Subgroups — #13323027

Order 1792.

610 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₈×#41	C₂	1	1	yes	maximal, characteristic
2	#11739960	C₂	1	2	yes	maximal
3	#11728505	C₂	1	4	yes	maximal, complemented
4	#11728504	C₂	1	2	yes	maximal, complemented
5	#11727486	C₂	1	2	yes	maximal, complemented
6	#11727484	C₂	1	1	yes	maximal, characteristic, complemented
7	#11727483	C₂	1	2	yes	maximal, complemented
8	C₂×#89252	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₇×#41	C₂²	1	1	yes	characteristic
10	C₇×C₈×Q₈	C₄	1	1	yes	characteristic
11	C₇×C₈×Q₈	C₄	1	1	yes	characteristic
12	C₇×C₈×D₄	C₄	1	1	yes	characteristic
13	C₇×C₈×D₄	C₄	1	2	yes	—
14	C₇×C₈×D₄	C₄	1	1	yes	characteristic
15	C₇×C₈×D₄	C₄	1	2	yes	—
16	C₇×#431	C₂²	1	1	yes	characteristic
17	C₇×#431	C₂²	1	2	yes	—
18	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
20	#90006	—	2	2	no	—
21	#89772	—	2	4	no	—
22	#89769	—	2	2	no	—
23	#89698	C₂²	1	4	yes	—
24	#89697	C₂²	1	4	yes	—
25	#89696	C₂²	1	2	yes	—
26	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
27	#89253	C₂²	1	4	yes	—
28	#89253	C₂²	1	4	yes	—
29	#89252	C₂²	1	2	yes	—
30	#89252	C₂²	1	1	yes	characteristic
31	#89252	C₂²	1	1	yes	characteristic
32	#89252	C₂²	1	2	yes	—
33	#89252	C₂²	1	2	yes	—
34	#17827	—	7	1	no	maximal, Sylow
35	C₂×C₇×#41	D₄	1	1	yes	characteristic
36	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
38	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
40	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
41	C₄×C₇×Q₈	Q₈	1	1	yes	characteristic
42	C₄×C₇×D₄	C₂×C₄	1	2	yes	—
43	C₄×C₇×D₄	C₂×C₄	1	1	yes	characteristic
44	C₄×C₇×D₄	Q₈	1	2	yes	—
45	C₄×C₇×D₄	Q₈	1	1	yes	characteristic
46	C₇×#117	D₄	1	1	yes	characteristic
47	C₇×#117	D₄	1	2	yes	—
48	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₇	C₂³	1	2	yes	—
50	C₂×#1136	—	2	2	no	—
51	C₂²×C₇⋊C₈	—	2	2	no	—
52	C₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
55	C₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
56	C₇×C₂²⋊C₈	C₂×C₄	1	2	yes	—
57	C₇×C₂²⋊C₈	C₂×C₄	1	2	yes	—
58	C₇×C₂²⋊C₈	C₂×C₄	1	2	yes	—
59	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
60	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
61	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
62	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
63	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
64	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
65	C₄×C₇×C₈	C₂³	1	2	yes	—
66	C₂²⋊(C₇⋊C₈)	—	2	4	no	—
67	C₂²⋊(C₇⋊C₈)	—	2	4	no	—
68	C₄⋊(C₇⋊C₈)	—	2	2	no	—
69	C₄⋊(C₇⋊C₈)	—	2	2	no	—
70	C₄⋊(C₇⋊C₈)	C₂³	1	2	yes	—
71	C₄⋊(C₇⋊C₈)	C₂³	1	2	yes	—
72	C₄⋊(C₇⋊C₈)	C₂³	1	4	yes	—
73	#6377	—	2	4	no	—
74	C₄×C₇⋊C₈	—	2	4	no	—
75	C₈×#41	D₇	1	1	yes	characteristic
76	#2925	—	7	2	no	—
77	#1572	—	7	4	no	—
78	#1571	—	7	2	no	—
