Subgroups — #13372720

Order 1792.

812 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#89796	C₂	1	2	yes	maximal
3	#11727239	C₂	1	4	yes	maximal
4	#11727238	C₂	1	2	yes	maximal
5	#11727146	C₂	1	1	yes	maximal, characteristic
6	#11727146	C₂	1	2	yes	maximal
7	#11727145	C₂	1	2	yes	maximal
8	C₄×C₄⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×C₇×Q₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₇×Q₈	C₄	1	1	yes	characteristic
11	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
12	C₄×C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
13	C₄×C₇×C₄⋊C₄	C₄	1	2	yes	—
14	C₄×C₇×C₄⋊C₄	C₄	1	2	yes	—
15	C₄×C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
16	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
18	C₄³×C₇	C₂²	1	1	yes	characteristic
19	C₄³×C₇	C₂²	1	2	yes	—
20	#89796	—	2	8	no	—
21	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
22	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	4	yes	—
23	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
24	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	4	yes	—
25	C₂×#6377	C₂²	1	4	yes	—
26	C₂×C₄×C₇⋊C₈	C₂²	1	2	yes	—
27	#89319	C₂²	1	1	yes	characteristic
28	#89319	C₂²	1	1	yes	characteristic
29	#89319	C₂²	1	2	yes	—
30	#89319	C₂²	1	4	yes	—
31	#89319	C₂²	1	2	yes	—
32	#22736	—	7	1	no	maximal, Sylow
33	C₂²×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
34	C₄×C₇×Q₈	—	2	4	no	—
35	C₄×C₇×Q₈	—	2	4	no	—
36	C₄×C₇×Q₈	D₄	1	4	yes	—
37	C₄×C₇×Q₈	Q₈	1	4	yes	—
38	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
39	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
41	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
42	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
43	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₇	C₂³	1	2	yes	—
54	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₇	C₂³	1	2	yes	—
57	C₂²×C₇⋊C₈	C₂³	1	2	yes	—
58	C₂²×C₇⋊C₈	C₂³	1	4	yes	—
59	C₂²×C₇⋊C₈	C₂³	1	2	yes	—
60	C₇×#95	C₂×C₄	1	2	yes	—
61	C₇×#95	C₂×C₄	1	2	yes	—
62	C₇×#95	C₂×C₄	1	2	yes	—
63	C₄⋊(C₇⋊C₈)	—	2	8	no	—
64	C₄⋊(C₇⋊C₈)	—	2	4	no	—
65	C₄⋊(C₇⋊C₈)	—	2	4	no	—
66	C₄⋊(C₇⋊C₈)	—	2	8	no	—
67	#6377	—	2	8	no	—
68	C₄×C₇⋊C₈	—	2	4	no	—
69	C₂×#446	—	7	2	no	—
70	C₄²×Q₈	D₇	1	1	yes	characteristic
71	#1940	—	7	4	no	—
72	#1939	—	7	2	no	—
73	#1831	—	7	2	no	—
74	#1831	—	7	1	no	—
75	#1767	—	7	2	no	—
76	C₄×C₄⋊C₈	—	7	1	no	—
77	C₂×C₇×Q₈	—	2	2	no	—
78	C₂×C₇×Q₈	—	2	2	no	—
79	C₂×C₇×Q₈	C₄⋊C₄	1	4	yes	—
80	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
91	C₇×C₄⋊C₄	—	2	4	no	—
92	C₇×C₄⋊C₄	—	2	4	no	—
93	C₇×C₄⋊C₄	—	2	2	no	—
94	C₇×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
95	C₇×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
