Subgroups — C₃×#90324 #13060965

Order 1344.

580 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×Q₈×D₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#1157	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₇×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6510	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₄×C₂²⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#6459	C₂	1	2	yes	maximal, complemented
7	C₃×#6458	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#6456	C₂	1	2	yes	maximal, complemented
9	C₃×#6450	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#6448	C₂	1	2	yes	maximal, complemented
11	C₃×#6447	C₂	1	2	yes	maximal, complemented
12	#90324	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₂²⋊D₇	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	—
17	C₃×Q₈×D₇	—	2	4	no	—
18	C₃×#1157	—	2	4	no	—
19	C₂×C₃×D₂₈	C₂²	1	2	yes	—
20	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₄×D₇	C₂²	1	2	yes	—
22	C₂×C₃×C₄×D₇	C₂²	1	2	yes	—
23	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
24	C₂×C₃×Dic₁₄	C₂²	1	2	yes	—
25	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic
26	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
28	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₃×C₂²⋊Dic₇	C₂²	1	1	yes	characteristic
30	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
31	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
32	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
33	C₃×(C₂×D₇)⋊C₄	C₂²	1	1	yes	characteristic
34	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
35	C₃×Dic₇⋊C₄	C₂²	1	1	yes	characteristic
36	C₃×C₄×Dic₇	C₂²	1	2	yes	—
37	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
38	C₂×Q₈×D₇	C₂×C₃	1	1	yes	characteristic, complemented
39	C₂×#1157	C₂×C₃	1	1	yes	characteristic, complemented
40	C₇×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
41	#6510	C₂×C₃	1	1	yes	characteristic, complemented
42	C₄×C₂²⋊D₇	C₂×C₃	1	1	yes	characteristic, complemented
43	#6459	C₂×C₃	1	2	yes	complemented
44	#6458	C₂×C₃	1	1	yes	characteristic, complemented
45	#6456	C₂×C₃	1	2	yes	complemented
46	#6450	C₂×C₃	1	1	yes	characteristic, complemented
47	#6448	C₂×C₃	1	2	yes	complemented
48	#6447	C₂×C₃	1	2	yes	complemented
49	C₃×#548	—	7	1	no	maximal, Hall
50	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
51	C₂²×C₃×D₇	C₂³	1	2	yes	—
52	C₂²×C₃×D₇	C₂³	1	1	yes	characteristic
53	C₃×C₇×Q₈	—	2	2	no	—
54	C₂×C₃×C₄×C₇	—	2	2	no	—
55	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
56	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
57	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
58	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
59	C₃×C₂²⋊D₇	—	2	2	no	—
60	C₃×C₂²⋊D₇	—	2	4	no	—
61	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
62	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
63	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
64	C₂×C₃×Dic₇	C₂³	1	2	yes	—
65	C₃×D₂₈	—	2	4	no	—
66	C₃×D₂₈	—	2	2	no	—
67	C₃×C₄×D₇	—	2	2	no	—
68	C₃×C₄×D₇	—	2	4	no	—
69	C₃×C₄×D₇	—	2	4	no	—
70	C₃×Dic₁₄	—	2	4	no	—
71	C₃×Dic₁₄	D₄	1	4	yes	complemented
72	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
