Subgroups — #13252003

Order 1792.

444 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#334	C₂	1	1	yes	maximal, characteristic, complemented
2	#11728224	C₂	1	1	yes	maximal, characteristic
3	#11728220	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#89278	C₂	1	1	yes	maximal, characteristic
5	C₄×#6395	C₂	1	1	yes	maximal, characteristic
6	#11725510	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₂₈	C₂²	1	1	yes	characteristic
9	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₂×#6395	C₄	1	1	yes	characteristic
13	C₂×#6395	C₄	1	1	yes	characteristic
14	C₂×(C₂×D₇)⋊C₈	—	2	1	no	—
15	C₂×(C₂×D₇)⋊C₈	—	2	1	no	—
16	C₂×#6389	—	2	1	no	—
17	C₄×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₇×#334	—	2	2	no	—
21	#89346	C₄	1	1	yes	characteristic
22	#89346	C₄	1	1	yes	characteristic
23	#89345	—	2	1	no	—
24	#89278	—	2	2	no	—
25	#9546	—	7	1	no	maximal, Sylow
26	C₂²×D₂₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×D₇	—	2	1	no	—
28	C₂²×C₇×C₈	—	2	1	no	—
29	C₂²×C₇×C₈	—	2	1	no	—
30	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
33	C₂×(C₂×D₇)⋊C₄	—	2	1	no	—
34	C₂×C₄⋊Dic₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊Dic₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×Dic₇	—	2	1	no	—
38	C₂²×C₇⋊C₈	—	2	1	no	—
39	C₂²×C₇⋊C₈	D₄	1	1	yes	characteristic
40	C₂²×C₇⋊C₈	D₄	1	1	yes	characteristic
41	C₄×D₂₈	—	2	1	no	—
42	C₄×D₂₈	—	2	1	no	—
43	C₄×D₂₈	—	2	1	no	—
44	C₄×D₂₈	—	2	1	no	—
45	C₇×C₄⋊C₈	—	2	1	no	—
46	C₇×C₄⋊C₈	—	2	1	no	—
47	C₇×C₄⋊C₈	—	2	1	no	—
48	C₇×C₄⋊C₈	D₄	1	2	yes	—
49	C₄×C₇×C₈	—	2	1	no	—
50	C₄×C₇×C₈	—	2	1	no	—
51	C₇⋊#95	—	2	1	no	—
52	#6395	—	2	2	no	—
53	#6395	—	2	2	no	—
54	(C₂×D₇)⋊C₈	—	4	2	no	—
55	(C₂×D₇)⋊C₈	—	4	2	no	—
56	#6389	—	4	2	no	—
57	C₄×C₇⋊C₈	—	2	1	no	—
58	C₄×C₇⋊C₈	—	2	1	no	—
59	C₄×C₇⋊C₈	—	2	1	no	—
60	C₄×C₇⋊C₈	—	2	1	no	—
61	#1954	—	7	1	no	—
62	#1770	—	7	1	no	—
63	C₄×#102	—	7	1	no	—
64	C₂×#334	D₇	1	1	yes	characteristic, complemented
65	C₂×D₄⋊C₈	—	7	1	no	—
66	#1279	—	7	1	no	—
67	#1273	—	7	1	no	—
68	C₂³×D₇	—	2	1	no	—
69	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₇	#41	1	1	yes	characteristic
72	C₂²×Dic₇	—	2	1	no	—
73	C₂²×Dic₇	—	2	1	no	—
74	C₂²×Dic₇	—	2	1	no	—
75	C₂×D₂₈	—	2	1	no	—
76	C₂×D₂₈	—	2	1	no	—
77	C₂×D₂₈	—	2	1	no	—
78	C₂×D₂₈	—	2	1	no	—
79	C₂×D₂₈	—	2	1	no	—
80	C₂×D₂₈	M₁₆	1	2	yes	—
81	C₂×C₄×D₇	—	4	1	no	—
82	C₂×C₄×D₇	—	4	1	no	—