96	C₇×C₄⋊C₄	M₁₆	1	4	yes	—
97	C₄²×C₇	—	2	4	no	—
98	C₄²×C₇	—	2	2	no	—
99	C₄²×C₇	—	2	4	no	—
100	C₄²×C₇	—	2	2	no	—
101	C₄²×C₇	—	2	4	no	—
102	C₄²×C₇	—	2	2	no	—
103	C₄²×C₇	—	2	4	no	—
104	C₄²×C₇	—	2	2	no	—
105	C₄²×C₇	—	4	2	no	—
106	C₄²×C₇	—	4	1	no	—
107	C₄²×C₇	M₁₆	1	4	yes	—
108	C₄²×C₇	C₂×D₄	1	2	yes	—
109	C₄²×C₇	C₂×D₄	1	4	yes	—
110	C₄²×C₇	C₂×Q₈	1	4	yes	—
111	C₄²×C₇	C₂×Q₈	1	2	yes	—
112	C₂×C₇⋊C₈	—	2	2	no	—
113	C₂×C₇⋊C₈	—	2	2	no	—
114	C₂×C₇⋊C₈	—	2	2	no	—
115	C₂×C₇⋊C₈	—	2	4	no	—
116	C₂×C₇⋊C₈	—	2	2	no	—
117	C₂×C₇⋊C₈	—	2	4	no	—
118	C₂×C₇⋊C₈	—	2	2	no	—
119	C₂×C₇⋊C₈	—	2	4	no	—
120	C₂×C₇⋊C₈	#41	1	4	yes	—
121	C₂×C₄×Q₈	Dic₇	1	1	yes	characteristic
122	C₂×C₄×Q₈	Dic₇	1	1	yes	characteristic
123	C₂×C₄×Q₈	C₂×D₇	1	1	yes	characteristic
124	#446	—	14	8	no	—
125	C₂×C₄⋊C₈	—	7	2	no	—
126	C₂×C₄⋊C₈	—	7	4	no	—
127	C₂×C₄⋊C₈	—	7	4	no	—
128	C₂×C₄⋊C₈	—	7	2	no	—
129	C₂×#97	—	7	4	no	—
130	C₂×C₄×C₈	—	7	2	no	—
131	C₄×C₄⋊C₄	Dic₇	1	1	yes	characteristic
132	C₄×C₄⋊C₄	Dic₇	1	2	yes	—
133	C₄×C₄⋊C₄	Dic₇	1	2	yes	—
134	C₄×C₄⋊C₄	Dic₇	1	1	yes	characteristic
135	C₄×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
136	C₄×C₄⋊C₄	C₂×D₇	1	2	yes	—
137	C₄³	C₂×D₇	1	1	yes	characteristic
138	C₄³	C₂×D₇	1	2	yes	—
139	#336	—	7	1	no	—
140	#336	—	7	2	no	—
141	#336	—	7	4	no	—
142	#336	—	7	2	no	—
143	#336	—	7	1	no	—
144	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
145	C₇×Q₈	—	2	8	no	—
146	C₂×C₄×C₇	—	2	1	no	—
147	C₂×C₄×C₇	—	2	2	no	—
148	C₂×C₄×C₇	—	2	1	no	—
149	C₂×C₄×C₇	—	2	1	no	—
150	C₂×C₄×C₇	—	2	1	no	—
151	C₂×C₄×C₇	—	2	2	no	—
152	C₂×C₄×C₇	—	2	2	no	—
153	C₂×C₄×C₇	—	2	1	no	—
154	C₂×C₄×C₇	—	2	2	no	—
155	C₂×C₄×C₇	—	2	1	no	—
156	C₂×C₄×C₇	—	2	1	no	—
157	C₂×C₄×C₇	—	2	2	no	—
158	C₂×C₄×C₇	—	2	1	no	—
159	C₂×C₄×C₇	—	2	2	no	—
160	C₂×C₄×C₇	—	2	2	no	—
161	C₂×C₄×C₇	—	2	1	no	—
162	C₂×C₄×C₇	—	2	2	no	—
163	C₂×C₄×C₇	—	2	1	no	—
164	C₂×C₄×C₇	—	2	2	no	—
165	C₂×C₄×C₇	—	2	1	no	—
166	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
167	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
168	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
169	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
170	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
171	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
172	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
173	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
174	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
175	C₂×C₄×C₇	#131	1	2	yes	—
176	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
178	C₂×C₄×C₇	C₂×#41	1	2	yes	—
179	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
180	C₇⋊C₈	—	2	4	no	—
181	C₇⋊C₈	—	4	2	no	—
182	C₇⋊C₈	—	4	4	no	—
