Subgroups — #13323122

Order 1792.

592 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11736738	C₂	1	1	yes	maximal, characteristic
2	#11736722	C₂	1	1	yes	maximal, characteristic
3	#11736497	C₂	1	1	yes	maximal, characteristic
4	C₇×#1574	C₂	1	1	yes	maximal, characteristic, complemented
5	#11728507	C₂	1	1	yes	maximal, characteristic, complemented
6	#11728499	C₂	1	1	yes	maximal, characteristic, complemented
7	#11728245	C₂	1	1	yes	maximal, characteristic, complemented
8	#11726326	C₂	1	2	yes	maximal, complemented
9	#11726322	C₂	1	2	yes	maximal, complemented
10	#11726287	C₂	1	2	yes	maximal, complemented
11	#11726285	C₂	1	2	yes	maximal, complemented
12	C₄×#1157	C₂²	1	1	yes	characteristic
13	C₇×#431	C₂²	1	1	yes	characteristic
14	C₇×#431	C₂²	1	1	yes	characteristic
15	C₇×#423	C₂²	1	1	yes	characteristic
16	#89900	—	2	2	no	—
17	#89697	C₂²	1	1	yes	characteristic
18	#89696	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
20	#89610	—	2	1	no	—
21	#89610	C₄	1	1	yes	characteristic
22	#89610	C₄	1	1	yes	characteristic
23	#89609	C₂²	1	2	yes	—
24	#89607	C₂²	1	2	yes	—
25	#89605	C₂²	1	2	yes	—
26	D₇×C₄⋊C₈	C₂²	1	2	yes	—
27	#89603	C₄	1	1	yes	characteristic
28	#89603	C₄	1	1	yes	characteristic
29	#89602	—	2	1	no	—
30	#89502	C₄	1	2	yes	—
31	#89502	C₄	1	2	yes	—
32	#89500	—	2	2	no	—
33	C₄×#1131	C₂²	1	2	yes	—
34	C₈×D₂₈	—	2	1	no	—
35	#89457	C₂²	1	2	yes	—
36	#89451	—	2	1	no	—
37	C₇×#335	C₂²	1	2	yes	—
38	C₇×#334	C₂²	1	2	yes	—
39	#89275	C₂²	1	1	yes	characteristic
40	#89275	C₂²	1	1	yes	characteristic
41	#89274	C₂²	1	1	yes	characteristic
42	#89274	C₂²	1	1	yes	characteristic
43	#89274	C₂²	1	2	yes	—
44	#89274	C₂²	1	2	yes	—
45	#89253	C₂²	1	2	yes	—
46	#89252	C₂²	1	2	yes	—
47	#17839	—	7	1	no	maximal, Sylow
48	C₂×#1157	Q₈	1	1	yes	characteristic
49	C₂×C₇×M₁₆	—	2	1	no	—
50	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
51	C₄×C₂²⋊D₇	C₂×C₄	1	2	yes	—
52	C₄×C₂²⋊D₇	D₄	1	2	yes	—
53	#6489	Q₈	1	1	yes	characteristic
54	C₂×#1136	—	2	1	no	—
55	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
57	C₂×#1131	—	2	2	no	—
58	C₂×#1131	C₂×C₄	1	2	yes	—
59	C₂×#1131	C₂×C₄	1	2	yes	—
60	C₂×C₈×D₇	—	2	2	no	—
61	C₄×D₂₈	C₂×C₄	1	1	yes	characteristic
62	C₄×D₂₈	D₄	1	1	yes	characteristic
63	#6435	Q₈	1	2	yes	—
64	C₄²×D₇	C₂³	1	2	yes	—
65	C₄×Dic₁₄	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₁₄	D₄	1	1	yes	characteristic
67	C₇×C₄⋊C₈	—	2	1	no	—
68	C₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
69	C₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
