Subgroups — #13324321

Order 1792.

682 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11735913	C₂	1	1	yes	maximal, characteristic
2	#11735903	C₂	1	1	yes	maximal, characteristic
3	#11735890	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₇×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
5	#11727492	C₂	1	1	yes	maximal, characteristic, complemented
6	#11727489	C₂	1	1	yes	maximal, characteristic
7	#11727484	C₂	1	1	yes	maximal, characteristic, complemented
8	#11725809	C₂	1	4	yes	maximal, complemented
9	C₈×#1132	C₂	1	4	yes	maximal, complemented
10	C₂×#6464	—	2	1	no	—
11	#90168	C₂²	1	1	yes	characteristic
12	#90161	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
16	#89886	C₄	1	4	yes	—
17	#89882	—	2	1	no	—
18	#89874	—	2	1	no	—
19	#89873	—	2	1	no	—
20	#89696	C₂²	1	1	yes	characteristic
21	#89696	C₂²	1	1	yes	characteristic
22	#89466	C₄	1	1	yes	characteristic
23	#89465	C₄	1	1	yes	characteristic
24	C₄×#1132	C₂²	1	4	yes	—
25	#89460	—	2	1	no	—
26	#89460	C₂²	1	2	yes	—
27	C₈×D₂₈	—	2	1	no	—
28	C₈×D₂₈	C₂²	1	2	yes	—
29	#89453	C₄	1	1	yes	characteristic
30	#89452	C₄	1	1	yes	characteristic
31	#89451	—	2	1	no	—
32	#89451	C₂²	1	2	yes	—
33	C₈×Dic₁₄	—	2	1	no	—
34	C₈×Dic₁₄	C₂²	1	2	yes	—
35	C₇×C₈²	C₂²	1	2	yes	—
36	C₇×C₈²	C₂²	1	2	yes	—
37	#89254	C₂²	1	4	yes	—
38	#89252	C₂²	1	4	yes	—
39	#89251	C₂²	1	4	yes	—
40	#17957	—	7	1	no	maximal, Sylow
41	C₂×#1157	—	2	1	no	—
42	C₂²×C₇×C₈	—	2	1	no	—
43	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
44	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
45	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
47	#6493	—	2	1	no	—
48	#6493	C₂×C₄	1	2	yes	—
49	#6489	—	2	1	no	—
50	#6487	—	2	1	no	—
51	#6487	C₂×C₄	1	2	yes	—
52	C₂×#1136	—	2	1	no	—
53	C₂×#1136	—	2	1	no	—
54	C₂×#1132	C₂×C₄	1	4	yes	—
55	#6464	—	4	4	no	—
56	C₂×#1131	—	2	2	no	—
57	C₂×C₈×D₇	—	2	2	no	—
58	#6438	—	2	1	no	—
59	#6437	C₂×C₄	1	1	yes	characteristic
60	C₄×D₂₈	—	2	1	no	—
61	C₄×D₂₈	C₂³	1	2	yes	—
62	#6433	—	2	1	no	—
63	#6432	C₂×C₄	1	1	yes	characteristic
64	#6432	C₂×C₄	1	1	yes	characteristic
65	#6432	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₁₄	—	2	1	no	—
67	C₄×Dic₁₄	C₂³	1	2	yes	—
68	C₄×C₇×C₈	—	2	1	no	—
69	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
70	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
71	C₄×C₇×C₈	D₄	1	1	yes	characteristic
72	C₄×C₇×C₈	D₄	1	1	yes	characteristic
73	C₄×C₇×C₈	C₂³	1	2	yes	—
74	C₄×C₇×C₈	C₂³	1	2	yes	—
75	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
76	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
77	#6395	C₂×C₄	1	4	yes	—
78	(C₂×D₇)⋊C₈	—	2	2	no	—
79	(C₂×D₇)⋊C₈	—	2	2	no	—
80	#6391	C₂×C₄	1	4	yes	—
81	#6390	C₂×C₄	1	4	yes	—
82	#6389	—	2	2	no	—
83	Dic₇⋊C₈	—	2	2	no	—
84	Dic₇⋊C₈	—	2	2	no	—
85	C₈×Dic₇	—	2	2	no	—
86	C₄⋊(C₇⋊C₈)	—	2	1	no	—
87	C₄⋊(C₇⋊C₈)	—	2	1	no	—
88	C₄⋊(C₇⋊C₈)	C₂³	1	2	yes	—
89	C₄⋊(C₇⋊C₈)	C₂³	1	2	yes	—
90	#3167	—	7	1	no	—
91	#2960	—	7	1	no	—
92	#2931	—	7	1	no	—
93	#1590	—	7	4	no	—
94	C₈×SD₁₆	—	7	4	no	—
95	#1566	—	7	1	no	—
