Subgroups — C₂²×C₇×#443 #13237393

Order 1792.

2750 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×#135	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇×#131	C₂	1	2	yes	maximal, complemented
3	C₂×C₇×#443	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₇×#108	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₇×#104	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₇×#41	C₂²	1	2	yes	—
8	C₂×C₇×#135	C₄	1	12	yes	complemented
9	C₂×C₇×#135	C₂²	1	12	yes	—
10	C₂²×C₇×Q₁₆	C₄	1	1	yes	characteristic, complemented
11	C₂²×C₇×SD₁₆	C₄	1	2	yes	complemented
12	C₂²×C₇×D₈	C₄	1	1	yes	characteristic, complemented
13	C₂×C₇×#131	—	2	12	no	—
14	C₂×C₇×#131	C₂²	1	24	yes	complemented
15	C₂²×C₇×M₁₆	—	2	2	no	—
16	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
17	C₂³×C₇×C₈	—	2	2	no	—
18	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
19	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
21	C₇×#443	C₂²	1	64	yes	direct product factor
22	C₂×C₇×#108	C₂²	1	12	yes	complemented
23	C₂×C₇×#104	C₂²	1	24	yes	complemented
24	C₂×C₄×C₇×C₈	C₂²	1	12	yes	complemented
25	C₂²×#443	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂×C₇×#41	—	2	6	no	—
27	C₂×C₇×#41	—	2	6	no	—
28	C₂×C₇×#41	C₂×C₄	1	12	yes	—
29	C₂×C₇×#41	C₂³	1	12	yes	—
30	C₂²×C₇×Q₈	C₂×C₄	1	2	yes	—
31	C₂²×C₇×D₄	—	2	2	no	—
32	C₂²×C₇×D₄	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₇	—	2	1	no	—
34	C₂³×C₄×C₇	—	2	2	no	—
35	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
36	C₇×#135	C₂×C₄	1	48	yes	complemented
37	C₇×#135	C₂³	1	16	yes	—
38	C₂×C₇×Q₁₆	C₂×C₄	1	12	yes	complemented
39	C₂×C₇×SD₁₆	C₂×C₄	1	24	yes	complemented
40	C₂×C₇×D₈	C₂×C₄	1	12	yes	complemented
41	C₇×#131	—	2	48	no	—
42	C₇×#131	C₂³	1	32	yes	complemented
43	C₂×C₇×M₁₆	—	2	6	no	—
44	C₂×C₇×M₁₆	—	2	24	no	—
45	C₂×C₇×M₁₆	C₂³	1	12	yes	—
46	C₂²×C₇×C₈	—	2	2	no	—
47	C₂²×C₇×C₈	—	2	24	no	—
48	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₇×C₈	C₂×C₄	1	6	yes	—
51	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
52	C₂²×C₇×C₈	D₄	1	6	yes	—
53	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
54	C₂²×C₇×C₈	C₂³	1	6	yes	—
55	C₂²×C₇×C₈	C₂³	1	6	yes	—
56	C₂×C₄²×C₇	—	2	3	no	—
57	C₂×C₄²×C₇	C₂³	1	6	yes	—
58	C₇×#108	C₂³	1	16	yes	complemented
59	C₇×#104	C₂³	1	32	yes	complemented
60	C₄×C₇×C₈	C₂³	1	16	yes	complemented
61	C₂²×#135	C₂×C₇	1	1	yes	characteristic
62	C₂²×#131	C₂×C₇	1	2	yes	complemented
63	C₂×#443	C₂×C₇	1	24	yes	direct product factor
64	C₂²×#108	C₂×C₇	1	1	yes	characteristic, complemented
65	C₂²×#104	C₂×C₇	1	2	yes	complemented
66	C₂²×C₄×C₈	C₂×C₇	1	1	yes	characteristic, complemented
67	C₂⁴×C₇	—	2	2	no	—
68	C₂⁴×C₇	#41	1	1	yes	characteristic
69	C₇×#41	—	2	24	no	—
70	C₇×#41	—	2	12	no	—
71	C₇×#41	—	2	12	no	—
72	C₇×#41	C₂²×C₄	1	24	yes	—
73	C₇×#41	C₂⁴	1	8	yes	—
74	C₂×C₇×Q₈	—	2	6	no	—
75	C₂×C₇×Q₈	C₂²×C₄	1	12	yes	—
76	C₂×C₇×D₄	—	2	12	no	—
77	C₂×C₇×D₄	—	2	12	no	—
78	C₂×C₇×D₄	—	2	6	no	—
79	C₂×C₇×D₄	C₂²×C₄	1	12	yes	—
