Subgroups — #13057795

Order 1344.

444 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₄₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₇×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#6435	C₂	1	1	yes	maximal, characteristic
4	C₄×#4248	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#4248	C₂	1	1	yes	maximal, characteristic, complemented
6	#10620993	C₂	1	1	yes	maximal, characteristic
7	#10620992	C₂	1	1	yes	maximal, characteristic
8	#10620984	C₂	1	1	yes	maximal, characteristic
9	#10620983	C₂	1	1	yes	maximal, characteristic
10	C₄×Dic₃×D₇	C₂	1	1	yes	maximal, characteristic
11	#10620919	C₂	1	1	yes	maximal, characteristic
12	#10620897	C₂	1	1	yes	maximal, characteristic, complemented
13	#10620896	C₂	1	1	yes	maximal, characteristic, complemented
14	#10620891	C₂	1	1	yes	maximal, characteristic, complemented
15	#10620890	C₂	1	1	yes	maximal, characteristic, complemented
16	#90269	—	3	1	no	maximal, Hall
17	C₂×Dic₄₂	C₂²	1	1	yes	characteristic
18	C₂×C₇×Dic₆	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
20	C₂×#4248	C₂²	1	1	yes	characteristic
21	C₂×#4248	C₂²	1	1	yes	characteristic
22	C₂×Dic₃×D₇	C₂²	1	1	yes	characteristic
23	C₂×Dic₃×D₇	C₂²	1	1	yes	characteristic
24	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
25	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
26	#66629	C₂²	1	1	yes	characteristic
27	#66629	C₂²	1	1	yes	characteristic
28	C₄×Dic₂₁	C₂²	1	1	yes	characteristic
29	C₄×Dic₂₁	C₂²	1	1	yes	characteristic
30	C₇×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
31	C₇×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
32	C₇×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
33	C₄×C₇×Dic₃	C₂²	1	1	yes	characteristic
34	C₄×C₇×Dic₃	C₂²	1	1	yes	characteristic
35	C₃×(C₂×D₇)⋊C₄	C₂²	1	1	yes	characteristic
36	C₃×(C₂×D₇)⋊C₄	C₂²	1	1	yes	characteristic
37	C₃×Dic₇⋊C₄	C₂²	1	1	yes	characteristic
38	C₃×Dic₇⋊C₄	C₂²	1	1	yes	characteristic
39	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
40	Dic₃⋊Dic₇	C₂²	1	1	yes	characteristic
41	Dic₃⋊Dic₇	C₂²	1	1	yes	characteristic
42	#66577	C₂²	1	1	yes	characteristic
43	#66577	C₂²	1	1	yes	characteristic
44	Dic₇⋊Dic₃	C₂²	1	1	yes	characteristic
45	Dic₇⋊Dic₃	C₂²	1	1	yes	characteristic
46	Dic₇⋊Dic₃	C₂²	1	1	yes	characteristic
47	Dic₇⋊Dic₃	C₂²	1	1	yes	characteristic
48	(C₂×D₇)⋊Dic₃	C₂²	1	1	yes	characteristic
49	(C₂×D₇)⋊Dic₃	C₂²	1	1	yes	characteristic
50	Dic₃×Dic₇	C₂²	1	1	yes	characteristic
51	Dic₃×Dic₇	C₂²	1	1	yes	characteristic
52	C₄×C₇×Q₈	—	3	1	no	—
53	#6509	—	3	1	no	—
54	#6507	—	3	1	no	—
