Subgroups — #13336893

Order 1792.

662 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×#498	C₂	1	1	yes	maximal, characteristic, complemented
2	#11740661	C₂	1	1	yes	maximal, characteristic
3	#11740658	C₂	1	1	yes	maximal, characteristic
4	#11735903	C₂	1	1	yes	maximal, characteristic
5	#11729529	C₂	1	1	yes	maximal, characteristic, complemented
6	#11729528	C₂	1	1	yes	maximal, characteristic, complemented
7	#11729099	C₂	1	4	yes	maximal, complemented
8	#11729098	C₂	1	4	yes	maximal, complemented
9	#11727483	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×#6464	—	2	1	no	—
13	#90168	C₂²	1	1	yes	characteristic
14	C₇×#498	—	2	1	no	—
15	C₇×#498	—	2	1	no	—
16	C₇×#498	C₂²	1	2	yes	—
17	C₇×#498	C₂²	1	2	yes	—
18	C₂×C₇×#106	—	2	1	no	—
19	C₂×C₇×#106	—	2	1	no	—
20	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
21	#89999	—	2	2	no	—
22	#89882	—	2	1	no	—
23	#89860	—	2	1	no	—
24	#89857	—	2	1	no	—
25	#89856	C₂²	1	4	yes	—
26	#89855	C₂²	1	4	yes	—
27	#89853	—	2	1	no	—
28	#89851	—	2	1	no	—
29	#89740	—	2	2	no	—
30	#89697	C₂²	1	1	yes	characteristic
31	#89697	C₂²	1	1	yes	characteristic
32	#89696	C₂²	1	1	yes	characteristic
33	#89460	—	2	1	no	—
34	#89460	C₂²	1	2	yes	—
35	C₈×D₂₈	—	2	1	no	—
36	C₈×D₂₈	C₂²	1	2	yes	—
37	#89342	C₂²	1	4	yes	—
38	#89341	C₂²	1	4	yes	—
39	#89253	C₂²	1	2	yes	—
40	#89252	C₂²	1	2	yes	—
41	#19191	—	7	1	no	maximal, Sylow
42	C₂²×C₇×Q₈	—	2	1	no	—
43	C₂²×C₇×Q₈	—	2	1	no	—
44	C₂×#1157	—	2	1	no	—
45	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
46	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
47	C₇×C₄⋊Q₈	—	2	1	no	—
48	C₇×C₄⋊Q₈	—	2	1	no	—
49	C₇×C₄⋊Q₈	—	2	1	no	—
50	C₇×C₄⋊Q₈	—	2	1	no	—
51	C₇×C₄⋊Q₈	—	2	1	no	—
52	C₇×C₄⋊Q₈	—	2	1	no	—
53	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
54	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
55	C₂×C₇×C₄⋊C₄	—	2	1	no	—
56	C₂×C₇×C₄⋊C₄	—	2	1	no	—
57	C₂×C₇×C₄⋊C₄	—	2	1	no	—
58	C₂×C₇×C₄⋊C₄	—	2	1	no	—
59	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
60	C₂×#1143	—	2	4	no	—
61	#6493	—	2	1	no	—
62	#6493	D₄	1	2	yes	—
63	C₂×#1136	—	2	1	no	—
64	#6464	—	4	4	no	—
65	C₂×#1131	—	2	2	no	—
66	C₂×C₈×D₇	—	2	2	no	—
67	#6438	—	2	1	no	—
68	#6437	D₄	1	1	yes	characteristic
69	C₄×D₂₈	—	2	1	no	—
70	C₄×D₂₈	C₂³	1	2	yes	—
71	#6432	D₄	1	1	yes	characteristic
72	C₇×#106	—	2	2	no	—
73	C₇×#106	—	2	2	no	—
74	C₇×#106	—	2	2	no	—
75	C₇×#106	—	2	2	no	—