73	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
74	C₂×C₂²⋊D₇	C₂²×C₃	1	1	yes	characteristic
75	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
76	Q₈×D₇	—	2	4	no	—
77	#1157	—	2	4	no	—
78	C₂×D₂₈	C₂²×C₃	1	2	yes	—
79	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
80	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
81	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
82	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
83	C₂×Dic₁₄	C₂²×C₃	1	2	yes	—
84	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
85	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
86	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
87	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
88	C₂²⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
89	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
90	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
91	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
92	(C₂×D₇)⋊C₄	C₂²×C₃	1	1	yes	characteristic
93	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
94	Dic₇⋊C₄	C₂²×C₃	1	1	yes	characteristic
95	C₄×Dic₇	C₂²×C₃	1	2	yes	—
96	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
97	C₂×C₃×#41	—	7	1	no	—
98	C₂²×C₃×Q₈	—	7	1	no	—
99	C₃×#124	—	7	1	no	—
100	C₃×#124	—	7	2	no	—
101	C₃×C₂²⋊Q₈	—	7	2	no	—
102	C₃×C₂²⋊Q₈	D₇	1	1	yes	characteristic, complemented
103	C₃×#121	—	7	1	no	—
104	C₃×#121	—	7	2	no	—
105	C₃×C₄×Q₈	—	7	1	no	—
106	C₃×C₄×D₄	—	7	1	no	—
107	C₃×C₄×D₄	—	7	2	no	—
108	C₂²×C₃×C₇	—	2	1	no	—
109	C₂²×C₃×C₇	—	2	1	no	—
110	C₂²×C₃×C₇	—	2	1	no	—
111	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
112	C₂×C₃×D₇	—	2	1	no	—
113	C₂×C₃×D₇	—	2	1	no	—
114	C₂×C₃×D₇	—	2	2	no	—
115	C₂×C₃×D₇	—	2	2	no	—
116	C₂×C₃×D₇	—	2	2	no	—
117	C₂×C₃×D₇	#41	1	2	yes	complemented
118	C₃×C₄×C₇	—	2	2	no	—
119	C₃×C₄×C₇	—	2	1	no	—
120	C₃×C₄×C₇	—	2	2	no	—
121	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
122	C₃×Dic₇	—	2	1	no	—
123	C₃×Dic₇	—	2	1	no	—
124	C₃×Dic₇	—	2	1	no	—
125	C₃×Dic₇	C₂×D₄	1	4	yes	—
126	#548	—	7	1	no	Sylow
127	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
128	C₂²×D₇	C₂³×C₃	1	2	yes	—
129	C₂²×D₇	C₂³×C₃	1	1	yes	characteristic
130	C₇×Q₈	—	2	2	no	—
131	C₂×C₄×C₇	—	2	2	no	—
132	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
133	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
134	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
135	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
136	C₂²⋊D₇	—	2	2	no	—
137	C₂²⋊D₇	—	2	4	no	—
138	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
139	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
140	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
141	C₂×Dic₇	C₂³×C₃	1	2	yes	—
142	D₂₈	—	2	4	no	—
143	D₂₈	—	2	2	no	—
144	C₄×D₇	—	2	2	no	—
145	C₄×D₇	—	2	4	no	—
146	C₄×D₇	—	2	4	no	—
147	Dic₁₄	—	2	4	no	—
148	Dic₁₄	C₃×D₄	1	4	yes	complemented
149	C₃×#41	—	14	4	no	—
150	C₂×C₃×Q₈	—	7	1	no	—
151	C₂×C₃×Q₈	—	7	2	no	—
152	C₂×C₃×Q₈	—	14	4	no	—
153	C₂×C₃×Q₈	C₂×D₇	1	1	yes	characteristic
154	C₂×C₃×D₄	—	7	1	no	—
155	C₂×C₃×D₄	—	7	2	no	—
156	C₂×C₃×D₄	—	7	2	no	—
157	C₂×C₃×D₄	—	7	1	no	—
158	C₂²×C₃×C₄	—	7	2	no	—
159	C₂²×C₃×C₄	—	7	1	no	—
160	C₂²×C₃×C₄	—	7	2	no	—
