Subgroups — #13324344

Order 1792.

666 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	C₇×#2960	C₂	1	1	yes	maximal, characteristic, complemented
2	#11740788	C₂	1	1	yes	maximal, characteristic
3	#11735903	C₂	1	1	yes	maximal, characteristic
4	#11729580	C₂	1	1	yes	maximal, characteristic, complemented
5	#11728709	C₂	1	4	yes	maximal, complemented
6	#11728691	C₂	1	4	yes	maximal, complemented
7	#11728252	C₂	1	1	yes	maximal, characteristic
8	#11728245	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#89249	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#533	C₂²	1	1	yes	characteristic
11	C₂×#6464	—	2	1	no	—
12	#90168	C₂²	1	1	yes	characteristic
13	C₇×#446	—	2	1	no	—
14	C₇×#446	C₂²	1	2	yes	—
15	C₇×C₈×Q₈	—	2	1	no	—
16	C₇×C₈×Q₈	C₂²	1	2	yes	—
17	C₇×#424	—	2	1	no	—
18	C₇×#423	C₂²	1	1	yes	characteristic
19	C₇×#405	—	2	1	no	—
20	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
21	#89882	—	2	1	no	—
22	#89860	C₄	1	1	yes	characteristic
23	#89857	C₄	1	1	yes	characteristic
24	#89853	C₄	1	1	yes	characteristic
25	#89851	C₄	1	1	yes	characteristic
26	#89820	C₄	1	4	yes	—
27	C₄×#1143	C₂²	1	4	yes	—
28	#89696	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
31	#89460	—	2	1	no	—
32	#89460	C₂²	1	2	yes	—
33	C₈×D₂₈	—	2	1	no	—
34	C₈×D₂₈	C₂²	1	2	yes	—
35	Q₈⋊(C₇⋊C₈)	C₂²	1	4	yes	—
36	#89278	C₂²	1	4	yes	—
37	#89274	C₂²	1	4	yes	—
38	#89249	C₂²	1	2	yes	—
39	#89249	C₂²	1	2	yes	—
40	#17960	—	7	1	no	maximal, Sylow
41	C₂×#1157	—	2	1	no	—
42	C₂×C₇×M₁₆	—	2	1	no	—
43	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
44	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
45	C₇×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
46	C₇×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
47	C₇×#125	—	2	1	no	—
48	C₇×C₂²⋊Q₈	—	2	1	no	—
49	C₇×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
50	C₄×C₇×Q₈	—	2	1	no	—
51	C₄×C₇×Q₈	C₂³	1	2	yes	—
52	C₇×#117	—	2	1	no	—
53	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
54	C₂×#1143	C₂×C₄	1	4	yes	—
55	#6493	—	2	1	no	—
56	#6493	C₂×C₄	1	2	yes	—
57	C₂×#1136	—	2	1	no	—
58	C₂²×C₇⋊C₈	—	2	1	no	—
59	C₂²×C₇⋊C₈	C₂×C₄	1	2	yes	—
60	#6464	—	4	4	no	—
61	C₂×#1131	—	2	2	no	—
62	C₂×C₈×D₇	—	2	2	no	—
63	#6438	—	2	1	no	—
64	#6437	C₂×C₄	1	1	yes	characteristic
65	C₄×D₂₈	—	2	1	no	—
66	C₄×D₂₈	C₂³	1	2	yes	—
67	#6432	C₂×C₄	1	1	yes	characteristic
68	C₇×C₄⋊C₈	—	2	2	no	—
69	C₇×C₄⋊C₈	—	2	2	no	—
70	C₇×C₄⋊C₈	—	2	1	no	—
71	C₇×C₄⋊C₈	C₂³	1	2	yes	—
72	C₇×#97	—	2	2	no	—
73	C₄×C₇×C₈	—	2	2	no	—
74	C₄×C₇×C₈	D₄	1	1	yes	characteristic
75	C₄×C₇×C₈	D₄	1	1	yes	characteristic
76	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
77	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
78	#6409	C₂×C₄	1	4	yes	—
79	(C₂×D₇)⋊C₈	—	2	2	no	—
80	(C₂×D₇)⋊C₈	—	2	2	no	—
81	#6383	C₂×C₄	1	4	yes	—
