Subgroups — #13324369

Order 1792.

614 conjugacy classes of subgroups.

The subgroup lattice has 378 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×#89250	C₂	1	1	yes	maximal, characteristic, complemented
6	#11726319	C₂	1	2	yes	maximal, complemented
7	#11726312	C₂	1	2	yes	maximal, complemented
8	#11726279	C₂	1	1	yes	maximal, characteristic, complemented
9	#11726278	C₂	1	1	yes	maximal, characteristic, complemented
10	#11726272	C₂	1	1	yes	maximal, characteristic, complemented
11	#11726271	C₂	1	1	yes	maximal, characteristic, 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	2	yes	—
35	C₇×#334	C₂²	1	1	yes	characteristic
36	C₇×#334	C₂²	1	1	yes	characteristic
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	#89250	C₂²	1	1	yes	characteristic
42	#89250	C₂²	1	1	yes	characteristic
43	#89250	C₂²	1	1	yes	characteristic
44	#89250	C₂²	1	1	yes	characteristic
45	#17963	—	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	2	yes	—
55	C₂×Dic₂₈	C₂×C₄	1	1	yes	characteristic
56	C₂×D₅₆	C₂×C₄	1	1	yes	characteristic
57	C₂×#1132	C₂×C₄	1	2	yes	—
58	C₂×#1131	—	2	2	no	—
59	C₂×C₈×D₇	—	2	2	no	—
60	#6439	—	2	2	no	—
61	#6438	C₂×C₄	1	2	yes	—
62	C₄×D₂₈	C₂³	1	2	yes	—
63	#6435	—	2	2	no	—
64	#6433	C₂×C₄	1	2	yes	—
65	C₄×Dic₁₄	C₂³	1	2	yes	—
66	C₇×C₄⋊C₈	C₂³	1	2	yes	—
67	C₇×C₄⋊C₈	C₂³	1	2	yes	—
68	C₇×C₂²⋊C₈	—	2	2	no	—
69	C₇×#97	—	2	2	no	—
70	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
71	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
72	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
73	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
74	#6395	C₂×C₄	1	2	yes	—
75	#6395	C₂×C₄	1	2	yes	—
76	(C₂×D₇)⋊C₈	—	2	2	no	—
77	(C₂×D₇)⋊C₈	—	2	2	no	—
78	#6392	C₂×C₄	1	1	yes	characteristic
79	#6392	C₂×C₄	1	1	yes	characteristic
80	#6391	C₂×C₄	1	2	yes	—
81	#6390	C₂×C₄	1	2	yes	—
82	#6390	C₂×C₄	1	2	yes	—
83	#6389	—	2	2	no	—
84	Dic₇⋊C₈	—	2	2	no	—
85	Dic₇⋊C₈	—	2	2	no	—
86	C₈×Dic₇	—	2	2	no	—
87	C₄×C₇⋊C₈	—	2	1	no	—
88	C₄×C₇⋊C₈	D₄	1	2	yes	—
89	C₄×C₇⋊C₈	C₂³	1	1	yes	characteristic
90	C₄×C₇⋊C₈	C₂³	1	1	yes	characteristic
91	C₄×C₇⋊C₈	C₂³	1	1	yes	characteristic
92	C₄×C₇⋊C₈	C₂³	1	1	yes	characteristic
93	#3168	—	7	1	no	—
94	#2966	—	7	2	no	—
95	#1594	—	7	1	no	—
96	#1593	—	7	1	no	—
97	#1592	—	7	2	no	—
98	#1580	—	7	1	no	—
99	#1579	—	7	1	no	—
100	SD₁₆⋊C₈	—	7	2	no	—
101	#1568	D₇	1	1	yes	characteristic, complemented
102	#1494	—	7	2	no	—
103	C₂×#322	—	7	1	no	—
104	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₇	#41	1	1	yes	characteristic
