Subgroups — #13324483

Order 1792.

592 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11736498	C₂	1	2	yes	maximal, complemented
2	#11735914	C₂	1	1	yes	maximal, characteristic
3	C₇×#1567	C₂	1	1	yes	maximal, characteristic, complemented
4	#11728253	C₂	1	2	yes	maximal, complemented
5	#11727478	C₂	1	1	yes	maximal, characteristic, complemented
6	#11726319	C₂	1	2	yes	maximal, complemented
7	#11726318	C₂	1	1	yes	maximal, characteristic, complemented
8	#11726317	C₂	1	1	yes	maximal, characteristic, complemented
9	#11726312	C₂	1	2	yes	maximal, complemented
10	#11726311	C₂	1	1	yes	maximal, characteristic, complemented
11	#11726310	C₂	1	1	yes	maximal, characteristic, complemented
12	#90170	C₂²	1	2	yes	—
13	C₇×#423	C₂²	1	2	yes	—
14	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
15	#89900	—	2	2	no	—
16	#89899	—	2	2	no	—
17	#89888	C₄	1	1	yes	characteristic
18	#89887	C₄	1	1	yes	characteristic
19	#89886	C₄	1	2	yes	—
20	#89697	C₂²	1	1	yes	characteristic
21	#89697	C₂²	1	1	yes	characteristic
22	#89612	—	2	2	no	—
23	#89611	—	2	2	no	—
24	#89610	C₂²	1	2	yes	—
25	#89606	C₂²	1	2	yes	—
26	#89603	C₂²	1	2	yes	—
27	#89602	C₂²	1	2	yes	—
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×#335	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	#89250	C₂²	1	1	yes	characteristic
42	#89250	C₂²	1	1	yes	characteristic
43	C₈×C₇⋊C₈	C₂²	1	1	yes	characteristic
44	C₈×C₇⋊C₈	C₂²	1	1	yes	characteristic
45	#17973	—	7	1	no	maximal, Sylow
46	C₂×C₇×M₁₆	—	2	2	no	—
47	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
50	#6493	C₂×C₄	1	2	yes	—
51	#6492	—	2	2	no	—
52	C₄×C₂²⋊D₇	—	2	2	no	—
53	#6487	C₂×C₄	1	2	yes	—
54	C₂×Dic₂₈	C₂×C₄	1	1	yes	characteristic
55	C₂×D₅₆	C₂×C₄	1	1	yes	characteristic
56	C₂×#1132	C₂×C₄	1	2	yes	—
57	C₂×#1131	—	2	2	no	—
58	C₂×C₈×D₇	—	2	2	no	—
59	#6439	—	2	2	no	—
60	#6438	C₂×C₄	1	2	yes	—
61	C₄×D₂₈	C₂³	1	2	yes	—
62	#6435	—	2	2	no	—
63	#6433	C₂×C₄	1	2	yes	—
64	C₄×Dic₁₄	C₂³	1	2	yes	—
65	C₇×C₄⋊C₈	—	2	2	no	—
66	C₇×C₄⋊C₈	C₂³	1	2	yes	—
67	C₇×C₄⋊C₈	C₂³	1	2	yes	—
68	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
69	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
70	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
71	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
72	C₂²⋊(C₇⋊C₈)	—	2	1	no	—
73	C₂²⋊(C₇⋊C₈)	D₄	1	1	yes	characteristic
74	C₂²⋊(C₇⋊C₈)	D₄	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	#2993	—	7	2	no	—
97	#1592	—	7	2	no	—
98	#1591	—	7	1	no	—
99	#1589	—	7	1	no	—
100	SD₁₆⋊C₈	—	7	2	no	—
101	C₈×Q₁₆	—	7	1	no	—
102	C₈×D₈	—	7	1	no	—
103	#1567	D₇	1	1	yes	characteristic, complemented
104	#1495	—	7	2	no	—
105	#1450	—	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	C₇×M₁₆	—	8	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	—	2	2	no	—