82	#6382	C₂×C₄	1	4	yes	—
83	C₄⋊(C₇⋊C₈)	—	2	1	no	—
84	C₄⋊(C₇⋊C₈)	C₂³	1	2	yes	—
85	C₄×C₇⋊C₈	—	2	1	no	—
86	C₄×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
87	C₄×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
88	C₄×C₇⋊C₈	C₂³	1	2	yes	—
89	C₄×C₇⋊C₈	C₂³	1	2	yes	—
90	#3167	—	7	1	no	—
91	#2960	D₇	1	1	yes	characteristic, complemented
92	#2931	—	7	1	no	—
93	#1592	—	7	4	no	—
94	SD₁₆⋊C₈	—	7	4	no	—
95	#1567	—	7	1	no	—
96	#1493	—	7	1	no	—
97	#1492	—	7	1	no	—
98	C₂×#322	—	7	1	no	—
99	C₂×C₇×Q₈	—	2	1	no	—
100	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₇	#41	1	1	yes	characteristic
104	C₂×C₂²⋊D₇	—	2	2	no	—
105	#1157	—	4	2	no	—
106	#1157	—	4	2	no	—
107	C₂×D₂₈	—	2	1	no	—
108	C₂×D₂₈	—	2	1	no	—
109	C₂×D₂₈	C₂²×C₄	1	2	yes	—
110	C₂×C₄×D₇	—	2	2	no	—
111	C₂×Dic₁₄	—	2	1	no	—
112	C₇×M₁₆	—	4	2	no	—
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	C₂×D₄	1	1	yes	characteristic
117	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
118	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
119	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
120	C₇×C₄⋊C₄	—	2	1	no	—
121	C₇×C₄⋊C₄	—	2	1	no	—
122	C₇×C₄⋊C₄	—	2	1	no	—
123	C₇×C₄⋊C₄	—	2	2	no	—
124	C₇×C₄⋊C₄	—	2	2	no	—
125	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
126	C₇×C₂²⋊C₄	—	2	2	no	—
127	C₄²×C₇	—	2	2	no	—
128	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
129	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₇	#41	1	1	yes	characteristic
131	C₄²×C₇	C₂⁴	1	1	yes	characteristic
132	#1143	—	2	8	no	—
133	(C₂×D₇)⋊C₄	—	2	2	no	—
134	(C₂×D₇)⋊C₄	—	2	2	no	—
135	C₄⋊Dic₇	—	2	1	no	—
136	C₄⋊Dic₇	C₂²×C₄	1	2	yes	—
137	#1136	—	4	2	no	—
138	C₂×C₇⋊C₈	—	2	4	no	—
139	C₂×C₇⋊C₈	—	2	2	no	—
140	C₂×C₇⋊C₈	—	2	2	no	—
141	C₂×C₇⋊C₈	—	4	2	no	—
142	C₂×C₇⋊C₈	C₂²×C₄	1	2	yes	—
143	C₂×C₇⋊C₈	C₂²×C₄	1	2	yes	—
144	#1131	—	4	4	no	—
145	C₈×D₇	—	4	4	no	—
146	C₂×#131	—	14	1	no	—
147	#533	C₂×D₇	1	1	yes	characteristic
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	—	2	1	no	—
155	#446	C₂×D₇	1	2	yes	—
156	C₈×Q₈	—	2	1	no	—
157	C₈×Q₈	C₂×D₇	1	2	yes	—
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	—	2	1	no	—
164	#423	—	7	1	no	—
165	#423	C₂×D₇	1	1	yes	characteristic
166	#408	—	14	1	no	—
167	#405	—	2	1	no	—
168	C₂×C₄×C₈	—	7	1	no	—
169	C₂×C₄×C₈	—	7	1	no	—
170	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
171	#335	—	7	4	no	—
172	Q₈⋊C₈	—	7	4	no	—
173	D₄⋊C₈	—	7	4	no	—
174	#322	—	7	2	no	—
175	#322	—	7	2	no	—
176	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
177	C₂²×D₇	—	2	2	no	—
178	C₇×Q₈	—	2	4	no	—
179	C₇×Q₈	—	4	2	no	—
180	C₂×C₄×C₇	—	2	2	no	—
181	C₂×C₄×C₇	—	2	2	no	—
182	C₂×C₄×C₇	—	2	2	no	—
183	C₂×C₄×C₇	—	2	2	no	—
184	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
185	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
