Subgroups — #13059518

Order 1344.

660 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	C₂²×C₄×D₃×C₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#66669	C₂	1	1	yes	maximal, characteristic, complemented
3	#10621407	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6493	C₂	1	1	yes	maximal, characteristic, complemented
5	#10621130	C₂	1	2	yes	maximal, complemented
6	Dic₃×C₂²⋊D₇	C₂	1	2	yes	maximal, complemented
7	#10621018	C₂	1	1	yes	maximal, characteristic, complemented
8	#10621014	C₂	1	2	yes	maximal, complemented
9	#10621003	C₂	1	2	yes	maximal, complemented
10	#10620927	C₂	1	1	yes	maximal, characteristic, complemented
11	#10620900	C₂	1	1	yes	maximal, characteristic, complemented
12	#90482	—	3	1	no	maximal, Hall
13	C₂³×D₃×C₇	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₄₂	C₂²	1	1	yes	characteristic, complemented
16	C₂²×C₇×Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₃×C₇	—	2	4	no	—
18	C₂×C₄×D₃×C₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₃×C₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₃×C₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₃×C₇	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	—
23	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic, complemented
24	C₂×#4245	C₂²	1	2	yes	—
25	C₂×#4245	C₂²	1	2	yes	—
26	C₂×Dic₃×D₇	C₂²	1	2	yes	complemented
27	#66669	—	2	4	no	—
28	#66636	C₂²	1	2	yes	—
29	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic, complemented
30	#66629	C₂²	1	2	yes	complemented
31	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	complemented
32	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic, complemented
33	Dic₇⋊Dic₃	C₂²	1	2	yes	complemented
34	(C₂×D₃)⋊Dic₇	C₂²	1	2	yes	—
35	(C₂×D₃)⋊Dic₇	C₂²	1	2	yes	—
36	(C₂×D₇)⋊Dic₃	C₂²	1	2	yes	complemented
37	Dic₃×Dic₇	C₂²	1	2	yes	complemented
38	C₂³×C₄×C₇	—	3	1	no	—
39	C₂×#1157	—	3	1	no	—
40	#6516	—	3	2	no	—
41	#6493	—	3	1	no	—
42	#6493	D₃	1	1	yes	characteristic, complemented
43	#6492	—	3	2	no	—
44	C₄×C₂²⋊D₇	—	3	2	no	—
45	C₄×C₂²⋊D₇	—	3	2	no	—
46	#6489	—	3	1	no	—
47	#6487	—	3	1	no	—
48	#6487	—	3	1	no	—
49	#5606	—	7	1	no	maximal, Hall
50	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
51	C₂²×D₃×C₇	—	2	2	no	—
52	C₂²×D₃×C₇	—	2	2	no	—
53	C₂²×D₃×C₇	—	2	2	no	—
54	C₂²×D₃×C₇	C₂³	1	1	yes	characteristic
55	C₂²×D₃×C₇	C₂³	1	1	yes	characteristic
56	C₂²×C₃×D₇	C₂³	1	2	yes	—
57	C₂×C₃×C₄×C₇	—	2	2	no	—
58	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