70	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
71	C₇×C₄⋊C₈	C₂³	1	2	yes	—
72	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
73	C₇×C₂²⋊C₈	—	2	1	no	—
74	C₇×C₂²⋊C₈	C₂×C₄	1	2	yes	—
75	C₄×C₇×C₈	—	2	1	no	—
76	C₄×C₇×C₈	C₂³	1	2	yes	—
77	C₂²⋊(C₇⋊C₈)	—	2	1	no	—
78	(C₂×D₇)⋊C₈	—	2	2	no	—
79	(C₂×D₇)⋊C₈	—	2	2	no	—
80	(C₂×D₇)⋊C₈	C₂×C₄	1	2	yes	—
81	(C₂×D₇)⋊C₈	C₂×C₄	1	2	yes	—
82	#6389	—	2	2	no	—
83	#6389	C₂×C₄	1	2	yes	—
84	#6389	C₂×C₄	1	2	yes	—
85	Dic₇⋊C₈	—	2	2	no	—
86	Dic₇⋊C₈	—	2	2	no	—
87	Dic₇⋊C₈	C₂×C₄	1	2	yes	—
88	Dic₇⋊C₈	C₂×C₄	1	2	yes	—
89	C₈×Dic₇	—	2	2	no	—
90	C₄⋊(C₇⋊C₈)	—	2	1	no	—
91	C₄⋊(C₇⋊C₈)	C₂³	1	2	yes	—
92	C₄⋊(C₇⋊C₈)	C₂³	1	1	yes	characteristic
93	C₄⋊(C₇⋊C₈)	C₂³	1	1	yes	characteristic
94	C₄×C₇⋊C₈	—	2	1	no	—
95	C₄×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
96	C₄×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
97	C₄×C₇⋊C₈	C₂³	1	1	yes	characteristic
98	C₄×C₇⋊C₈	C₂³	1	1	yes	characteristic
99	#2991	—	7	1	no	—
100	#2990	—	7	1	no	—
101	#2965	—	7	1	no	—
102	#1574	—	7	1	no	—
103	#1574	D₇	1	1	yes	characteristic, complemented
104	#1572	—	7	2	no	—
105	#1571	—	7	2	no	—
106	#1569	—	7	2	no	—
107	#1567	—	7	1	no	—
108	#1565	—	7	2	no	—
109	#1564	—	7	1	no	—
110	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
113	C₂×C₂²⋊D₇	C₄⋊C₄	1	2	yes	—
114	#1157	—	4	2	no	—
115	C₂×D₂₈	C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×D₇	C₂²×C₄	1	2	yes	—
117	C₂×C₄×D₇	C₂×D₄	1	2	yes	—
118	C₂×C₄×D₇	C₂×Q₈	1	2	yes	—
119	C₂×Dic₁₄	C₄⋊C₄	1	1	yes	characteristic
120	C₇×M₁₆	—	8	1	no	—
121	C₂×C₇×C₈	—	2	1	no	—
122	C₂×C₇×C₈	—	2	1	no	—
123	C₂×C₇×C₈	—	2	2	no	—
124	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
125	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
126	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
127	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
129	C₄²×C₇	C₂⁴	1	1	yes	characteristic
130	C₂²⋊Dic₇	C₄⋊C₄	1	2	yes	—
131	(C₂×D₇)⋊C₄	C₄⋊C₄	1	2	yes	—
132	(C₂×D₇)⋊C₄	C₄⋊C₄	1	2	yes	—
133	C₄⋊Dic₇	C₄⋊C₄	1	1	yes	characteristic
134	C₄⋊Dic₇	C₄⋊C₄	1	1	yes	characteristic
135	Dic₇⋊C₄	C₄⋊C₄	1	2	yes	—
136	Dic₇⋊C₄	C₄⋊C₄	1	2	yes	—
137	C₄×Dic₇	C₂²×C₄	1	2	yes	—
138	C₄×Dic₇	C₂×D₄	1	2	yes	—
139	C₄×Dic₇	C₂×Q₈	1	2	yes	—
140	#1136	—	8	1	no	—
141	C₂×C₇⋊C₈	—	2	2	no	—
142	C₂×C₇⋊C₈	—	2	1	no	—
143	C₂×C₇⋊C₈	—	2	1	no	—
144	C₂×C₇⋊C₈	—	2	2	no	—
145	C₂×C₇⋊C₈	—	4	1	no	—
146	C₂×C₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
147	C₂×C₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
148	C₂×C₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
