Subgroups — #13324335

Order 1792.

630 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11736770	C₂	1	2	yes	maximal, complemented
2	#11735914	C₂	1	1	yes	maximal, characteristic
3	C₇×#1568	C₂	1	1	yes	maximal, characteristic, complemented
4	#11728554	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#11726318	C₂	1	1	yes	maximal, characteristic, complemented
7	#11726317	C₂	1	1	yes	maximal, characteristic, complemented
8	#11726311	C₂	1	1	yes	maximal, characteristic, complemented
9	#11726310	C₂	1	1	yes	maximal, characteristic, complemented
10	#11726277	C₂	1	2	yes	maximal, complemented
11	#11726270	C₂	1	2	yes	maximal, complemented
12	#90170	C₂²	1	2	yes	—
13	C₇×#432	C₂²	1	2	yes	—
14	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
15	#89888	C₄	1	1	yes	characteristic
16	#89887	C₄	1	1	yes	characteristic
17	#89886	C₄	1	2	yes	—
18	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
20	#89610	C₂²	1	2	yes	—
21	#89606	C₂²	1	2	yes	—
22	#89603	C₂²	1	2	yes	—
23	#89602	C₂²	1	2	yes	—
24	#89502	—	2	2	no	—
25	(C₂²⋊D₇)⋊C₈	—	2	2	no	—
26	#89481	—	2	2	no	—
27	#89480	—	2	2	no	—
28	C₄×Dic₂₈	C₂²	1	1	yes	characteristic
29	#89469	C₄	1	2	yes	—
30	#89468	C₄	1	1	yes	characteristic
31	C₄×D₅₆	C₂²	1	1	yes	characteristic
32	C₄×#1132	C₂²	1	2	yes	—
33	#89455	C₄	1	1	yes	characteristic
34	C₇×#335	C₂²	1	1	yes	characteristic
35	C₇×#335	C₂²	1	1	yes	characteristic
36	C₇×#334	C₂²	1	2	yes	—
37	#89279	C₂²	1	2	yes	—
38	#89279	C₂²	1	2	yes	—
39	#89278	C₂²	1	2	yes	—
40	#89278	C₂²	1	2	yes	—
41	C₈×C₇⋊C₈	C₂²	1	1	yes	characteristic
42	C₈×C₇⋊C₈	C₂²	1	1	yes	characteristic
43	C₈×C₇⋊C₈	C₂²	1	1	yes	characteristic
44	C₈×C₇⋊C₈	C₂²	1	1	yes	characteristic
45	#17959	—	7	1	no	maximal, Sylow
46	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
49	#6493	C₂×C₄	1	2	yes	—
50	#6492	—	2	2	no	—
51	C₄×C₂²⋊D₇	—	2	2	no	—
52	#6487	C₂×C₄	1	2	yes	—
53	C₂²×C₇⋊C₈	—	2	1	no	—
54	C₂²×C₇⋊C₈	D₄	1	1	yes	characteristic
55	C₂²×C₇⋊C₈	D₄	1	1	yes	characteristic
56	C₂×Dic₂₈	C₂×C₄	1	1	yes	characteristic
57	C₂×D₅₆	C₂×C₄	1	1	yes	characteristic
58	C₂×#1132	C₂×C₄	1	2	yes	—
59	C₂×#1131	—	2	2	no	—
60	C₂×C₈×D₇	—	2	2	no	—
61	#6439	—	2	2	no	—
62	#6438	C₂×C₄	1	2	yes	—
63	C₄×D₂₈	C₂³	1	2	yes	—
64	#6435	—	2	2	no	—
65	#6433	C₂×C₄	1	2	yes	—
66	C₄×Dic₁₄	C₂³	1	2	yes	—
67	C₇×C₄⋊C₈	C₂³	1	2	yes	—
68	C₇×C₄⋊C₈	C₂³	1	2	yes	—
69	C₇×C₂²⋊C₈	—	2	2	no	—
70	C₇×#97	—	2	2	no	—
71	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
