Subgroups — #13244011

Order 1792.

650 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#432	C₂	1	1	yes	maximal, characteristic
2	#11739505	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#1136	C₂	1	1	yes	maximal, characteristic
4	#11725554	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
6	C₂²×#1136	—	2	1	no	—
7	C₇×#432	—	2	1	no	—
8	C₇×#432	—	2	1	no	—
9	C₇×#432	C₄	1	2	yes	complemented
10	C₇×#432	C₂²	1	2	yes	—
11	C₂×C₇×C₄⋊C₈	C₄	1	2	yes	complemented
12	C₂×C₇×C₂²⋊C₈	—	2	1	no	—
13	C₂×C₇×#97	—	2	1	no	—
14	#89984	C₄	1	2	yes	—
15	#89983	—	2	1	no	—
16	C₄×C₂²⋊Dic₇	—	2	1	no	—
17	#89709	C₄	1	2	yes	—
18	#89708	—	2	1	no	—
19	C₄×C₄⋊Dic₇	—	2	1	no	—
20	#89704	C₂²	1	2	yes	—
21	C₄×#1136	—	2	1	no	—
22	C₄×#1136	—	2	1	no	—
23	C₄×#1136	C₄	1	2	yes	complemented
24	C₄×#1136	C₂²	1	2	yes	—
25	C₂×#6377	C₄	1	2	yes	complemented
26	C₂×C₄×C₇⋊C₈	—	2	1	no	—
27	#8073	—	7	1	no	maximal, Sylow
28	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
31	C₂²×C₇×C₈	—	2	2	no	—
32	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₇	Q₈	1	1	yes	characteristic
42	C₂×C₄²×C₇	Q₈	1	1	yes	characteristic
43	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
44	C₂×C₂²⋊Dic₇	—	2	2	no	—
45	C₂×C₄⋊Dic₇	—	2	2	no	—
46	C₂×C₄×Dic₇	—	2	2	no	—
47	C₂×#1136	—	2	8	no	—
48	C₂×#1136	—	2	2	no	—
49	C₂×#1136	—	2	2	no	—
50	C₂²×C₇⋊C₈	—	2	2	no	—
51	C₇×C₄⋊C₈	—	2	2	no	—
52	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
53	C₇×C₂²⋊C₈	—	2	2	no	—
54	C₇×C₂²⋊C₈	—	2	2	no	—
55	C₇×#97	—	2	2	no	—
56	C₇×#97	—	2	2	no	—
57	C₇⋊#95	—	2	2	no	—
58	C₇⋊#95	—	2	2	no	—
59	C₇⋊#95	—	2	2	no	—
60	C₇⋊#95	C₂×C₄	1	4	yes	—
61	#6377	—	2	2	no	—
62	#6377	C₂×C₄	1	4	yes	—
63	C₄×C₇⋊C₈	—	2	2	no	—
64	C₄×C₇⋊C₈	—	2	2	no	—
65	C₂×#432	D₇	1	1	yes	characteristic
66	C₂×C₄×M₁₆	—	7	1	no	—
67	#2283	—	7	1	no	—
68	#1300	—	7	4	no	—
69	C₂⁴×C₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇	—	2	4	no	—
71	C₂²×C₄×C₇	—	4	2	no	—
72	C₂²×C₄×C₇	—	4	2	no	—
73	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
74	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
75	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×Dic₇	—	2	4	no	—
92	C₂×C₇×C₈	—	2	4	no	—
93	C₂×C₇×C₈	—	4	4	no	—
94	C₄²×C₇	C₄²	1	1	yes	characteristic
95	C₄²×C₇	C₄²	1	1	yes	characteristic
96	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
97	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
98	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
99	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
100	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
101	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
102	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
103	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