79	#1568	—	7	2	no	—
80	#1566	—	7	1	no	—
81	#1564	—	7	2	no	—
82	C₂×#334	—	7	1	no	—
83	C₇×#41	—	2	4	no	—
84	C₂×C₇×Q₈	C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₇×D₄	C₄⋊C₄	1	2	yes	—
86	C₂×C₇×D₄	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
92	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
93	C₂×C₇×C₈	—	2	2	no	—
94	C₂×C₇×C₈	—	2	4	no	—
95	C₂×C₇×C₈	—	4	1	no	—
96	C₂×C₇×C₈	—	4	2	no	—
97	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
100	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
102	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₇×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
104	C₇×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
105	C₇×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
106	C₇×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
107	C₇×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
108	C₇×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
109	C₇×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
110	C₄²×C₇	C₂²×C₄	1	2	yes	—
111	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₇	C₂×D₄	1	2	yes	—
113	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₇	C₂×Q₈	1	2	yes	—
115	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
116	C₄²×C₇	C₂⁴	1	1	yes	characteristic
117	#1136	—	4	4	no	—
118	C₂×C₇⋊C₈	—	2	2	no	—
119	C₂×C₇⋊C₈	—	2	2	no	—
120	C₂×C₇⋊C₈	—	2	4	no	—
121	C₂×C₇⋊C₈	—	4	4	no	—
122	C₄×#41	C₂×D₇	1	1	yes	characteristic
123	C₈×Q₈	Dic₇	1	1	yes	characteristic
124	C₈×Q₈	Dic₇	1	1	yes	characteristic
125	C₈×D₄	Dic₇	1	1	yes	characteristic
126	C₈×D₄	Dic₇	1	2	yes	—
127	C₈×D₄	Dic₇	1	1	yes	characteristic
128	C₈×D₄	Dic₇	1	2	yes	—
129	#433	—	14	4	no	—
130	#432	—	7	4	no	—
131	#431	—	7	4	no	—
132	#431	C₂×D₇	1	1	yes	characteristic
133	#431	C₂×D₇	1	2	yes	—
134	#423	—	7	2	no	—
135	C₂×C₄⋊C₈	—	7	2	no	—
136	#408	—	14	2	no	—
137	#405	—	14	2	no	—
138	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
139	C₂×C₄×C₈	C₂×D₇	1	2	yes	—
140	#335	—	7	4	no	—
141	#335	—	7	4	no	—
142	#334	—	7	1	no	—
143	#334	—	7	1	no	—
144	#334	—	7	2	no	—
145	#334	—	7	2	no	—
146	#334	—	7	2	no	—
147	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
148	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₇×Q₈	—	2	2	no	—
150	C₇×D₄	—	2	4	no	—
151	C₇×D₄	—	2	2	no	—
152	C₂×C₄×C₇	—	2	1	no	—
153	C₂×C₄×C₇	—	2	2	no	—
154	C₂×C₄×C₇	—	2	1	no	—
155	C₂×C₄×C₇	—	2	2	no	—
156	C₂×C₄×C₇	—	2	4	no	—
157	C₂×C₄×C₇	—	4	1	no	—
158	C₂×C₄×C₇	—	4	2	no	—
159	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
160	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
162	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
163	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₂×C₄×C₇	#135	1	4	yes	—