72	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
73	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
74	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
75	#6395	C₂×C₄	1	2	yes	—
76	#6395	C₂×C₄	1	2	yes	—
77	(C₂×D₇)⋊C₈	—	2	2	no	—
78	(C₂×D₇)⋊C₈	—	2	2	no	—
79	#6392	C₂×C₄	1	1	yes	characteristic
80	#6392	C₂×C₄	1	1	yes	characteristic
81	#6391	C₂×C₄	1	2	yes	—
82	#6390	C₂×C₄	1	2	yes	—
83	#6390	C₂×C₄	1	2	yes	—
84	#6389	—	2	2	no	—
85	Dic₇⋊C₈	—	2	2	no	—
86	Dic₇⋊C₈	—	2	2	no	—
87	C₈×Dic₇	—	2	2	no	—
88	C₄×C₇⋊C₈	—	2	1	no	—
89	C₄×C₇⋊C₈	D₄	1	1	yes	characteristic
90	C₄×C₇⋊C₈	D₄	1	1	yes	characteristic
91	C₄×C₇⋊C₈	C₂³	1	1	yes	characteristic
92	C₄×C₇⋊C₈	C₂³	1	1	yes	characteristic
93	C₄×C₇⋊C₈	C₂³	1	1	yes	characteristic
94	C₄×C₇⋊C₈	C₂³	1	1	yes	characteristic
95	#3168	—	7	1	no	—
96	#2966	—	7	2	no	—
97	#1591	—	7	1	no	—
98	#1590	—	7	2	no	—
99	#1589	—	7	1	no	—
100	C₈×Q₁₆	—	7	1	no	—
101	C₈×SD₁₆	—	7	2	no	—
102	C₈×D₈	—	7	1	no	—
103	#1568	D₇	1	1	yes	characteristic, complemented
104	#1494	—	7	2	no	—
105	C₂×C₈²	—	7	1	no	—
106	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₇	#41	1	1	yes	characteristic
109	C₂×C₂²⋊D₇	—	2	2	no	—
110	C₂×D₂₈	C₂²×C₄	1	2	yes	—
111	C₂×C₄×D₇	—	2	2	no	—
112	C₂×Dic₁₄	C₂²×C₄	1	2	yes	—
113	C₂×C₇×C₈	—	2	2	no	—
114	C₂×C₇×C₈	—	2	2	no	—
115	C₂×C₇×C₈	—	2	2	no	—
116	C₂×C₇×C₈	—	2	2	no	—
117	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
121	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
122	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₇	#41	1	1	yes	characteristic
124	C₄²×C₇	C₂⁴	1	1	yes	characteristic
125	C₂²⋊Dic₇	—	2	2	no	—
126	(C₂×D₇)⋊C₄	—	2	2	no	—
127	(C₂×D₇)⋊C₄	—	2	2	no	—
128	(C₂×D₇)⋊C₄	—	2	2	no	—
129	(C₂×D₇)⋊C₄	—	2	2	no	—
130	C₄⋊Dic₇	C₂²×C₄	1	2	yes	—
131	C₄⋊Dic₇	C₂²×C₄	1	2	yes	—
132	Dic₇⋊C₄	—	2	2	no	—
133	Dic₇⋊C₄	—	2	2	no	—
134	Dic₇⋊C₄	—	2	2	no	—
135	Dic₇⋊C₄	—	2	2	no	—
136	C₄×Dic₇	—	2	2	no	—
137	C₂×C₇⋊C₈	—	2	4	no	—
138	C₂×C₇⋊C₈	—	2	1	no	—
139	C₂×C₇⋊C₈	—	2	1	no	—
140	C₂×C₇⋊C₈	—	2	2	no	—
141	C₂×C₇⋊C₈	—	2	2	no	—
142	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
143	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
144	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
145	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
146	Dic₂₈	—	2	2	no	—
147	D₅₆	—	2	2	no	—
148	#1132	—	2	4	no	—
149	#1131	—	4	4	no	—
150	C₈×D₇	—	4	4	no	—
151	#529	—	7	2	no	—
152	#499	—	7	1	no	—
153	#495	—	7	1	no	—
