Subgroups — #13323014

Order 1792.

572 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2922	C₂	1	1	yes	maximal, characteristic
2	#11739966	C₂	1	2	yes	maximal
3	#11728507	C₂	1	2	yes	maximal, complemented
4	#11728506	C₂	1	2	yes	maximal, complemented
5	#11728505	C₂	1	2	yes	maximal, complemented
6	#11728504	C₂	1	2	yes	maximal, complemented
7	#11727605	C₂	1	2	yes	maximal, complemented
8	#11727486	C₂	1	1	yes	maximal, characteristic, complemented
9	#11727485	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₇×#41	C₂²	1	1	yes	characteristic
11	C₇×#433	C₄	1	4	yes	—
12	C₇×#431	C₂²	1	2	yes	—
13	C₇×#431	C₂²	1	2	yes	—
14	C₇×#424	C₄	1	2	yes	—
15	C₇×#408	C₄	1	2	yes	—
16	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
18	#89796	—	2	2	no	—
19	#89784	—	2	2	no	—
20	#89783	—	2	2	no	—
21	#89783	—	2	2	no	—
22	#89698	C₂²	1	2	yes	—
23	#89698	C₂²	1	2	yes	—
24	#89697	C₂²	1	2	yes	—
25	#89696	C₂²	1	2	yes	—
26	#89696	C₂²	1	2	yes	—
27	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
28	#89253	C₂²	1	1	yes	characteristic
29	#89253	C₂²	1	2	yes	—
30	#89253	C₂²	1	2	yes	—
31	#89253	C₂²	1	2	yes	—
32	#89253	C₂²	1	1	yes	characteristic
33	#89253	C₂²	1	1	yes	characteristic
34	#89253	C₂²	1	1	yes	characteristic
35	#89252	C₂²	1	2	yes	—
36	#89252	C₂²	1	2	yes	—
37	#89252	C₂²	1	2	yes	—
38	#17825	—	7	1	no	maximal, Sylow
39	C₂×C₇×#41	C₂×C₄	1	1	yes	characteristic
40	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
41	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
42	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
43	C₄×C₇×Q₈	Q₈	1	1	yes	characteristic
44	C₄×C₇×D₄	D₄	1	1	yes	characteristic
45	C₄×C₇×D₄	D₄	1	2	yes	—
46	C₄×C₇×D₄	Q₈	1	2	yes	—
47	C₄×C₇×D₄	Q₈	1	1	yes	characteristic
48	C₇×#117	C₂×C₄	1	1	yes	characteristic
49	C₇×#117	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₇	C₂³	1	2	yes	—
51	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
52	C₂×#1136	—	2	2	no	—
53	C₂×#1136	—	2	2	no	—
54	C₂²×C₇⋊C₈	—	2	2	no	—
55	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
56	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
57	C₇×C₂²⋊C₈	C₂×C₄	1	4	yes	—
58	C₇×C₂²⋊C₈	C₂×C₄	1	4	yes	—
59	C₇×#97	C₂×C₄	1	2	yes	—
60	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
61	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
62	C₄×C₇×C₈	C₂³	1	2	yes	—
63	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
64	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
65	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
66	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
67	C₄⋊(C₇⋊C₈)	—	2	2	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	2	yes	—
73	C₄⋊(C₇⋊C₈)	C₂³	1	2	yes	—
74	#6377	—	2	2	no	—
75	#6377	—	2	2	no	—
76	C₄×C₇⋊C₈	—	2	2	no	—
77	#2989	—	7	2	no	—
78	#2922	D₇	1	1	yes	characteristic
79	#1574	—	7	2	no	—
80	#1573	—	7	2	no	—
81	#1572	—	7	2	no	—
82	#1571	—	7	2	no	—
83	#1569	—	7	2	no	—
84	#1568	—	7	1	no	—
85	#1567	—	7	1	no	—
86	C₇×#41	—	4	2	no	—
87	C₂×C₇×Q₈	C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₇×D₄	C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₇×D₄	C₄⋊C₄	1	2	yes	—
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	2	yes	—
93	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
95	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
96	C₇×M₁₆	—	4	2	no	—
97	C₂×C₇×C₈	—	4	2	no	—
98	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
99	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
100	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
101	C₇×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