161	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
162	C₃×C₄⋊C₄	—	7	1	no	—
163	C₃×C₄⋊C₄	—	7	2	no	—
164	C₃×C₄⋊C₄	C₂×D₇	1	2	yes	—
165	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
166	C₃×C₂²⋊C₄	—	7	1	no	—
167	C₃×C₂²⋊C₄	—	7	1	no	—
168	C₃×C₂²⋊C₄	—	7	2	no	—
169	C₃×C₂²⋊C₄	—	7	2	no	—
170	C₃×C₂²⋊C₄	—	7	2	no	—
171	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
172	C₃×C₄²	—	7	2	no	—
173	C₃×C₄²	—	7	1	no	—
174	C₂×C₃×C₇	—	4	1	no	—
175	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
177	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
178	C₃×D₇	—	2	2	no	—
179	C₃×D₇	—	4	2	no	—
180	C₂×#41	—	7	1	no	—
181	C₂²×Q₈	—	7	1	no	—
182	#124	—	7	1	no	—
183	#124	—	7	2	no	—
184	C₂²⋊Q₈	—	7	2	no	—
185	C₂²⋊Q₈	C₃×D₇	1	1	yes	characteristic, complemented
186	#121	—	7	1	no	—
187	#121	—	7	2	no	—
188	C₄×Q₈	—	7	1	no	—
189	C₄×D₄	—	7	1	no	—
190	C₄×D₄	—	7	2	no	—
191	C₂²×C₇	—	2	1	no	—
192	C₂²×C₇	—	2	1	no	—
193	C₂²×C₇	—	2	1	no	—
194	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
195	C₂×D₇	—	2	1	no	—
196	C₂×D₇	—	2	1	no	—
197	C₂×D₇	—	2	2	no	—
198	C₂×D₇	—	2	2	no	—
199	C₂×D₇	—	2	2	no	—
200	C₂×D₇	C₃×#41	1	2	yes	complemented
201	C₄×C₇	—	2	2	no	—
202	C₄×C₇	—	2	1	no	—
203	C₄×C₇	—	2	2	no	—
204	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
205	Dic₇	—	2	1	no	—
206	Dic₇	—	2	1	no	—
207	Dic₇	—	2	1	no	—
208	Dic₇	C₂×C₃×D₄	1	4	yes	—
209	C₂³×C₃	—	7	1	no	—
210	C₂³×C₃	—	7	2	no	—
211	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
212	C₃×Q₈	—	2	2	no	—
213	C₃×Q₈	—	7	4	no	—
214	C₃×Q₈	—	14	4	no	—
215	C₃×D₄	—	14	4	no	—
216	C₃×D₄	—	14	4	no	—
217	C₃×D₄	—	14	2	no	—
218	C₃×D₄	—	14	2	no	—
219	C₂×C₃×C₄	—	2	2	no	—
220	C₂×C₃×C₄	—	7	2	no	—
221	C₂×C₃×C₄	—	7	1	no	—
222	C₂×C₃×C₄	—	7	1	no	—
223	C₂×C₃×C₄	—	7	1	no	—
224	C₂×C₃×C₄	—	14	4	no	—
225	C₂×C₃×C₄	—	14	4	no	—
226	C₂×C₃×C₄	—	14	2	no	—
227	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
228	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
229	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
231	C₃×C₇	#548	1	1	yes	characteristic, complemented, Hall
232	#41	—	14	4	no	—
233	C₂×Q₈	—	7	1	no	—
234	C₂×Q₈	—	7	2	no	—
235	C₂×Q₈	—	14	4	no	—
236	C₂×Q₈	C₂×C₃×D₇	1	1	yes	characteristic
237	C₂×D₄	—	7	1	no	—
238	C₂×D₄	—	7	2	no	—
239	C₂×D₄	—	7	2	no	—
240	C₂×D₄	—	7	1	no	—
241	C₂²×C₄	—	7	2	no	—
242	C₂²×C₄	—	7	1	no	—
243	C₂²×C₄	—	7	2	no	—
244	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
245	C₄⋊C₄	—	7	1	no	—
246	C₄⋊C₄	—	7	2	no	—
247	C₄⋊C₄	C₂×C₃×D₇	1	2	yes	—
248	C₄⋊C₄	C₂×C₃×D₇	1	1	yes	characteristic
249	C₂²⋊C₄	—	7	1	no	—
250	C₂²⋊C₄	—	7	1	no	—
251	C₂²⋊C₄	—	7	2	no	—
252	C₂²⋊C₄	—	7	2	no	—
253	C₂²⋊C₄	—	7	2	no	—
254	C₂²⋊C₄	C₂×C₃×D₇	1	2	yes	—
255	C₄²	—	7	2	no	—
256	C₄²	—	7	1	no	—
257	C₂×C₇	—	4	1	no	—
258	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
259	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
260	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
261	D₇	—	2	2	no	—
262	D₇	—	4	2	no	—
263	C₂²×C₃	—	2	1	no	—
264	C₂²×C₃	—	2	1	no	—
265	C₂²×C₃	—	2	1	no	—
266	C₂²×C₃	—	7	2	no	—
267	C₂²×C₃	—	14	1	no	—
268	C₂²×C₃	—	14	1	no	—
269	C₂²×C₃	—	14	2	no	—
270	C₂²×C₃	—	14	2	no	—
271	C₂²×C₃	—	14	2	no	—
272	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