154	#488	—	7	2	no	—
155	#467	—	7	1	no	—
156	#466	—	7	1	no	—
157	#465	—	7	2	no	—
158	#446	—	7	2	no	—
159	C₈×Q₈	—	7	2	no	—
160	C₄×Q₁₆	—	7	1	no	—
161	C₄×SD₁₆	—	7	2	no	—
162	C₄×D₈	—	7	1	no	—
163	#436	—	7	2	no	—
164	#435	—	14	2	no	—
165	C₈×D₄	—	7	2	no	—
166	C₈×D₄	—	14	2	no	—
167	#433	—	14	2	no	—
168	#432	C₂×D₇	1	2	yes	—
169	#405	—	14	2	no	—
170	C₂×C₄×C₈	—	7	1	no	—
171	C₂×C₄×C₈	—	7	1	no	—
172	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
173	#335	C₂×D₇	1	1	yes	characteristic
174	#335	C₂×D₇	1	1	yes	characteristic
175	#334	C₂×D₇	1	2	yes	—
176	Q₈⋊C₈	—	7	2	no	—
177	Q₈⋊C₈	—	7	2	no	—
178	D₄⋊C₈	—	7	2	no	—
179	D₄⋊C₈	—	7	2	no	—
180	C₈²	—	7	1	no	—
181	C₈²	—	7	1	no	—
182	C₈²	—	7	1	no	—
183	C₈²	—	7	1	no	—
184	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
185	C₂²×D₇	—	2	2	no	—
186	C₂×C₄×C₇	—	2	2	no	—
187	C₂×C₄×C₇	—	2	1	no	—
188	C₂×C₄×C₇	—	2	1	no	—
189	C₂×C₄×C₇	—	2	1	no	—
190	C₂×C₄×C₇	—	2	1	no	—
191	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
192	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
193	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
194	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
195	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
197	C₂²⋊D₇	—	4	4	no	—
198	C₂×Dic₇	—	2	2	no	—
199	C₂×Dic₇	—	2	2	no	—
200	C₂×Dic₇	—	2	2	no	—
201	D₂₈	—	2	2	no	—
202	D₂₈	—	2	2	no	—
203	C₄×D₇	—	4	2	no	—
204	C₄×D₇	—	4	2	no	—
205	Dic₁₄	—	2	2	no	—
206	Dic₁₄	—	2	2	no	—
207	C₇×C₈	—	2	1	no	—
208	C₇×C₈	—	2	1	no	—
209	C₇×C₈	—	4	2	no	—
210	C₇×C₈	—	4	2	no	—
211	C₇×C₈	#131	1	2	yes	—
212	C₇×C₈	#131	1	2	yes	—
213	C₇⋊C₈	—	2	2	no	—
214	C₇⋊C₈	—	2	2	no	—
215	C₇⋊C₈	—	2	1	no	—
216	C₇⋊C₈	—	2	1	no	—
217	C₇⋊C₈	#135	1	2	yes	—
218	C₇⋊C₈	#135	1	2	yes	—
219	C₂×Q₁₆	—	7	1	no	—
220	C₂×SD₁₆	—	7	2	no	—
221	C₂×D₈	—	7	1	no	—
222	C₂×M₁₆	—	14	2	no	—
223	C₂²×C₈	—	7	1	no	—
224	C₂²×C₈	—	7	1	no	—
225	C₂²×C₈	—	14	2	no	—
226	C₂²×C₈	—	14	1	no	—
227	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
228	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
229	#126	—	14	2	no	—
230	#125	—	7	2	no	—
231	#124	—	7	2	no	—
232	#123	—	14	2	no	—
233	C₂²⋊Q₈	—	7	2	no	—
234	#121	—	7	2	no	—
235	C₄×Q₈	—	7	2	no	—
236	C₄×D₄	—	7	2	no	—
237	C₄×D₄	—	14	2	no	—
238	#117	—	14	2	no	—
239	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
240	#107	—	7	1	no	—
241	#107	—	7	1	no	—
242	#106	—	7	2	no	—
243	C₄⋊C₈	—	14	2	no	—
244	C₄⋊C₈	—	14	2	no	—
245	C₄⋊C₈	C₂²×D₇	1	2	yes	—