96	#1493	—	7	1	no	—
97	#1492	—	7	1	no	—
98	C₂×C₈²	D₇	1	1	yes	characteristic, complemented
99	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₇	#41	1	1	yes	characteristic
102	C₂×C₂²⋊D₇	—	2	2	no	—
103	#1157	—	4	2	no	—
104	#1157	—	4	2	no	—
105	C₂×D₂₈	—	2	1	no	—
106	C₂×D₂₈	—	2	1	no	—
107	C₂×D₂₈	C₂²×C₄	1	2	yes	—
108	C₂×C₄×D₇	—	2	2	no	—
109	C₂×Dic₁₄	—	2	1	no	—
110	C₂×Dic₁₄	—	2	1	no	—
111	C₂×Dic₁₄	C₂²×C₄	1	2	yes	—
112	C₂×C₇×C₈	—	2	4	no	—
113	C₂×C₇×C₈	—	2	2	no	—
114	C₂×C₇×C₈	—	2	2	no	—
115	C₂×C₇×C₈	—	2	4	no	—
116	C₂×C₇×C₈	SD₁₆	1	4	yes	complemented
117	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
118	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
119	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
120	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
121	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
122	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
124	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₇	#41	1	1	yes	characteristic
126	C₄²×C₇	C₂⁴	1	1	yes	characteristic
127	C₂²⋊Dic₇	—	2	2	no	—
128	(C₂×D₇)⋊C₄	—	2	2	no	—
129	(C₂×D₇)⋊C₄	—	2	2	no	—
130	C₄⋊Dic₇	—	2	1	no	—
131	C₄⋊Dic₇	—	2	1	no	—
132	C₄⋊Dic₇	—	2	1	no	—
133	C₄⋊Dic₇	—	2	1	no	—
134	C₄⋊Dic₇	C₂²×C₄	1	2	yes	—
135	C₄⋊Dic₇	C₂²×C₄	1	2	yes	—
136	Dic₇⋊C₄	—	2	2	no	—
137	Dic₇⋊C₄	—	2	2	no	—
138	C₄×Dic₇	—	2	2	no	—
139	#1136	—	4	2	no	—
140	#1136	—	4	2	no	—
141	C₂×C₇⋊C₈	—	2	2	no	—
142	C₂×C₇⋊C₈	—	2	2	no	—
143	#1132	—	2	8	no	—
144	#1131	—	4	4	no	—
145	C₈×D₇	—	4	4	no	—
146	C₂×#131	—	14	1	no	—
147	#533	—	7	1	no	—
148	#532	—	7	1	no	—
149	#498	—	7	1	no	—
150	#492	—	7	1	no	—
151	#487	—	7	1	no	—
152	#486	—	7	1	no	—
153	#465	—	7	4	no	—
154	#446	—	7	2	no	—
155	#446	—	14	1	no	—
156	C₈×Q₈	—	7	2	no	—
157	C₈×Q₈	—	14	1	no	—
158	C₄×SD₁₆	—	7	4	no	—
159	#436	—	7	2	no	—
160	#436	—	14	1	no	—
161	C₈×D₄	—	7	2	no	—
162	C₈×D₄	—	14	1	no	—
163	#424	—	14	1	no	—
164	#423	—	7	1	no	—
165	#423	—	7	1	no	—
166	#408	—	14	1	no	—
167	#405	—	14	1	no	—
168	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
169	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
170	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
171	#334	—	7	4	no	—
172	Q₈⋊C₈	—	7	4	no	—
173	D₄⋊C₈	—	7	4	no	—
174	C₈²	C₂×D₇	1	2	yes	—
175	C₈²	C₂×D₇	1	2	yes	—
176	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
177	C₂²×D₇	—	2	2	no	—
178	C₂×C₄×C₇	—	2	2	no	—
179	C₂×C₄×C₇	—	2	2	no	—
180	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
181	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
182	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
183	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
184	C₂×C₄×C₇	#135	1	2	yes	—
185	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
186	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
188	C₂²⋊D₇	—	4	4	no	—
189	C₂×Dic₇	—	2	2	no	—
190	C₂×Dic₇	—	2	2	no	—
191	C₂×Dic₇	—	2	2	no	—
192	D₂₈	—	2	4	no	—
193	D₂₈	—	4	2	no	—
194	D₂₈	—	4	2	no	—
195	C₄×D₇	—	4	2	no	—
196	C₄×D₇	—	4	2	no	—
197	Dic₁₄	—	2	4	no	—