102	C₇×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
103	C₇×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
104	C₇×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
105	C₇×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
106	C₇×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
107	C₇×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
108	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₇	C₂²×C₄	1	2	yes	—
110	C₄²×C₇	C₂×D₄	1	2	yes	—
111	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₇	C₂×Q₈	1	2	yes	—
113	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
114	C₄²×C₇	C₂⁴	1	1	yes	characteristic
115	#1136	—	8	2	no	—
116	#1136	—	8	2	no	—
117	C₂×C₇⋊C₈	—	2	2	no	—
118	C₂×C₇⋊C₈	—	2	2	no	—
119	C₂×C₇⋊C₈	—	2	2	no	—
120	C₂×C₇⋊C₈	—	2	2	no	—
121	C₂×C₇⋊C₈	—	8	2	no	—
122	C₄×#41	C₂×D₇	1	1	yes	characteristic
123	#446	—	14	2	no	—
124	#436	—	14	2	no	—
125	#435	—	14	2	no	—
126	#435	—	14	2	no	—
127	#433	Dic₇	1	4	yes	—
128	#432	—	7	2	no	—
129	#432	—	7	2	no	—
130	#431	—	7	2	no	—
131	#431	C₂×D₇	1	2	yes	—
132	#431	C₂×D₇	1	2	yes	—
133	#424	Dic₇	1	2	yes	—
134	#423	—	7	2	no	—
135	#423	—	7	2	no	—
136	C₂×C₄⋊C₈	—	7	2	no	—
137	#408	Dic₇	1	2	yes	—
138	C₄×M₁₆	C₂×D₇	1	1	yes	characteristic
139	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
140	#335	—	7	2	no	—
141	#335	—	7	1	no	—
142	#335	—	7	1	no	—
143	#335	—	7	1	no	—
144	#335	—	7	1	no	—
145	#335	—	7	2	no	—
146	#335	—	7	2	no	—
147	#334	—	7	2	no	—
148	#334	—	7	2	no	—
149	#334	—	7	2	no	—
150	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
151	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₇×Q₈	—	4	1	no	—
153	C₇×D₄	—	4	1	no	—
154	C₇×D₄	—	4	2	no	—
155	C₂×C₄×C₇	—	2	2	no	—
156	C₂×C₄×C₇	—	2	4	no	—
157	C₂×C₄×C₇	—	2	2	no	—
158	C₂×C₄×C₇	—	2	1	no	—
159	C₂×C₄×C₇	—	2	2	no	—
160	C₂×C₄×C₇	—	2	1	no	—
161	C₂×C₄×C₇	—	4	2	no	—
162	C₂×C₄×C₇	—	4	1	no	—
163	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
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	2	yes	—
168	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
169	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
170	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
172	C₇×C₈	—	2	2	no	—
173	C₇×C₈	—	2	2	no	—
174	C₇×C₈	—	2	4	no	—
175	C₇⋊C₈	—	4	2	no	—
176	C₇⋊C₈	—	4	2	no	—
177	C₇⋊C₈	—	4	2	no	—
178	C₇⋊C₈	—	4	2	no	—
179	C₂×#41	C₂×Dic₇	1	1	yes	characteristic
180	C₂×M₁₆	—	14	2	no	—
181	C₂×M₁₆	—	14	2	no	—
182	C₂×M₁₆	C₂×Dic₇	1	2	yes	—
183	C₂²×C₈	—	14	2	no	—
184	C₂²×C₈	C₂×Dic₇	1	2	yes	—
185	C₄×Q₈	Dic₁₄	1	1	yes	characteristic
186	C₄×Q₈	D₂₈	1	1	yes	characteristic
187	C₄×D₄	Dic₁₄	1	1	yes	characteristic
188	C₄×D₄	Dic₁₄	1	2	yes	—
189	C₄×D₄	D₂₈	1	1	yes	characteristic
190	C₄×D₄	D₂₈	1	2	yes	—
191	#117	C₂×Dic₇	1	1	yes	characteristic
192	#117	C₂×Dic₇	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	2	no	—
198	C₄⋊C₈	—	7	2	no	—
199	C₄⋊C₈	—	14	2	no	—
200	C₄⋊C₈	—	14	2	no	—
201	C₄⋊C₈	—	14	2	no	—
202	C₄⋊C₈	C₂×Dic₇	1	4	yes	—
203	C₄⋊C₈	C₂×Dic₇	1	4	yes	—
204	C₂²⋊C₈	—	14	2	no	—
205	C₂²⋊C₈	—	14	2	no	—
206	C₂²⋊C₈	—	14	2	no	—
207	C₂²⋊C₈	C₂×Dic₇	1	4	yes	—
208	C₂²⋊C₈	C₂×Dic₇	1	4	yes	—
209	#97	—	14	2	no	—
210	#97	—	14	2	no	—
211	#97	C₂×Dic₇	1	2	yes	—
212	C₄×C₈	—	14	2	no	—
213	C₄×C₈	C₂×Dic₇	1	2	yes	—
214	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
215	C₄×C₈	C₂²×D₇	1	2	yes	—
216	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
217	C₂²×C₇	—	2	1	no	—
218	C₂²×C₇	—	2	2	no	—
219	C₂²×C₇	—	2	2	no	—
220	C₂²×C₇	—	2	1	no	—
221	C₂²×C₇	—	2	2	no	—
222	C₂²×C₇	—	2	1	no	—