183	C₂²×Q₈	C₂×Dic₇	1	1	yes	characteristic
184	C₂²×C₈	—	7	2	no	—
185	C₂²×C₈	—	7	2	no	—
186	C₂²×C₈	—	7	4	no	—
187	C₄×Q₈	—	2	4	no	—
188	C₄×Q₈	—	2	4	no	—
189	C₄×Q₈	Dic₁₄	1	4	yes	—
190	C₄×Q₈	D₂₈	1	4	yes	—
191	C₂×C₄⋊C₄	C₂×Dic₇	1	1	yes	characteristic
192	C₂×C₄⋊C₄	C₂×Dic₇	1	2	yes	—
193	C₂×C₄⋊C₄	C₂×Dic₇	1	2	yes	—
194	C₂×C₄⋊C₄	C₂×Dic₇	1	1	yes	characteristic
195	C₂×C₄⋊C₄	C₂×Dic₇	1	2	yes	—
196	C₂×C₄⋊C₄	C₂×Dic₇	1	1	yes	characteristic
197	C₂×C₄²	C₂×Dic₇	1	2	yes	—
198	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
199	C₂×C₄²	C₂×Dic₇	1	2	yes	—
200	C₂×C₄²	C₂×Dic₇	1	2	yes	—
201	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
202	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
203	C₂×C₄²	C₂×Dic₇	1	2	yes	—
204	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
205	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
206	C₂×C₄²	C₂²×D₇	1	2	yes	—
207	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
208	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
209	C₂×C₄²	C₂²×D₇	1	2	yes	—
210	C₄⋊C₈	—	14	8	no	—
211	C₄⋊C₈	—	14	4	no	—
212	C₄⋊C₈	—	14	4	no	—
213	C₄⋊C₈	—	14	8	no	—
214	#97	—	14	8	no	—
215	C₄×C₈	—	14	4	no	—
216	#95	C₂×Dic₇	1	2	yes	—
217	#95	C₂×Dic₇	1	2	yes	—
218	#95	C₂×Dic₇	1	2	yes	—
219	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
220	C₂²×C₇	C₄×#41	1	1	yes	characteristic
221	C₂²×C₇	#519	1	1	yes	characteristic
222	C₂²×C₇	C₂²×M₁₆	1	1	yes	characteristic
223	C₂²×C₇	C₂²×M₁₆	1	1	yes	characteristic
224	C₂²×C₇	C₂×#131	1	1	yes	characteristic
225	C₂²×C₇	#568	1	1	yes	characteristic
226	C₄×C₇	—	2	4	no	—
227	C₄×C₇	—	2	2	no	—
228	C₄×C₇	—	2	4	no	—
229	C₄×C₇	—	2	2	no	—
230	C₄×C₇	—	2	4	no	—
231	C₄×C₇	—	2	2	no	—
232	C₄×C₇	—	2	2	no	—
233	C₄×C₇	—	4	2	no	—
234	C₄×C₇	—	4	1	no	—
235	C₄×C₇	#423	1	4	yes	—
236	C₄×C₇	#548	1	2	yes	—
237	C₄×C₇	#557	1	2	yes	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	C₄⋊Dic₇	1	4	yes	—
241	C₂²×C₄	C₂²×Dic₇	1	2	yes	—
242	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
243	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
244	C₂²×C₄	C₂²×Dic₇	1	2	yes	—
245	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
246	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
247	C₂²×C₄	C₂²×Dic₇	1	2	yes	—
248	C₂²×C₄	C₂²×Dic₇	1	2	yes	—
249	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
250	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
251	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
252	C₂×C₈	—	7	4	no	—
253	C₂×C₈	—	14	2	no	—
254	C₂×C₈	—	14	2	no	—
255	C₂×C₈	—	14	2	no	—
256	C₂×C₈	—	14	4	no	—
257	C₂×C₈	—	14	2	no	—
258	C₂×C₈	—	14	4	no	—
259	C₂×C₈	—	14	2	no	—
260	C₂×C₈	—	14	4	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	#1136	1	4	yes	—
