Subgroups — #13323017

Order 1792.

702 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11736457	C₂	1	2	yes	maximal
2	C₈×#1157	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#334	C₂	1	1	yes	maximal, characteristic, complemented
4	#11728175	C₂	1	2	yes	maximal, complemented
5	#11727484	C₂	1	1	yes	maximal, characteristic, complemented
6	#11726286	C₂	1	4	yes	maximal, complemented
7	#11726283	C₂	1	2	yes	maximal, complemented
8	D₇×#334	C₂	1	2	yes	maximal, complemented
9	C₂×#6464	—	2	2	no	—
10	C₄×#1157	C₂²	1	1	yes	characteristic
11	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
13	C₈×C₂²⋊D₇	C₄	1	2	yes	—
14	C₈×C₂²⋊D₇	C₄	1	2	yes	—
15	#89874	—	2	2	no	—
16	#89697	C₂²	1	1	yes	characteristic
17	#89696	C₂²	1	2	yes	—
18	#89611	—	2	4	no	—
19	#89609	C₂²	1	4	yes	—
20	D₇×C₄⋊C₈	C₂²	1	4	yes	—
21	C₈×D₂₈	C₄	1	1	yes	characteristic
22	C₈×D₂₈	C₄	1	1	yes	characteristic
23	#89457	C₂²	1	2	yes	—
24	C₄×C₈×D₇	C₂²	1	2	yes	—
25	C₈×Dic₁₄	C₄	1	1	yes	characteristic
26	C₈×Dic₁₄	C₄	1	1	yes	characteristic
27	C₇×#334	C₂²	1	1	yes	characteristic
28	C₇×#334	C₂²	1	1	yes	characteristic
29	C₇×#334	C₂²	1	2	yes	—
30	#89275	C₂²	1	4	yes	—
31	#89275	C₂²	1	4	yes	—
32	#89252	C₂²	1	2	yes	—
33	#89252	C₂²	1	2	yes	—
34	#17826	—	7	1	no	maximal, Sylow
35	C₂×#1157	D₄	1	1	yes	characteristic
36	C₂²×C₇×C₈	—	2	2	no	—
37	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
40	C₄×C₂²⋊D₇	C₂×C₄	1	2	yes	—
41	C₄×C₂²⋊D₇	Q₈	1	2	yes	—
42	#6489	D₄	1	1	yes	characteristic
43	C₂×#1136	—	2	2	no	—
44	#6464	—	4	8	no	—
45	C₂×#1131	—	2	4	no	—
46	C₂×C₈×D₇	—	2	4	no	—
47	C₂×C₈×D₇	C₂×C₄	1	2	yes	—
48	C₂×C₈×D₇	C₂×C₄	1	2	yes	—
49	C₄×D₂₈	C₂×C₄	1	1	yes	characteristic
50	C₄×D₂₈	Q₈	1	1	yes	characteristic
51	#6435	D₄	1	2	yes	—
52	C₄²×D₇	C₂³	1	2	yes	—
53	C₄×Dic₁₄	C₂×C₄	1	1	yes	characteristic
54	C₄×Dic₁₄	Q₈	1	1	yes	characteristic
55	C₇×C₄⋊C₈	—	2	2	no	—
56	C₇×C₄⋊C₈	C₂³	1	2	yes	—
57	C₇×C₄⋊C₈	C₂³	1	2	yes	—
58	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
59	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
60	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
61	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
62	C₂²⋊(C₇⋊C₈)	C₂×C₄	1	2	yes	—
63	(C₂×D₇)⋊C₈	C₂×C₄	1	2	yes	—
64	(C₂×D₇)⋊C₈	C₂×C₄	1	2	yes	—
65	Dic₇⋊C₈	—	2	4	no	—
66	Dic₇⋊C₈	—	2	4	no	—
67	Dic₇⋊C₈	C₂×C₄	1	2	yes	—
68	Dic₇⋊C₈	C₂×C₄	1	2	yes	—
69	C₈×Dic₇	C₂×C₄	1	2	yes	—
70	C₈×Dic₇	C₂×C₄	1	2	yes	—
71	C₄⋊(C₇⋊C₈)	—	2	2	no	—
72	C₄⋊(C₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
73	C₄⋊(C₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
74	C₄⋊(C₇⋊C₈)	C₂³	1	4	yes	—
75	C₄×C₇⋊C₈	C₂³	1	2	yes	—