149	C₂×C₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
150	#1131	—	2	4	no	—
151	#1131	—	4	2	no	—
152	#1131	—	8	2	no	—
153	C₈×D₇	—	8	2	no	—
154	C₄×#41	—	7	1	no	—
155	#446	—	7	1	no	—
156	#446	—	7	1	no	—
157	#446	—	14	1	no	—
158	C₈×Q₈	—	14	1	no	—
159	#436	—	7	1	no	—
160	#436	—	7	1	no	—
161	#436	—	14	1	no	—
162	#436	—	14	2	no	—
163	#435	—	7	2	no	—
164	#435	—	7	2	no	—
165	#435	—	14	2	no	—
166	C₈×D₄	—	14	1	no	—
167	#432	—	7	2	no	—
168	#431	—	7	2	no	—
169	#431	—	7	1	no	—
170	#431	—	7	2	no	—
171	#431	C₂×D₇	1	1	yes	characteristic
172	#431	C₂×D₇	1	1	yes	characteristic
173	#423	—	7	1	no	—
174	#423	—	7	2	no	—
175	#423	C₂×D₇	1	1	yes	characteristic
176	C₂×C₄⋊C₈	—	7	2	no	—
177	C₄×M₁₆	—	7	2	no	—
178	C₂×C₄×C₈	—	7	1	no	—
179	#335	—	7	1	no	—
180	#335	—	7	1	no	—
181	#335	—	7	2	no	—
182	#335	—	7	2	no	—
183	#335	—	7	2	no	—
184	#335	C₂×D₇	1	2	yes	—
185	#334	—	7	1	no	—
186	#334	—	7	1	no	—
187	#334	—	7	2	no	—
188	#334	C₂×D₇	1	2	yes	—
189	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
190	C₂²×D₇	C₂×C₄⋊C₄	1	2	yes	—
191	C₂×C₄×C₇	—	2	1	no	—
192	C₂×C₄×C₇	—	2	1	no	—
193	C₂×C₄×C₇	—	2	1	no	—
194	C₂×C₄×C₇	—	2	1	no	—
195	C₂×C₄×C₇	—	2	2	no	—
196	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
197	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
198	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
199	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
200	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
201	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
202	C₂²⋊D₇	—	4	2	no	—
203	C₂×Dic₇	C₂×C₄⋊C₄	1	2	yes	—
204	C₂×Dic₇	C₂×C₄⋊C₄	1	2	yes	—
205	C₂×Dic₇	C₂×C₄⋊C₄	1	2	yes	—
206	D₂₈	—	2	2	no	—
207	C₄×D₇	—	2	2	no	—
208	C₄×D₇	—	2	2	no	—
209	C₄×D₇	—	2	2	no	—
210	C₄×D₇	—	2	2	no	—
211	C₄×D₇	—	4	2	no	—
212	Dic₁₄	—	2	2	no	—
213	C₇×C₈	—	2	2	no	—
214	C₇×C₈	—	2	2	no	—
215	C₇×C₈	—	4	1	no	—
216	C₇×C₈	—	4	2	no	—
217	C₇×C₈	—	4	1	no	—
218	C₇⋊C₈	—	2	1	no	—
219	C₇⋊C₈	—	2	1	no	—
220	C₇⋊C₈	—	2	1	no	—
221	C₇⋊C₈	—	4	2	no	—
222	C₇⋊C₈	—	4	1	no	—
223	C₇⋊C₈	—	4	1	no	—
224	C₇⋊C₈	#131	1	2	yes	—
225	C₂×#41	—	7	1	no	—
226	C₂×M₁₆	—	2	1	no	—
227	C₂×M₁₆	—	7	2	no	—
228	C₂×M₁₆	—	7	2	no	—
229	C₂×M₁₆	—	14	2	no	—
230	C₂×M₁₆	—	14	1	no	—
231	C₂²×C₈	—	7	1	no	—
232	C₂²×C₈	—	7	1	no	—
233	C₂²×C₈	—	14	2	no	—
234	C₄×Q₈	—	7	1	no	—
235	C₄×Q₈	—	7	1	no	—
236	C₄×D₄	—	7	2	no	—
237	C₄×D₄	—	7	2	no	—
238	C₄×D₄	—	7	1	no	—
239	C₄×D₄	—	7	1	no	—
240	#117	—	7	1	no	—
241	#117	—	7	2	no	—
242	C₂×C₄²	—	7	2	no	—
243	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