83	C₂×C₄×D₇	—	4	1	no	—
84	C₂×C₄×D₇	—	4	1	no	—
85	C₂×C₄×D₇	—	4	1	no	—
86	C₂×C₄×D₇	—	4	1	no	—
87	C₂×C₇×C₈	—	2	1	no	—
88	C₂×C₇×C₈	—	2	1	no	—
89	C₂×C₇×C₈	—	2	1	no	—
90	C₂×C₇×C₈	—	2	2	no	—
91	C₂×C₇×C₈	—	2	1	no	—
92	C₂×C₇×C₈	—	2	1	no	—
93	C₂×C₇×C₈	—	4	1	no	—
94	C₂×C₇×C₈	—	4	1	no	—
95	C₂×C₇×C₈	—	4	1	no	—
96	C₂×C₇×C₈	—	4	1	no	—
97	C₂×C₇×C₈	—	4	1	no	—
98	C₂×C₇×C₈	M₁₆	1	2	yes	—
99	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₇	#41	1	1	yes	characteristic
101	C₄²×C₇	#41	1	1	yes	characteristic
102	C₄²×C₇	#41	1	1	yes	characteristic
103	(C₂×D₇)⋊C₄	—	4	1	no	—
104	(C₂×D₇)⋊C₄	—	4	1	no	—
105	C₄⋊Dic₇	—	2	1	no	—
106	C₄⋊Dic₇	—	2	1	no	—
107	C₄⋊Dic₇	—	2	1	no	—
108	C₄⋊Dic₇	—	2	1	no	—
109	C₄⋊Dic₇	—	2	1	no	—
110	C₄⋊Dic₇	—	2	1	no	—
111	C₄×Dic₇	—	4	1	no	—
112	C₄×Dic₇	—	4	1	no	—
113	C₂×C₇⋊C₈	—	2	1	no	—
114	C₂×C₇⋊C₈	—	2	1	no	—
115	C₂×C₇⋊C₈	—	2	2	no	—
116	C₂×C₇⋊C₈	—	2	1	no	—
117	C₂×C₇⋊C₈	—	2	1	no	—
118	C₂×C₇⋊C₈	—	2	1	no	—
119	C₂×C₇⋊C₈	—	4	1	no	—
120	C₂×C₇⋊C₈	—	4	1	no	—
121	C₂×C₇⋊C₈	SD₁₆	1	2	yes	—
122	C₂×C₄×D₄	—	7	1	no	—
123	C₂×C₄⋊C₈	C₂×D₇	1	1	yes	characteristic
124	C₂×C₄⋊C₈	C₂×D₇	1	1	yes	characteristic
125	C₂×#102	—	7	1	no	—
126	C₂×#102	—	7	1	no	—
127	C₂×C₂²⋊C₈	—	14	1	no	—
128	C₂×C₂²⋊C₈	—	14	1	no	—
129	C₂×#97	—	14	1	no	—
130	C₂×C₄×C₈	—	7	1	no	—
131	C₂×C₄×C₈	—	7	1	no	—
132	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
133	C₄×C₄⋊C₄	—	7	1	no	—
134	#340	—	7	1	no	—
135	#340	—	7	1	no	—
136	#336	—	14	1	no	—
137	#334	—	2	2	no	—
138	D₄⋊C₈	—	14	2	no	—
139	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
140	C₂²×D₇	—	2	2	no	—
141	C₂²×D₇	—	4	1	no	—
142	C₂²×D₇	—	4	1	no	—
143	C₂²×D₇	—	4	1	no	—
144	C₂²×D₇	—	4	1	no	—
145	C₂²×D₇	—	4	1	no	—
146	C₂²×D₇	—	4	1	no	—
147	C₂×C₄×C₇	—	2	1	no	—
148	C₂×C₄×C₇	—	2	1	no	—
149	C₂×C₄×C₇	#117	1	1	yes	characteristic
150	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₇	#121	1	1	yes	characteristic
152	C₂×C₄×C₇	#123	1	1	yes	characteristic
153	C₂×C₄×C₇	#124	1	1	yes	characteristic
154	C₂×C₄×C₇	#126	1	1	yes	characteristic
155	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
156	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
157	C₂×C₄×C₇	#131	1	1	yes	characteristic
158	C₂×C₄×C₇	#131	1	1	yes	characteristic
159	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
160	C₂×C₄×C₇	#135	1	1	yes	characteristic