55	#6460	—	3	1	no	—
56	#6459	—	3	1	no	—
57	#6459	—	3	1	no	—
58	#6452	—	3	1	no	—
59	#6452	—	3	1	no	—
60	#6451	—	3	1	no	—
61	#6435	—	3	1	no	—
62	#6435	D₃	1	1	yes	characteristic
63	C₄²×D₇	—	3	1	no	—
64	C₄×Dic₁₄	—	3	1	no	—
65	C₄×Dic₁₄	—	3	1	no	—
66	C₄×Dic₁₄	—	3	1	no	—
67	#5517	—	7	1	no	maximal, Hall
68	C₂²×C₃×D₇	C₂³	1	1	yes	characteristic
69	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
70	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
71	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
72	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
73	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
74	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
75	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
76	Dic₄₂	—	2	2	no	—
77	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
78	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
79	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
80	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
81	C₇×Dic₆	—	2	2	no	—
82	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
83	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
84	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
85	C₃×C₄×D₇	Q₈	1	4	yes	complemented
86	#4248	—	2	2	no	—
87	#4248	—	2	2	no	—
88	Dic₃×D₇	—	2	2	no	—
89	Dic₃×D₇	—	2	2	no	—
90	C₂×C₇×Q₈	—	3	1	no	—
91	C₂×C₄×D₇	—	3	1	no	—
92	C₂×C₄×D₇	—	3	1	no	—
93	C₂×C₄×D₇	C₂×D₃	1	1	yes	characteristic
94	C₂×Dic₁₄	—	3	1	no	—
95	C₂×Dic₁₄	—	3	1	no	—
96	C₂×Dic₁₄	—	3	1	no	—
97	C₇×C₄⋊C₄	—	3	1	no	—
98	C₇×C₄⋊C₄	—	3	1	no	—
99	C₇×C₄⋊C₄	—	3	1	no	—
100	C₄²×C₇	—	3	1	no	—
101	C₄²×C₇	—	3	1	no	—
102	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
103	(C₂×D₇)⋊C₄	—	3	1	no	—
104	(C₂×D₇)⋊C₄	—	3	1	no	—
105	(C₂×D₇)⋊C₄	C₂×D₃	1	1	yes	characteristic
106	(C₂×D₇)⋊C₄	C₂×D₃	1	1	yes	characteristic
107	C₄⋊Dic₇	—	3	1	no	—
108	C₄⋊Dic₇	—	3	1	no	—
109	C₄⋊Dic₇	—	3	1	no	—
110	Dic₇⋊C₄	—	3	1	no	—
111	Dic₇⋊C₄	—	3	1	no	—
112	Dic₇⋊C₄	—	3	1	no	—
113	Dic₇⋊C₄	—	3	1	no	—
114	Dic₇⋊C₄	—	3	1	no	—
115	Dic₇⋊C₄	—	3	1	no	—
116	Dic₇⋊C₄	—	3	1	no	—
117	Dic₇⋊C₄	—	3	1	no	—
118	Dic₇⋊C₄	C₂×D₃	1	1	yes	characteristic
119	Dic₇⋊C₄	C₂×D₃	1	1	yes	characteristic
120	C₄×Dic₇	—	3	1	no	—
121	C₄×Dic₇	—	3	1	no	—
122	C₄×Dic₇	—	3	1	no	—
123	C₄×Dic₇	—	3	1	no	—
124	C₄×Dic₇	C₂×D₃	1	1	yes	characteristic
125	C₃×#117	—	7	1	no	—
126	#926	—	7	1	no	—
127	C₂×C₄×Dic₃	—	7	1	no	—
128	#890	—	7	1	no	—
129	#890	—	7	1	no	—
130	#888	—	7	1	no	—
131	#888	—	7	1	no	—
132	#878	—	7	1	no	—
133	#878	—	7	1	no	—
134	#878	—	7	1	no	—
135	#878	—	7	1	no	—
136	C₄×Dic₆	—	7	1	no	—
