Subgroups — #13323117

Order 1792.

594 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11736785	C₂	1	2	yes	maximal
2	#11735892	C₂	1	1	yes	maximal, characteristic
3	C₇×#1568	C₂	1	1	yes	maximal, characteristic, complemented
4	#11728551	C₂	1	2	yes	maximal, complemented
5	C₂×#89252	C₂	1	1	yes	maximal, characteristic, complemented
6	#11726325	C₂	1	2	yes	maximal, complemented
7	#11726322	C₂	1	2	yes	maximal, complemented
8	#11726287	C₂	1	2	yes	maximal, complemented
9	#11726286	C₂	1	2	yes	maximal, complemented
10	C₄×#1157	C₂²	1	1	yes	characteristic
11	C₇×#432	C₂²	1	2	yes	—
12	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
13	#89883	C₄	1	2	yes	—
14	#89883	C₄	1	2	yes	—
15	#89697	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
17	#89609	C₂²	1	2	yes	—
18	#89607	C₂²	1	2	yes	—
19	#89605	C₂²	1	2	yes	—
20	D₇×C₄⋊C₈	C₂²	1	2	yes	—
21	#89502	—	2	2	no	—
22	(C₂²⋊D₇)⋊C₈	—	2	2	no	—
23	#89478	—	2	2	no	—
24	#89473	—	2	2	no	—
25	#89460	C₄	1	1	yes	characteristic
26	#89460	C₄	1	1	yes	characteristic
27	C₄×#1131	C₂²	1	2	yes	—
28	#89457	C₂²	1	2	yes	—
29	#89451	C₄	1	1	yes	characteristic
30	#89451	C₄	1	1	yes	characteristic
31	C₇×#335	C₂²	1	2	yes	—
32	C₇×#334	C₂²	1	1	yes	characteristic
33	C₇×#334	C₂²	1	1	yes	characteristic
34	#89275	C₂²	1	2	yes	—
35	#89275	C₂²	1	2	yes	—
36	#89274	C₂²	1	2	yes	—
37	#89274	C₂²	1	2	yes	—
38	#89252	C₂²	1	2	yes	—
39	#89252	C₂²	1	1	yes	characteristic
40	#89252	C₂²	1	1	yes	characteristic
41	#17838	—	7	1	no	maximal, Sylow
42	C₂×#1157	Q₈	1	1	yes	characteristic
43	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
46	C₄×C₂²⋊D₇	C₂×C₄	1	2	yes	—
47	C₄×C₂²⋊D₇	D₄	1	2	yes	—
48	#6489	Q₈	1	1	yes	characteristic
49	C₂²×C₇⋊C₈	—	2	2	no	—
50	C₂×#1131	—	2	2	no	—
51	C₂×#1131	C₂×C₄	1	2	yes	—
52	C₂×#1131	C₂×C₄	1	2	yes	—
53	C₂×C₈×D₇	—	2	2	no	—
54	C₄×D₂₈	C₂×C₄	1	1	yes	characteristic
55	C₄×D₂₈	D₄	1	1	yes	characteristic
56	#6435	Q₈	1	2	yes	—
57	C₄²×D₇	C₂³	1	2	yes	—
58	C₄×Dic₁₄	C₂×C₄	1	1	yes	characteristic
59	C₄×Dic₁₄	D₄	1	1	yes	characteristic
60	C₇×C₄⋊C₈	C₂³	1	2	yes	—
61	C₇×C₄⋊C₈	C₂³	1	2	yes	—
62	C₇×C₂²⋊C₈	—	2	2	no	—
63	C₇×#97	—	2	2	no	—
64	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
65	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
66	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
67	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
68	C₂²⋊(C₇⋊C₈)	C₂×C₄	1	2	yes	—
69	(C₂×D₇)⋊C₈	—	2	2	no	—
70	(C₂×D₇)⋊C₈	—	2	2	no	—
71	(C₂×D₇)⋊C₈	C₂×C₄	1	2	yes	—
72	(C₂×D₇)⋊C₈	C₂×C₄	1	2	yes	—
73	#6389	—	2	2	no	—
74	#6389	C₂×C₄	1	2	yes	—
75	#6389	C₂×C₄	1	2	yes	—
76	Dic₇⋊C₈	—	2	2	no	—
77	Dic₇⋊C₈	—	2	2	no	—
78	Dic₇⋊C₈	C₂×C₄	1	2	yes	—
79	Dic₇⋊C₈	C₂×C₄	1	2	yes	—
80	C₈×Dic₇	—	2	2	no	—
81	C₄⋊(C₇⋊C₈)	—	2	2	no	—
