Subgroups — #13507364

Order 1792.

552 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#1159	C₂	1	1	yes	maximal, characteristic
2	C₇×D₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#11729161	C₂	1	2	yes	maximal, complemented
4	C₂×#89297	C₂	1	2	yes	maximal, complemented
5	D₄×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#11727035	C₂	1	1	yes	maximal, characteristic, complemented
7	#11727029	C₂	1	1	yes	maximal, characteristic, complemented
8	#11726856	C₂	1	2	yes	maximal, complemented
9	#11726847	C₂	1	2	yes	maximal, complemented
10	C₄×#6372	C₂	1	1	yes	maximal, characteristic, complemented
11	#11726802	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#1159	C₄	1	1	yes	characteristic
13	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
14	C₇×C₈×D₄	C₂²	1	1	yes	characteristic
15	C₈×C₂²⋊D₇	C₂²	1	2	yes	—
16	C₂×C₈×Dic₇	C₂²	1	2	yes	—
17	C₂²×C₇⋊C₁₆	C₂²	1	2	yes	—
18	D₄×C₇⋊C₈	C₄	1	1	yes	characteristic
19	C₂×#6372	C₂²	1	1	yes	characteristic
20	C₂×#6372	C₂²	1	1	yes	characteristic
21	#89605	C₄	1	1	yes	characteristic
22	#89602	C₄	1	1	yes	characteristic
23	(C₂²⋊D₇)⋊C₈	C₄	1	2	yes	—
24	#89490	C₄	1	2	yes	—
25	C₄×C₈×D₇	C₂²	1	1	yes	characteristic
26	C₈×Dic₁₄	C₂²	1	1	yes	characteristic
27	C₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
28	C₇×C₂²⋊C₁₆	C₂²	1	2	yes	—
29	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
30	C₂²⋊(C₇⋊C₁₆)	C₂²	1	2	yes	—
31	(C₂×D₇)⋊C₁₆	C₂²	1	2	yes	—
32	#89297	C₂²	1	1	yes	characteristic
33	#89297	C₂²	1	2	yes	—
34	#89297	C₂²	1	2	yes	—
35	#89297	C₂²	1	2	yes	—
36	#89297	C₂²	1	1	yes	characteristic
37	Dic₇⋊C₁₆	C₂²	1	2	yes	—
38	Dic₇⋊C₁₆	C₂²	1	2	yes	—
39	C₄⋊(C₇⋊C₁₆)	C₂²	1	1	yes	characteristic
40	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
41	#33484	—	7	1	no	maximal, Sylow
42	C₂×#1159	C₈	1	1	yes	characteristic
43	C₂²×C₇×C₈	C₂³	1	2	yes	—
44	C₄×C₇×D₄	C₂×C₄	1	1	yes	characteristic
45	D₄×Dic₇	C₈	1	1	yes	characteristic
46	C₄×C₂²⋊D₇	C₂×C₄	1	2	yes	—
47	C₂×C₄×Dic₇	C₂×C₄	1	2	yes	—
48	C₂²×C₇⋊C₈	C₂×C₄	1	2	yes	—
49	C₂×C₈×D₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₇	C₂³	1	1	yes	characteristic
51	#6455	C₈	1	1	yes	characteristic
52	#6450	C₈	1	1	yes	characteristic
53	(C₂²⋊D₇)⋊C₄	C₈	1	2	yes	—
54	#6440	C₈	1	2	yes	—
55	C₄²×D₇	C₂×C₄	1	1	yes	characteristic
56	C₄×Dic₁₄	C₂×C₄	1	1	yes	characteristic
57	C₂×C₇×C₁₆	—	2	4	no	—