59	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
60	C₂×Dic₂₁	C₂³	1	2	yes	—
61	C₂×Dic₂₁	C₂³	1	2	yes	—
62	Dic₄₂	—	2	2	no	—
63	C₂×C₇×Dic₃	—	2	2	no	—
64	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
65	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
66	C₄×D₃×C₇	—	2	2	no	—
67	C₄×D₃×C₇	—	2	2	no	—
68	C₄×D₃×C₇	—	2	2	no	—
69	C₄×D₃×C₇	D₄	1	4	yes	complemented
70	C₃×C₂²⋊D₇	—	2	4	no	—
71	C₂×C₃×Dic₇	C₂³	1	2	yes	—
72	C₃×D₂₈	—	2	2	no	—
73	#4245	—	2	4	no	—
74	#4245	—	2	4	no	—
75	Dic₃×D₇	—	2	4	no	—
76	C₂⁴×C₇	—	3	1	no	—
77	C₂²×C₄×C₇	—	3	1	no	—
78	C₂²×C₄×C₇	—	3	1	no	—
79	C₂²×C₄×C₇	—	3	1	no	—
80	C₂²×C₄×C₇	—	3	1	no	—
81	C₂²×C₄×C₇	—	3	1	no	—
82	C₂²×C₄×C₇	—	6	4	no	—
83	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
84	C₂×C₂²⋊D₇	—	3	2	no	—
85	C₂×C₂²⋊D₇	—	3	2	no	—
86	C₂×C₂²⋊D₇	C₂×D₃	1	2	yes	—
87	#1157	—	6	4	no	—
88	C₂×D₂₈	C₂×D₃	1	1	yes	characteristic, complemented
89	C₂×C₄×D₇	—	3	2	no	—
90	C₂×Dic₁₄	—	3	1	no	—
91	C₂²⋊Dic₇	—	3	2	no	—
92	C₂²⋊Dic₇	—	3	2	no	—
93	C₂²⋊Dic₇	—	3	2	no	—
94	(C₂×D₇)⋊C₄	—	3	2	no	—
95	(C₂×D₇)⋊C₄	C₂×D₃	1	2	yes	complemented
96	C₄⋊Dic₇	—	3	1	no	—
97	C₄⋊Dic₇	C₂×D₃	1	1	yes	characteristic, complemented
98	Dic₇⋊C₄	—	3	2	no	—
99	Dic₇⋊C₄	—	3	2	no	—
100	C₄×Dic₇	—	3	2	no	—
101	C₂×#237	—	7	1	no	—
102	C₂²×C₄×D₃	D₇	1	1	yes	characteristic, complemented
103	C₃×#121	—	7	1	no	—
104	#938	—	7	1	no	—
105	#937	—	7	2	no	—
106	D₄×Dic₃	—	7	2	no	—
107	#924	—	7	1	no	—
108	#897	—	7	1	no	—
109	#892	—	7	1	no	—
110	#883	—	7	2	no	—
111	#67⋊C₄	—	7	2	no	—
112	C₂²×C₃×C₇	—	2	1	no	—
113	C₂²×C₃×C₇	—	2	1	no	—
114	C₂²×C₃×C₇	#41	1	2	yes	complemented
115	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
116	C₂×D₃×C₇	—	2	8	no	—
117	C₂×D₃×C₇	—	2	1	no	—
118	C₂×D₃×C₇	—	2	1	no	—
119	C₂×D₃×C₇	—	2	1	no	—
120	C₂×D₃×C₇	—	2	1	no	—
121	C₂×D₃×C₇	C₂×D₄	1	2	yes	—
122	C₂×D₃×C₇	#41	1	2	yes	complemented
123	C₂×C₃×D₇	—	2	2	no	—
124	C₂×C₃×D₇	—	2	2	no	—
125	C₂×C₃×D₇	—	2	2	no	—
126	C₃×C₄×C₇	—	2	1	no	—
127	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
128	Dic₂₁	—	2	2	no	—
129	Dic₂₁	—	2	2	no	—
130	C₃×Dic₇	—	2	2	no	—
131	C₇×Dic₃	—	2	1	no	—
132	C₇×Dic₃	C₂×D₄	1	2	yes	—
133	#525	—	21	1	no	Sylow
134	C₂³×C₇	—	3	1	no	—
135	C₂³×C₇	—	3	1	no	—
136	C₂³×C₇	—	6	2	no	—
137	C₂³×C₇	—	6	2	no	—
138	C₂³×C₇	—	6	2	no	—
139	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
140	C₂²×D₇	C₂²×D₃	1	2	yes	—
141	C₂×C₄×C₇	—	2	2	no	—
142	C₂×C₄×C₇	—	3	1	no	—
143	C₂×C₄×C₇	—	3	4	no	—
144	C₂×C₄×C₇	—	3	1	no	—
145	C₂×C₄×C₇	—	6	2	no	—