76	C₄×C₇×C₈	D₄	1	1	yes	characteristic
77	C₄×C₇×C₈	D₄	1	1	yes	characteristic
78	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
79	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
80	#6409	—	2	4	no	—
81	C₂²⋊(C₇⋊C₈)	—	2	1	no	—
82	C₂²⋊(C₇⋊C₈)	—	2	1	no	—
83	(C₂×D₇)⋊C₈	—	2	2	no	—
84	(C₂×D₇)⋊C₈	—	2	2	no	—
85	#6383	—	2	4	no	—
86	#6382	—	2	4	no	—
87	C₄⋊(C₇⋊C₈)	—	2	1	no	—
88	C₄⋊(C₇⋊C₈)	C₂³	1	2	yes	—
89	C₄⋊(C₇⋊C₈)	C₂³	1	2	yes	—
90	C₄⋊(C₇⋊C₈)	C₂³	1	2	yes	—
91	C₄×C₇⋊C₈	—	2	1	no	—
92	C₄×C₇⋊C₈	—	2	1	no	—
93	#3227	—	7	1	no	—
94	#3206	—	7	1	no	—
95	C₂×#498	D₇	1	1	yes	characteristic, complemented
96	#2931	—	7	1	no	—
97	#1691	—	7	4	no	—
98	#1671	—	7	4	no	—
99	#1564	—	7	1	no	—
100	#1549	—	7	1	no	—
101	#1548	—	7	1	no	—
102	C₂×C₇×Q₈	—	2	2	no	—
103	C₂×C₇×Q₈	—	2	2	no	—
104	C₂×C₇×Q₈	—	4	1	no	—
105	C₂×C₇×Q₈	—	4	1	no	—
106	C₂×C₇×Q₈	—	8	2	no	—
107	C₂×C₇×Q₈	—	8	2	no	—
108	C₂²×C₄×C₇	—	4	1	no	—
109	C₂²×C₄×C₇	—	4	1	no	—
110	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₇	#41	1	1	yes	characteristic
113	C₂×C₂²⋊D₇	—	2	2	no	—
114	#1157	—	4	2	no	—
115	#1157	—	4	2	no	—
116	C₂×D₂₈	—	2	1	no	—
117	C₂×D₂₈	—	2	1	no	—
118	C₂×D₂₈	C₂×D₄	1	2	yes	—
119	C₂×C₄×D₇	—	2	2	no	—
120	C₂×Dic₁₄	—	2	1	no	—
121	C₂×C₇×C₈	—	2	2	no	—
122	C₂×C₇×C₈	SD₁₆	1	4	yes	complemented
123	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
124	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
125	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
126	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
127	C₇×C₄⋊C₄	—	2	2	no	—
128	C₇×C₄⋊C₄	—	2	2	no	—
129	C₇×C₄⋊C₄	—	2	2	no	—
130	C₇×C₄⋊C₄	—	2	2	no	—
131	C₇×C₄⋊C₄	—	4	1	no	—
132	C₇×C₄⋊C₄	—	4	1	no	—
133	C₇×C₄⋊C₄	—	4	1	no	—
134	C₇×C₄⋊C₄	—	4	1	no	—
135	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
136	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
137	C₄²×C₇	#41	1	1	yes	characteristic
138	C₄²×C₇	C₂⁴	1	1	yes	characteristic
139	#1143	—	8	4	no	—
140	(C₂×D₇)⋊C₄	—	2	2	no	—
141	(C₂×D₇)⋊C₄	—	2	2	no	—
142	C₄⋊Dic₇	—	2	1	no	—
143	C₄⋊Dic₇	C₂×D₄	1	2	yes	—
144	#1136	—	4	2	no	—
145	C₂×C₇⋊C₈	—	2	2	no	—
146	C₂×C₇⋊C₈	—	2	2	no	—
147	C₂×C₇⋊C₈	—	2	2	no	—
148	#1131	—	4	4	no	—
149	C₈×D₇	—	4	4	no	—
150	C₂×#131	—	14	1	no	—
151	#532	—	7	1	no	—
152	C₂×C₄⋊Q₈	C₂×D₇	1	1	yes	characteristic
153	C₂×C₄⋊Q₈	C₂×D₇	1	1	yes	characteristic
154	#498	—	2	1	no	—
155	#498	—	2	1	no	—
156	#498	—	14	1	no	—
157	#498	C₂×D₇	1	2	yes	—