107	C₂×C₂²⋊D₇	—	2	2	no	—
108	C₂×D₂₈	C₂²×C₄	1	2	yes	—
109	C₂×C₄×D₇	—	2	2	no	—
110	C₂×Dic₁₄	C₂²×C₄	1	2	yes	—
111	C₂×C₇×C₈	—	2	2	no	—
112	C₂×C₇×C₈	—	2	2	no	—
113	C₂×C₇×C₈	—	2	2	no	—
114	C₂×C₇×C₈	—	2	2	no	—
115	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
119	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
120	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₇	#41	1	1	yes	characteristic
122	C₄²×C₇	C₂⁴	1	1	yes	characteristic
123	C₂²⋊Dic₇	—	2	2	no	—
124	(C₂×D₇)⋊C₄	—	2	2	no	—
125	(C₂×D₇)⋊C₄	—	2	2	no	—
126	(C₂×D₇)⋊C₄	—	2	2	no	—
127	(C₂×D₇)⋊C₄	—	2	2	no	—
128	C₄⋊Dic₇	C₂²×C₄	1	2	yes	—
129	C₄⋊Dic₇	C₂²×C₄	1	2	yes	—
130	Dic₇⋊C₄	—	2	2	no	—
131	Dic₇⋊C₄	—	2	2	no	—
132	Dic₇⋊C₄	—	2	2	no	—
133	Dic₇⋊C₄	—	2	2	no	—
134	C₄×Dic₇	—	2	2	no	—
135	C₂×C₇⋊C₈	—	2	1	no	—
136	C₂×C₇⋊C₈	—	2	1	no	—
137	C₂×C₇⋊C₈	—	4	2	no	—
138	C₂×C₇⋊C₈	—	4	2	no	—
139	C₂×C₇⋊C₈	C₂×D₄	1	2	yes	—
140	C₂×C₇⋊C₈	C₂×D₄	1	2	yes	—
141	Dic₂₈	—	2	2	no	—
142	D₅₆	—	2	2	no	—
143	#1132	—	2	4	no	—
144	#1131	—	4	4	no	—
145	C₈×D₇	—	4	4	no	—
146	#529	—	7	2	no	—
147	#499	—	7	1	no	—
148	#495	—	7	1	no	—
149	#488	—	7	2	no	—
150	#467	—	7	1	no	—
151	#466	—	7	1	no	—
152	#465	—	7	2	no	—
153	#446	—	7	2	no	—
154	C₈×Q₈	—	7	2	no	—
155	C₄×Q₁₆	—	7	1	no	—
156	C₄×SD₁₆	—	7	2	no	—
157	C₄×D₈	—	7	1	no	—
158	#436	—	7	2	no	—
159	#435	—	14	2	no	—
160	C₈×D₄	—	7	2	no	—
161	C₈×D₄	—	14	2	no	—
162	#433	—	14	2	no	—
163	#432	C₂×D₇	1	2	yes	—
164	#405	—	14	2	no	—
165	C₂×C₄×C₈	—	7	1	no	—
166	C₂×C₄×C₈	—	7	1	no	—
167	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
168	#335	C₂×D₇	1	2	yes	—
169	#334	C₂×D₇	1	1	yes	characteristic
170	#334	C₂×D₇	1	1	yes	characteristic
171	Q₈⋊C₈	—	7	2	no	—
172	Q₈⋊C₈	—	7	2	no	—
173	D₄⋊C₈	—	7	2	no	—
174	D₄⋊C₈	—	7	2	no	—
175	#322	—	7	1	no	—
176	#322	—	7	1	no	—
177	#322	—	7	1	no	—
178	#322	—	7	1	no	—
179	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
180	C₂²×D₇	—	2	2	no	—
181	C₂×C₄×C₇	—	2	2	no	—
182	C₂×C₄×C₇	—	2	1	no	—
183	C₂×C₄×C₇	—	2	1	no	—
184	C₂×C₄×C₇	—	2	2	no	—
185	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
186	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
187	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
188	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
189	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
191	C₂²⋊D₇	—	4	4	no	—
192	C₂×Dic₇	—	2	2	no	—
193	C₂×Dic₇	—	2	2	no	—
194	C₂×Dic₇	—	2	2	no	—
195	D₂₈	—	2	2	no	—
196	D₂₈	—	2	2	no	—
197	C₄×D₇	—	4	2	no	—