273	C₃×C₄	—	2	1	no	—
274	C₃×C₄	—	2	2	no	—
275	C₃×C₄	—	2	2	no	—
276	C₃×C₄	—	7	4	no	—
277	C₃×C₄	—	14	1	no	—
278	C₃×C₄	—	14	1	no	—
279	C₃×C₄	—	14	1	no	—
280	C₃×C₄	D₄×D₇	1	2	yes	—
281	C₂³	—	7	1	no	—
282	C₂³	—	7	2	no	—
283	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
284	Q₈	—	2	2	no	—
285	Q₈	—	7	4	no	—
286	Q₈	—	14	4	no	—
287	D₄	—	14	4	no	—
288	D₄	—	14	4	no	—
289	D₄	—	14	2	no	—
290	D₄	—	14	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	7	2	no	—
293	C₂×C₄	—	7	1	no	—
294	C₂×C₄	—	7	1	no	—
295	C₂×C₄	—	7	1	no	—
296	C₂×C₄	—	14	4	no	—
297	C₂×C₄	—	14	4	no	—
298	C₂×C₄	—	14	2	no	—
299	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
300	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
301	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
302	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
303	C₇	C₃×#548	1	1	yes	characteristic, complemented, Sylow
304	C₂×C₃	—	4	1	no	—
305	C₂×C₃	—	14	2	no	—
306	C₂×C₃	—	28	2	no	—
307	C₂×C₃	C₂×D₄×D₇	1	1	yes	characteristic
308	C₂×C₃	#6551	1	1	yes	characteristic
309	C₂×C₃	D₇×#41	1	1	yes	characteristic
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	7	2	no	—
314	C₂²	—	14	1	no	—
315	C₂²	—	14	1	no	—
316	C₂²	—	14	2	no	—
317	C₂²	—	14	2	no	—
318	C₂²	—	14	2	no	—
319	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	2	no	—
322	C₄	—	2	2	no	—
323	C₄	—	7	4	no	—
324	C₄	—	14	1	no	—
325	C₄	—	14	1	no	—
326	C₄	—	14	1	no	—
327	C₄	C₃×D₄×D₇	1	2	yes	—
328	C₃	#90324	1	1	yes	characteristic, direct product factor, Sylow
329	C₂	—	4	1	no	—
330	C₂	—	14	2	no	—
331	C₂	—	28	2	no	—
332	C₂	C₂×C₃×D₄×D₇	1	1	yes	characteristic
333	C₂	C₃×#6551	1	1	yes	characteristic
334	C₂	C₃×D₇×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₇
commutator, cosocle	C₂²×C₇	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₇
Fitting	C₃×C₇×C₂²⋊Q₈	C₂
socle	C₂²×C₃×C₇	C₂⁴
hypercenter	C₃×C₂²⋊Q₈	D₇
hypocenter	C₇	C₃×#548
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×Q₈×D₇	C₂	semidirect
2. C₂×C₃×#1157	C₂	semidirect
3. C₃×C₇×C₂²⋊Q₈	C₂	semidirect
4. C₃×#6510	C₂	semidirect
5. C₃×C₄×C₂²⋊D₇	C₂	semidirect
6. C₃×#6459	C₂	semidirect
7. C₃×#6458	C₂	semidirect
8. C₃×#6456	C₂	semidirect
9. C₃×#6450	C₂	semidirect
10. C₃×#6448	C₂	semidirect
11. C₃×#6447	C₂	semidirect
12. #90324	C₃	direct (N × H)
38. C₂×Q₈×D₇	C₂×C₃	semidirect
39. C₂×#1157	C₂×C₃	semidirect
40. C₇×C₂²⋊Q₈	C₂×C₃	semidirect
41. #6510	C₂×C₃	semidirect
42. C₄×C₂²⋊D₇	C₂×C₃	semidirect
43. #6459	C₂×C₃	semidirect
44. #6458	C₂×C₃	semidirect
45. #6456	C₂×C₃	semidirect
46. #6450	C₂×C₃	semidirect
47. #6448	C₂×C₃	semidirect
48. #6447	C₂×C₃	semidirect
71. C₃×Dic₁₄	D₄	semidirect
102. C₃×C₂²⋊Q₈	D₇	semidirect
117. C₂×C₃×D₇	#41	semidirect
148. Dic₁₄	C₃×D₄	semidirect
185. C₂²⋊Q₈	C₃×D₇	semidirect
200. C₂×D₇	C₃×#41	semidirect
231. C₃×C₇	#548	semidirect
303. C₇	C₃×#548	semidirect
328. C₃	#90324	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×Q₈×D₇	2
C₂×C₃×#1157	2
C₃×C₇×C₂²⋊Q₈	2
C₃×#6510	2
C₃×C₄×C₂²⋊D₇	2
C₃×#6459	2
C₃×#6458	2
C₃×#6456	2
C₃×#6450	2
C₃×#6448	2
C₃×#6447	2
#90324	3
C₃×#548	7

Subgroups — C3×#90324 #13060965

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90324 #13060965