161	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
163	C₂×Dic₇	—	4	1	no	—
164	C₂×Dic₇	—	4	1	no	—
165	C₂×Dic₇	—	4	1	no	—
166	C₂×Dic₇	—	4	1	no	—
167	C₂×Dic₇	—	4	1	no	—
168	C₂×Dic₇	—	4	1	no	—
169	C₂×Dic₇	—	4	1	no	—
170	C₂×Dic₇	—	4	1	no	—
171	C₂×Dic₇	—	4	1	no	—
172	D₂₈	—	2	2	no	—
173	D₂₈	—	2	2	no	—
174	D₂₈	—	4	1	no	—
175	D₂₈	—	4	1	no	—
176	C₄×D₇	—	8	1	no	—
177	C₄×D₇	—	8	1	no	—
178	C₄×D₇	—	8	1	no	—
179	C₄×D₇	—	8	1	no	—
180	C₇×C₈	—	2	2	no	—
181	C₇×C₈	—	4	1	no	—
182	C₇×C₈	—	4	2	no	—
183	C₇×C₈	—	8	1	no	—
184	C₇⋊C₈	—	2	2	no	—
185	C₇⋊C₈	—	4	2	no	—
186	C₇⋊C₈	—	4	1	no	—
187	C₂²×D₄	—	7	1	no	—
188	C₂³×C₄	—	14	1	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	—	7	1	no	—
192	C₂²×C₈	—	7	1	no	—
193	C₂²×C₈	—	14	1	no	—
194	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
195	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
196	C₄×D₄	—	14	1	no	—
197	C₄×D₄	—	14	1	no	—
198	C₄×D₄	—	14	1	no	—
199	C₄×D₄	—	14	1	no	—
200	C₂×C₄⋊C₄	—	7	1	no	—
201	C₂×C₄⋊C₄	—	7	1	no	—
202	C₂×C₄⋊C₄	—	7	1	no	—
203	C₂×C₂²⋊C₄	—	14	1	no	—
204	C₂×C₄²	—	14	1	no	—
205	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
206	C₄⋊C₈	—	2	1	no	—
207	C₄⋊C₈	—	2	1	no	—
208	C₄⋊C₈	—	2	1	no	—
209	C₄⋊C₈	C₂²⋊D₇	1	2	yes	—
210	#102	—	14	2	no	—
211	#102	—	14	2	no	—
212	C₂²⋊C₈	—	28	2	no	—
213	C₂²⋊C₈	—	28	2	no	—
214	#97	—	28	2	no	—
215	C₄×C₈	—	2	1	no	—
216	C₄×C₈	—	2	1	no	—
217	C₄×C₈	—	14	1	no	—
218	C₄×C₈	—	14	1	no	—
219	C₄×C₈	—	14	1	no	—
220	C₄×C₈	—	14	1	no	—
221	#95	—	14	1	no	—
222	C₂²×C₇	#388	1	1	yes	characteristic
223	C₂²×C₇	#435	1	1	yes	characteristic
224	C₂²×C₇	#436	1	1	yes	characteristic
225	C₂²×C₇	C₄×SD₁₆	1	1	yes	characteristic
226	C₂²×C₇	#442	1	1	yes	characteristic
227	C₂²×C₇	#443	1	1	yes	characteristic
228	C₂²×C₇	#444	1	1	yes	characteristic
229	C₂×D₇	—	4	2	no	—
230	C₂×D₇	—	4	2	no	—
231	C₂×D₇	—	4	2	no	—
232	C₂×D₇	—	8	1	no	—
233	C₂×D₇	—	8	1	no	—
234	C₂×D₇	—	8	1	no	—
235	C₂×D₇	—	8	1	no	—
236	C₄×C₇	—	2	1	no	—
237	C₄×C₇	—	2	1	no	—
238	C₄×C₇	#432	1	1	yes	characteristic
239	C₄×C₇	#433	1	1	yes	characteristic
240	C₄×C₇	#435	1	1	yes	characteristic
241	C₄×C₇	#435	1	1	yes	characteristic
242	C₄×C₇	#460	1	1	yes	characteristic
243	C₄×C₇	#463	1	1	yes	characteristic
244	C₄×C₇	#481	1	1	yes	characteristic
245	C₄×C₇	#482	1	1	yes	characteristic
246	Dic₇	—	8	1	no	—
247	Dic₇	—	8	1	no	—
248	Dic₇	—	8	1	no	—
249	C₂⁴	—	14	1	no	—
250	C₂×D₄	—	7	2	no	—