246	C₄⋊C₈	C₂²×D₇	1	2	yes	—
247	#103	—	7	2	no	—
248	#103	—	7	2	no	—
249	#102	—	7	2	no	—
250	#102	—	7	2	no	—
251	C₂²⋊C₈	—	2	2	no	—
252	C₂²⋊C₈	—	14	2	no	—
253	C₂²⋊C₈	—	14	2	no	—
254	#97	—	2	2	no	—
255	#97	—	14	2	no	—
256	C₄×C₈	—	7	1	no	—
257	C₄×C₈	—	7	1	no	—
258	C₄×C₈	—	7	1	no	—
259	C₄×C₈	—	7	1	no	—
260	C₄×C₈	—	7	1	no	—
261	C₄×C₈	—	7	1	no	—
262	C₄×C₈	—	14	2	no	—
263	C₄×C₈	—	14	1	no	—
264	C₄×C₈	C₄×D₇	1	1	yes	characteristic
265	C₄×C₈	C₄×D₇	1	1	yes	characteristic
266	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
268	C₂²×C₇	—	2	1	no	—
269	C₂²×C₇	—	2	1	no	—
270	C₂²×C₇	#443	1	2	yes	—
271	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
272	C₂×D₇	—	4	2	no	—
273	C₂×D₇	—	4	2	no	—
274	C₂×D₇	—	4	2	no	—
275	C₄×C₇	—	2	1	no	—
276	C₄×C₇	—	2	1	no	—
277	C₄×C₇	—	2	1	no	—
278	C₄×C₇	—	2	1	no	—
279	C₄×C₇	C₂×#131	1	1	yes	characteristic
280	C₄×C₇	C₂×#131	1	1	yes	characteristic
281	C₄×C₇	C₂×#135	1	1	yes	characteristic
282	C₄×C₇	C₂×#135	1	1	yes	characteristic
283	Dic₇	—	4	2	no	—
284	Dic₇	—	4	2	no	—
285	Dic₇	—	4	2	no	—
286	C₂×Q₈	—	7	2	no	—
287	C₂×D₄	—	7	2	no	—
288	C₂×D₄	—	14	2	no	—
289	C₂²×C₄	—	14	2	no	—
290	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
291	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
292	C₂²×C₄	#1161	1	1	yes	characteristic
293	Q₁₆	—	14	2	no	—
294	SD₁₆	—	14	4	no	—
295	D₈	—	14	2	no	—
296	M₁₆	—	28	4	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	7	1	no	—
302	C₂×C₈	—	7	1	no	—
303	C₂×C₈	—	7	1	no	—
304	C₂×C₈	—	7	1	no	—
305	C₂×C₈	—	14	2	no	—
306	C₂×C₈	—	14	2	no	—
307	C₂×C₈	—	14	4	no	—
308	C₂×C₈	—	14	1	no	—
309	C₂×C₈	—	14	1	no	—
310	C₂×C₈	—	28	4	no	—
311	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
312	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
313	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
314	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
315	C₄⋊C₄	—	7	2	no	—
316	C₄⋊C₄	—	7	2	no	—
317	C₄⋊C₄	—	14	2	no	—
318	C₄⋊C₄	—	14	2	no	—
319	C₄⋊C₄	—	14	2	no	—
320	C₄⋊C₄	—	14	2	no	—
321	C₂²⋊C₄	—	14	2	no	—
322	C₂²⋊C₄	—	14	2	no	—
323	C₂²⋊C₄	—	14	2	no	—
324	C₂²⋊C₄	—	14	2	no	—
325	C₂²⋊C₄	—	14	2	no	—
326	C₄²	—	14	2	no	—
327	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
328	C₄²	D₄×D₇	1	1	yes	characteristic
329	C₄²	#1161	1	1	yes	characteristic
330	C₄²	C₂³×D₇	1	1	yes	characteristic
331	C₂×C₇	—	2	2	no	—
332	C₂×C₇	#2931	1	1	yes	characteristic
333	C₂×C₇	C₄×#135	1	1	yes	characteristic