166	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
167	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
169	C₇×C₈	—	2	1	no	—
170	C₇×C₈	—	2	2	no	—
171	C₇×C₈	—	2	1	no	—
172	C₇×C₈	—	2	1	no	—
173	C₇×C₈	—	2	2	no	—
174	C₇×C₈	#131	1	2	yes	—
175	C₇⋊C₈	—	4	2	no	—
176	C₇⋊C₈	—	4	2	no	—
177	C₇⋊C₈	—	4	4	no	—
178	C₂×#41	D₂₈	1	1	yes	characteristic
179	C₂×M₁₆	—	14	2	no	—
180	C₂²×C₈	—	14	2	no	—
181	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
182	C₂²×C₈	C₂×Dic₇	1	2	yes	—
183	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
184	C₂²×C₈	C₂×Dic₇	1	2	yes	—
185	C₄×Q₈	Dic₁₄	1	1	yes	characteristic
186	C₄×Q₈	C₂×Dic₇	1	1	yes	characteristic
187	C₄×D₄	Dic₁₄	1	2	yes	—
188	C₄×D₄	Dic₁₄	1	1	yes	characteristic
189	C₄×D₄	C₂×Dic₇	1	2	yes	—
190	C₄×D₄	C₂×Dic₇	1	1	yes	characteristic
191	#117	D₂₈	1	1	yes	characteristic
192	#117	D₂₈	1	2	yes	—
193	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
194	C₂×C₄²	C₂²×D₇	1	2	yes	—
195	C₄⋊C₈	—	7	2	no	—
196	C₄⋊C₈	—	7	2	no	—
197	C₄⋊C₈	—	7	4	no	—
198	C₄⋊C₈	—	14	2	no	—
199	C₄⋊C₈	—	14	2	no	—
200	C₄⋊C₈	C₂×Dic₇	1	1	yes	characteristic
201	C₄⋊C₈	C₂×Dic₇	1	1	yes	characteristic
202	C₄⋊C₈	C₂×Dic₇	1	2	yes	—
203	C₄⋊C₈	C₂×Dic₇	1	2	yes	—
204	C₂²⋊C₈	—	14	4	no	—
205	C₂²⋊C₈	—	14	4	no	—
206	C₂²⋊C₈	C₂×Dic₇	1	2	yes	—
207	C₂²⋊C₈	C₂×Dic₇	1	2	yes	—
208	C₂²⋊C₈	C₂×Dic₇	1	2	yes	—
209	#97	—	14	4	no	—
210	C₄×C₈	—	14	4	no	—
211	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
212	C₄×C₈	C₂×Dic₇	1	2	yes	—
213	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
214	C₄×C₈	C₂×Dic₇	1	2	yes	—
215	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
216	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
217	C₄×C₈	C₂²×D₇	1	2	yes	—
218	C₂²×C₇	—	2	1	no	—
219	C₂²×C₇	—	2	1	no	—
220	C₂²×C₇	—	2	2	no	—
221	C₂²×C₇	—	2	2	no	—
222	C₂²×C₇	—	2	2	no	—
223	C₂²×C₇	#427	1	2	yes	—
224	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
225	C₄×C₇	—	2	1	no	—
226	C₄×C₇	—	2	2	no	—
227	C₄×C₇	—	2	2	no	—
228	C₄×C₇	—	2	1	no	—
229	C₄×C₇	—	2	1	no	—
230	C₄×C₇	—	2	2	no	—
231	C₄×C₇	#427	1	2	yes	—
232	C₄×C₇	C₂×#131	1	1	yes	characteristic
233	C₄×C₇	#568	1	1	yes	characteristic
234	C₄×C₇	C₂×#135	1	2	yes	—
235	#41	—	2	4	no	—
236	C₂×Q₈	C₄⋊Dic₇	1	1	yes	characteristic
237	C₂×D₄	C₄⋊Dic₇	1	2	yes	—
238	C₂×D₄	C₄⋊Dic₇	1	1	yes	characteristic
239	C₂²×C₄	C₂×Dic₁₄	1	2	yes	—
240	C₂²×C₄	C₂×Dic₁₄	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₂₈	1	2	yes	—
242	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
243	C₂²×C₄	C₂²×Dic₇	1	2	yes	—
244	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
245	M₁₆	—	28	4	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	4	2	no	—
250	C₂×C₈	—	14	2	no	—
251	C₂×C₈	—	14	4	no	—
252	C₂×C₈	—	14	2	no	—
253	C₂×C₈	—	28	4	no	—
254	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