58	C₂×C₇×C₁₆	C₂³	1	2	yes	—
59	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
61	C₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
62	C₇×C₂²⋊C₈	C₂×C₄	1	2	yes	—
63	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
64	C₂²⋊(C₇⋊C₈)	C₂×C₄	1	2	yes	—
65	(C₂×D₇)⋊C₈	C₂×C₄	1	2	yes	—
66	Dic₇⋊C₈	C₂×C₄	1	2	yes	—
67	Dic₇⋊C₈	C₂×C₄	1	2	yes	—
68	C₈×Dic₇	C₂×C₄	1	2	yes	—
69	C₈×Dic₇	C₂×C₄	1	2	yes	—
70	C₈×Dic₇	C₂×C₄	1	1	yes	characteristic
71	C₈×Dic₇	C₂³	1	1	yes	characteristic
72	C₈×Dic₇	C₂³	1	2	yes	—
73	C₂×C₇⋊C₁₆	—	2	4	no	—
74	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
75	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
76	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
77	C₄⋊(C₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
78	C₄×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
79	#6372	—	2	2	no	—
80	#6372	—	2	2	no	—
81	C₈×#41	—	7	1	no	—
82	#2183	—	7	1	no	—
83	#2183	—	7	1	no	—
84	#2168	—	7	2	no	—
85	#2168	—	7	2	no	—
86	D₄×C₁₆	—	7	1	no	—
87	D₄×C₁₆	D₇	1	1	yes	characteristic, complemented
88	#2163	—	7	1	no	—
89	#2163	—	7	2	no	—
90	C₄×M₃₂	—	7	1	no	—
91	C₂×#346	—	7	2	no	—
92	C₂×C₇×D₄	C₂×C₈	1	1	yes	characteristic
93	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
94	C₂×C₂²⋊D₇	C₂×C₈	1	2	yes	—
95	C₂²×Dic₇	C₂×C₈	1	2	yes	—
96	C₂²×Dic₇	C₂×C₈	1	2	yes	—
97	#1159	—	2	4	no	—
98	C₂×C₄×D₇	C₂×C₈	1	1	yes	characteristic
99	C₂×C₄×D₇	C₂×C₈	1	1	yes	characteristic
100	C₂×C₄×D₇	C₂²×C₄	1	1	yes	characteristic
101	C₂×Dic₁₄	C₂×C₈	1	1	yes	characteristic
102	C₂×C₇×C₈	—	2	2	no	—
103	C₂×C₇×C₈	—	2	2	no	—
104	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
106	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
107	C₇×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
108	C₇×C₂²⋊C₄	C₂×C₈	1	2	yes	—
109	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
110	C₂²⋊Dic₇	C₂×C₈	1	2	yes	—
111	(C₂×D₇)⋊C₄	C₂×C₈	1	2	yes	—
112	C₄⋊Dic₇	C₂×C₈	1	1	yes	characteristic
113	Dic₇⋊C₄	C₂×C₈	1	2	yes	—
114	Dic₇⋊C₄	C₂×C₈	1	2	yes	—
115	C₄×Dic₇	C₂×C₈	1	1	yes	characteristic
116	C₄×Dic₇	C₂×C₈	1	1	yes	characteristic
117	C₄×Dic₇	C₂×C₈	1	2	yes	—
118	C₄×Dic₇	C₂×C₈	1	2	yes	—
119	C₄×Dic₇	C₂²×C₄	1	2	yes	—
120	C₄×Dic₇	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₇⋊C₈	—	2	4	no	—
122	C₂×C₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₇⋊C₈	C₂²×C₄	1	2	yes	—
