Subgroups — #13322839

Order 1792.

666 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	C₄×C₇×#131	C₂	1	1	yes	maximal, characteristic
2	#11739966	C₂	1	2	yes	maximal
3	#11727605	C₂	1	4	yes	maximal, complemented
4	#11727604	C₂	1	2	yes	maximal, complemented
5	#11727486	C₂	1	2	yes	maximal, complemented
6	#11727484	C₂	1	2	yes	maximal, complemented
7	#11727483	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#89252	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₇×#131	C₄	1	1	yes	characteristic
10	C₂×C₇×#131	C₄	1	1	yes	characteristic
11	C₄×C₇×#41	C₂²	1	1	yes	characteristic
12	C₇×#405	C₄	1	1	yes	characteristic
13	C₇×#405	C₄	1	2	yes	—
14	C₇×#405	C₄	1	2	yes	—
15	C₇×#405	C₄	1	1	yes	characteristic
16	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
17	C₄×C₇×M₁₆	C₂²	1	2	yes	—
18	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
20	#89796	—	2	2	no	—
21	#89784	—	2	2	no	—
22	#89783	—	2	4	no	—
23	#89698	C₂²	1	4	yes	—
24	#89697	C₂²	1	2	yes	—
25	#89696	C₂²	1	4	yes	—
26	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
27	#89253	C₂²	1	2	yes	—
28	#89253	C₂²	1	4	yes	—
29	#89252	C₂²	1	2	yes	—
30	#89252	C₂²	1	2	yes	—
31	#89252	C₂²	1	1	yes	characteristic
32	#89252	C₂²	1	1	yes	characteristic
33	#89252	C₂²	1	4	yes	—
34	#17798	—	7	1	no	maximal, Sylow
35	C₂×C₇×#41	C₂×C₄	1	1	yes	characteristic
36	C₇×#131	—	2	4	no	—
37	C₇×#131	—	2	4	no	—
38	C₂×C₇×M₁₆	C₂×C₄	1	1	yes	characteristic
39	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
40	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
41	C₂×C₇×M₁₆	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
44	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
45	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
46	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
47	C₄×C₇×Q₈	Q₈	1	1	yes	characteristic
48	C₄×C₇×D₄	D₄	1	2	yes	—
49	C₄×C₇×D₄	D₄	1	1	yes	characteristic
50	C₄×C₇×D₄	Q₈	1	2	yes	—
51	C₄×C₇×D₄	Q₈	1	1	yes	characteristic
52	C₇×#117	C₂×C₄	1	2	yes	—
53	C₇×#117	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₇	C₂³	1	2	yes	—
56	C₂×#1136	—	2	4	no	—
57	C₂²×C₇⋊C₈	—	2	2	no	—
58	C₇×#97	C₂×C₄	1	2	yes	—
59	C₇×#97	C₂×C₄	1	2	yes	—
60	C₇×#97	C₂×C₄	1	2	yes	—
61	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
62	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
63	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
64	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
65	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
66	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
67	C₄×C₇×C₈	C₂³	1	2	yes	—
68	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
69	C₂²⋊(C₇⋊C₈)	—	2	4	no	—
70	C₄⋊(C₇⋊C₈)	—	2	2	no	—
71	C₄⋊(C₇⋊C₈)	—	2	4	no	—
72	C₄⋊(C₇⋊C₈)	C₂³	1	4	yes	—
73	C₄⋊(C₇⋊C₈)	C₂³	1	2	yes	—
74	C₄⋊(C₇⋊C₈)	C₂³	1	2	yes	—
75	#6377	—	2	4	no	—
76	C₄×C₇⋊C₈	—	2	2	no	—
77	#2989	—	7	2	no	—
78	C₄×#131	D₇	1	1	yes	characteristic
79	#1569	—	7	4	no	—
80	#1568	—	7	2	no	—