82	C₄⋊(C₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
83	C₄⋊(C₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
84	C₄⋊(C₇⋊C₈)	C₂³	1	2	yes	—
85	C₄⋊(C₇⋊C₈)	C₂³	1	2	yes	—
86	C₄×C₇⋊C₈	C₂³	1	2	yes	—
87	#2990	—	7	2	no	—
88	#2965	—	7	1	no	—
89	#1573	—	7	2	no	—
90	#1572	—	7	2	no	—
91	#1570	—	7	2	no	—
92	#1569	—	7	2	no	—
93	#1568	D₇	1	1	yes	characteristic, complemented
94	#1565	—	7	2	no	—
95	C₂×#334	—	7	1	no	—
96	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
99	C₂×C₂²⋊D₇	C₄⋊C₄	1	2	yes	—
100	#1157	—	4	2	no	—
101	C₂×D₂₈	C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×D₇	C₂²×C₄	1	2	yes	—
103	C₂×C₄×D₇	C₂×D₄	1	2	yes	—
104	C₂×C₄×D₇	C₂×Q₈	1	2	yes	—
105	C₂×Dic₁₄	C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₇×C₈	—	2	2	no	—
107	C₂×C₇×C₈	—	2	2	no	—
108	C₂×C₇×C₈	—	2	2	no	—
109	C₂×C₇×C₈	—	4	1	no	—
110	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
115	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
117	C₄²×C₇	C₂⁴	1	1	yes	characteristic
118	C₂²⋊Dic₇	C₄⋊C₄	1	2	yes	—
119	(C₂×D₇)⋊C₄	C₄⋊C₄	1	2	yes	—
120	(C₂×D₇)⋊C₄	C₄⋊C₄	1	2	yes	—
121	C₄⋊Dic₇	C₄⋊C₄	1	1	yes	characteristic
122	C₄⋊Dic₇	C₄⋊C₄	1	1	yes	characteristic
123	Dic₇⋊C₄	C₄⋊C₄	1	2	yes	—
124	Dic₇⋊C₄	C₄⋊C₄	1	2	yes	—
125	C₄×Dic₇	C₂²×C₄	1	2	yes	—
126	C₄×Dic₇	C₂×D₄	1	2	yes	—
127	C₄×Dic₇	C₂×Q₈	1	2	yes	—
128	C₂×C₇⋊C₈	—	2	2	no	—
129	C₂×C₇⋊C₈	—	2	2	no	—
130	C₂×C₇⋊C₈	—	8	2	no	—
131	C₂×C₇⋊C₈	C₂²×C₄	1	2	yes	—
132	C₂×C₇⋊C₈	C₂²×C₄	1	2	yes	—
133	#1131	—	2	4	no	—
134	#1131	—	4	2	no	—
135	#1131	—	8	2	no	—
136	C₈×D₇	—	8	2	no	—
137	C₄×#41	—	7	1	no	—
138	#446	—	7	1	no	—
139	#446	—	7	1	no	—
140	#446	—	14	2	no	—
141	#436	—	7	1	no	—
142	#436	—	7	1	no	—
143	#435	—	7	2	no	—
144	#435	—	7	2	no	—
145	#435	—	14	2	no	—
146	#435	—	14	2	no	—
147	C₈×D₄	—	14	2	no	—
148	#432	—	7	2	no	—
149	#432	C₂×D₇	1	2	yes	—
150	#431	—	7	1	no	—
151	#431	—	7	2	no	—
152	#431	—	7	2	no	—
153	#423	—	7	2	no	—
154	C₂×C₄⋊C₈	—	7	2	no	—
155	C₂×C₄⋊C₈	—	7	2	no	—
156	C₄×M₁₆	—	7	2	no	—
157	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
158	#335	—	7	2	no	—
159	#335	—	7	2	no	—
160	#335	C₂×D₇	1	2	yes	—
161	#334	—	7	2	no	—
162	#334	—	7	2	no	—
163	#334	—	7	1	no	—
164	#334	—	7	1	no	—
165	#334	—	7	2	no	—
166	#334	C₂×D₇	1	1	yes	characteristic
167	#334	C₂×D₇	1	1	yes	characteristic
168	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
169	C₂²×D₇	C₂×C₄⋊C₄	1	2	yes	—
170	C₂×C₄×C₇	—	2	2	no	—
171	C₂×C₄×C₇	—	2	1	no	—
172	C₂×C₄×C₇	—	2	1	no	—
173	C₂×C₄×C₇	—	2	2	no	—
174	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
175	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
176	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
177	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
178	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