104	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
105	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
106	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
110	C₂²⋊Dic₇	—	4	4	no	—
111	C₄⋊Dic₇	—	4	4	no	—
112	C₄×Dic₇	—	4	4	no	—
113	#1136	—	2	8	no	—
114	#1136	—	2	8	no	—
115	C₂×C₇⋊C₈	—	2	8	no	—
116	C₂×C₇⋊C₈	—	2	4	no	—
117	C₂²×M₁₆	—	14	1	no	—
118	C₂²×C₄²	C₂×D₇	1	1	yes	characteristic
119	#432	—	2	1	no	—
120	#432	—	2	1	no	—
121	#432	Dic₇	1	2	yes	complemented
122	#432	C₂×D₇	1	2	yes	—
123	C₂×C₄⋊C₈	Dic₇	1	2	yes	complemented
124	C₂×C₂²⋊C₈	—	2	1	no	—
125	C₄×M₁₆	—	7	2	no	—
126	C₄×M₁₆	—	7	2	no	—
127	C₄×M₁₆	—	14	1	no	—
128	C₄×M₁₆	—	14	1	no	—
129	C₂×#97	—	2	1	no	—
130	C₂×#97	—	7	2	no	—
131	C₂×C₄×C₈	—	14	1	no	—
132	#383	—	7	2	no	—
133	#382	—	14	1	no	—
134	#381	—	14	1	no	—
135	#380	—	7	2	no	—
136	C₄×C₄⋊C₄	—	14	1	no	—
137	C₄×C₂²⋊C₄	—	14	1	no	—
138	#376	—	7	2	no	—
139	C₂³×C₇	—	4	1	no	—
140	C₂³×C₇	—	4	1	no	—
141	C₂³×C₇	#95	1	1	yes	characteristic
142	C₂³×C₇	#95	1	1	yes	characteristic
143	C₂³×C₇	#104	1	4	yes	—
144	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₇	—	2	2	no	—
146	C₂×C₄×C₇	—	2	2	no	—
147	C₂×C₄×C₇	—	2	2	no	—
148	C₂×C₄×C₇	—	2	2	no	—
149	C₂×C₄×C₇	—	2	2	no	—
150	C₂×C₄×C₇	—	2	2	no	—
151	C₂×C₄×C₇	—	2	2	no	—
152	C₂×C₄×C₇	—	2	2	no	—
153	C₂×C₄×C₇	—	2	2	no	—
154	C₂×C₄×C₇	—	2	2	no	—
155	C₂×C₄×C₇	—	2	2	no	—
156	C₂×C₄×C₇	—	2	2	no	—
157	C₂×C₄×C₇	—	4	1	no	—
158	C₂×C₄×C₇	—	4	1	no	—
159	C₂×C₄×C₇	—	4	1	no	—
160	C₂×C₄×C₇	—	4	1	no	—
161	C₂×C₄×C₇	—	4	1	no	—
162	C₂×C₄×C₇	—	4	1	no	—
163	C₂×C₄×C₇	#95	1	1	yes	characteristic
164	C₂×C₄×C₇	#95	1	1	yes	characteristic
165	C₂×C₄×C₇	#95	1	1	yes	characteristic
166	C₂×C₄×C₇	#95	1	1	yes	characteristic
167	C₂×C₄×C₇	#95	1	1	yes	characteristic
168	C₂×C₄×C₇	#95	1	1	yes	characteristic
169	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
170	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
171	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
172	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
175	C₂×Dic₇	—	4	4	no	—
176	C₂×Dic₇	—	4	4	no	—
177	C₂×Dic₇	—	4	4	no	—
178	C₇×C₈	—	4	4	no	—
179	C₇⋊C₈	—	2	8	no	—
180	C₂³×C₄	C₄×D₇	1	1	yes	characteristic
181	C₂³×C₄	D₂₈	1	1	yes	characteristic
182	C₂³×C₄	C₂²⋊D₇	1	1	yes	characteristic
183	C₂×M₁₆	—	14	8	no	—
184	C₂×M₁₆	—	14	2	no	—
185	C₂×M₁₆	—	14	2	no	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	14	2	no	—
188	C₂×C₄⋊C₄	—	14	2	no	—
189	C₂×C₂²⋊C₄	—	14	2	no	—
190	C₂×C₄²	—	14	2	no	—
191	C₂×C₄²	Dic₁₄	1	1	yes	characteristic
192	C₂×C₄²	Dic₁₄	1	1	yes	characteristic
193	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
194	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
195	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