186	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
187	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
188	C₂×C₄×C₇	#137	1	2	yes	—
189	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
190	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
192	C₂²⋊D₇	—	4	4	no	—
193	C₂×Dic₇	—	2	2	no	—
194	D₂₈	—	2	4	no	—
195	D₂₈	—	4	2	no	—
196	D₂₈	—	4	2	no	—
197	C₄×D₇	—	4	2	no	—
198	C₄×D₇	—	4	2	no	—
199	Dic₁₄	—	4	2	no	—
200	C₇×C₈	—	2	1	no	—
201	C₇×C₈	—	2	1	no	—
202	C₇×C₈	—	2	1	no	—
203	C₇×C₈	—	2	1	no	—
204	C₇×C₈	—	4	2	no	—
205	C₇⋊C₈	—	2	2	no	—
206	C₇⋊C₈	—	4	2	no	—
207	C₇⋊C₈	—	4	2	no	—
208	C₇⋊C₈	#131	1	4	yes	complemented
209	C₂×#41	—	14	1	no	—
210	C₂×SD₁₆	—	7	4	no	—
211	#131	—	28	4	no	—
212	C₂×M₁₆	—	2	1	no	—
213	C₂×M₁₆	—	14	1	no	—
214	C₂×M₁₆	—	14	2	no	—
215	C₂²×C₈	—	7	2	no	—
216	C₂²×C₈	—	14	1	no	—
217	C₂²×C₈	—	14	2	no	—
218	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
219	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
220	C₄⋊Q₈	—	7	1	no	—
221	C₄⋊Q₈	C₄×D₇	1	1	yes	characteristic
222	C₄⋊Q₈	C₄×D₇	1	1	yes	characteristic
223	#127	—	7	1	no	—
224	#125	—	2	1	no	—
225	#124	—	14	1	no	—
226	C₂²⋊Q₈	—	2	1	no	—
227	C₂²⋊Q₈	C₄×D₇	1	2	yes	—
228	#121	—	7	2	no	—
229	#121	—	14	1	no	—
230	C₄×Q₈	—	2	1	no	—
231	C₄×Q₈	C₂²×D₇	1	2	yes	—
232	C₄×D₄	—	7	2	no	—
233	C₄×D₄	—	14	1	no	—
234	#117	—	2	1	no	—
235	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
236	#106	—	7	4	no	—
237	C₄⋊C₈	—	2	2	no	—
238	C₄⋊C₈	—	2	2	no	—
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	—	7	2	no	—
241	C₄⋊C₈	—	14	1	no	—
242	C₄⋊C₈	C₂²×D₇	1	2	yes	—
243	#103	—	7	4	no	—
244	#102	—	7	4	no	—
245	C₂²⋊C₈	—	14	2	no	—
246	C₂²⋊C₈	—	14	2	no	—
247	#97	—	2	2	no	—
248	C₄×C₈	—	2	2	no	—
249	C₄×C₈	—	7	1	no	—
250	C₄×C₈	—	7	1	no	—
251	C₄×C₈	—	7	2	no	—
252	C₄×C₈	—	7	2	no	—
253	C₄×C₈	—	14	1	no	—
254	C₄×C₈	C₂²⋊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	1	yes	characteristic
258	C₂²×C₇	—	2	1	no	—
259	C₂²×C₇	—	2	1	no	—
260	C₂²×C₇	#440	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	—	4	2	no	—
268	C₄×C₇	—	4	2	no	—
269	C₄×C₇	#440	1	2	yes	—
270	C₄×C₇	#444	1	2	yes	—
271	C₄×C₇	C₂×#131	1	2	yes	—
272	C₄×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
273	C₄×C₇	C₂×#137	1	1	yes	characteristic
274	Dic₇	—	4	2	no	—
275	#41	—	28	2	no	—
276	#41	—	28	2	no	—
277	C₂×Q₈	—	2	1	no	—
278	C₂×Q₈	—	14	1	no	—
279	C₂×Q₈	C₂×C₄×D₇	1	2	yes	—
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	#1157	1	1	yes	characteristic
287	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
288	SD₁₆	—	14	8	no	—
289	M₁₆	—	4	2	no	—
290	M₁₆	—	28	4	no	—
291	M₁₆	—	28	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	7	2	no	—
296	C₂×C₈	—	7	2	no	—
297	C₂×C₈	—	14	4	no	—
298	C₂×C₈	—	14	2	no	—
299	C₂×C₈	—	14	2	no	—
300	C₂×C₈	—	28	4	no	—
301	C₂×C₈	—	28	2	no	—