244	C₄⋊C₈	—	2	1	no	—
245	C₄⋊C₈	—	7	2	no	—
246	C₄⋊C₈	—	7	2	no	—
247	C₄⋊C₈	—	7	1	no	—
248	C₄⋊C₈	—	7	2	no	—
249	C₄⋊C₈	—	7	1	no	—
250	C₄⋊C₈	—	14	2	no	—
251	C₄⋊C₈	—	14	2	no	—
252	C₄⋊C₈	—	14	1	no	—
253	C₄⋊C₈	C₄×D₇	1	1	yes	characteristic
254	C₄⋊C₈	C₄×D₇	1	1	yes	characteristic
255	C₄⋊C₈	C₂²×D₇	1	1	yes	characteristic
256	C₄⋊C₈	C₂²×D₇	1	1	yes	characteristic
257	C₄⋊C₈	C₂²×D₇	1	2	yes	—
258	C₂²⋊C₈	—	2	1	no	—
259	C₂²⋊C₈	—	7	2	no	—
260	C₂²⋊C₈	—	7	2	no	—
261	C₂²⋊C₈	—	14	2	no	—
262	C₂²⋊C₈	—	14	2	no	—
263	C₂²⋊C₈	—	14	1	no	—
264	C₂²⋊C₈	C₄×D₇	1	2	yes	—
265	#97	—	7	2	no	—
266	#97	—	7	2	no	—
267	#97	—	14	2	no	—
268	C₄×C₈	—	2	1	no	—
269	C₄×C₈	—	7	1	no	—
270	C₄×C₈	—	7	1	no	—
271	C₄×C₈	—	7	1	no	—
272	C₄×C₈	—	7	1	no	—
273	C₄×C₈	—	14	2	no	—
274	C₄×C₈	—	14	1	no	—
275	C₄×C₈	C₂²×D₇	1	2	yes	—
276	C₂²×C₇	—	2	1	no	—
277	C₂²×C₇	—	2	1	no	—
278	C₂²×C₇	—	2	1	no	—
279	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
280	C₂×D₇	—	2	2	no	—
281	C₂×D₇	—	2	2	no	—
282	C₂×D₇	—	2	2	no	—
283	C₄×C₇	—	2	1	no	—
284	C₄×C₇	—	2	1	no	—
285	C₄×C₇	—	2	1	no	—
286	C₄×C₇	#430	1	2	yes	—
287	C₄×C₇	C₂×#131	1	1	yes	characteristic
288	C₄×C₇	#568	1	1	yes	characteristic
289	C₄×C₇	#576	1	1	yes	characteristic
290	C₄×C₇	#578	1	1	yes	characteristic
291	Dic₇	—	2	2	no	—
292	Dic₇	—	2	2	no	—
293	Dic₇	—	2	2	no	—
294	#41	—	28	2	no	—
295	C₂×Q₈	—	7	1	no	—
296	C₂×D₄	—	7	2	no	—
297	C₂×D₄	—	7	1	no	—
298	C₂²×C₄	—	7	2	no	—
299	C₂²×C₄	—	7	2	no	—
300	C₂²×C₄	—	7	2	no	—
301	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
302	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
303	C₂²×C₄	Q₈×D₇	1	1	yes	characteristic
304	M₁₆	—	8	1	no	—
305	M₁₆	—	14	4	no	—
306	M₁₆	—	28	2	no	—
307	M₁₆	—	56	2	no	—
308	M₁₆	—	56	1	no	—
309	C₂×C₈	—	2	1	no	—
310	C₂×C₈	—	2	1	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	7	1	no	—
313	C₂×C₈	—	7	1	no	—
314	C₂×C₈	—	7	1	no	—
315	C₂×C₈	—	7	1	no	—
316	C₂×C₈	—	14	2	no	—
317	C₂×C₈	—	14	1	no	—
318	C₂×C₈	—	14	1	no	—
319	C₂×C₈	—	14	2	no	—
320	C₂×C₈	—	28	1	no	—
321	C₂×C₈	—	56	2	no	—
322	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
323	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
324	C₄⋊C₄	—	7	2	no	—
325	C₄⋊C₄	—	7	2	no	—
326	C₄⋊C₄	—	7	1	no	—
327	C₄⋊C₄	—	7	1	no	—
328	C₂²⋊C₄	—	7	2	no	—
329	C₂²⋊C₄	—	7	2	no	—
330	C₂²⋊C₄	—	7	2	no	—
331	C₄²	—	7	2	no	—
332	C₄²	—	7	2	no	—
333	C₄²	—	7	2	no	—
334	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
335	C₄²	D₄×D₇	1	1	yes	characteristic
336	C₄²	Q₈×D₇	1	1	yes	characteristic