196	C₂×C₄²	D₂₈	1	1	yes	characteristic
197	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
198	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
199	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
200	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
201	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
202	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
203	C₄⋊C₈	—	2	2	no	—
204	C₄⋊C₈	C₂×Dic₇	1	4	yes	—
205	C₂²⋊C₈	—	2	2	no	—
206	C₂²⋊C₈	—	2	2	no	—
207	#97	—	2	2	no	—
208	#97	—	2	2	no	—
209	#97	—	7	4	no	—
210	#97	—	14	2	no	—
211	C₄×C₈	—	14	2	no	—
212	C₄×C₈	—	14	2	no	—
213	#95	—	7	4	no	—
214	#95	—	14	2	no	—
215	#95	—	14	2	no	—
216	#95	—	14	2	no	—
217	C₂²×C₇	—	2	1	no	—
218	C₂²×C₇	—	2	1	no	—
219	C₂²×C₇	—	2	1	no	—
220	C₂²×C₇	—	2	1	no	—
221	C₂²×C₇	—	2	1	no	—
222	C₂²×C₇	—	2	1	no	—
223	C₂²×C₇	—	4	4	no	—
224	C₂²×C₇	#339	1	2	yes	—
225	C₂²×C₇	#339	1	2	yes	—
226	C₂²×C₇	C₂×#95	1	1	yes	characteristic
227	C₂²×C₇	C₂×#104	1	2	yes	—
228	C₄×C₇	—	2	1	no	—
229	C₄×C₇	—	2	1	no	—
230	C₄×C₇	—	2	1	no	—
231	C₄×C₇	—	2	1	no	—
232	C₄×C₇	—	2	1	no	—
233	C₄×C₇	—	2	1	no	—
234	C₄×C₇	—	2	1	no	—
235	C₄×C₇	—	2	1	no	—
236	C₄×C₇	—	2	1	no	—
237	C₄×C₇	—	2	1	no	—
238	C₄×C₇	#344	1	2	yes	—
239	C₄×C₇	#421	1	2	yes	—
240	Dic₇	—	8	4	no	—
241	C₂⁴	(C₂×D₇)⋊C₄	1	1	yes	characteristic
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	4	2	no	—
244	C₂²×C₄	—	4	2	no	—
245	C₂²×C₄	—	14	4	no	—
246	C₂²×C₄	C₄×Dic₇	1	1	yes	characteristic
247	C₂²×C₄	C₄×Dic₇	1	1	yes	characteristic
248	C₂²×C₄	Dic₇⋊C₄	1	1	yes	characteristic
249	C₂²×C₄	Dic₇⋊C₄	1	1	yes	characteristic
250	C₂²×C₄	Dic₇⋊C₄	1	1	yes	characteristic
251	C₂²×C₄	Dic₇⋊C₄	1	1	yes	characteristic
252	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
253	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
254	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
255	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
256	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
257	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
258	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
259	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
260	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
263	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
264	M₁₆	—	14	8	no	—
265	M₁₆	—	14	8	no	—
266	C₂×C₈	—	2	4	no	—
267	C₂×C₈	—	4	4	no	—
268	C₂×C₈	—	14	8	no	—
269	C₂×C₈	—	14	4	no	—
270	C₄⋊C₄	—	28	4	no	—
271	C₂²⋊C₄	—	28	4	no	—
272	C₄²	—	28	4	no	—
273	C₄²	C₄×Dic₇	1	1	yes	characteristic
274	C₄²	C₄×Dic₇	1	1	yes	characteristic
275	C₄²	Dic₇⋊C₄	1	1	yes	characteristic
276	C₄²	Dic₇⋊C₄	1	1	yes	characteristic
277	C₄²	Dic₇⋊C₄	1	1	yes	characteristic