124	C₂×C₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
125	C₈×D₇	—	2	1	no	—
126	C₈×D₇	—	2	1	no	—
127	C₈×D₇	—	2	2	no	—
128	C₇×C₁₆	—	2	2	no	—
129	C₇×C₁₆	—	2	1	no	—
130	C₇×C₁₆	#41	1	2	yes	complemented
131	C₇⋊C₁₆	—	2	1	no	—
132	C₇⋊C₁₆	—	2	2	no	—
133	C₇⋊C₁₆	#41	1	2	yes	complemented
134	C₄×#41	—	7	1	no	—
135	C₂×M₃₂	—	7	1	no	—
136	C₂×M₃₂	—	7	1	no	—
137	C₂²×C₁₆	—	7	2	no	—
138	C₂²×C₁₆	C₂×D₇	1	2	yes	—
139	C₈×Q₈	—	7	1	no	—
140	C₈×Q₈	—	7	1	no	—
141	C₈×D₄	—	7	2	no	—
142	C₈×D₄	—	7	1	no	—
143	C₈×D₄	—	7	2	no	—
144	C₈×D₄	C₂×D₇	1	1	yes	characteristic
145	#431	—	7	1	no	—
146	#431	—	7	2	no	—
147	C₂×C₄×C₈	—	7	2	no	—
148	C₂×C₄×C₈	—	7	1	no	—
149	C₄⋊C₁₆	—	7	2	no	—
150	C₄⋊C₁₆	—	7	2	no	—
151	C₄⋊C₁₆	—	7	1	no	—
152	C₄⋊C₁₆	C₂×D₇	1	1	yes	characteristic
153	C₂²⋊C₁₆	—	7	2	no	—
154	C₂²⋊C₁₆	—	7	2	no	—
155	C₂²⋊C₁₆	C₂×D₇	1	2	yes	—
156	#346	—	7	1	no	—
157	#346	—	7	2	no	—
158	#346	—	7	2	no	—
159	#346	—	7	2	no	—
160	#346	—	7	1	no	—
161	C₄×C₁₆	—	7	1	no	—
162	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
163	C₂³×C₇	C₂²×C₈	1	2	yes	—
164	C₂²×D₇	C₂²×C₈	1	1	yes	characteristic
165	C₇×D₄	M₃₂	1	4	yes	complemented
166	C₂×C₄×C₇	—	2	2	no	—
167	C₂×C₄×C₇	—	2	2	no	—
168	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
169	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
170	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
171	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
172	C₂²⋊D₇	—	2	4	no	—
173	C₂×Dic₇	—	2	4	no	—
174	C₂×Dic₇	—	4	2	no	—
175	C₂×Dic₇	C₂²×C₈	1	1	yes	characteristic
176	C₂×Dic₇	C₂²×C₈	1	1	yes	characteristic
177	C₂×Dic₇	C₂²×C₈	1	1	yes	characteristic
178	C₂×Dic₇	C₂²×C₈	1	2	yes	—
179	C₂×Dic₇	C₂²×C₈	1	2	yes	—
180	C₄×D₇	—	2	1	no	—
181	C₄×D₇	—	2	1	no	—
182	C₄×D₇	—	2	2	no	—
183	C₄×D₇	—	4	1	no	—
184	Dic₁₄	—	2	2	no	—
185	C₇×C₈	—	2	2	no	—
186	C₇×C₈	—	2	1	no	—
187	C₇×C₈	C₂×#41	1	1	yes	characteristic
188	C₇×C₈	C₂×#41	1	1	yes	characteristic
189	C₇⋊C₈	—	2	1	no	—
190	C₇⋊C₈	—	2	2	no	—
191	C₇⋊C₈	#131	1	2	yes	—
192	C₂×#41	—	7	1	no	—
193	C₂²×C₈	—	7	2	no	—
194	C₂²×C₈	—	7	1	no	—
195	C₂²×C₈	—	7	1	no	—
196	C₂²×C₈	C₂²×D₇	1	2	yes	—
197	C₄×Q₈	—	7	1	no	—
198	C₄×Q₈	—	7	1	no	—
199	C₄×D₄	—	7	1	no	—
200	C₄×D₄	—	7	2	no	—
201	C₄×D₄	—	7	2	no	—
202	C₄×D₄	C₄×D₇	1	1	yes	characteristic
203	#117	—	7	1	no	—
204	#117	—	7	2	no	—
205	C₂×C₄²	—	7	2	no	—
206	C₂×C₄²	—	7	1	no	—
207	M₃₂	—	14	2	no	—
208	M₃₂	—	14	2	no	—