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	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
122	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
123	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₇	#41	1	1	yes	characteristic
125	C₄²×C₇	C₂⁴	1	1	yes	characteristic
126	C₂²⋊Dic₇	—	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	(C₂×D₇)⋊C₄	—	2	2	no	—
131	C₄⋊Dic₇	C₂²×C₄	1	2	yes	—
132	C₄⋊Dic₇	C₂²×C₄	1	2	yes	—
133	Dic₇⋊C₄	—	2	2	no	—
134	Dic₇⋊C₄	—	2	2	no	—
135	Dic₇⋊C₄	—	2	2	no	—
136	Dic₇⋊C₄	—	2	2	no	—
137	C₄×Dic₇	—	2	2	no	—
138	C₂×C₇⋊C₈	—	2	1	no	—
139	C₂×C₇⋊C₈	—	2	1	no	—
140	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
141	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
142	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
143	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
144	Dic₂₈	—	2	2	no	—
145	D₅₆	—	2	2	no	—
146	#1132	—	2	4	no	—
147	#1131	—	8	2	no	—
148	C₈×D₇	—	8	2	no	—
149	#529	—	7	2	no	—
150	#499	—	7	1	no	—
151	#495	—	7	1	no	—
152	#488	—	7	2	no	—
153	#467	—	7	1	no	—
154	#466	—	7	1	no	—
155	#465	—	7	2	no	—
156	#446	—	7	2	no	—
157	C₈×Q₈	—	7	2	no	—
158	C₄×Q₁₆	—	7	1	no	—
159	C₄×SD₁₆	—	7	2	no	—
160	C₄×D₈	—	7	1	no	—
161	#436	—	7	2	no	—
162	#436	—	14	2	no	—
163	#435	—	14	2	no	—
164	C₈×D₄	—	7	2	no	—
165	#433	—	14	2	no	—
166	#431	—	7	1	no	—
167	#431	—	7	1	no	—
168	#424	—	14	2	no	—
169	#423	C₂×D₇	1	2	yes	—
170	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
171	#335	C₂×D₇	1	1	yes	characteristic
172	#335	C₂×D₇	1	1	yes	characteristic
173	#335	C₂×D₇	1	2	yes	—
174	Q₈⋊C₈	—	7	2	no	—
175	Q₈⋊C₈	—	7	2	no	—
176	D₄⋊C₈	—	7	2	no	—
177	D₄⋊C₈	—	7	2	no	—
178	#322	—	7	1	no	—
179	#322	—	7	1	no	—
180	C₈²	—	7	1	no	—
181	C₈²	—	7	1	no	—
182	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
183	C₂²×D₇	—	2	2	no	—
184	C₂×C₄×C₇	—	2	1	no	—
185	C₂×C₄×C₇	—	2	1	no	—
186	C₂×C₄×C₇	—	2	1	no	—
187	C₂×C₄×C₇	—	2	1	no	—
188	C₂×C₄×C₇	—	4	1	no	—
189	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
190	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
191	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
192	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
193	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
195	C₂²⋊D₇	—	8	2	no	—
196	C₂×Dic₇	—	2	2	no	—
197	C₂×Dic₇	—	2	2	no	—
198	C₂×Dic₇	—	2	2	no	—
199	D₂₈	—	2	2	no	—
200	D₂₈	—	2	2	no	—
201	C₄×D₇	—	4	2	no	—
202	C₄×D₇	—	4	2	no	—
203	Dic₁₄	—	2	2	no	—
204	Dic₁₄	—	2	2	no	—
205	C₇×C₈	—	2	1	no	—
206	C₇×C₈	—	2	1	no	—
207	C₇×C₈	—	4	2	no	—
208	C₇×C₈	—	4	2	no	—
209	C₇×C₈	#131	1	2	yes	—
210	C₇×C₈	#131	1	2	yes	—
211	C₇⋊C₈	—	2	1	no	—
212	C₇⋊C₈	—	2	1	no	—