223	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
224	C₄×C₇	—	2	1	no	—
225	C₄×C₇	—	2	2	no	—
226	C₄×C₇	—	2	2	no	—
227	C₄×C₇	—	2	1	no	—
228	C₄×C₇	—	2	1	no	—
229	C₄×C₇	—	2	1	no	—
230	C₄×C₇	—	2	2	no	—
231	C₄×C₇	#568	1	2	yes	—
232	C₄×C₇	#576	1	1	yes	characteristic
233	C₄×C₇	#578	1	1	yes	characteristic
234	#41	—	4	2	no	—
235	C₂×Q₈	C₄⋊Dic₇	1	1	yes	characteristic
236	C₂×D₄	C₄⋊Dic₇	1	1	yes	characteristic
237	C₂×D₄	C₄⋊Dic₇	1	2	yes	—
238	C₂²×C₄	C₂×Dic₁₄	1	1	yes	characteristic
239	C₂²×C₄	C₂×Dic₁₄	1	2	yes	—
240	C₂²×C₄	C₂×D₂₈	1	2	yes	—
241	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
242	C₂²×C₄	C₂²×Dic₇	1	2	yes	—
243	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
244	M₁₆	—	4	2	no	—
245	M₁₆	—	56	2	no	—
246	M₁₆	—	56	2	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	—	14	2	no	—
249	C₂×C₈	—	14	2	no	—
250	C₂×C₈	—	14	2	no	—
251	C₂×C₈	—	14	2	no	—
252	C₂×C₈	—	56	2	no	—
253	C₂×C₈	C₂²×Dic₇	1	2	yes	—
254	C₂×C₈	C₂²×Dic₇	1	2	yes	—
255	C₂×C₈	C₂²×Dic₇	1	4	yes	—
256	C₄⋊C₄	C₄⋊Dic₇	1	1	yes	characteristic
257	C₄⋊C₄	C₄⋊Dic₇	1	2	yes	—
258	C₄⋊C₄	C₄⋊Dic₇	1	1	yes	characteristic
259	C₄⋊C₄	C₄⋊Dic₇	1	2	yes	—
260	C₂²⋊C₄	C₄⋊Dic₇	1	2	yes	—
261	C₂²⋊C₄	C₄⋊Dic₇	1	2	yes	—
262	C₂²⋊C₄	C₄⋊Dic₇	1	2	yes	—
263	C₄²	C₂×Dic₁₄	1	1	yes	characteristic
264	C₄²	C₂×Dic₁₄	1	2	yes	—
265	C₄²	C₂×D₂₈	1	2	yes	—
266	C₄²	C₂×D₂₈	1	1	yes	characteristic
267	C₄²	C₂²×Dic₇	1	1	yes	characteristic
268	C₄²	C₂²×Dic₇	1	2	yes	—
269	C₄²	C₂³×D₇	1	1	yes	characteristic
270	C₂×C₇	—	4	1	no	—
271	C₂×C₇	—	4	2	no	—
272	C₂×C₇	#2905	1	1	yes	characteristic
273	C₂×C₇	#2913	1	1	yes	characteristic
274	C₂×C₇	#2916	1	1	yes	characteristic
275	C₂³	C₂×C₄⋊Dic₇	1	2	yes	—
276	C₂³	C₂×C₄⋊Dic₇	1	1	yes	characteristic
277	Q₈	—	4	1	no	—
278	D₄	—	4	1	no	—
279	D₄	—	4	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	C₂×C₄⋊Dic₇	1	2	yes	—
289	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
290	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
291	C₂×C₄	C₂×C₄⋊Dic₇	1	2	yes	—
292	C₂×C₄	C₂×C₄⋊Dic₇	1	2	yes	—
293	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
294	C₂×C₄	C₂²×Dic₁₄	1	1	yes	characteristic
295	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
296	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
297	C₈	—	2	2	no	—
298	C₈	—	2	2	no	—
299	C₈	—	2	4	no	—
300	C₈	—	28	2	no	—
301	C₈	—	28	2	no	—
302	C₈	—	28	2	no	—
303	C₈	—	28	2	no	—
304	C₇	#17825	1	1	yes	characteristic, complemented, Sylow
305	C₂²	—	2	1	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	1	no	—
311	C₂²	C₂²×C₄⋊Dic₇	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	1	no	—
319	C₄	#90443	1	1	yes	characteristic
320	C₄	#90444	1	1	yes	characteristic
321	C₄	#90510	1	2	yes	—
322	C₂	—	4	1	no	—
323	C₂	—	4	2	no	—
324	C₂	#11739649	1	1	yes	characteristic
325	C₂	#11741230	1	1	yes	characteristic
326	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×#2922	C₂
socle	C₂²×C₇	C₂²×C₄⋊C₄
prime radical	C₂×C₇×#41	C₂×C₄
cosocle	C₄²×C₇	C₂⁴
hypercenter	#2922	D₇
hypocenter	C₇	#17825
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ #2922

Complements

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

N	H ≅ G/N	type
3. #11728507	C₂	semidirect
4. #11728506	C₂	semidirect
5. #11728505	C₂	semidirect
6. #11728504	C₂	semidirect
7. #11727605	C₂	semidirect
8. #11727486	C₂	semidirect
9. #11727485	C₂	semidirect
304. C₇	#17825	semidirect

Maximal subgroups

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

subgroup	index
C₇×#2922	2
#11739966	2
#11728507	2
#11728506	2
#11728505	2
#11728504	2
#11727605	2
#11727486	2
#11727485	2
#17825	7