137	C₄×Dic₆	—	7	1	no	—
138	C₄×Dic₆	—	7	1	no	—
139	C₄×Dic₆	D₇	1	1	yes	characteristic, complemented
140	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
141	C₂×C₃×D₇	—	2	1	no	—
142	C₂×C₃×D₇	—	2	1	no	—
143	C₂×C₃×D₇	C₂×Q₈	1	2	yes	—
144	C₃×C₄×C₇	—	2	1	no	—
145	C₃×C₄×C₇	—	2	1	no	—
146	C₃×C₄×C₇	C₂×Q₈	1	2	yes	—
147	Dic₂₁	—	2	1	no	—
148	Dic₂₁	—	2	1	no	—
149	Dic₂₁	—	2	1	no	—
150	Dic₂₁	#41	1	2	yes	—
151	C₃×Dic₇	—	2	1	no	—
152	C₃×Dic₇	—	2	1	no	—
153	C₃×Dic₇	C₂×Q₈	1	2	yes	—
154	C₇×Dic₃	—	2	1	no	—
155	C₇×Dic₃	—	2	1	no	—
156	C₇×Dic₃	—	2	1	no	—
157	C₇×Dic₃	#41	1	2	yes	—
158	#533	—	21	1	no	Sylow
159	C₂²×D₇	C₂²×D₃	1	1	yes	characteristic
160	C₇×Q₈	—	6	2	no	—
161	C₂×C₄×C₇	—	3	1	no	—
162	C₂×C₄×C₇	—	3	1	no	—
163	C₂×C₄×C₇	—	3	1	no	—
164	C₂×C₄×C₇	—	3	1	no	—
165	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
166	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
167	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
168	C₂×Dic₇	—	3	1	no	—
169	C₂×Dic₇	—	3	1	no	—
170	C₂×Dic₇	—	3	1	no	—
171	C₂×Dic₇	—	3	1	no	—
172	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
173	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
174	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
175	C₄×D₇	—	6	2	no	—
176	C₄×D₇	—	6	2	no	—
177	C₄×D₇	Dic₆	1	4	yes	complemented
178	Dic₁₄	—	6	2	no	—
179	Dic₁₄	—	6	2	no	—
180	Dic₁₄	—	6	2	no	—
181	C₂²×C₃×C₄	—	7	1	no	—
182	C₂²×Dic₃	—	7	1	no	—
183	C₂²×Dic₃	—	7	1	no	—
184	C₂×Dic₆	—	7	1	no	—
185	C₂×Dic₆	—	7	1	no	—
186	C₂×Dic₆	—	7	1	no	—
187	C₂×Dic₆	C₂×D₇	1	1	yes	characteristic
188	C₃×C₄⋊C₄	—	7	1	no	—
189	C₃×C₄⋊C₄	—	7	1	no	—
190	C₃×C₂²⋊C₄	—	7	1	no	—
191	C₃×C₂²⋊C₄	—	7	1	no	—
192	C₃×C₄²	—	7	1	no	—
193	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
194	#217	—	7	1	no	—
195	#217	—	7	1	no	—
196	C₄⋊Dic₃	—	7	1	no	—
197	C₄⋊Dic₃	—	7	1	no	—
198	C₄⋊Dic₃	—	7	1	no	—
199	C₄⋊Dic₃	—	7	1	no	—
200	C₄⋊Dic₃	—	7	1	no	—
201	C₄⋊Dic₃	C₂×D₇	1	1	yes	characteristic
202	Dic₃⋊C₄	—	7	1	no	—
203	Dic₃⋊C₄	—	7	1	no	—
204	Dic₃⋊C₄	—	7	1	no	—
205	Dic₃⋊C₄	—	7	1	no	—
206	Dic₃⋊C₄	—	7	1	no	—
207	Dic₃⋊C₄	—	7	1	no	—
208	Dic₃⋊C₄	C₂×D₇	1	1	yes	characteristic
209	Dic₃⋊C₄	C₂×D₇	1	1	yes	characteristic
210	C₄×Dic₃	—	7	1	no	—
211	C₄×Dic₃	—	7	1	no	—
212	C₄×Dic₃	—	7	1	no	—
213	C₄×Dic₃	—	7	1	no	—
214	C₄×Dic₃	C₂×D₇	1	1	yes	characteristic
215	C₄×Dic₃	C₂×D₇	1	1	yes	characteristic
216	C₂×C₃×C₇	C₂²×Q₈	1	1	yes	characteristic
217	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
218	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
219	C₃×D₇	—	2	2	no	—
220	C₄⋊Q₈	—	21	1	no	—
221	C₄⋊Q₈	—	21	1	no	—