213	C₇⋊C₈	—	2	1	no	—
214	C₇⋊C₈	—	2	1	no	—
215	C₇⋊C₈	—	4	1	no	—
216	C₇⋊C₈	—	4	1	no	—
217	C₂×Q₁₆	—	7	1	no	—
218	C₂×SD₁₆	—	7	2	no	—
219	C₂×D₈	—	7	1	no	—
220	C₂×M₁₆	—	2	2	no	—
221	C₂×M₁₆	—	14	2	no	—
222	C₂²×C₈	—	14	2	no	—
223	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
224	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
225	#126	—	14	2	no	—
226	#125	—	7	2	no	—
227	#124	—	7	2	no	—
228	#123	—	14	2	no	—
229	C₂²⋊Q₈	—	7	2	no	—
230	#121	—	7	2	no	—
231	C₄×Q₈	—	7	2	no	—
232	C₄×D₄	—	7	2	no	—
233	C₄×D₄	—	14	2	no	—
234	#117	—	14	2	no	—
235	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
236	#107	—	7	1	no	—
237	#107	—	7	1	no	—
238	#106	—	7	2	no	—
239	C₄⋊C₈	—	2	2	no	—
240	C₄⋊C₈	—	7	1	no	—
241	C₄⋊C₈	—	7	1	no	—
242	C₄⋊C₈	—	14	2	no	—
243	C₄⋊C₈	—	14	2	no	—
244	C₄⋊C₈	—	14	1	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	—	7	1	no	—
252	C₂²⋊C₈	—	7	1	no	—
253	C₂²⋊C₈	—	14	2	no	—
254	C₂²⋊C₈	—	14	2	no	—
255	C₂²⋊C₈	—	14	1	no	—
256	#97	—	14	2	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	—	14	2	no	—
262	C₄×C₈	C₄×D₇	1	1	yes	characteristic
263	C₄×C₈	C₄×D₇	1	1	yes	characteristic
264	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
265	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
266	C₂²×C₇	—	2	1	no	—
267	C₂²×C₇	—	2	1	no	—
268	C₂²×C₇	—	2	1	no	—
269	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
270	C₂×D₇	—	4	2	no	—
271	C₂×D₇	—	4	2	no	—
272	C₂×D₇	—	4	2	no	—
273	C₄×C₇	—	2	1	no	—
274	C₄×C₇	—	2	1	no	—
275	C₄×C₇	—	2	1	no	—
276	C₄×C₇	—	2	1	no	—
277	C₄×C₇	C₂×#131	1	1	yes	characteristic
278	C₄×C₇	C₂×#131	1	1	yes	characteristic
279	C₄×C₇	#576	1	1	yes	characteristic
280	C₄×C₇	#578	1	1	yes	characteristic
281	Dic₇	—	4	2	no	—
282	Dic₇	—	4	2	no	—
283	Dic₇	—	4	2	no	—
284	C₂×Q₈	—	7	2	no	—
285	C₂×D₄	—	7	2	no	—
286	C₂×D₄	—	14	2	no	—
287	C₂²×C₄	—	14	2	no	—
288	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
289	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
290	C₂²×C₄	#1161	1	1	yes	characteristic
291	Q₁₆	—	14	2	no	—
292	SD₁₆	—	14	4	no	—
293	D₈	—	14	2	no	—
294	M₁₆	—	8	2	no	—
295	M₁₆	—	56	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	7	1	no	—
301	C₂×C₈	—	7	1	no	—
302	C₂×C₈	—	7	1	no	—
303	C₂×C₈	—	7	1	no	—
304	C₂×C₈	—	14	1	no	—
305	C₂×C₈	—	14	1	no	—
306	C₂×C₈	—	56	2	no	—
307	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
308	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
309	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
310	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
311	C₄⋊C₄	—	7	2	no	—
312	C₄⋊C₄	—	7	2	no	—
313	C₄⋊C₄	—	14	2	no	—
314	C₄⋊C₄	—	14	2	no	—
315	C₄⋊C₄	—	14	2	no	—
316	C₄⋊C₄	—	14	2	no	—
317	C₂²⋊C₄	—	14	2	no	—