198	C₄×D₇	—	4	2	no	—
199	Dic₁₄	—	2	2	no	—
200	Dic₁₄	—	2	2	no	—
201	C₇×C₈	—	2	1	no	—
202	C₇×C₈	—	2	1	no	—
203	C₇×C₈	—	4	2	no	—
204	C₇×C₈	—	4	2	no	—
205	C₇×C₈	#131	1	2	yes	—
206	C₇×C₈	#131	1	2	yes	—
207	C₇⋊C₈	—	2	2	no	—
208	C₇⋊C₈	—	2	2	no	—
209	C₇⋊C₈	—	4	1	no	—
210	C₇⋊C₈	—	4	1	no	—
211	C₂×Q₁₆	—	7	1	no	—
212	C₂×SD₁₆	—	7	2	no	—
213	C₂×D₈	—	7	1	no	—
214	C₂×M₁₆	—	14	2	no	—
215	C₂²×C₈	—	7	2	no	—
216	C₂²×C₈	—	14	2	no	—
217	C₂²×C₈	—	14	1	no	—
218	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
219	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
220	#126	—	14	2	no	—
221	#125	—	7	2	no	—
222	#124	—	7	2	no	—
223	#123	—	14	2	no	—
224	C₂²⋊Q₈	—	7	2	no	—
225	#121	—	7	2	no	—
226	C₄×Q₈	—	7	2	no	—
227	C₄×D₄	—	7	2	no	—
228	C₄×D₄	—	14	2	no	—
229	#117	—	14	2	no	—
230	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
231	#107	—	7	1	no	—
232	#107	—	7	1	no	—
233	#106	—	7	2	no	—
234	C₄⋊C₈	—	14	2	no	—
235	C₄⋊C₈	—	14	2	no	—
236	C₄⋊C₈	C₂²×D₇	1	2	yes	—
237	C₄⋊C₈	C₂²×D₇	1	2	yes	—
238	#103	—	7	2	no	—
239	#103	—	7	2	no	—
240	#102	—	7	2	no	—
241	#102	—	7	2	no	—
242	C₂²⋊C₈	—	2	2	no	—
243	C₂²⋊C₈	—	14	2	no	—
244	C₂²⋊C₈	—	14	2	no	—
245	#97	—	2	2	no	—
246	#97	—	14	2	no	—
247	C₄×C₈	—	7	2	no	—
248	C₄×C₈	—	7	1	no	—
249	C₄×C₈	—	7	1	no	—
250	C₄×C₈	—	7	1	no	—
251	C₄×C₈	—	7	1	no	—
252	C₄×C₈	—	14	2	no	—
253	C₄×C₈	—	14	1	no	—
254	C₄×C₈	C₄×D₇	1	1	yes	characteristic
255	C₄×C₈	C₄×D₇	1	1	yes	characteristic
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	#444	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	—	2	1	no	—
268	C₄×C₇	—	2	1	no	—
269	C₄×C₇	C₂×#131	1	1	yes	characteristic
270	C₄×C₇	C₂×#131	1	1	yes	characteristic
271	C₄×C₇	#575	1	2	yes	—
272	Dic₇	—	4	2	no	—
273	Dic₇	—	4	2	no	—
274	Dic₇	—	4	2	no	—
275	C₂×Q₈	—	7	2	no	—
276	C₂×D₄	—	7	2	no	—
277	C₂×D₄	—	14	2	no	—
278	C₂²×C₄	—	14	2	no	—
279	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
280	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
281	C₂²×C₄	#1161	1	1	yes	characteristic
282	Q₁₆	—	14	2	no	—
283	SD₁₆	—	14	4	no	—
284	D₈	—	14	2	no	—
285	M₁₆	—	28	4	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	7	2	no	—
291	C₂×C₈	—	7	2	no	—
292	C₂×C₈	—	14	1	no	—
293	C₂×C₈	—	14	1	no	—
294	C₂×C₈	—	28	4	no	—
295	C₂×C₈	—	28	2	no	—
296	C₂×C₈	—	28	2	no	—
297	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
298	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
299	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