222	#125	—	21	1	no	—
223	#125	—	21	1	no	—
224	C₂²⋊Q₈	—	21	1	no	—
225	C₂²⋊Q₈	—	21	1	no	—
226	C₂²⋊Q₈	—	21	1	no	—
227	C₂²⋊Q₈	—	21	1	no	—
228	C₄×Q₈	—	3	1	no	—
229	C₄×Q₈	—	21	1	no	—
230	C₄×Q₈	—	21	1	no	—
231	C₄×Q₈	—	21	1	no	—
232	#117	—	7	1	no	—
233	#117	—	21	1	no	—
234	C₂×C₄²	—	21	1	no	—
235	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
236	C₂×D₇	—	2	1	no	—
237	C₂×D₇	—	2	1	no	—
238	C₂×D₇	C₂×Dic₆	1	2	yes	—
239	C₄×C₇	—	2	1	no	—
240	C₄×C₇	—	2	1	no	—
241	C₄×C₇	—	3	2	no	—
242	C₄×C₇	—	6	1	no	—
243	C₄×C₇	—	6	1	no	—
244	C₄×C₇	—	6	1	no	—
245	C₄×C₇	C₂×Dic₆	1	2	yes	—
246	Dic₇	—	2	1	no	—
247	Dic₇	—	2	1	no	—
248	Dic₇	—	3	2	no	—
249	Dic₇	—	6	1	no	—
250	Dic₇	—	6	1	no	—
251	Dic₇	—	6	1	no	—
252	Dic₇	C₂×Dic₆	1	2	yes	—
253	C₂³×C₃	—	7	1	no	—
254	C₂×C₃×C₄	—	7	1	no	—
255	C₂×C₃×C₄	—	7	4	no	—
256	C₂×C₃×C₄	—	7	1	no	—
257	C₂×C₃×C₄	—	7	1	no	—
258	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
259	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
260	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
261	C₂×Dic₃	—	7	1	no	—
262	C₂×Dic₃	—	7	1	no	—
263	C₂×Dic₃	—	7	1	no	—
264	C₂×Dic₃	—	7	1	no	—
265	C₂×Dic₃	—	14	2	no	—
266	C₂×Dic₃	—	14	2	no	—
267	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
268	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
269	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
270	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
271	Dic₆	—	2	2	no	—
272	Dic₆	—	14	2	no	—
273	Dic₆	—	14	2	no	—
274	Dic₆	—	14	2	no	—
275	C₃×C₇	#533	1	1	yes	characteristic, complemented, Hall
276	C₂×Q₈	—	3	1	no	—
277	C₂×Q₈	—	21	1	no	—
278	C₂×Q₈	—	21	1	no	—
279	C₂×Q₈	—	21	1	no	—
280	C₂²×C₄	—	7	1	no	—
281	C₂²×C₄	—	21	1	no	—
282	C₂²×C₄	—	21	1	no	—
283	C₄⋊C₄	—	3	1	no	—
284	C₄⋊C₄	—	3	1	no	—
285	C₄⋊C₄	—	3	1	no	—
286	C₄⋊C₄	—	7	1	no	—
287	C₄⋊C₄	—	7	1	no	—
288	C₄⋊C₄	—	21	1	no	—
289	C₄⋊C₄	—	21	1	no	—
290	C₄⋊C₄	—	21	1	no	—
291	C₄⋊C₄	—	21	1	no	—
292	C₄⋊C₄	—	21	1	no	—
293	C₄⋊C₄	—	21	1	no	—
294	C₄⋊C₄	—	21	1	no	—
295	C₄⋊C₄	—	21	1	no	—
296	C₄⋊C₄	—	21	1	no	—
297	C₄⋊C₄	—	21	1	no	—
298	C₄⋊C₄	—	21	1	no	—
299	C₂²⋊C₄	—	7	1	no	—
300	C₂²⋊C₄	—	7	1	no	—
301	C₂²⋊C₄	—	21	1	no	—
302	C₂²⋊C₄	—	21	1	no	—
303	C₄²	—	3	1	no	—
304	C₄²	—	3	1	no	—
305	C₄²	—	7	1	no	—
306	C₄²	—	21	1	no	—
307	C₄²	—	21	1	no	—
308	C₄²	—	21	1	no	—
309	C₄²	—	21	1	no	—
310	C₄²	D₃×D₇	1	1	yes	characteristic
311	C₂×C₇	C₂²×Dic₆	1	1	yes	characteristic
312	C₂×C₇	D₃×#41	1	1	yes	characteristic
313	C₂×C₇	D₃×#41	1	1	yes	characteristic
314	D₇	—	2	2	no	—
315	C₂²×C₃	—	7	2	no	—
316	C₂²×C₃	—	14	1	no	—
317	C₂²×C₃	—	14	1	no	—