158	#498	C₂×D₇	1	2	yes	—
159	#492	—	14	1	no	—
160	#487	—	14	1	no	—
161	#486	—	14	1	no	—
162	#481	—	14	2	no	—
163	#476	—	7	4	no	—
164	#461	—	7	4	no	—
165	#448	—	14	2	no	—
166	#436	—	7	2	no	—
167	#436	—	14	1	no	—
168	C₈×D₄	—	7	2	no	—
169	C₈×D₄	—	14	1	no	—
170	#431	—	7	1	no	—
171	#431	—	7	1	no	—
172	C₂×#106	—	2	1	no	—
173	C₂×#106	—	2	1	no	—
174	#423	—	7	1	no	—
175	#408	—	14	1	no	—
176	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
177	#335	—	7	2	no	—
178	#334	—	7	2	no	—
179	#333	—	7	4	no	—
180	#332	—	7	4	no	—
181	C₂³×C₇	#121	1	1	yes	characteristic
182	C₂²×D₇	—	2	2	no	—
183	C₇×Q₈	—	4	4	no	—
184	C₇×Q₈	—	8	2	no	—
185	C₇×Q₈	—	8	2	no	—
186	C₇×Q₈	—	8	2	no	—
187	C₂×C₄×C₇	—	2	2	no	—
188	C₂×C₄×C₇	—	2	2	no	—
189	C₂×C₄×C₇	—	4	2	no	—
190	C₂×C₄×C₇	—	4	2	no	—
191	C₂×C₄×C₇	—	8	2	no	—
192	C₂×C₄×C₇	—	8	2	no	—
193	C₂×C₄×C₇	#121	1	1	yes	characteristic
194	C₂×C₄×C₇	#121	1	1	yes	characteristic
195	C₂×C₄×C₇	#121	1	1	yes	characteristic
196	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
197	C₂×C₄×C₇	#137	1	2	yes	—
198	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
199	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
200	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
201	C₂²⋊D₇	—	4	4	no	—
202	C₂×Dic₇	—	2	2	no	—
203	D₂₈	—	2	4	no	—
204	D₂₈	—	4	2	no	—
205	D₂₈	—	4	2	no	—
206	C₄×D₇	—	4	2	no	—
207	C₄×D₇	—	4	2	no	—
208	Dic₁₄	—	4	2	no	—
209	C₇×C₈	—	2	1	no	—
210	C₇×C₈	—	2	1	no	—
211	C₇×C₈	C₂×SD₁₆	1	2	yes	—
212	C₇×C₈	C₂×SD₁₆	1	2	yes	—
213	C₇⋊C₈	—	4	2	no	—
214	C₇⋊C₈	—	4	2	no	—
215	C₇⋊C₈	—	4	2	no	—
216	C₂×#41	—	14	1	no	—
217	C₂²×Q₈	—	2	1	no	—
218	C₂²×Q₈	—	2	1	no	—
219	C₂×SD₁₆	—	14	4	no	—
220	#131	—	28	4	no	—
221	C₂×M₁₆	—	14	1	no	—
222	C₂×M₁₆	—	14	2	no	—
223	C₂²×C₈	—	14	2	no	—
224	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
225	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
226	C₄⋊Q₈	—	2	1	no	—
227	C₄⋊Q₈	—	2	1	no	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	—	2	1	no	—
230	C₄⋊Q₈	—	2	1	no	—
231	C₄⋊Q₈	—	2	1	no	—
232	C₄⋊Q₈	—	7	1	no	—
233	C₄⋊Q₈	C₂²×D₇	1	2	yes	—
234	C₄⋊Q₈	C₂²×D₇	1	2	yes	—
235	#127	—	7	1	no	—
236	#124	—	14	1	no	—
237	#121	—	7	2	no	—
238	#121	—	14	1	no	—
239	C₄×D₄	—	7	2	no	—
240	C₄×D₄	—	14	1	no	—
241	C₂×C₄⋊C₄	—	2	1	no	—
242	C₂×C₄⋊C₄	—	2	1	no	—
243	C₂×C₄⋊C₄	—	2	1	no	—
244	C₂×C₄⋊C₄	—	2	1	no	—
245	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
246	#106	—	2	2	no	—