198	Dic₁₄	—	4	2	no	—
199	Dic₁₄	—	4	2	no	—
200	C₇×C₈	—	2	4	no	—
201	C₇×C₈	—	2	2	no	—
202	C₇×C₈	—	2	1	no	—
203	C₇×C₈	—	2	1	no	—
204	C₇×C₈	#131	1	4	yes	complemented
205	C₇×C₈	C₂×SD₁₆	1	2	yes	—
206	C₇×C₈	C₂×SD₁₆	1	2	yes	—
207	C₇⋊C₈	—	4	2	no	—
208	C₇⋊C₈	—	4	2	no	—
209	C₂×#41	—	14	1	no	—
210	C₂×SD₁₆	—	7	4	no	—
211	#131	—	28	4	no	—
212	C₂×M₁₆	—	14	1	no	—
213	C₂×M₁₆	—	14	1	no	—
214	C₂×M₁₆	—	14	2	no	—
215	C₂²×C₈	—	2	1	no	—
216	C₂²×C₈	—	14	2	no	—
217	C₂²×C₈	C₄×D₇	1	2	yes	—
218	C₂²×C₈	D₂₈	1	1	yes	characteristic
219	C₂²×C₈	D₂₈	1	1	yes	characteristic
220	C₄⋊Q₈	—	7	1	no	—
221	C₄⋊Q₈	—	7	1	no	—
222	C₄⋊Q₈	—	7	1	no	—
223	#127	—	7	1	no	—
224	#125	—	14	1	no	—
225	#124	—	14	1	no	—
226	C₂²⋊Q₈	—	7	2	no	—
227	C₂²⋊Q₈	—	14	1	no	—
228	#121	—	7	2	no	—
229	#121	—	14	1	no	—
230	C₄×Q₈	—	7	2	no	—
231	C₄×Q₈	—	14	1	no	—
232	C₄×D₄	—	7	2	no	—
233	C₄×D₄	—	14	1	no	—
234	#117	—	14	1	no	—
235	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
236	#106	—	7	4	no	—
237	C₄⋊C₈	—	7	2	no	—
238	C₄⋊C₈	—	7	2	no	—
239	C₄⋊C₈	—	14	2	no	—
240	C₄⋊C₈	—	14	2	no	—
241	C₄⋊C₈	—	14	1	no	—
242	C₄⋊C₈	—	14	1	no	—
243	#103	—	7	4	no	—
244	#102	—	7	4	no	—
245	C₂²⋊C₈	—	14	2	no	—
246	C₂²⋊C₈	—	14	2	no	—
247	#97	—	14	2	no	—
248	C₄×C₈	—	2	1	no	—
249	C₄×C₈	—	14	2	no	—
250	C₄×C₈	C₄×D₇	1	1	yes	characteristic
251	C₄×C₈	C₄×D₇	1	1	yes	characteristic
252	C₄×C₈	D₂₈	1	1	yes	characteristic
253	C₄×C₈	D₂₈	1	1	yes	characteristic
254	C₄×C₈	C₂²×D₇	1	2	yes	—
255	C₄×C₈	C₂²×D₇	1	2	yes	—
256	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
257	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
258	C₂²×C₇	—	2	1	no	—
259	C₂²×C₇	—	2	1	no	—
260	C₂²×C₇	C₄×SD₁₆	1	2	yes	—
261	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
262	C₂×D₇	—	4	2	no	—
263	C₂×D₇	—	4	2	no	—
264	C₂×D₇	—	4	2	no	—
265	C₄×C₇	—	2	1	no	—
266	C₄×C₇	—	2	1	no	—
267	C₄×C₇	C₄×SD₁₆	1	2	yes	—
268	C₄×C₇	#443	1	2	yes	—
269	C₄×C₇	C₂×#131	1	2	yes	—
270	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
271	C₄×C₇	C₂×#135	1	1	yes	characteristic
272	Dic₇	—	4	2	no	—
273	Dic₇	—	4	2	no	—
274	Dic₇	—	4	2	no	—
275	#41	—	28	2	no	—
276	#41	—	28	2	no	—
277	C₂×Q₈	—	7	2	no	—
278	C₂×Q₈	—	14	1	no	—
279	C₂×Q₈	—	14	1	no	—
280	C₂×D₄	—	7	2	no	—
281	C₂×D₄	—	14	2	no	—
282	C₂×D₄	—	14	1	no	—
283	C₂×D₄	—	14	1	no	—
284	C₂²×C₄	—	14	2	no	—
285	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
286	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
287	C₂²×C₄	#1157	1	1	yes	characteristic
288	SD₁₆	—	14	8	no	—
289	M₁₆	—	28	2	no	—
290	M₁₆	—	28	2	no	—
291	M₁₆	—	28	4	no	—
292	C₂×C₈	—	2	4	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	4	no	—
296	C₂×C₈	—	14	2	no	—
297	C₂×C₈	—	14	2	no	—
298	C₂×C₈	—	28	4	no	—
299	C₂×C₈	#1132	1	4	yes	complemented
300	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
301	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
302	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
303	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
304	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