334	C₂×C₇	C₂×#443	1	1	yes	characteristic
335	D₇	—	8	2	no	—
336	C₂³	—	14	2	no	—
337	C₂³	#6456	1	1	yes	characteristic
338	Q₈	—	14	2	no	—
339	Q₈	—	14	2	no	—
340	D₄	—	14	2	no	—
341	D₄	—	14	2	no	—
342	D₄	—	28	4	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	14	2	no	—
349	C₂×C₄	—	14	2	no	—
350	C₂×C₄	—	14	2	no	—
351	C₂×C₄	—	28	2	no	—
352	C₂×C₄	—	28	2	no	—
353	C₂×C₄	#6456	1	1	yes	characteristic
354	C₂×C₄	#6456	1	1	yes	characteristic
355	C₂×C₄	#6456	1	1	yes	characteristic
356	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
357	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
358	C₂×C₄	C₂×#1161	1	1	yes	characteristic
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	4	2	no	—
362	C₈	—	4	2	no	—
363	C₈	—	7	2	no	—
364	C₈	—	7	2	no	—
365	C₈	—	14	2	no	—
366	C₈	—	14	2	no	—
367	C₈	—	14	1	no	—
368	C₈	—	14	1	no	—
369	C₈	#6470	1	2	yes	—
370	C₈	#6470	1	2	yes	—
371	C₇	#17959	1	1	yes	characteristic, complemented, Sylow
372	C₂²	—	2	1	no	—
373	C₂²	—	2	1	no	—
374	C₂²	—	28	2	no	—
375	C₂²	—	28	2	no	—
376	C₂²	—	28	2	no	—
377	C₂²	#89667	1	2	yes	—
378	C₂²	C₂×#6456	1	1	yes	characteristic
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	2	1	no	—
382	C₄	—	2	1	no	—
383	C₄	—	28	2	no	—
384	C₄	—	28	2	no	—
385	C₄	—	28	2	no	—
386	C₄	C₂×#6470	1	1	yes	characteristic
387	C₄	C₂×#6470	1	1	yes	characteristic
388	C₄	D₇×#135	1	1	yes	characteristic
389	C₄	D₇×#135	1	1	yes	characteristic
390	C₂	—	2	2	no	—
391	C₂	—	56	2	no	—
392	C₂	#11736486	1	1	yes	characteristic
393	C₂	#11736610	1	1	yes	characteristic
394	C₂	C₂×#89667	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇
commutator	C₂×C₄×C₇	C₂³×C₄
Frattini	C₄²	C₂³×D₇
Fitting	C₇×#1568	C₂
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	#89887	C₄
cosocle	C₄²×C₇	C₂⁴
hypercenter	#1568	D₇
hypocenter	C₇	#17959
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
1. #11736770	C₂	semidirect
3. C₇×#1568	C₂	semidirect
4. #11728554	C₂	semidirect
5. C₂×C₈×C₇⋊C₈	C₂	semidirect
6. #11726318	C₂	semidirect
7. #11726317	C₂	semidirect
8. #11726311	C₂	semidirect
9. #11726310	C₂	semidirect
10. #11726277	C₂	semidirect
11. #11726270	C₂	semidirect
103. #1568	D₇	semidirect
371. C₇	#17959	semidirect

Maximal subgroups

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

subgroup	index
#11736770	2
#11735914	2
C₇×#1568	2
#11728554	2
C₂×C₈×C₇⋊C₈	2
#11726318	2
#11726317	2
#11726311	2
#11726310	2
#11726277	2
#11726270	2
#17959	7