247	#106	—	2	2	no	—
248	#106	—	2	2	no	—
249	#106	—	2	2	no	—
250	#106	—	14	4	no	—
251	C₄⋊C₈	—	7	2	no	—
252	C₄⋊C₈	—	7	2	no	—
253	C₄⋊C₈	—	7	2	no	—
254	C₄⋊C₈	—	14	1	no	—
255	#103	—	14	4	no	—
256	#102	—	14	4	no	—
257	C₂²⋊C₈	—	14	1	no	—
258	C₂²⋊C₈	—	14	1	no	—
259	C₂²⋊C₈	—	14	2	no	—
260	C₂²⋊C₈	—	14	2	no	—
261	C₄×C₈	—	14	1	no	—
262	C₄×C₈	—	14	1	no	—
263	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
264	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
265	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
266	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
267	C₂²×C₇	—	2	1	no	—
268	C₂²×C₇	—	2	1	no	—
269	C₂²×C₇	#461	1	2	yes	—
270	C₂²×C₇	C₂×#121	1	1	yes	characteristic
271	C₂×D₇	—	4	2	no	—
272	C₂×D₇	—	4	2	no	—
273	C₂×D₇	—	4	2	no	—
274	C₄×C₇	—	2	1	no	—
275	C₄×C₇	—	2	1	no	—
276	C₄×C₇	—	8	2	no	—
277	C₄×C₇	—	8	2	no	—
278	C₄×C₇	#461	1	2	yes	—
279	C₄×C₇	#471	1	2	yes	—
280	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
281	C₄×C₇	C₂×#137	1	1	yes	characteristic
282	C₄×C₇	#577	1	2	yes	—
283	Dic₇	—	4	2	no	—
284	#41	—	28	2	no	—
285	#41	—	28	2	no	—
286	C₂×Q₈	—	2	2	no	—
287	C₂×Q₈	—	2	2	no	—
288	C₂×Q₈	—	4	1	no	—
289	C₂×Q₈	—	4	1	no	—
290	C₂×Q₈	—	8	2	no	—
291	C₂×Q₈	—	8	2	no	—
292	C₂×Q₈	—	14	1	no	—
293	C₂×D₄	—	7	2	no	—
294	C₂×D₄	—	14	2	no	—
295	C₂×D₄	—	14	1	no	—
296	C₂×D₄	—	14	1	no	—
297	C₂²×C₄	—	4	1	no	—
298	C₂²×C₄	—	4	1	no	—
299	C₂²×C₄	—	14	2	no	—
300	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
301	C₂²×C₄	#1159	1	1	yes	characteristic
302	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
303	SD₁₆	—	56	4	no	—
304	M₁₆	—	28	2	no	—
305	M₁₆	—	28	4	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	14	2	no	—
308	C₂×C₈	—	14	2	no	—
309	C₂×C₈	—	14	2	no	—
310	C₂×C₈	—	28	4	no	—
311	C₂×C₈	#1143	1	4	yes	complemented
312	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
313	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
314	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
315	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	4	1	no	—
321	C₄⋊C₄	—	4	1	no	—
322	C₄⋊C₄	—	4	1	no	—
323	C₄⋊C₄	—	4	1	no	—
324	C₄⋊C₄	—	7	2	no	—
325	C₄⋊C₄	—	14	1	no	—
326	C₂²⋊C₄	—	14	2	no	—
327	C₂²⋊C₄	—	14	2	no	—
328	C₄²	D₄×D₇	1	1	yes	characteristic
329	C₄²	#1159	1	1	yes	characteristic
330	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
331	C₄²	C₂³×D₇	1	1	yes	characteristic