146	C₂×C₄×C₇	—	6	2	no	—
147	C₂×C₄×C₇	—	6	2	no	—
148	C₂×C₄×C₇	—	6	2	no	—
149	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
150	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
151	C₂²⋊D₇	—	2	4	no	—
152	C₂²⋊D₇	—	6	4	no	—
153	C₂²⋊D₇	—	6	4	no	—
154	C₂×Dic₇	—	3	2	no	—
155	C₂×Dic₇	—	3	2	no	—
156	C₂×Dic₇	C₂²×D₃	1	2	yes	—
157	D₂₈	—	2	2	no	—
158	C₄×D₇	—	6	4	no	—
159	Dic₁₄	—	6	2	no	—
160	C₂³×D₃	C₂×D₇	1	1	yes	characteristic
161	C₂×C₃×D₄	—	7	2	no	—
162	C₂×C₃×D₄	—	7	1	no	—
163	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
164	C₂×#67	—	7	2	no	—
165	C₂×#67	—	7	2	no	—
166	C₂²×Dic₃	—	7	2	no	—
167	C₂²×Dic₃	C₂×D₇	1	1	yes	characteristic
168	#237	—	14	4	no	—
169	C₂×C₄×D₃	—	2	4	no	—
170	C₂×C₄×D₃	C₂×D₇	1	1	yes	characteristic
171	C₂×C₄×D₃	C₂×D₇	1	1	yes	characteristic
172	C₂×C₄×D₃	C₂×D₇	1	1	yes	characteristic
173	C₂×C₄×D₃	C₂×D₇	1	1	yes	characteristic
174	C₂×Dic₆	—	7	1	no	—
175	C₃×C₄⋊C₄	—	7	1	no	—
176	C₃×C₂²⋊C₄	—	7	2	no	—
177	#217	—	7	2	no	—
178	#217	—	7	2	no	—
179	(C₂×D₃)⋊C₄	—	7	2	no	—
180	(C₂×D₃)⋊C₄	—	7	2	no	—
181	C₄⋊Dic₃	—	7	1	no	—
182	C₄⋊Dic₃	—	7	2	no	—
183	Dic₃⋊C₄	—	7	2	no	—
184	C₄×Dic₃	—	7	2	no	—
185	C₂×C₃×C₇	—	2	2	no	—
186	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
188	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
189	C₃×D₇	—	4	2	no	—
190	D₃×C₇	—	2	2	no	—
191	D₃×C₇	—	2	2	no	—
192	C₂×#41	—	21	1	no	—
193	C₂³×C₄	—	3	1	no	—
194	#123	—	21	2	no	—
195	C₂²⋊Q₈	—	21	1	no	—
196	C₂²⋊Q₈	—	21	1	no	—
197	#121	—	7	1	no	—
198	#121	—	21	1	no	—
199	#120	—	21	2	no	—
200	C₄×D₄	—	21	2	no	—
201	C₄×D₄	—	21	2	no	—
202	#117	—	21	1	no	—
203	C₂²×C₇	—	2	1	no	—
204	C₂²×C₇	—	2	1	no	—
205	C₂²×C₇	—	3	2	no	—
206	C₂²×C₇	—	3	2	no	—
207	C₂²×C₇	—	6	8	no	—
208	C₂²×C₇	—	6	1	no	—
209	C₂²×C₇	—	6	1	no	—
210	C₂²×C₇	—	6	1	no	—
211	C₂²×C₇	—	6	1	no	—
212	C₂²×C₇	#237	1	2	yes	complemented
213	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
214	C₂×D₇	—	2	2	no	—
215	C₂×D₇	—	2	2	no	—
216	C₂×D₇	—	2	2	no	—
217	C₄×C₇	—	2	1	no	—
218	C₄×C₇	—	3	2	no	—
219	C₄×C₇	—	6	1	no	—
220	C₄×C₇	D₃×D₄	1	2	yes	—
221	Dic₇	—	2	2	no	—
222	Dic₇	—	6	2	no	—
223	Dic₇	—	6	2	no	—
224	C₂³×C₃	—	7	2	no	—
225	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
226	C₂²×D₃	—	2	2	no	—
227	C₂²×D₃	—	2	2	no	—
228	C₂²×D₃	—	2	2	no	—
229	C₂²×D₃	C₂²×D₇	1	1	yes	characteristic
230	C₂²×D₃	C₂²×D₇	1	1	yes	characteristic
231	C₃×D₄	—	14	4	no	—
232	C₃×D₄	—	14	2	no	—
233	C₂×C₃×C₄	—	2	2	no	—
234	C₂×C₃×C₄	—	7	2	no	—
235	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