209	C₂×C₁₆	—	2	4	no	—
210	C₂×C₁₆	—	7	1	no	—
211	C₂×C₁₆	—	7	1	no	—
212	C₂×C₁₆	—	7	2	no	—
213	C₂×C₁₆	—	14	4	no	—
214	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
215	C₂×C₁₆	C₂²×D₇	1	2	yes	—
216	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
217	C₄⋊C₈	—	7	2	no	—
218	C₄⋊C₈	—	7	2	no	—
219	C₄⋊C₈	—	7	1	no	—
220	C₄⋊C₈	C₄×D₇	1	1	yes	characteristic
221	C₂²⋊C₈	—	7	2	no	—
222	C₂²⋊C₈	—	7	2	no	—
223	C₂²⋊C₈	C₄×D₇	1	2	yes	—
224	C₄×C₈	—	7	2	no	—
225	C₄×C₈	—	7	1	no	—
226	C₄×C₈	—	7	2	no	—
227	C₄×C₈	—	7	1	no	—
228	C₄×C₈	—	7	2	no	—
229	C₄×C₈	—	7	1	no	—
230	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
231	C₂²×C₇	—	2	2	no	—
232	C₂²×C₇	—	2	2	no	—
233	C₂²×C₇	C₂×M₃₂	1	4	yes	—
234	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
235	C₂×D₇	—	2	1	no	—
236	C₂×D₇	—	2	1	no	—
237	C₂×D₇	—	2	1	no	—
238	C₄×C₇	—	2	1	no	—
239	C₄×C₇	—	2	2	no	—
240	C₄×C₇	C₂×M₃₂	1	2	yes	—
241	C₄×C₇	C₄×#41	1	1	yes	characteristic
242	C₄×C₇	C₂×#131	1	1	yes	characteristic
243	Dic₇	—	2	2	no	—
244	Dic₇	—	2	1	no	—
245	Dic₇	—	2	1	no	—
246	Dic₇	—	2	2	no	—
247	Dic₇	—	2	1	no	—
248	#41	—	14	4	no	—
249	C₂×Q₈	—	7	1	no	—
250	C₂×D₄	—	7	2	no	—
251	C₂×D₄	C₈×D₇	1	1	yes	characteristic
252	C₂²×C₄	—	7	2	no	—
253	C₂²×C₄	—	7	2	no	—
254	C₂²×C₄	—	7	1	no	—
255	C₂²×C₄	—	7	1	no	—
256	C₂²×C₄	—	7	1	no	—
257	C₂²×C₄	C₂×C₄×D₇	1	2	yes	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	7	1	no	—
261	C₂×C₈	—	7	2	no	—
262	C₂×C₈	—	7	1	no	—
263	C₂×C₈	—	14	2	no	—
264	C₂×C₈	—	14	4	no	—
265	C₂×C₈	—	14	1	no	—
266	C₂×C₈	—	14	1	no	—
267	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
268	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
269	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
270	C₄⋊C₄	—	7	2	no	—
271	C₄⋊C₄	—	7	2	no	—
272	C₄⋊C₄	—	7	1	no	—
273	C₄⋊C₄	C₈×D₇	1	1	yes	characteristic
274	C₂²⋊C₄	—	7	2	no	—
275	C₂²⋊C₄	—	7	2	no	—
276	C₂²⋊C₄	C₈×D₇	1	2	yes	—
277	C₄²	—	7	2	no	—
278	C₄²	—	7	1	no	—
279	C₄²	—	7	1	no	—
280	C₄²	—	7	2	no	—
281	C₄²	—	7	1	no	—
282	C₄²	—	7	2	no	—
283	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
284	C₁₆	—	2	2	no	—
285	C₁₆	—	2	1	no	—
286	C₁₆	—	7	2	no	—
287	C₁₆	—	14	1	no	—
288	C₁₆	—	14	2	no	—
289	C₁₆	#1159	1	2	yes	complemented
290	C₂×C₇	—	2	4	no	—
291	C₂×C₇	C₈×#41	1	1	yes	characteristic
292	C₂×C₇	C₂²×M₃₂	1	1	yes	characteristic
293	C₂×C₇	#3407	1	1	yes	characteristic