318	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
319	C₃×C₄	—	2	1	no	—
320	C₃×C₄	—	2	1	no	—
321	C₃×C₄	—	7	2	no	—
322	C₃×C₄	—	14	1	no	—
323	C₃×C₄	—	14	1	no	—
324	C₃×C₄	Q₈×D₇	1	2	yes	—
325	Dic₃	—	2	1	no	—
326	Dic₃	—	2	1	no	—
327	Dic₃	—	2	1	no	—
328	Dic₃	—	7	2	no	—
329	Dic₃	—	14	1	no	—
330	Dic₃	—	14	1	no	—
331	Dic₃	—	14	1	no	—
332	Dic₃	#1157	1	2	yes	—
333	C₂³	—	7	1	no	—
334	Q₈	—	6	2	no	—
335	Q₈	—	42	2	no	—
336	Q₈	—	42	2	no	—
337	Q₈	—	42	2	no	—
338	C₂×C₄	—	3	1	no	—
339	C₂×C₄	—	3	1	no	—
340	C₂×C₄	—	3	1	no	—
341	C₂×C₄	—	3	1	no	—
342	C₂×C₄	—	7	1	no	—
343	C₂×C₄	—	7	4	no	—
344	C₂×C₄	—	7	1	no	—
345	C₂×C₄	—	7	1	no	—
346	C₂×C₄	—	21	1	no	—
347	C₂×C₄	—	21	1	no	—
348	C₂×C₄	—	21	1	no	—
349	C₂×C₄	—	21	1	no	—
350	C₂×C₄	—	42	2	no	—
351	C₂×C₄	—	42	2	no	—
352	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
353	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
354	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
355	C₇	#5517	1	1	yes	characteristic, complemented, Sylow
356	C₂×C₃	—	14	2	no	—
357	C₂×C₃	C₂×#1157	1	1	yes	characteristic
358	C₂×C₃	C₂×Q₈×D₇	1	1	yes	characteristic
359	C₂×C₃	D₇×#41	1	1	yes	characteristic
360	C₂²	—	7	2	no	—
361	C₂²	—	14	1	no	—
362	C₂²	—	14	1	no	—
363	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	3	2	no	—
367	C₄	—	6	1	no	—
368	C₄	—	6	1	no	—
369	C₄	—	6	1	no	—
370	C₄	—	7	2	no	—
371	C₄	—	14	1	no	—
372	C₄	—	14	1	no	—
373	C₄	—	21	2	no	—
374	C₄	—	42	1	no	—
375	C₄	—	42	1	no	—
376	C₄	—	42	1	no	—
377	C₄	D₇×Dic₆	1	2	yes	—
378	C₃	#90269	1	1	yes	characteristic, complemented, Sylow
379	C₂	—	14	2	no	—
380	C₂	C₂×D₇×Dic₆	1	1	yes	characteristic
381	C₂	D₃×#1157	1	1	yes	characteristic
382	C₂	#10621563	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃×D₇
commutator, socle, cosocle	C₂²×C₃×C₇	C₂⁴
Frattini	C₂²	C₂²×D₃×D₇
Fitting	C₃×C₄²×C₇	C₂²
prime radical	C₂²×C₃×D₇	C₂³
hypercenter	C₄²	D₃×D₇
hypocenter	C₃×C₇	#533
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
1. C₄×Dic₄₂	C₂	semidirect
2. C₄×C₇×Dic₆	C₂	semidirect
4. C₄×#4248	C₂	semidirect
5. C₄×#4248	C₂	semidirect
12. #10620897	C₂	semidirect
13. #10620896	C₂	semidirect
14. #10620891	C₂	semidirect
15. #10620890	C₂	semidirect
85. C₃×C₄×D₇	Q₈	semidirect
139. C₄×Dic₆	D₇	semidirect
177. C₄×D₇	Dic₆	semidirect
275. C₃×C₇	#533	semidirect
355. C₇	#5517	semidirect
378. C₃	#90269	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₄₂	2
C₄×C₇×Dic₆	2
C₃×#6435	2
C₄×#4248	2
#10620993	2
#10620992	2
#10620984	2
#10620983	2
C₄×Dic₃×D₇	2
#10620919	2
#10620897	2
#10620896	2
#10620891	2
#10620890	2
#90269	3
#5517	7