81	#1566	—	7	2	no	—
82	#1565	—	7	2	no	—
83	#1564	—	7	1	no	—
84	C₂×#334	—	7	1	no	—
85	C₇×#41	—	2	2	no	—
86	C₇×#41	—	2	2	no	—
87	C₂×C₇×Q₈	C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₇×D₄	C₄⋊C₄	1	2	yes	—
89	C₂×C₇×D₄	C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
95	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
96	C₇×M₁₆	—	2	2	no	—
97	C₇×M₁₆	—	2	4	no	—
98	C₇×M₁₆	—	2	4	no	—
99	C₇×M₁₆	M₁₆	1	4	yes	complemented
100	C₂×C₇×C₈	—	2	2	no	—
101	C₂×C₇×C₈	—	2	4	no	—
102	C₂×C₇×C₈	—	2	4	no	—
103	C₂×C₇×C₈	M₁₆	1	4	yes	complemented
104	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
107	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
109	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
110	C₇×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
111	C₇×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
112	C₇×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
113	C₇×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
114	C₇×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
115	C₇×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
116	C₇×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
117	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
118	C₄²×C₇	C₂²×C₄	1	2	yes	—
119	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₇	C₂×D₄	1	2	yes	—
121	C₄²×C₇	C₂×Q₈	1	2	yes	—
122	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
123	C₄²×C₇	C₂⁴	1	1	yes	characteristic
124	#1136	—	8	4	no	—
125	C₂×C₇⋊C₈	—	2	2	no	—
126	C₂×C₇⋊C₈	—	2	4	no	—
127	C₂×C₇⋊C₈	—	2	2	no	—
128	C₂×C₇⋊C₈	—	8	2	no	—
129	C₂×#131	Dic₇	1	1	yes	characteristic
130	C₂×#131	Dic₇	1	1	yes	characteristic
131	C₄×#41	C₂×D₇	1	1	yes	characteristic
132	#446	—	14	2	no	—
133	#436	—	14	2	no	—
134	#435	—	14	4	no	—
135	#432	—	7	4	no	—
136	#431	—	7	2	no	—
137	#423	—	7	4	no	—
138	C₂×C₄⋊C₈	—	7	2	no	—
139	#405	Dic₇	1	1	yes	characteristic
140	#405	Dic₇	1	2	yes	—
141	#405	Dic₇	1	2	yes	—
142	#405	Dic₇	1	1	yes	characteristic
143	C₄×M₁₆	C₂×D₇	1	1	yes	characteristic
144	C₄×M₁₆	C₂×D₇	1	2	yes	—
145	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
146	C₂×C₄×C₈	C₂×D₇	1	2	yes	—
147	#335	—	7	2	no	—
148	#335	—	7	4	no	—
149	#334	—	7	2	no	—
150	#334	—	7	2	no	—
151	#334	—	7	1	no	—
152	#334	—	7	1	no	—
153	#334	—	7	4	no	—
154	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
155	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₇×Q₈	—	2	2	no	—
157	C₇×D₄	—	2	4	no	—
158	C₇×D₄	—	2	2	no	—
159	C₂×C₄×C₇	—	2	2	no	—
160	C₂×C₄×C₇	—	2	2	no	—
161	C₂×C₄×C₇	—	2	2	no	—
162	C₂×C₄×C₇	—	2	4	no	—
163	C₂×C₄×C₇	—	4	1	no	—
164	C₂×C₄×C₇	—	4	2	no	—
165	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
166	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
167	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
169	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
170	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
171	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