278	C₄²	Dic₇⋊C₄	1	1	yes	characteristic
279	C₄²	C₄⋊Dic₇	1	1	yes	characteristic
280	C₄²	C₄⋊Dic₇	1	1	yes	characteristic
281	C₄²	(C₂×D₇)⋊C₄	1	1	yes	characteristic
282	C₄²	(C₂×D₇)⋊C₄	1	1	yes	characteristic
283	C₄²	C₂²⋊Dic₇	1	1	yes	characteristic
284	C₄²	C₂²⋊Dic₇	1	1	yes	characteristic
285	C₄²	C₂×Dic₁₄	1	1	yes	characteristic
286	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
287	C₄²	C₂²×Dic₇	1	1	yes	characteristic
288	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
289	C₂×C₇	—	2	2	no	—
290	C₂×C₇	—	2	2	no	—
291	C₂×C₇	—	2	2	no	—
292	C₂×C₇	C₂×#339	1	1	yes	characteristic
293	C₂×C₇	#1733	1	1	yes	characteristic
294	C₂×C₇	C₂×#344	1	1	yes	characteristic
295	C₂³	—	4	1	no	—
296	C₂³	—	4	1	no	—
297	C₂³	#6399	1	4	yes	—
298	C₂³	C₇⋊#95	1	1	yes	characteristic
299	C₂³	C₇⋊#95	1	1	yes	characteristic
300	C₂³	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	28	4	no	—
320	C₂×C₄	—	28	4	no	—
321	C₂×C₄	—	28	4	no	—
322	C₂×C₄	C₇⋊#95	1	1	yes	characteristic
323	C₂×C₄	C₇⋊#95	1	1	yes	characteristic
324	C₂×C₄	C₇⋊#95	1	1	yes	characteristic
325	C₂×C₄	C₇⋊#95	1	1	yes	characteristic
326	C₂×C₄	C₇⋊#95	1	1	yes	characteristic
327	C₂×C₄	C₇⋊#95	1	1	yes	characteristic
328	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
329	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
330	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
331	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
332	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
333	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
334	C₈	—	4	4	no	—
335	C₈	—	14	8	no	—
336	C₇	#8073	1	1	yes	characteristic, complemented, Sylow
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	4	4	no	—
344	C₂²	#89350	1	2	yes	—
345	C₂²	#89350	1	2	yes	—
346	C₂²	C₂×#6399	1	2	yes	—
347	C₂²	C₂×C₇⋊#95	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	56	4	no	—
359	C₄	#89351	1	2	yes	—
360	C₄	#90008	1	2	yes	—
361	C₂	—	2	2	no	—
362	C₂	—	2	2	no	—
363	C₂	—	2	2	no	—
364	C₂	#11729328	1	1	yes	characteristic
365	C₂	C₂×#89350	1	1	yes	characteristic
366	C₂	C₂×#89351	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₇
commutator	C₂×C₄×C₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₂×C₇×#432	C₂
socle	C₂²×C₇	C₂×#95
prime radical	C₂⁴×C₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	C₂×#432	D₇
hypocenter	C₇	#8073
LCS	G ▷ C₂×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
4. #11725554	C₂	semidirect
9. C₇×#432	C₄	semidirect
11. C₂×C₇×C₄⋊C₈	C₄	semidirect
23. C₄×#1136	C₄	semidirect
25. C₂×#6377	C₄	semidirect
121. #432	Dic₇	semidirect
123. C₂×C₄⋊C₈	Dic₇	semidirect
336. C₇	#8073	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#432	2
#11739505	2
C₂×C₄×#1136	2
#11725554	2
#8073	7