332	C₂×C₇	—	2	2	no	—
333	C₂×C₇	C₂×#461	1	1	yes	characteristic
334	C₂×C₇	#3057	1	1	yes	characteristic
335	C₂×C₇	C₂×#471	1	1	yes	characteristic
336	D₇	—	8	2	no	—
337	C₂³	—	14	2	no	—
338	C₂³	#6501	1	1	yes	characteristic
339	Q₈	—	4	4	no	—
340	Q₈	—	8	2	no	—
341	Q₈	—	8	2	no	—
342	Q₈	—	8	2	no	—
343	Q₈	—	28	2	no	—
344	D₄	—	14	4	no	—
345	D₄	—	28	4	no	—
346	D₄	—	28	2	no	—
347	D₄	—	28	2	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	—	8	2	no	—
353	C₂×C₄	—	8	2	no	—
354	C₂×C₄	—	14	2	no	—
355	C₂×C₄	—	28	2	no	—
356	C₂×C₄	—	28	2	no	—
357	C₂×C₄	#6501	1	1	yes	characteristic
358	C₂×C₄	#6501	1	1	yes	characteristic
359	C₂×C₄	#6501	1	1	yes	characteristic
360	C₂×C₄	C₂×#1143	1	2	yes	—
361	C₂×C₄	#6505	1	2	yes	—
362	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
363	C₂×C₄	C₂×#1159	1	1	yes	characteristic
364	C₂×C₄	C₂²×C₂²⋊D₇	1	1	yes	characteristic
365	C₈	—	2	1	no	—
366	C₈	—	2	1	no	—
367	C₈	—	28	2	no	—
368	C₈	—	28	2	no	—
369	C₈	—	28	2	no	—
370	C₈	C₂×#1143	1	2	yes	—
371	C₈	C₂×#1143	1	2	yes	—
372	C₇	#19191	1	1	yes	characteristic, complemented, Sylow
373	C₂²	—	2	1	no	—
374	C₂²	—	2	1	no	—
375	C₂²	—	28	2	no	—
376	C₂²	—	28	2	no	—
377	C₂²	—	28	2	no	—
378	C₂²	#89855	1	2	yes	—
379	C₂²	C₂×#6501	1	1	yes	characteristic
380	C₄	—	2	1	no	—
381	C₄	—	2	1	no	—
382	C₄	—	8	2	no	—
383	C₄	—	8	2	no	—
384	C₄	—	28	2	no	—
385	C₄	#89855	1	2	yes	—
386	C₄	#89949	1	2	yes	—
387	C₄	#90453	1	2	yes	—
388	C₄	C₂²×#1143	1	1	yes	characteristic
389	C₄	C₂×#6505	1	1	yes	characteristic
390	C₂	—	2	2	no	—
391	C₂	—	56	2	no	—
392	C₂	C₂×#89855	1	1	yes	characteristic
393	C₂	#11741308	1	1	yes	characteristic
394	C₂	C₂×#89949	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6501
commutator, cosocle	C₄²×C₇	C₂⁴
Frattini	C₄²	C₂³×D₇
Fitting	C₂×C₇×#498	C₂
socle	C₂²×C₇	C₂×#121
prime radical	#90168	C₂²
hypercenter	C₂×#498	D₇
hypocenter	C₇	#19191
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#498

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#498	C₂	semidirect
5. #11729529	C₂	semidirect
6. #11729528	C₂	semidirect
7. #11729099	C₂	semidirect
8. #11729098	C₂	semidirect
9. #11727483	C₂	semidirect
95. C₂×#498	D₇	semidirect
122. C₂×C₇×C₈	SD₁₆	semidirect
311. C₂×C₈	#1143	semidirect
372. C₇	#19191	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#498	2
#11740661	2
#11740658	2
#11735903	2
#11729529	2
#11729528	2
#11729099	2
#11729098	2
#11727483	2
#19191	7