180	C₂²⋊D₇	—	2	4	no	—
181	C₂×Dic₇	C₂×C₄⋊C₄	1	2	yes	—
182	C₂×Dic₇	C₂×C₄⋊C₄	1	2	yes	—
183	C₂×Dic₇	C₂×C₄⋊C₄	1	2	yes	—
184	D₂₈	—	4	1	no	—
185	C₄×D₇	—	2	2	no	—
186	C₄×D₇	—	2	2	no	—
187	C₄×D₇	—	2	2	no	—
188	C₄×D₇	—	2	2	no	—
189	C₄×D₇	—	4	2	no	—
190	Dic₁₄	—	4	1	no	—
191	C₇×C₈	—	2	1	no	—
192	C₇×C₈	—	2	1	no	—
193	C₇×C₈	—	2	1	no	—
194	C₇×C₈	—	4	2	no	—
195	C₇×C₈	—	4	2	no	—
196	C₇×C₈	#131	1	2	yes	—
197	C₇⋊C₈	—	2	2	no	—
198	C₇⋊C₈	—	2	2	no	—
199	C₇⋊C₈	—	4	2	no	—
200	C₇⋊C₈	—	4	2	no	—
201	C₂×#41	—	7	1	no	—
202	C₂×M₁₆	—	7	2	no	—
203	C₂×M₁₆	—	7	2	no	—
204	C₂×M₁₆	—	14	2	no	—
205	C₂²×C₈	—	14	2	no	—
206	C₂²×C₈	—	14	2	no	—
207	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
208	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
209	C₄×Q₈	—	7	1	no	—
210	C₄×Q₈	—	7	1	no	—
211	C₄×D₄	—	7	2	no	—
212	C₄×D₄	—	7	2	no	—
213	C₄×D₄	—	7	1	no	—
214	C₄×D₄	—	7	1	no	—
215	#117	—	7	1	no	—
216	#117	—	7	2	no	—
217	C₂×C₄²	—	7	2	no	—
218	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
219	C₄⋊C₈	—	7	2	no	—
220	C₄⋊C₈	—	7	2	no	—
221	C₄⋊C₈	—	7	1	no	—
222	C₄⋊C₈	—	7	1	no	—
223	C₄⋊C₈	—	7	2	no	—
224	C₄⋊C₈	—	7	2	no	—
225	C₄⋊C₈	—	14	2	no	—
226	C₄⋊C₈	—	14	2	no	—
227	C₄⋊C₈	—	14	2	no	—
228	C₄⋊C₈	C₂²×D₇	1	2	yes	—
229	C₄⋊C₈	C₂²×D₇	1	2	yes	—
230	C₂²⋊C₈	—	2	2	no	—
231	C₂²⋊C₈	—	7	2	no	—
232	C₂²⋊C₈	—	7	2	no	—
233	C₂²⋊C₈	—	7	2	no	—
234	C₂²⋊C₈	—	14	2	no	—
235	C₂²⋊C₈	—	14	2	no	—
236	#97	—	2	2	no	—
237	#97	—	7	2	no	—
238	#97	—	7	2	no	—
239	#97	—	14	2	no	—
240	C₄×C₈	—	7	2	no	—
241	C₄×C₈	—	14	2	no	—
242	C₄×C₈	C₄×D₇	1	1	yes	characteristic
243	C₄×C₈	C₄×D₇	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
246	C₂²×C₇	—	2	1	no	—
247	C₂²×C₇	—	2	1	no	—
248	C₂²×C₇	#430	1	2	yes	—
249	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
250	C₂×D₇	—	2	2	no	—
251	C₂×D₇	—	2	2	no	—
252	C₂×D₇	—	2	2	no	—
253	C₄×C₇	—	2	1	no	—
254	C₄×C₇	—	2	1	no	—
255	C₄×C₇	—	2	1	no	—
256	C₄×C₇	—	2	1	no	—
257	C₄×C₇	C₂×#131	1	1	yes	characteristic
258	C₄×C₇	#568	1	1	yes	characteristic
259	C₄×C₇	#577	1	2	yes	—
260	Dic₇	—	2	2	no	—
261	Dic₇	—	2	2	no	—
262	Dic₇	—	2	2	no	—
263	#41	—	28	2	no	—
264	C₂×Q₈	—	7	1	no	—
265	C₂×D₄	—	7	2	no	—
266	C₂×D₄	—	7	1	no	—
267	C₂²×C₄	—	7	2	no	—
268	C₂²×C₄	—	7	2	no	—
269	C₂²×C₄	—	7	2	no	—
270	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
271	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
272	C₂²×C₄	Q₈×D₇	1	1	yes	characteristic
273	M₁₆	—	14	4	no	—
274	M₁₆	—	28	2	no	—
275	M₁₆	—	56	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	4	1	no	—
280	C₂×C₈	—	7	2	no	—
281	C₂×C₈	—	7	2	no	—
282	C₂×C₈	—	14	2	no	—
283	C₂×C₈	—	14	2	no	—
284	C₂×C₈	—	56	2	no	—
285	C₂×C₈	—	56	2	no	—
286	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