251	C₂×D₄	—	14	1	no	—
252	C₂×D₄	—	14	1	no	—
253	C₂×D₄	—	14	1	no	—
254	C₂×D₄	—	14	1	no	—
255	C₂×D₄	—	14	1	no	—
256	C₂²×C₄	—	14	1	no	—
257	C₂²×C₄	—	14	1	no	—
258	C₂²×C₄	—	14	1	no	—
259	C₂²×C₄	—	28	1	no	—
260	C₂²×C₄	—	28	1	no	—
261	C₂²×C₄	—	28	1	no	—
262	C₂²×C₄	—	28	1	no	—
263	C₂²×C₄	—	28	1	no	—
264	C₂²×C₄	—	28	1	no	—
265	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
266	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
267	C₂²×C₄	#1161	1	1	yes	characteristic
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	2	1	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	4	1	no	—
275	C₂×C₈	—	4	1	no	—
276	C₂×C₈	—	4	1	no	—
277	C₂×C₈	—	4	1	no	—
278	C₂×C₈	—	4	1	no	—
279	C₂×C₈	—	7	2	no	—
280	C₂×C₈	—	14	1	no	—
281	C₂×C₈	—	14	1	no	—
282	C₂×C₈	—	14	2	no	—
283	C₂×C₈	—	14	1	no	—
284	C₂×C₈	—	14	1	no	—
285	C₂×C₈	—	14	1	no	—
286	C₂×C₈	—	28	1	no	—
287	C₂×C₈	—	28	1	no	—
288	C₂×C₈	#1131	1	2	yes	—
289	C₄⋊C₄	—	14	1	no	—
290	C₄⋊C₄	—	14	1	no	—
291	C₄⋊C₄	—	14	1	no	—
292	C₄⋊C₄	—	14	1	no	—
293	C₄⋊C₄	—	14	1	no	—
294	C₄⋊C₄	—	14	1	no	—
295	C₂²⋊C₄	—	28	1	no	—
296	C₂²⋊C₄	—	28	1	no	—
297	C₄²	—	28	1	no	—
298	C₄²	—	28	1	no	—
299	C₄²	#1157	1	1	yes	characteristic
300	C₄²	#1159	1	1	yes	characteristic
301	C₄²	#1161	1	1	yes	characteristic
302	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
303	C₂×C₇	#1582	1	1	yes	characteristic
304	C₂×C₇	#1586	1	1	yes	characteristic
305	C₂×C₇	#1590	1	1	yes	characteristic
306	C₂×C₇	#1594	1	1	yes	characteristic
307	C₂×C₇	#1924	1	1	yes	characteristic
308	C₂×C₇	#1926	1	1	yes	characteristic
309	C₂×C₇	#1929	1	1	yes	characteristic
310	D₇	—	8	2	no	—
311	C₂³	—	14	2	no	—
312	C₂³	—	28	1	no	—
313	C₂³	—	28	1	no	—
314	C₂³	—	28	1	no	—
315	C₂³	—	28	1	no	—
316	C₂³	—	28	1	no	—
317	C₂³	—	28	1	no	—
318	C₂³	#6456	1	1	yes	characteristic
319	D₄	—	14	2	no	—
320	D₄	—	14	2	no	—
321	D₄	—	28	1	no	—
322	D₄	—	28	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	28	1	no	—
326	C₂×C₄	—	28	1	no	—
327	C₂×C₄	—	28	1	no	—
328	C₂×C₄	—	28	1	no	—
329	C₂×C₄	—	28	1	no	—
330	C₂×C₄	—	28	1	no	—
331	C₂×C₄	—	28	1	no	—
332	C₂×C₄	—	28	1	no	—
333	C₂×C₄	—	28	1	no	—
334	C₂×C₄	—	56	1	no	—
335	C₂×C₄	—	56	1	no	—
336	C₂×C₄	—	56	1	no	—
337	C₂×C₄	—	56	1	no	—
338	C₂×C₄	#6455	1	1	yes	characteristic
339	C₂×C₄	#6457	1	1	yes	characteristic
340	C₂×C₄	#6461	1	1	yes	characteristic