318	C₂²⋊C₄	—	14	2	no	—
319	C₂²⋊C₄	—	14	2	no	—
320	C₂²⋊C₄	—	14	2	no	—
321	C₂²⋊C₄	—	14	2	no	—
322	C₄²	—	14	2	no	—
323	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
324	C₄²	D₄×D₇	1	1	yes	characteristic
325	C₄²	#1161	1	1	yes	characteristic
326	C₄²	C₂³×D₇	1	1	yes	characteristic
327	C₂×C₇	—	4	1	no	—
328	C₂×C₇	#2931	1	1	yes	characteristic
329	C₂×C₇	#2951	1	1	yes	characteristic
330	C₂×C₇	#2957	1	1	yes	characteristic
331	D₇	—	8	2	no	—
332	C₂³	—	14	2	no	—
333	C₂³	#6456	1	1	yes	characteristic
334	Q₈	—	14	2	no	—
335	Q₈	—	14	2	no	—
336	D₄	—	14	2	no	—
337	D₄	—	14	2	no	—
338	D₄	—	56	2	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	14	2	no	—
345	C₂×C₄	—	14	2	no	—
346	C₂×C₄	—	14	2	no	—
347	C₂×C₄	—	28	2	no	—
348	C₂×C₄	—	28	2	no	—
349	C₂×C₄	#6456	1	1	yes	characteristic
350	C₂×C₄	#6456	1	1	yes	characteristic
351	C₂×C₄	#6456	1	1	yes	characteristic
352	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
353	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
354	C₂×C₄	C₂×#1161	1	1	yes	characteristic
355	C₈	—	2	1	no	—
356	C₈	—	2	1	no	—
357	C₈	—	4	2	no	—
358	C₈	—	4	2	no	—
359	C₈	—	14	1	no	—
360	C₈	—	14	1	no	—
361	C₈	—	14	1	no	—
362	C₈	—	14	1	no	—
363	C₈	—	28	1	no	—
364	C₈	—	28	1	no	—
365	C₈	#6470	1	2	yes	—
366	C₈	#6470	1	2	yes	—
367	C₇	#17973	1	1	yes	characteristic, complemented, Sylow
368	C₂²	—	2	1	no	—
369	C₂²	—	2	1	no	—
370	C₂²	—	2	1	no	—
371	C₂²	—	28	2	no	—
372	C₂²	—	28	2	no	—
373	C₂²	—	28	2	no	—
374	C₂²	C₂×#6456	1	1	yes	characteristic
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	28	2	no	—
380	C₄	—	28	2	no	—
381	C₄	—	28	2	no	—
382	C₄	C₂×#6470	1	1	yes	characteristic
383	C₄	C₂×#6470	1	1	yes	characteristic
384	C₄	#90448	1	1	yes	characteristic
385	C₄	#90457	1	1	yes	characteristic
386	C₂	—	4	1	no	—
387	C₂	—	56	2	no	—
388	C₂	#11736486	1	1	yes	characteristic
389	C₂	#11736581	1	1	yes	characteristic
390	C₂	#11736694	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×#1567	C₂
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	#89887	C₄
cosocle	C₄²×C₇	C₂⁴
hypercenter	#1567	D₇
hypocenter	C₇	#17973
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
1. #11736498	C₂	semidirect
3. C₇×#1567	C₂	semidirect
4. #11728253	C₂	semidirect
5. #11727478	C₂	semidirect
6. #11726319	C₂	semidirect
7. #11726318	C₂	semidirect
8. #11726317	C₂	semidirect
9. #11726312	C₂	semidirect
10. #11726311	C₂	semidirect
11. #11726310	C₂	semidirect
103. #1567	D₇	semidirect
367. C₇	#17973	semidirect

Maximal subgroups

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

subgroup	index
#11736498	2
#11735914	2
C₇×#1567	2
#11728253	2
#11727478	2
#11726319	2
#11726318	2
#11726317	2
#11726312	2
#11726311	2
#11726310	2
#17973	7