302	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
303	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
304	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
305	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
306	C₄⋊C₄	—	2	1	no	—
307	C₄⋊C₄	—	2	1	no	—
308	C₄⋊C₄	—	2	1	no	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	7	2	no	—
312	C₄⋊C₄	—	14	1	no	—
313	C₄⋊C₄	C₂×C₄×D₇	1	2	yes	—
314	C₂²⋊C₄	—	2	2	no	—
315	C₂²⋊C₄	—	14	2	no	—
316	C₂²⋊C₄	—	14	2	no	—
317	C₄²	—	2	2	no	—
318	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
319	C₄²	#1157	1	1	yes	characteristic
320	C₄²	C₂×C₂²⋊D₇	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×#440	1	1	yes	characteristic
325	C₂×C₇	C₂×#444	1	1	yes	characteristic
326	D₇	—	8	2	no	—
327	C₂³	—	14	2	no	—
328	C₂³	C₄×C₂²⋊D₇	1	1	yes	characteristic
329	Q₈	—	2	4	no	—
330	Q₈	—	4	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	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	14	2	no	—
341	C₂×C₄	—	28	2	no	—
342	C₂×C₄	—	28	2	no	—
343	C₂×C₄	C₄×C₂²⋊D₇	1	1	yes	characteristic
344	C₂×C₄	C₄×C₂²⋊D₇	1	1	yes	characteristic
345	C₂×C₄	C₄×C₂²⋊D₇	1	1	yes	characteristic
346	C₂×C₄	#6505	1	2	yes	—
347	C₂×C₄	#6512	1	2	yes	—
348	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
349	C₂×C₄	C₂×#1157	1	1	yes	characteristic
350	C₂×C₄	C₂²×C₂²⋊D₇	1	1	yes	characteristic
351	C₈	—	2	1	no	—
352	C₈	—	2	1	no	—
353	C₈	—	2	1	no	—
354	C₈	—	2	1	no	—
355	C₈	—	4	2	no	—
356	C₈	—	7	4	no	—
357	C₈	—	14	2	no	—
358	C₈	—	28	2	no	—
359	C₈	—	28	2	no	—
360	C₇	#17960	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	#89798	1	2	yes	—
367	C₂²	C₂×C₄×C₂²⋊D₇	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	4	2	no	—
371	C₄	—	4	2	no	—
372	C₄	—	28	2	no	—
373	C₄	#89798	1	2	yes	—
374	C₄	#89917	1	2	yes	—
375	C₄	D₇×#131	1	2	yes	—
376	C₄	C₂×#6505	1	1	yes	characteristic
377	C₄	C₂×#6512	1	1	yes	characteristic
378	C₂	—	2	2	no	—
379	C₂	—	56	2	no	—
380	C₂	C₂×#89798	1	1	yes	characteristic
381	C₂	#11741231	1	1	yes	characteristic
382	C₂	C₂×#89917	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₇
commutator	C₂×C₄×C₇	C₂³×C₄
Frattini	C₄²	C₂³×D₇
Fitting	C₇×#2960	C₂
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	#90168	C₂²
cosocle	C₄²×C₇	C₂⁴
hypercenter	#2960	D₇
hypocenter	C₇	#17960
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
1. C₇×#2960	C₂	semidirect
4. #11729580	C₂	semidirect
5. #11728709	C₂	semidirect
6. #11728691	C₂	semidirect
8. #11728245	C₂	semidirect
9. C₂×#89249	C₂	semidirect
91. #2960	D₇	semidirect
208. C₇⋊C₈	#131	semidirect
360. C₇	#17960	semidirect

Maximal subgroups

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

subgroup	index
C₇×#2960	2
#11740788	2
#11735903	2
#11729580	2
#11728709	2
#11728691	2
#11728252	2
#11728245	2
C₂×#89249	2
#17960	7