287	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
288	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
289	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
290	C₄⋊C₄	—	7	2	no	—
291	C₄⋊C₄	—	7	2	no	—
292	C₄⋊C₄	—	7	1	no	—
293	C₄⋊C₄	—	7	1	no	—
294	C₂²⋊C₄	—	7	2	no	—
295	C₂²⋊C₄	—	7	2	no	—
296	C₂²⋊C₄	—	7	2	no	—
297	C₄²	—	7	2	no	—
298	C₄²	—	7	2	no	—
299	C₄²	—	7	2	no	—
300	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
301	C₄²	D₄×D₇	1	1	yes	characteristic
302	C₄²	Q₈×D₇	1	1	yes	characteristic
303	C₄²	C₂³×D₇	1	1	yes	characteristic
304	C₂×C₇	—	2	2	no	—
305	C₂×C₇	#2906	1	1	yes	characteristic
306	C₂×C₇	#2912	1	1	yes	characteristic
307	C₂×C₇	C₂×#430	1	1	yes	characteristic
308	D₇	—	4	2	no	—
309	C₂³	—	7	2	no	—
310	C₂³	D₇×C₄⋊C₄	1	1	yes	characteristic
311	Q₈	—	28	1	no	—
312	D₄	—	14	4	no	—
313	D₄	—	28	1	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	7	2	no	—
319	C₂×C₄	—	7	2	no	—
320	C₂×C₄	—	7	2	no	—
321	C₂×C₄	—	14	2	no	—
322	C₂×C₄	—	14	2	no	—
323	C₂×C₄	—	14	2	no	—
324	C₂×C₄	—	14	2	no	—
325	C₂×C₄	—	28	2	no	—
326	C₂×C₄	D₇×C₄⋊C₄	1	1	yes	characteristic
327	C₂×C₄	D₇×C₄⋊C₄	1	1	yes	characteristic
328	C₂×C₄	D₇×C₄⋊C₄	1	1	yes	characteristic
329	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
330	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
331	C₂×C₄	C₂×Q₈×D₇	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	4	2	no	—
336	C₈	—	4	2	no	—
337	C₈	—	14	2	no	—
338	C₈	—	14	2	no	—
339	C₈	—	28	2	no	—
340	C₈	—	28	2	no	—
341	C₈	#6470	1	2	yes	—
342	C₇	#17838	1	1	yes	characteristic, complemented, Sylow
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	14	2	no	—
346	C₂²	—	14	2	no	—
347	C₂²	—	14	2	no	—
348	C₂²	#89665	1	2	yes	—
349	C₂²	C₂×D₇×C₄⋊C₄	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	14	2	no	—
355	C₄	—	14	2	no	—
356	C₄	—	14	2	no	—
357	C₄	C₂×#6470	1	1	yes	characteristic
358	C₄	#90436	1	1	yes	characteristic
359	C₄	#90461	1	2	yes	—
360	C₂	—	2	2	no	—
361	C₂	—	28	2	no	—
362	C₂	#11736494	1	1	yes	characteristic
363	C₂	#11736623	1	1	yes	characteristic
364	C₂	C₂×#89665	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	C₂²×C₄⋊C₄
prime radical	C₂×#1157	Q₈
cosocle	C₄²×C₇	C₂⁴
hypercenter	#1568	D₇
hypocenter	C₇	#17838
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
3. C₇×#1568	C₂	semidirect
4. #11728551	C₂	semidirect
5. C₂×#89252	C₂	semidirect
6. #11726325	C₂	semidirect
7. #11726322	C₂	semidirect
8. #11726287	C₂	semidirect
9. #11726286	C₂	semidirect
93. #1568	D₇	semidirect
342. C₇	#17838	semidirect

Maximal subgroups

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

subgroup	index
#11736785	2
#11735892	2
C₇×#1568	2
#11728551	2
C₂×#89252	2
#11726325	2
#11726322	2
#11726287	2
#11726286	2
#17838	7