172	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
173	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
174	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
176	C₇×C₈	—	2	1	no	—
177	C₇×C₈	—	2	1	no	—
178	C₇×C₈	—	2	2	no	—
179	C₇×C₈	C₂×M₁₆	1	2	yes	—
180	C₇×C₈	C₂×M₁₆	1	4	yes	—
181	C₇×C₈	C₂×M₁₆	1	2	yes	—
182	C₇⋊C₈	—	4	2	no	—
183	C₇⋊C₈	—	4	4	no	—
184	C₇⋊C₈	—	4	2	no	—
185	C₂×#41	C₂×Dic₇	1	1	yes	characteristic
186	#131	—	2	4	no	—
187	#131	—	2	4	no	—
188	C₂×M₁₆	—	14	4	no	—
189	C₂×M₁₆	C₂×Dic₇	1	1	yes	characteristic
190	C₂×M₁₆	C₂×Dic₇	1	2	yes	—
191	C₂×M₁₆	C₂×Dic₇	1	2	yes	—
192	C₂×M₁₆	C₂×Dic₇	1	1	yes	characteristic
193	C₂²×C₈	—	14	2	no	—
194	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
195	C₂²×C₈	C₂×Dic₇	1	2	yes	—
196	C₂²×C₈	C₂×Dic₇	1	2	yes	—
197	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
198	C₄×Q₈	Dic₁₄	1	1	yes	characteristic
199	C₄×Q₈	D₂₈	1	1	yes	characteristic
200	C₄×D₄	Dic₁₄	1	2	yes	—
201	C₄×D₄	Dic₁₄	1	1	yes	characteristic
202	C₄×D₄	D₂₈	1	2	yes	—
203	C₄×D₄	D₂₈	1	1	yes	characteristic
204	#117	C₂×Dic₇	1	2	yes	—
205	#117	C₂×Dic₇	1	1	yes	characteristic
206	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
207	C₂×C₄²	C₂²×D₇	1	2	yes	—
208	C₄⋊C₈	—	7	4	no	—
209	C₄⋊C₈	—	7	2	no	—
210	C₄⋊C₈	—	7	2	no	—
211	C₄⋊C₈	—	14	2	no	—
212	C₄⋊C₈	—	14	4	no	—
213	C₂²⋊C₈	—	14	2	no	—
214	C₂²⋊C₈	—	14	4	no	—
215	#97	—	14	4	no	—
216	#97	C₂×Dic₇	1	2	yes	—
217	#97	C₂×Dic₇	1	2	yes	—
218	#97	C₂×Dic₇	1	2	yes	—
219	C₄×C₈	—	14	2	no	—
220	C₄×C₈	C₂×Dic₇	1	2	yes	—
221	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
222	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
223	C₄×C₈	C₂×Dic₇	1	2	yes	—
224	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
225	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
226	C₄×C₈	C₂²×D₇	1	2	yes	—
227	C₂²×C₇	—	2	1	no	—
228	C₂²×C₇	—	2	1	no	—
229	C₂²×C₇	—	2	2	no	—
230	C₂²×C₇	—	2	2	no	—
231	C₂²×C₇	—	2	2	no	—
232	C₂²×C₇	#423	1	2	yes	—
233	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
234	C₄×C₇	—	2	2	no	—
235	C₄×C₇	—	2	1	no	—
236	C₄×C₇	—	2	2	no	—
237	C₄×C₇	—	2	1	no	—
238	C₄×C₇	—	2	1	no	—
239	C₄×C₇	—	2	2	no	—
240	C₄×C₇	#423	1	2	yes	—
241	C₄×C₇	C₂²×M₁₆	1	1	yes	characteristic
242	C₄×C₇	C₂²×M₁₆	1	1	yes	characteristic
243	C₄×C₇	#577	1	2	yes	—
244	#41	—	2	2	no	—
245	#41	—	2	2	no	—
246	C₂×Q₈	C₄⋊Dic₇	1	1	yes	characteristic
247	C₂×D₄	C₄⋊Dic₇	1	2	yes	—
248	C₂×D₄	C₄⋊Dic₇	1	1	yes	characteristic
249	C₂²×C₄	C₂×Dic₁₄	1	2	yes	—
250	C₂²×C₄	C₂×Dic₁₄	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₂₈	1	2	yes	—
253	C₂²×C₄	C₂²×Dic₇	1	2	yes	—
254	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
255	M₁₆	—	2	2	no	—
256	M₁₆	—	2	4	no	—
257	M₁₆	—	2	4	no	—
258	M₁₆	—	56	4	no	—
259	M₁₆	#1136	1	4	yes	complemented
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	14	2	no	—
264	C₂×C₈	—	14	2	no	—
265	C₂×C₈	—	14	4	no	—
266	C₂×C₈	—	56	2	no	—
267	C₂×C₈	#1136	1	4	yes	complemented