305	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
306	C₄⋊C₄	—	7	2	no	—
307	C₄⋊C₄	—	7	2	no	—
308	C₄⋊C₄	—	14	2	no	—
309	C₄⋊C₄	—	14	1	no	—
310	C₄⋊C₄	—	14	1	no	—
311	C₄⋊C₄	—	14	1	no	—
312	C₄⋊C₄	—	14	1	no	—
313	C₄⋊C₄	—	14	2	no	—
314	C₂²⋊C₄	—	14	2	no	—
315	C₂²⋊C₄	—	14	2	no	—
316	C₂²⋊C₄	—	14	2	no	—
317	C₄²	—	14	2	no	—
318	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
319	C₄²	C₂×D₂₈	1	1	yes	characteristic
320	C₄²	#1157	1	1	yes	characteristic
321	C₄²	C₂³×D₇	1	1	yes	characteristic
322	C₂×C₇	—	2	2	no	—
323	C₂×C₇	#2931	1	1	yes	characteristic
324	C₂×C₇	C₂×C₄×SD₁₆	1	1	yes	characteristic
325	C₂×C₇	C₂×#443	1	1	yes	characteristic
326	D₇	—	8	2	no	—
327	C₂³	—	14	2	no	—
328	C₂³	C₄×D₂₈	1	1	yes	characteristic
329	Q₈	—	14	4	no	—
330	Q₈	—	28	2	no	—
331	Q₈	—	28	2	no	—
332	D₄	—	14	4	no	—
333	D₄	—	28	4	no	—
334	D₄	—	28	2	no	—
335	D₄	—	28	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	14	2	no	—
339	C₂×C₄	—	14	2	no	—
340	C₂×C₄	—	14	2	no	—
341	C₂×C₄	—	28	2	no	—
342	C₂×C₄	—	28	2	no	—
343	C₂×C₄	C₄×D₂₈	1	1	yes	characteristic
344	C₂×C₄	C₄×D₂₈	1	1	yes	characteristic
345	C₂×C₄	C₄×D₂₈	1	1	yes	characteristic
346	C₂×C₄	C₂×#1132	1	2	yes	—
347	C₂×C₄	#6467	1	2	yes	—
348	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
349	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
350	C₂×C₄	C₂×#1157	1	1	yes	characteristic
351	C₈	—	2	4	no	—
352	C₈	—	2	2	no	—
353	C₈	—	2	1	no	—
354	C₈	—	2	1	no	—
355	C₈	—	28	2	no	—
356	C₈	—	28	2	no	—
357	C₈	#6464	1	4	yes	complemented
358	C₈	C₂×#1132	1	2	yes	—
359	C₈	C₂×#1132	1	2	yes	—
360	C₇	#17957	1	1	yes	characteristic, complemented, Sylow
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	28	2	no	—
364	C₂²	—	28	2	no	—
365	C₂²	—	28	2	no	—
366	C₂²	C₄×#1132	1	2	yes	—
367	C₂²	C₂×C₄×D₂₈	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	28	2	no	—
371	C₄	—	28	2	no	—
372	C₄	—	28	2	no	—
373	C₄	C₄×#1132	1	2	yes	—
374	C₄	#89472	1	2	yes	—
375	C₄	C₂×#6464	1	2	yes	—
376	C₄	C₂²×#1132	1	1	yes	characteristic
377	C₄	C₂×#6467	1	1	yes	characteristic
378	C₂	—	2	2	no	—
379	C₂	—	56	2	no	—
380	C₂	#11735903	1	1	yes	characteristic
381	C₂	C₂×C₄×#1132	1	1	yes	characteristic
382	C₂	C₂×#89472	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₈
commutator	C₂×C₄×C₇	C₂³×C₄
Frattini	C₄²	C₂³×D₇
Fitting	C₂×C₇×C₈²	C₂
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	#90168	C₂²
cosocle	C₄²×C₇	C₂⁴
hypercenter	C₂×C₈²	D₇
hypocenter	C₇	#17957
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
3. #11735890	C₂	semidirect
4. C₂×C₇×C₈²	C₂	semidirect
5. #11727492	C₂	semidirect
7. #11727484	C₂	semidirect
8. #11725809	C₂	semidirect
9. C₈×#1132	C₂	semidirect
98. C₂×C₈²	D₇	semidirect
116. C₂×C₇×C₈	SD₁₆	semidirect
204. C₇×C₈	#131	semidirect
299. C₂×C₈	#1132	semidirect
357. C₈	#6464	semidirect
360. C₇	#17957	semidirect

Maximal subgroups

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

subgroup	index
#11735913	2
#11735903	2
#11735890	2
C₂×C₇×C₈²	2
#11727492	2
#11727489	2
#11727484	2
#11725809	2
C₈×#1132	2
#17957	7