236	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
237	#67	—	14	4	no	—
238	#67	—	14	4	no	—
239	C₂×Dic₃	—	2	2	no	—
240	C₂×Dic₃	—	7	2	no	—
241	C₂×Dic₃	—	7	2	no	—
242	C₂×Dic₃	—	14	4	no	—
243	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
244	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
245	C₄×D₃	—	2	2	no	—
246	C₄×D₃	—	2	2	no	—
247	C₄×D₃	—	2	2	no	—
248	C₄×D₃	C₂²⋊D₇	1	4	yes	complemented
249	Dic₆	—	14	2	no	—
250	C₃×C₇	#525	1	1	yes	characteristic, complemented, Hall
251	C₂⁴	—	3	1	no	—
252	#41	—	42	4	no	—
253	C₂×Q₈	—	21	1	no	—
254	C₂×D₄	—	7	2	no	—
255	C₂×D₄	—	7	1	no	—
256	C₂×D₄	—	21	2	no	—
257	C₂×D₄	—	21	2	no	—
258	C₂²×C₄	—	3	1	no	—
259	C₂²×C₄	—	3	1	no	—
260	C₂²×C₄	—	3	1	no	—
261	C₂²×C₄	—	3	1	no	—
262	C₂²×C₄	—	3	1	no	—
263	C₂²×C₄	—	6	4	no	—
264	C₂²×C₄	—	21	2	no	—
265	C₂²×C₄	D₃×D₇	1	1	yes	characteristic
266	C₄⋊C₄	—	7	1	no	—
267	C₄⋊C₄	—	21	2	no	—
268	C₄⋊C₄	—	21	2	no	—
269	C₄⋊C₄	—	21	1	no	—
270	C₂²⋊C₄	—	7	2	no	—
271	C₂²⋊C₄	—	21	2	no	—
272	C₂²⋊C₄	—	21	2	no	—
273	C₂²⋊C₄	—	21	2	no	—
274	C₂²⋊C₄	—	21	2	no	—
275	C₄²	—	21	2	no	—
276	C₂×C₇	—	2	2	no	—
277	C₂×C₇	—	6	2	no	—
278	C₂×C₇	—	6	2	no	—
279	C₂×C₇	C₂×D₃×D₄	1	1	yes	characteristic
280	C₂×C₇	C₂×#237	1	1	yes	characteristic
281	C₂×C₇	D₃×#41	1	1	yes	characteristic
282	D₇	—	4	2	no	—
283	C₂²×C₃	—	2	1	no	—
284	C₂²×C₃	—	2	1	no	—
285	C₂²×C₃	—	14	2	no	—
286	C₂²×C₃	—	14	2	no	—
287	C₂²×C₃	—	14	2	no	—
288	C₂²×C₃	#1157	1	2	yes	complemented
289	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
290	C₂×D₃	—	2	8	no	—
291	C₂×D₃	—	2	1	no	—
292	C₂×D₃	—	2	1	no	—
293	C₂×D₃	—	2	1	no	—
294	C₂×D₃	—	2	1	no	—
295	C₂×D₃	#1157	1	2	yes	complemented
296	C₂×D₃	C₂×C₂²⋊D₇	1	2	yes	—
297	C₃×C₄	—	2	1	no	—
298	C₃×C₄	—	14	2	no	—
299	C₃×C₄	C₂×C₂²⋊D₇	1	2	yes	—
300	Dic₃	—	2	1	no	—
301	Dic₃	—	14	2	no	—
302	Dic₃	—	14	2	no	—
303	Dic₃	C₂×C₂²⋊D₇	1	2	yes	—
304	C₂³	—	3	1	no	—
305	C₂³	—	3	1	no	—
306	C₂³	—	6	2	no	—
307	C₂³	—	6	2	no	—
308	C₂³	—	6	2	no	—
309	C₂³	—	7	2	no	—
310	C₂³	C₂×D₃×D₇	1	1	yes	characteristic
311	Q₈	—	42	2	no	—
312	D₄	—	14	4	no	—
313	D₄	—	14	2	no	—
314	D₄	—	42	4	no	—
315	D₄	—	42	4	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	3	1	no	—
318	C₂×C₄	—	3	4	no	—
319	C₂×C₄	—	3	1	no	—
320	C₂×C₄	—	6	2	no	—
321	C₂×C₄	—	6	2	no	—
322	C₂×C₄	—	6	2	no	—
323	C₂×C₄	—	6	2	no	—
324	C₂×C₄	—	7	2	no	—
325	C₂×C₄	—	21	2	no	—
326	C₂×C₄	—	21	2	no	—
327	C₂×C₄	—	42	4	no	—
328	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