76	C₈×#41	—	7	1	no	—
77	#2906	—	7	2	no	—
78	#1570	—	7	2	no	—
79	#1570	—	7	4	no	—
80	#1566	—	7	1	no	—
81	#1566	—	7	2	no	—
82	C₂×#334	—	7	2	no	—
83	C₂×#334	D₇	1	1	yes	characteristic, complemented
84	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₂²⋊D₇	C₄⋊C₄	1	2	yes	—
88	#1157	SD₁₆	1	8	yes	—
89	C₂×D₂₈	C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×D₇	C₂²×C₄	1	2	yes	—
91	C₂×C₄×D₇	C₂×D₄	1	2	yes	—
92	C₂×C₄×D₇	C₂×Q₈	1	2	yes	—
93	C₂×Dic₁₄	C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₇×C₈	—	2	2	no	—
95	C₂×C₇×C₈	—	2	2	no	—
96	C₂×C₇×C₈	—	2	2	no	—
97	C₂×C₇×C₈	—	4	1	no	—
98	C₂×C₇×C₈	—	4	4	no	—
99	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
106	C₄²×C₇	C₂⁴	1	1	yes	characteristic
107	C₂²⋊Dic₇	C₄⋊C₄	1	2	yes	—
108	(C₂×D₇)⋊C₄	C₄⋊C₄	1	2	yes	—
109	(C₂×D₇)⋊C₄	C₄⋊C₄	1	2	yes	—
110	C₄⋊Dic₇	C₄⋊C₄	1	1	yes	characteristic
111	C₄⋊Dic₇	C₄⋊C₄	1	1	yes	characteristic
112	Dic₇⋊C₄	C₄⋊C₄	1	2	yes	—
113	Dic₇⋊C₄	C₄⋊C₄	1	2	yes	—
114	C₄×Dic₇	C₂²×C₄	1	2	yes	—
115	C₄×Dic₇	C₂×D₄	1	2	yes	—
116	C₄×Dic₇	C₂×Q₈	1	2	yes	—
117	#1136	—	4	4	no	—
118	C₂×C₇⋊C₈	—	2	4	no	—
119	C₂×C₇⋊C₈	C₂²×C₄	1	2	yes	—
120	C₂×C₇⋊C₈	C₂²×C₄	1	2	yes	—
121	#1131	—	4	8	no	—
122	C₈×D₇	—	2	4	no	—
123	C₈×D₇	—	4	2	no	—
124	C₈×D₇	—	4	8	no	—
125	C₂×#131	—	14	2	no	—
126	C₄×#41	—	7	1	no	—
127	C₈×Q₈	—	7	1	no	—
128	C₈×Q₈	—	7	1	no	—
129	C₈×D₄	—	7	2	no	—
130	C₈×D₄	—	7	2	no	—
131	C₈×D₄	—	7	1	no	—
132	C₈×D₄	—	7	1	no	—
133	#431	—	7	1	no	—
134	#431	—	7	2	no	—
135	#424	—	14	4	no	—
136	#424	—	14	2	no	—
137	#423	—	7	2	no	—
138	#423	—	7	4	no	—
139	C₂×C₄⋊C₈	—	7	4	no	—
140	C₂×C₄⋊C₈	C₂×D₇	1	2	yes	—
141	C₂×C₄×C₈	—	7	2	no	—
142	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
143	#334	—	7	4	no	—
144	#334	—	7	4	no	—
145	#334	—	7	2	no	—
146	#334	—	7	2	no	—
147	#334	C₂×D₇	1	1	yes	characteristic
148	#334	C₂×D₇	1	1	yes	characteristic
149	#334	C₂×D₇	1	2	yes	—
150	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂²×D₇	C₂×C₄⋊C₄	1	2	yes	—
152	C₂×C₄×C₇	—	2	1	no	—
153	C₂×C₄×C₇	—	2	1	no	—
154	C₂×C₄×C₇	—	4	1	no	—
155	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
158	C₂×C₄×C₇	C₂×SD₁₆	1	4	yes	—
159	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
160	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
162	C₂²⋊D₇	#106	1	8	yes	—
163	C₂×Dic₇	C₂×C₄⋊C₄	1	2	yes	—
164	C₂×Dic₇	C₂×C₄⋊C₄	1	2	yes	—
165	C₂×Dic₇	C₂×C₄⋊C₄	1	2	yes	—
166	D₂₈	#106	1	4	yes	—
167	C₄×D₇	—	2	2	no	—
168	C₄×D₇	—	2	2	no	—
169	C₄×D₇	—	4	2	no	—
170	C₄×D₇	C₂×SD₁₆	1	8	yes	—
171	Dic₁₄	#106	1	4	yes	—
172	C₇×C₈	—	2	1	no	—
173	C₇×C₈	—	2	1	no	—
174	C₇×C₈	—	2	1	no	—
175	C₇×C₈	—	4	2	no	—
176	C₇×C₈	—	4	2	no	—