341	C₂×C₄	C₂×#1131	1	1	yes	characteristic
342	C₂×C₄	#6464	1	1	yes	characteristic
343	C₂×C₄	D₇×M₁₆	1	1	yes	characteristic
344	C₂×C₄	#6470	1	1	yes	characteristic
345	C₂×C₄	D₇×SD₁₆	1	1	yes	characteristic
346	C₂×C₄	#6477	1	1	yes	characteristic
347	C₂×C₄	#6477	1	1	yes	characteristic
348	C₂×C₄	#6479	1	1	yes	characteristic
349	C₂×C₄	C₄×C₂²⋊D₇	1	1	yes	characteristic
350	C₂×C₄	#6502	1	1	yes	characteristic
351	C₂×C₄	#6510	1	1	yes	characteristic
352	C₈	—	2	2	no	—
353	C₈	—	4	1	no	—
354	C₈	—	4	2	no	—
355	C₈	—	8	1	no	—
356	C₈	—	14	2	no	—
357	C₈	—	28	2	no	—
358	C₈	—	28	1	no	—
359	C₇	#9546	1	1	yes	characteristic, complemented, Sylow
360	C₂²	—	28	2	no	—
361	C₂²	—	28	2	no	—
362	C₂²	—	28	2	no	—
363	C₂²	—	56	1	no	—
364	C₂²	—	56	1	no	—
365	C₂²	—	56	1	no	—
366	C₂²	—	56	1	no	—
367	C₂²	#89608	1	1	yes	characteristic
368	C₂²	#89610	1	1	yes	characteristic
369	C₂²	#89624	1	1	yes	characteristic
370	C₂²	#89641	1	1	yes	characteristic
371	C₂²	#89666	1	1	yes	characteristic
372	C₂²	#89667	1	1	yes	characteristic
373	C₂²	#89762	1	1	yes	characteristic
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	56	1	no	—
377	C₄	—	56	1	no	—
378	C₄	—	56	1	no	—
379	C₄	#89607	1	1	yes	characteristic
380	C₄	#89612	1	1	yes	characteristic
381	C₄	#89635	1	1	yes	characteristic
382	C₄	#89638	1	1	yes	characteristic
383	C₄	#89883	1	1	yes	characteristic
384	C₄	#89899	1	1	yes	characteristic
385	C₄	#89935	1	1	yes	characteristic
386	C₄	#89937	1	1	yes	characteristic
387	C₂	—	56	2	no	—
388	C₂	#11726273	1	1	yes	characteristic
389	C₂	#11726275	1	1	yes	characteristic
390	C₂	#11726277	1	1	yes	characteristic
391	C₂	#11726279	1	1	yes	characteristic
392	C₂	#11728204	1	1	yes	characteristic
393	C₂	#11728323	1	1	yes	characteristic
394	C₂	#11728453	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇
commutator	C₂²×C₄×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₂×C₇×#334	C₂
socle	C₂³×C₇	C₄×D₄
prime radical	C₂²×D₂₈	C₂×C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	C₂×#334	D₇
hypocenter	C₇	#9546
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#334	C₂	semidirect
3. #11728220	C₂	semidirect
6. #11725510	C₂	semidirect
7. C₇⋊#1273	C₂	semidirect
64. C₂×#334	D₇	semidirect
359. C₇	#9546	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#334	2
#11728224	2
#11728220	2
C₂×#89278	2
C₄×#6395	2
#11725510	2
C₇⋊#1273	2
#9546	7