268	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
269	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
270	C₂×C₈	C₂²×Dic₇	1	2	yes	—
271	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
272	C₂×C₈	C₂²×Dic₇	1	2	yes	—
273	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
274	C₄⋊C₄	C₄⋊Dic₇	1	2	yes	—
275	C₄⋊C₄	C₄⋊Dic₇	1	1	yes	characteristic
276	C₄⋊C₄	C₄⋊Dic₇	1	2	yes	—
277	C₄⋊C₄	C₄⋊Dic₇	1	1	yes	characteristic
278	C₂²⋊C₄	C₄⋊Dic₇	1	2	yes	—
279	C₂²⋊C₄	C₄⋊Dic₇	1	2	yes	—
280	C₂²⋊C₄	C₄⋊Dic₇	1	2	yes	—
281	C₄²	C₂×Dic₁₄	1	2	yes	—
282	C₄²	C₂×Dic₁₄	1	1	yes	characteristic
283	C₄²	C₂×D₂₈	1	1	yes	characteristic
284	C₄²	C₂×D₂₈	1	2	yes	—
285	C₄²	C₂²×Dic₇	1	1	yes	characteristic
286	C₄²	C₂²×Dic₇	1	2	yes	—
287	C₄²	C₂³×D₇	1	1	yes	characteristic
288	C₂×C₇	—	2	2	no	—
289	C₂×C₇	—	4	2	no	—
290	C₂×C₇	C₂×#423	1	1	yes	characteristic
291	C₂×C₇	#2912	1	1	yes	characteristic
292	C₂×C₇	#2916	1	1	yes	characteristic
293	C₂³	C₂×C₄⋊Dic₇	1	2	yes	—
294	C₂³	C₂×C₄⋊Dic₇	1	1	yes	characteristic
295	Q₈	—	2	2	no	—
296	D₄	—	2	4	no	—
297	D₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	C₂×#1136	1	2	yes	—
305	C₂×C₄	C₂×#1136	1	2	yes	—
306	C₂×C₄	C₂×C₄⋊Dic₇	1	2	yes	—
307	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
308	C₂×C₄	C₂×C₄⋊Dic₇	1	2	yes	—
309	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
310	C₂×C₄	C₂×C₄⋊Dic₇	1	2	yes	—
311	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
312	C₂×C₄	C₂²×Dic₁₄	1	1	yes	characteristic
313	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
314	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	2	1	no	—
317	C₈	—	2	2	no	—
318	C₈	—	28	2	no	—
319	C₈	—	28	4	no	—
320	C₈	—	28	2	no	—
321	C₈	C₂×#1136	1	2	yes	—
322	C₈	C₂×#1136	1	4	yes	—
323	C₈	C₂×#1136	1	2	yes	—
324	C₇	#17798	1	1	yes	characteristic, complemented, Sylow
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	2	2	no	—
330	C₂²	#89696	1	2	yes	—
331	C₂²	C₂²×C₄⋊Dic₇	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	2	no	—
337	C₄	—	2	2	no	—
338	C₄	#89696	1	2	yes	—
339	C₄	#90442	1	2	yes	—
340	C₄	C₂²×#1136	1	1	yes	characteristic
341	C₄	C₂²×#1136	1	1	yes	characteristic
342	C₂	—	2	2	no	—
343	C₂	—	4	2	no	—
344	C₂	C₂×#89696	1	1	yes	characteristic
345	C₂	#11741229	1	1	yes	characteristic
346	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	C₄×C₇×#131	C₂
socle	C₂²×C₇	C₂²×C₄⋊C₄
prime radical	C₂×C₇×#41	C₂×C₄
cosocle	C₄²×C₇	C₂⁴
hypercenter	C₄×#131	D₇
hypocenter	C₇	#17798
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
3. #11727605	C₂	semidirect
4. #11727604	C₂	semidirect
5. #11727486	C₂	semidirect
6. #11727484	C₂	semidirect
7. #11727483	C₂	semidirect
8. C₂×#89252	C₂	semidirect
99. C₇×M₁₆	M₁₆	semidirect
103. C₂×C₇×C₈	M₁₆	semidirect
259. M₁₆	#1136	semidirect
267. C₂×C₈	#1136	semidirect
324. C₇	#17798	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₇×#131	2
#11739966	2
#11727605	2
#11727604	2
#11727486	2
#11727484	2
#11727483	2
C₂×#89252	2
#17798	7