255	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
256	C₂×C₈	C₂²×Dic₇	1	2	yes	—
257	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
258	C₂×C₈	C₂²×Dic₇	1	2	yes	—
259	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
260	C₄⋊C₄	C₄⋊Dic₇	1	2	yes	—
261	C₄⋊C₄	C₄⋊Dic₇	1	1	yes	characteristic
262	C₄⋊C₄	C₄⋊Dic₇	1	2	yes	—
263	C₄⋊C₄	C₄⋊Dic₇	1	1	yes	characteristic
264	C₂²⋊C₄	C₄⋊Dic₇	1	2	yes	—
265	C₂²⋊C₄	C₄⋊Dic₇	1	2	yes	—
266	C₂²⋊C₄	C₄⋊Dic₇	1	2	yes	—
267	C₄²	C₂×Dic₁₄	1	2	yes	—
268	C₄²	C₂×Dic₁₄	1	1	yes	characteristic
269	C₄²	C₂×D₂₈	1	2	yes	—
270	C₄²	C₂×D₂₈	1	1	yes	characteristic
271	C₄²	C₂²×Dic₇	1	2	yes	—
272	C₄²	C₂²×Dic₇	1	1	yes	characteristic
273	C₄²	C₂³×D₇	1	1	yes	characteristic
274	C₂×C₇	—	2	2	no	—
275	C₂×C₇	—	4	2	no	—
276	C₂×C₇	#2906	1	1	yes	characteristic
277	C₂×C₇	C₂×#427	1	1	yes	characteristic
278	C₂×C₇	#2916	1	1	yes	characteristic
279	C₂³	C₂×C₄⋊Dic₇	1	2	yes	—
280	C₂³	C₂×C₄⋊Dic₇	1	1	yes	characteristic
281	Q₈	—	2	2	no	—
282	D₄	—	2	4	no	—
283	D₄	—	2	2	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	#6467	1	4	yes	—
292	C₂×C₄	C₂×C₄⋊Dic₇	1	2	yes	—
293	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
294	C₂×C₄	C₂×C₄⋊Dic₇	1	2	yes	—
295	C₂×C₄	C₂×C₄⋊Dic₇	1	2	yes	—
296	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
297	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
298	C₂×C₄	C₂²×Dic₁₄	1	1	yes	characteristic
299	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
300	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	2	no	—
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	—	2	2	no	—
306	C₈	—	28	2	no	—
307	C₈	—	28	2	no	—
308	C₈	—	28	4	no	—
309	C₈	#6511	1	2	yes	—
310	C₇	#17827	1	1	yes	characteristic, complemented, Sylow
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	#89878	1	2	yes	—
317	C₂²	C₂²×C₄⋊Dic₇	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	#89878	1	2	yes	—
325	C₄	C₂×#6467	1	2	yes	—
326	C₄	C₂×#6511	1	1	yes	characteristic
327	C₄	#90510	1	1	yes	characteristic
328	C₂	—	2	2	no	—
329	C₂	—	4	2	no	—
330	C₂	#11739957	1	1	yes	characteristic
331	C₂	C₂×#89878	1	1	yes	characteristic
332	C₂	#11741234	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₈
commutator	C₂×C₄×C₇	C₂³×C₄
Frattini	C₄²	C₂³×D₇
Fitting	C₇×C₈×#41	C₂
socle	C₂²×C₇	C₂²×C₄⋊C₄
prime radical	C₂×C₇×#41	D₄
cosocle	C₄²×C₇	C₂⁴
hypercenter	C₈×#41	D₇
hypocenter	C₇	#17827
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
3. #11728505	C₂	semidirect
4. #11728504	C₂	semidirect
5. #11727486	C₂	semidirect
6. #11727484	C₂	semidirect
7. #11727483	C₂	semidirect
8. C₂×#89252	C₂	semidirect
310. C₇	#17827	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₈×#41	2
#11739960	2
#11728505	2
#11728504	2
#11727486	2
#11727484	2
#11727483	2
C₂×#89252	2
#17827	7