329	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
330	C₇	#5606	1	1	yes	characteristic, complemented, Sylow
331	C₂×C₃	—	2	2	no	—
332	C₂×C₃	—	28	2	no	—
333	C₂×C₃	C₂×#1157	1	1	yes	characteristic
334	C₂×C₃	C₂×#1157	1	1	yes	characteristic
335	C₂×C₃	C₂²×C₂²⋊D₇	1	1	yes	characteristic
336	D₃	—	2	2	no	—
337	D₃	—	2	2	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	3	2	no	—
341	C₂²	—	3	2	no	—
342	C₂²	—	6	8	no	—
343	C₂²	—	6	1	no	—
344	C₂²	—	6	1	no	—
345	C₂²	—	6	1	no	—
346	C₂²	—	6	1	no	—
347	C₂²	—	14	2	no	—
348	C₂²	—	14	2	no	—
349	C₂²	—	14	2	no	—
350	C₂²	#66669	1	2	yes	complemented
351	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	3	2	no	—
354	C₄	—	6	1	no	—
355	C₄	—	14	2	no	—
356	C₄	—	42	2	no	—
357	C₄	—	42	2	no	—
358	C₄	D₃×C₂²⋊D₇	1	2	yes	—
359	C₃	#90482	1	1	yes	characteristic, complemented, Sylow
360	C₂	—	2	2	no	—
361	C₂	—	6	2	no	—
362	C₂	—	6	2	no	—
363	C₂	—	28	2	no	—
364	C₂	C₂×#66669	1	1	yes	characteristic
365	C₂	D₃×#1157	1	1	yes	characteristic
366	C₂	C₂×D₃×C₂²⋊D₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₇
commutator, socle, cosocle	C₂²×C₃×C₇	C₂⁴
Fitting	C₂²×C₃×C₄×C₇	C₂²
hypercenter	C₂²×C₄	D₃×D₇
hypocenter	C₃×C₇	#525
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄×D₃×C₇	C₂	semidirect
2. C₂×#66669	C₂	semidirect
3. #10621407	C₂	semidirect
4. C₃×#6493	C₂	semidirect
5. #10621130	C₂	semidirect
6. Dic₃×C₂²⋊D₇	C₂	semidirect
7. #10621018	C₂	semidirect
8. #10621014	C₂	semidirect
9. #10621003	C₂	semidirect
10. #10620927	C₂	semidirect
11. #10620900	C₂	semidirect
15. C₂×Dic₄₂	C₂²	semidirect
23. C₂×C₃×D₂₈	C₂²	semidirect
26. C₂×Dic₃×D₇	C₂²	semidirect
29. C₄⋊Dic₂₁	C₂²	semidirect
30. #66629	C₂²	semidirect
31. C₃×(C₂×D₇)⋊C₄	C₂²	semidirect
32. C₃×C₄⋊Dic₇	C₂²	semidirect
33. Dic₇⋊Dic₃	C₂²	semidirect
36. (C₂×D₇)⋊Dic₃	C₂²	semidirect
37. Dic₃×Dic₇	C₂²	semidirect
42. #6493	D₃	semidirect
69. C₄×D₃×C₇	D₄	semidirect
88. C₂×D₂₈	C₂×D₃	semidirect
95. (C₂×D₇)⋊C₄	C₂×D₃	semidirect
97. C₄⋊Dic₇	C₂×D₃	semidirect
102. C₂²×C₄×D₃	D₇	semidirect
114. C₂²×C₃×C₇	#41	semidirect
122. C₂×D₃×C₇	#41	semidirect
212. C₂²×C₇	#237	semidirect
248. C₄×D₃	C₂²⋊D₇	semidirect
250. C₃×C₇	#525	semidirect
288. C₂²×C₃	#1157	semidirect
295. C₂×D₃	#1157	semidirect
330. C₇	#5606	semidirect
350. C₂²	#66669	semidirect
359. C₃	#90482	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₃×C₇	2
C₂×#66669	2
#10621407	2
C₃×#6493	2
#10621130	2
Dic₃×C₂²⋊D₇	2
#10621018	2
#10621014	2
#10621003	2
#10620927	2
#10620900	2
#90482	3
#5606	7