265	C₄⋊C₄	C₄⋊Dic₇	1	8	yes	—
266	C₄⋊C₄	C₄⋊Dic₇	1	4	yes	—
267	C₄²	—	2	4	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	4	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	4	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	4	no	—
275	C₄²	—	4	2	no	—
276	C₄²	—	4	1	no	—
277	C₄²	#1136	1	4	yes	—
278	C₄²	C₂×Dic₁₄	1	4	yes	—
279	C₄²	C₂×Dic₁₄	1	2	yes	—
280	C₄²	C₂×D₂₈	1	2	yes	—
281	C₄²	C₂×D₂₈	1	4	yes	—
282	C₂×C₇	#2351	1	1	yes	characteristic
283	C₂×C₇	C₂×#423	1	1	yes	characteristic
284	C₂×C₇	#2906	1	1	yes	characteristic
285	C₂×C₇	#2965	1	1	yes	characteristic
286	C₂×C₇	#2970	1	1	yes	characteristic
287	C₂×C₇	#2983	1	1	yes	characteristic
288	C₂×C₇	#2984	1	1	yes	characteristic
289	C₂³	C₂³×Dic₇	1	1	yes	characteristic
290	Q₈	—	2	8	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	C₂×#1136	1	2	yes	—
312	C₂×C₄	C₂×#1136	1	4	yes	—
313	C₂×C₄	C₂×#1136	1	2	yes	—
314	C₂×C₄	C₂×#1136	1	2	yes	—
315	C₂×C₄	C₂×#1136	1	2	yes	—
316	C₂×C₄	C₂×C₄⋊Dic₇	1	2	yes	—
317	C₂×C₄	C₂×C₄⋊Dic₇	1	4	yes	—
318	C₂×C₄	C₂×C₄⋊Dic₇	1	4	yes	—
319	C₂×C₄	C₂×C₄⋊Dic₇	1	2	yes	—
320	C₂×C₄	#6511	1	2	yes	—
321	C₂×C₄	C₂²×Dic₁₄	1	1	yes	characteristic
322	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
323	C₂×C₄	#6551	1	2	yes	—
324	C₂×C₄	D₇×#41	1	2	yes	—
325	C₈	—	14	4	no	—
326	C₈	—	28	2	no	—
327	C₈	—	28	4	no	—
328	C₇	#22736	1	1	yes	characteristic, complemented, Sylow
329	C₂²	C₂²×#1136	1	1	yes	characteristic
330	C₂²	C₂²×#1136	1	1	yes	characteristic
331	C₂²	C₂²×C₄⋊Dic₇	1	1	yes	characteristic
332	C₂²	#90503	1	1	yes	characteristic
333	C₂²	C₂×#6511	1	1	yes	characteristic
334	C₂²	#90510	1	1	yes	characteristic
335	C₂²	#41×Dic₇	1	1	yes	characteristic
336	C₄	—	2	2	no	—
337	C₄	—	2	4	no	—
338	C₄	—	2	2	no	—
339	C₄	—	2	4	no	—
340	C₄	—	2	4	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	2	no	—
343	C₄	—	4	2	no	—
344	C₄	—	4	1	no	—
345	C₄	#89696	1	4	yes	—
346	C₄	#90260	1	2	yes	—
347	C₄	#90267	1	2	yes	—
348	C₂	C₂×#89696	1	1	yes	characteristic
349	C₂	#11739881	1	1	yes	characteristic
350	C₂	#11739882	1	1	yes	characteristic
351	C₂	#11739905	1	1	yes	characteristic
352	C₂	#11739957	1	1	yes	characteristic
353	C₂	#11739962	1	1	yes	characteristic
354	C₂	#11739963	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇
commutator, socle, prime radical	C₂³×C₇	C₂³×C₄
Fitting	C₄²×C₇×Q₈	C₂
cosocle	C₂²×C₄×C₇	C₂⁴
hypercenter	C₄²×Q₈	D₇
hypocenter	C₇	#22736
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
328. C₇	#22736	semidirect

Maximal subgroups

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

subgroup	index
C₄²×C₇×Q₈	2
C₂×#89796	2
#11727239	2
#11727238	2
#11727146	2
#11727145	2
C₄×C₄⋊(C₇⋊C₈)	2
#22736	7