
192	C₂²⋊Q₈	—	28	1	no	—
193	#121	—	14	2	no	—
194	#121	—	28	1	no	—
195	C₄×Q₈	—	14	2	no	—
196	C₄×Q₈	—	28	1	no	—
197	C₄×D₄	—	14	2	no	—
198	C₄×D₄	—	28	1	no	—
199	#117	—	28	1	no	—
200	C₂×C₄²	D₂₈	1	1	yes	characteristic
201	SD₃₂	—	14	8	no	—
202	C₂×C₁₆	—	2	4	no	—
203	C₂×C₁₆	C₂²×D₇	1	2	yes	—
204	C₂×C₁₆	C₂²×D₇	1	2	yes	—
205	#107	—	7	2	no	—
206	#107	—	7	2	no	—
207	#107	—	14	1	no	—
208	#107	—	14	1	no	—
209	#103	—	14	2	no	—
210	#103	—	14	2	no	—
211	#102	—	14	2	no	—
212	#102	—	14	2	no	—
213	C₄×C₈	D₂₈	1	1	yes	characteristic
214	C₄×C₈	D₂₈	1	1	yes	characteristic
215	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
216	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
217	C₂²×C₇	—	2	1	no	—
218	C₂²×C₇	—	2	1	no	—
219	C₂²×C₇	C₂×SD₃₂	1	2	yes	—
220	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
221	C₂×D₇	—	8	2	no	—
222	C₂×D₇	—	8	2	no	—
223	C₂×D₇	—	8	2	no	—
224	C₄×C₇	—	2	1	no	—
225	C₄×C₇	—	2	1	no	—
226	C₄×C₇	C₂×SD₃₂	1	2	yes	—
227	C₄×C₇	C₂×SD₃₂	1	2	yes	—
228	C₄×C₇	#508	1	2	yes	—
229	C₄×C₇	#532	1	1	yes	characteristic
230	C₄×C₇	C₂×#135	1	1	yes	characteristic
231	Dic₇	—	8	2	no	—
232	Dic₇	—	8	2	no	—
233	Dic₇	—	8	2	no	—
234	#41	—	56	4	no	—
235	C₂×Q₈	—	14	2	no	—
236	C₂×Q₈	—	28	1	no	—
237	C₂×Q₈	—	28	1	no	—
238	C₂×D₄	—	14	2	no	—
239	C₂×D₄	—	28	2	no	—
240	C₂×D₄	—	28	1	no	—
241	C₂×D₄	—	28	1	no	—
242	C₂²×C₄	—	28	2	no	—
243	C₂²×C₄	D₅₆	1	1	yes	characteristic
244	C₂²×C₄	D₅₆	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
246	Q₁₆	—	14	4	no	—
247	Q₁₆	—	28	2	no	—
248	D₈	—	14	4	no	—
249	D₈	—	28	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
253	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
254	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
255	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
256	C₄⋊C₄	—	14	2	no	—
257	C₄⋊C₄	—	14	2	no	—
258	C₄⋊C₄	—	28	2	no	—
259	C₄⋊C₄	—	28	2	no	—
260	C₄⋊C₄	—	28	1	no	—
261	C₄⋊C₄	—	28	1	no	—
262	C₄⋊C₄	—	28	1	no	—
263	C₄⋊C₄	—	28	1	no	—
264	C₂²⋊C₄	—	28	2	no	—
265	C₂²⋊C₄	—	28	2	no	—
266	C₂²⋊C₄	—	28	2	no	—
267	C₄²	—	28	2	no	—
268	C₄²	D₅₆	1	1	yes	characteristic
269	C₄²	D₅₆	1	1	yes	characteristic
270	C₄²	C₂×D₂₈	1	1	yes	characteristic
271	C₄²	C₂×D₂₈	1	1	yes	characteristic
272	C₁₆	—	2	2	no	—
273	C₁₆	#1157	1	4	yes	complemented
274	C₂×C₇	—	2	2	no	—
275	C₂×C₇	#3169	1	1	yes	characteristic
276	C₂×C₇	C₂²×SD₃₂	1	1	yes	characteristic
277	C₂×C₇	C₂×#508	1	1	yes	characteristic
278	D₇	—	16	2	no	—
279	C₂³	—	28	2	no	—
280	C₂³	C₂×D₅₆	1	1	yes	characteristic
281	Q₈	—	28	2	no	—
282	Q₈	—	28	2	no	—
283	Q₈	—	56	2	no	—
284	Q₈	—	56	2	no	—
285	D₄	—	28	2	no	—
286	D₄	—	28	2	no	—
287	D₄	—	56	4	no	—
288	D₄	—	56	2	no	—
289	D₄	—	56	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	28	2	no	—
294	C₂×C₄	—	28	2	no	—
295	C₂×C₄	—	28	2	no	—
296	C₂×C₄	—	56	4	no	—
297	C₂×C₄	#6374	1	4	yes	complemented
298	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
299	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
300	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
301	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
302	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	#6467	1	2	yes	—
307	C₈	C₂×#1157	1	2	yes	—
308	C₇	#33736	1	1	yes	characteristic, complemented, Sylow
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	56	2	no	—
312	C₂²	—	56	2	no	—
313	C₂²	—	56	2	no	—
314	C₂²	C₂×#6374	1	2	yes	—
315	C₂²	C₂²×D₅₆	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	56	2	no	—
319	C₄	—	56	2	no	—
320	C₄	—	56	2	no	—
321	C₄	C₂×#6374	1	2	yes	—
322	C₄	C₂×#6374	1	2	yes	—
323	C₄	#89676	1	2	yes	—
324	C₄	#90168	1	1	yes	characteristic
325	C₄	C₂×#6467	1	1	yes	characteristic
326	C₂	—	2	2	no	—
327	C₂	—	112	2	no	—
328	C₂	#11735915	1	1	yes	characteristic
329	C₂	C₂²×#6374	1	1	yes	characteristic
330	C₂	C₂×#89676	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅₆
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂×C₄×C₇×C₁₆	C₂
socle	C₂²×C₇	C₂²×D₈
prime radical	#11735915	C₂
hypercenter	C₂×C₄×C₁₆	D₇
hypocenter	C₇	#33736
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. #11735920	C₂	semidirect
3. C₂×C₄×C₇×C₁₆	C₂	semidirect
4. #11729168	C₂	semidirect
5. #11726819	C₂	semidirect
6. C₄×#6374	C₂	semidirect
7. #11726815	C₂	semidirect
8. #11726812	C₂	semidirect
9. #11726800	C₂	semidirect
83. C₂×C₄×C₁₆	D₇	semidirect
123. C₇×C₁₆	#41	semidirect
154. C₂×C₄×C₇	SD₃₂	semidirect
273. C₁₆	#1157	semidirect
297. C₂×C₄	#6374	semidirect
308. C₇	#33736	semidirect

Maximal subgroups

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

subgroup	index
#11735920	2
#11735915	2
C₂×C₄×C₇×C₁₆	2
#11729168	2
#11726819	2
C₄×#6374	2
#11726815	2
#11726812	2
#11726800	2
#33736	7