177	C₇×C₈	#131	1	2	yes	complemented
178	C₇⋊C₈	—	2	2	no	—
179	C₇⋊C₈	—	2	2	no	—
180	C₇⋊C₈	—	4	4	no	—
181	C₂×#41	—	7	1	no	—
182	#131	—	28	8	no	—
183	C₂×M₁₆	—	14	4	no	—
184	C₂×M₁₆	—	14	2	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	7	2	no	—
187	C₂²×C₈	—	7	2	no	—
188	C₂²×C₈	—	14	4	no	—
189	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
190	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
191	C₄×Q₈	—	7	1	no	—
192	C₄×Q₈	—	7	1	no	—
193	C₄×D₄	—	7	2	no	—
194	C₄×D₄	—	7	2	no	—
195	C₄×D₄	—	7	1	no	—
196	C₄×D₄	—	7	1	no	—
197	#117	—	7	1	no	—
198	#117	—	7	2	no	—
199	C₂×C₄²	—	7	2	no	—
200	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
201	C₄⋊C₈	—	2	2	no	—
202	C₄⋊C₈	—	7	2	no	—
203	C₄⋊C₈	—	7	2	no	—
204	C₄⋊C₈	—	7	1	no	—
205	C₄⋊C₈	—	7	1	no	—
206	C₄⋊C₈	—	7	4	no	—
207	C₄⋊C₈	—	14	4	no	—
208	C₄⋊C₈	—	14	4	no	—
209	C₄⋊C₈	—	14	2	no	—
210	C₄⋊C₈	C₂²×D₇	1	2	yes	—
211	C₄⋊C₈	C₂²×D₇	1	2	yes	—
212	C₂²⋊C₈	—	7	2	no	—
213	C₂²⋊C₈	—	7	2	no	—
214	C₂²⋊C₈	—	7	2	no	—
215	C₄×C₈	—	7	2	no	—
216	C₄×C₈	—	7	2	no	—
217	C₄×C₈	—	7	2	no	—
218	C₄×C₈	C₄×D₇	1	1	yes	characteristic
219	C₄×C₈	C₄×D₇	1	1	yes	characteristic
220	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
221	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
222	C₂²×C₇	—	2	1	no	—
223	C₂²×C₇	—	2	1	no	—
224	C₂²×C₇	C₂×#106	1	2	yes	—
225	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
226	C₂×D₇	—	2	2	no	—
227	C₂×D₇	—	2	2	no	—
228	C₂×D₇	C₂×#106	1	4	yes	—
229	C₄×C₇	—	2	1	no	—
230	C₄×C₇	—	2	1	no	—
231	C₄×C₇	—	2	1	no	—
232	C₄×C₇	C₂×#106	1	2	yes	—
233	C₄×C₇	C₂×#131	1	1	yes	characteristic
234	C₄×C₇	#568	1	1	yes	characteristic
235	C₄×C₇	C₂²×SD₁₆	1	2	yes	—
236	Dic₇	—	2	2	no	—
237	Dic₇	—	2	2	no	—
238	Dic₇	C₂×#106	1	4	yes	—
239	#41	—	7	8	no	—
240	C₂×Q₈	—	7	1	no	—
241	C₂×D₄	—	7	2	no	—
242	C₂×D₄	—	7	1	no	—
243	C₂²×C₄	—	7	2	no	—
244	C₂²×C₄	—	7	2	no	—
245	C₂²×C₄	—	7	2	no	—
246	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
247	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
248	C₂²×C₄	Q₈×D₇	1	1	yes	characteristic
249	M₁₆	—	28	8	no	—
250	M₁₆	—	28	4	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	4	no	—
256	C₂×C₈	—	7	2	no	—
257	C₂×C₈	—	7	2	no	—
258	C₂×C₈	—	14	4	no	—
259	C₂×C₈	—	14	4	no	—
260	C₂×C₈	—	28	2	no	—
261	C₂×C₈	—	28	8	no	—
262	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
263	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
264	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
265	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
266	C₄⋊C₄	—	7	2	no	—
267	C₄⋊C₄	—	7	2	no	—
268	C₄⋊C₄	—	7	1	no	—
269	C₄⋊C₄	—	7	1	no	—