294	D₇	—	4	1	no	—
295	C₂³	—	7	1	no	—
296	C₂³	C₂×C₈×D₇	1	2	yes	—
297	Q₈	—	14	2	no	—
298	D₄	—	14	4	no	—
299	D₄	#6372	1	4	yes	complemented
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	7	1	no	—
303	C₂×C₄	—	7	1	no	—
304	C₂×C₄	—	7	1	no	—
305	C₂×C₄	—	7	2	no	—
306	C₂×C₄	—	7	2	no	—
307	C₂×C₄	—	14	4	no	—
308	C₂×C₄	—	14	2	no	—
309	C₂×C₄	—	14	1	no	—
310	C₂×C₄	—	14	1	no	—
311	C₂×C₄	—	28	1	no	—
312	C₂×C₄	—	28	2	no	—
313	C₂×C₄	C₂×C₈×D₇	1	2	yes	—
314	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
315	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
316	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	2	2	no	—
319	C₈	—	7	2	no	—
320	C₈	—	14	2	no	—
321	C₈	—	14	1	no	—
322	C₈	C₂×#1159	1	1	yes	characteristic
323	C₈	D₇×#41	1	1	yes	characteristic
324	C₇	#33484	1	1	yes	characteristic, complemented, Sylow
325	C₂²	—	2	2	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	14	1	no	—
328	C₂²	—	14	1	no	—
329	C₂²	—	14	1	no	—
330	C₂²	C₂×#6372	1	4	yes	—
331	C₂²	C₂²×C₈×D₇	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	1	no	—
334	C₄	—	14	1	no	—
335	C₄	—	14	1	no	—
336	C₄	—	14	2	no	—
337	C₄	—	14	1	no	—
338	C₄	—	14	2	no	—
339	C₄	C₂×#6372	1	2	yes	—
340	C₄	C₄×#1159	1	1	yes	characteristic
341	C₄	D₇×#131	1	1	yes	characteristic
342	C₂	—	2	4	no	—
343	C₂	—	28	1	no	—
344	C₂	C₈×#1159	1	1	yes	characteristic
345	C₂	C₂²×#6372	1	1	yes	characteristic
346	C₂	#11739457	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇
commutator, socle	C₂²×C₇	C₂³×C₈
Fitting	C₇×D₄×C₁₆	C₂
prime radical	C₂×#1159	C₈
cosocle	C₂×C₇×C₈	C₂⁴
hypercenter	D₄×C₁₆	D₇
hypocenter	C₇	#33484
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₇×D₄×C₁₆	C₂	semidirect
3. #11729161	C₂	semidirect
4. C₂×#89297	C₂	semidirect
5. D₄×C₇⋊C₁₆	C₂	semidirect
6. #11727035	C₂	semidirect
7. #11727029	C₂	semidirect
8. #11726856	C₂	semidirect
9. #11726847	C₂	semidirect
10. C₄×#6372	C₂	semidirect
11. #11726802	C₂	semidirect
87. D₄×C₁₆	D₇	semidirect
130. C₇×C₁₆	#41	semidirect
133. C₇⋊C₁₆	#41	semidirect
165. C₇×D₄	M₃₂	semidirect
289. C₁₆	#1159	semidirect
299. D₄	#6372	semidirect
324. C₇	#33484	semidirect

Maximal subgroups

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

subgroup	index
C₈×#1159	2
C₇×D₄×C₁₆	2
#11729161	2
C₂×#89297	2
D₄×C₇⋊C₁₆	2
#11727035	2
#11727029	2
#11726856	2
#11726847	2
C₄×#6372	2
#11726802	2
#33484	7