300	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
301	C₄⋊C₄	—	7	2	no	—
302	C₄⋊C₄	—	7	2	no	—
303	C₄⋊C₄	—	14	2	no	—
304	C₄⋊C₄	—	14	2	no	—
305	C₄⋊C₄	—	14	2	no	—
306	C₄⋊C₄	—	14	2	no	—
307	C₂²⋊C₄	—	14	2	no	—
308	C₂²⋊C₄	—	14	2	no	—
309	C₂²⋊C₄	—	14	2	no	—
310	C₂²⋊C₄	—	14	2	no	—
311	C₂²⋊C₄	—	14	2	no	—
312	C₄²	—	14	2	no	—
313	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
314	C₄²	D₄×D₇	1	1	yes	characteristic
315	C₄²	#1161	1	1	yes	characteristic
316	C₄²	C₂³×D₇	1	1	yes	characteristic
317	C₂×C₇	—	2	2	no	—
318	C₂×C₇	#2931	1	1	yes	characteristic
319	C₂×C₇	#2943	1	1	yes	characteristic
320	C₂×C₇	C₂×#444	1	1	yes	characteristic
321	D₇	—	8	2	no	—
322	C₂³	—	14	2	no	—
323	C₂³	#6456	1	1	yes	characteristic
324	Q₈	—	14	2	no	—
325	Q₈	—	14	2	no	—
326	D₄	—	14	2	no	—
327	D₄	—	14	2	no	—
328	D₄	—	28	4	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	14	2	no	—
334	C₂×C₄	—	14	2	no	—
335	C₂×C₄	—	14	2	no	—
336	C₂×C₄	—	28	2	no	—
337	C₂×C₄	—	28	2	no	—
338	C₂×C₄	#6456	1	1	yes	characteristic
339	C₂×C₄	#6456	1	1	yes	characteristic
340	C₂×C₄	#6456	1	1	yes	characteristic
341	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
342	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
343	C₂×C₄	C₂×#1161	1	1	yes	characteristic
344	C₈	—	2	1	no	—
345	C₈	—	2	1	no	—
346	C₈	—	4	2	no	—
347	C₈	—	4	2	no	—
348	C₈	—	14	2	no	—
349	C₈	—	14	2	no	—
350	C₈	—	28	1	no	—
351	C₈	—	28	1	no	—
352	C₈	#6470	1	2	yes	—
353	C₈	#6470	1	2	yes	—
354	C₇	#17963	1	1	yes	characteristic, complemented, Sylow
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	28	2	no	—
358	C₂²	—	28	2	no	—
359	C₂²	—	28	2	no	—
360	C₂²	#89666	1	2	yes	—
361	C₂²	C₂×#6456	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	28	2	no	—
367	C₄	—	28	2	no	—
368	C₄	—	28	2	no	—
369	C₄	C₂×#6470	1	1	yes	characteristic
370	C₄	C₂×#6470	1	1	yes	characteristic
371	C₄	#90459	1	2	yes	—
372	C₂	—	2	2	no	—
373	C₂	—	56	2	no	—
374	C₂	#11736486	1	1	yes	characteristic
375	C₂	#11736611	1	1	yes	characteristic
376	C₂	C₂×#89666	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	#17963
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×#89250	C₂	semidirect
6. #11726319	C₂	semidirect
7. #11726312	C₂	semidirect
8. #11726279	C₂	semidirect
9. #11726278	C₂	semidirect
10. #11726272	C₂	semidirect
11. #11726271	C₂	semidirect
101. #1568	D₇	semidirect
354. C₇	#17963	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×#89250	2
#11726319	2
#11726312	2
#11726279	2
#11726278	2
#11726272	2
#11726271	2
#17963	7