270	C₂²⋊C₄	—	7	2	no	—
271	C₂²⋊C₄	—	7	2	no	—
272	C₂²⋊C₄	—	7	2	no	—
273	C₄²	—	7	2	no	—
274	C₄²	—	7	2	no	—
275	C₄²	—	7	2	no	—
276	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
277	C₄²	D₄×D₇	1	1	yes	characteristic
278	C₄²	Q₈×D₇	1	1	yes	characteristic
279	C₄²	C₂³×D₇	1	1	yes	characteristic
280	C₂×C₇	—	2	2	no	—
281	C₂×C₇	#2906	1	1	yes	characteristic
282	C₂×C₇	C₂²×#106	1	1	yes	characteristic
283	C₂×C₇	#2916	1	1	yes	characteristic
284	D₇	—	2	4	no	—
285	C₂³	—	7	2	no	—
286	C₂³	D₇×C₄⋊C₄	1	1	yes	characteristic
287	Q₈	—	7	4	no	—
288	D₄	—	7	8	no	—
289	D₄	—	7	4	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	7	2	no	—
294	C₂×C₄	—	7	8	no	—
295	C₂×C₄	—	7	2	no	—
296	C₂×C₄	—	7	2	no	—
297	C₂×C₄	—	14	2	no	—
298	C₂×C₄	—	14	2	no	—
299	C₂×C₄	—	28	2	no	—
300	C₂×C₄	D₇×C₄⋊C₄	1	1	yes	characteristic
301	C₂×C₄	D₇×C₄⋊C₄	1	1	yes	characteristic
302	C₂×C₄	D₇×C₄⋊C₄	1	1	yes	characteristic
303	C₂×C₄	D₇×SD₁₆	1	4	yes	—
304	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
305	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
306	C₂×C₄	C₂×Q₈×D₇	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	4	2	no	—
311	C₈	—	4	2	no	—
312	C₈	—	14	2	no	—
313	C₈	—	14	2	no	—
314	C₈	—	28	4	no	—
315	C₈	#6464	1	2	yes	complemented
316	C₇	#17826	1	1	yes	characteristic, complemented, Sylow
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	7	4	no	—
320	C₂²	—	14	2	no	—
321	C₂²	—	14	2	no	—
322	C₂²	D₇×#106	1	2	yes	—
323	C₂²	C₂×D₇×C₄⋊C₄	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	7	4	no	—
328	C₄	—	14	2	no	—
329	C₄	—	14	2	no	—
330	C₄	D₇×#106	1	2	yes	—
331	C₄	C₂×#6464	1	1	yes	characteristic
332	C₄	#90436	1	1	yes	characteristic
333	C₄	C₂×D₇×SD₁₆	1	2	yes	—
334	C₂	—	2	2	no	—
335	C₂	—	14	4	no	—
336	C₂	#11736457	1	1	yes	characteristic
337	C₂	C₂×D₇×#106	1	1	yes	characteristic
338	C₂	#11736697	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇
commutator	C₂×C₄×C₇	C₂³×C₄
Frattini	C₄²	C₂³×D₇
Fitting	C₂×C₇×#334	C₂
socle	C₂²×C₇	C₂²×C₄⋊C₄
prime radical	C₂×#1157	D₄
cosocle	C₄²×C₇	C₂⁴
hypercenter	C₂×#334	D₇
hypocenter	C₇	#17826
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
3. C₂×C₇×#334	C₂	semidirect
4. #11728175	C₂	semidirect
5. #11727484	C₂	semidirect
6. #11726286	C₂	semidirect
7. #11726283	C₂	semidirect
8. D₇×#334	C₂	semidirect
83. C₂×#334	D₇	semidirect
177. C₇×C₈	#131	semidirect
315. C₈	#6464	semidirect
316. C₇	#17826	semidirect

Maximal subgroups

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

subgroup	index
#11736457	2
C₈×#1157	2
C₂×C₇×#334	2
#11728175	2
#11727484	2
#11726286	2
#11726283	2
D₇×#334	2
#17826	7