337	C₄²	C₂³×D₇	1	1	yes	characteristic
338	C₂×C₇	—	4	1	no	—
339	C₂×C₇	#2906	1	1	yes	characteristic
340	C₂×C₇	#2913	1	1	yes	characteristic
341	C₂×C₇	C₂×#430	1	1	yes	characteristic
342	D₇	—	4	2	no	—
343	C₂³	—	7	2	no	—
344	C₂³	D₇×C₄⋊C₄	1	1	yes	characteristic
345	Q₈	—	14	2	no	—
346	D₄	—	14	2	no	—
347	D₄	—	28	2	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	7	2	no	—
354	C₂×C₄	—	7	2	no	—
355	C₂×C₄	—	7	2	no	—
356	C₂×C₄	—	14	2	no	—
357	C₂×C₄	—	14	2	no	—
358	C₂×C₄	—	14	2	no	—
359	C₂×C₄	—	14	2	no	—
360	C₂×C₄	—	28	2	no	—
361	C₂×C₄	D₇×C₄⋊C₄	1	1	yes	characteristic
362	C₂×C₄	D₇×C₄⋊C₄	1	1	yes	characteristic
363	C₂×C₄	D₇×C₄⋊C₄	1	1	yes	characteristic
364	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
365	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
366	C₂×C₄	C₂×Q₈×D₇	1	1	yes	characteristic
367	C₈	—	2	2	no	—
368	C₈	—	2	2	no	—
369	C₈	—	4	1	no	—
370	C₈	—	4	2	no	—
371	C₈	—	4	1	no	—
372	C₈	—	7	2	no	—
373	C₈	—	14	1	no	—
374	C₈	—	14	1	no	—
375	C₈	—	14	1	no	—
376	C₈	—	28	2	no	—
377	C₈	—	28	1	no	—
378	C₈	—	28	1	no	—
379	C₇	#17839	1	1	yes	characteristic, complemented, Sylow
380	C₂²	—	2	1	no	—
381	C₂²	—	2	1	no	—
382	C₂²	—	2	1	no	—
383	C₂²	—	14	2	no	—
384	C₂²	—	14	2	no	—
385	C₂²	—	14	2	no	—
386	C₂²	C₂×D₇×C₄⋊C₄	1	1	yes	characteristic
387	C₄	—	2	1	no	—
388	C₄	—	2	1	no	—
389	C₄	—	2	1	no	—
390	C₄	—	14	2	no	—
391	C₄	—	14	2	no	—
392	C₄	—	14	2	no	—
393	C₄	#89665	1	2	yes	—
394	C₄	D₇×#131	1	1	yes	characteristic
395	C₄	#90441	1	1	yes	characteristic
396	C₄	#90465	1	1	yes	characteristic
397	C₄	#90470	1	1	yes	characteristic
398	C₂	—	4	1	no	—
399	C₂	—	28	2	no	—
400	C₂	#11736526	1	1	yes	characteristic
401	C₂	#11736655	1	1	yes	characteristic
402	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×#1574	C₂
socle	C₂²×C₇	C₂²×C₄⋊C₄
prime radical	C₂×#1157	Q₈
cosocle	C₄²×C₇	C₂⁴
hypercenter	#1574	D₇
hypocenter	C₇	#17839
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1574

Complements

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

N	H ≅ G/N	type
4. C₇×#1574	C₂	semidirect
5. #11728507	C₂	semidirect
6. #11728499	C₂	semidirect
7. #11728245	C₂	semidirect
8. #11726326	C₂	semidirect
9. #11726322	C₂	semidirect
10. #11726287	C₂	semidirect
11. #11726285	C₂	semidirect
103. #1574	D₇	semidirect
379. C₇	#17839	semidirect

Maximal subgroups

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

subgroup	index
#11736738	2
#11736722	2
#11736497	2
C₇×#1574	2
#11728507	2
#11728499	2
#11728245	2
#11726326	2
#11726322	2
#11726287	2
#11726285	2
#17839	7