80	C₂²×C₄×C₇	—	2	8	no	—
81	C₂²×C₄×C₇	—	2	2	no	—
82	C₂²×C₄×C₇	—	2	6	no	—
83	C₂²×C₄×C₇	—	2	12	no	—
84	C₂²×C₄×C₇	—	2	12	no	—
85	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
87	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
89	C₂²×C₄×C₇	#41	1	3	yes	—
90	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
91	C₇×Q₁₆	C₂²×C₄	1	16	yes	complemented
92	C₇×SD₁₆	C₂²×C₄	1	32	yes	complemented
93	C₇×D₈	C₂²×C₄	1	16	yes	complemented
94	C₇×M₁₆	—	2	12	no	—
95	C₇×M₁₆	—	2	32	no	—
96	C₇×M₁₆	C₂⁴	1	8	yes	—
97	C₂×C₇×C₈	—	2	12	no	—
98	C₂×C₇×C₈	—	2	32	no	—
99	C₂×C₇×C₈	C₂²×C₄	1	6	yes	—
100	C₂×C₇×C₈	C₂²×C₄	1	6	yes	—
101	C₂×C₇×C₈	C₂²×C₄	1	12	yes	—
102	C₂×C₇×C₈	C₂×D₄	1	6	yes	—
103	C₂×C₇×C₈	C₂×D₄	1	6	yes	—
104	C₂×C₇×C₈	C₂×D₄	1	12	yes	—
105	C₂×C₇×C₈	C₂⁴	1	4	yes	—
106	C₂×C₇×C₈	C₂⁴	1	4	yes	—
107	C₄²×C₇	—	2	6	no	—
108	C₄²×C₇	C₂⁴	1	4	yes	—
109	C₂²×#41	C₂²×C₇	1	2	yes	—
110	C₂×#135	C₄×C₇	1	12	yes	complemented
111	C₂×#135	C₂²×C₇	1	12	yes	—
112	C₂²×Q₁₆	C₄×C₇	1	1	yes	characteristic, complemented
113	C₂²×SD₁₆	C₄×C₇	1	2	yes	complemented
114	C₂²×D₈	C₄×C₇	1	1	yes	characteristic, complemented
115	C₂×#131	—	2	12	no	—
116	C₂×#131	C₂²×C₇	1	24	yes	complemented
117	C₂²×M₁₆	—	2	2	no	—
118	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
119	C₂³×C₈	—	2	2	no	—
120	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
121	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
122	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
123	#443	C₂²×C₇	1	64	yes	direct product factor
124	C₂×#108	C₂²×C₇	1	12	yes	complemented
125	C₂×#104	C₂²×C₇	1	24	yes	complemented
126	C₂×C₄×C₈	C₂²×C₇	1	12	yes	complemented
127	C₂³×C₇	—	2	4	no	—
128	C₂³×C₇	—	2	12	no	—
129	C₂³×C₇	—	4	8	no	—
130	C₂³×C₇	C₄×D₄	1	3	yes	—
131	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
132	C₂³×C₇	C₂×#41	1	3	yes	—
133	C₇×Q₈	—	2	12	no	—
134	C₇×Q₈	C₂³×C₄	1	8	yes	—
135	C₇×D₄	—	2	24	no	—
136	C₇×D₄	—	2	8	no	—
137	C₇×D₄	—	2	12	no	—
138	C₇×D₄	C₂³×C₄	1	8	yes	—
139	C₂×C₄×C₇	—	2	24	no	—
140	C₂×C₄×C₇	—	2	12	no	—
141	C₂×C₄×C₇	—	2	4	no	—
142	C₂×C₄×C₇	—	2	24	no	—
143	C₂×C₄×C₇	—	2	8	no	—
144	C₂×C₄×C₇	C₄×D₄	1	3	yes	—
145	C₂×C₄×C₇	C₄×D₄	1	3	yes	—
146	C₂×C₄×C₇	C₄×D₄	1	6	yes	—
147	C₂×C₄×C₇	C₂³×C₄	1	3	yes	—
148	C₂×C₄×C₇	C₂³×C₄	1	3	yes	—
149	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
150	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
151	C₂×C₄×C₇	C₂×#41	1	3	yes	—
152	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
153	C₇×C₈	—	2	8	no	—
154	C₇×C₈	C₂³×C₄	1	4	yes	—
155	C₇×C₈	C₂³×C₄	1	4	yes	—
156	C₇×C₈	C₂²×D₄	1	4	yes	—
157	C₇×C₈	C₂²×D₄	1	4	yes	—
158	C₂×#41	—	2	6	no	—
159	C₂×#41	—	2	6	no	—
160	C₂×#41	C₂×C₄×C₇	1	12	yes	—
161	C₂×#41	C₂³×C₇	1	12	yes	—
162	C₂²×Q₈	C₂×C₄×C₇	1	2	yes	—
163	C₂²×D₄	—	2	2	no	—
164	C₂²×D₄	C₂×C₄×C₇	1	2	yes	—
165	C₂³×C₄	—	2	1	no	—
166	C₂³×C₄	—	2	2	no	—
167	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
168	#135	C₂×C₄×C₇	1	48	yes	complemented
169	#135	C₂³×C₇	1	16	yes	—
170	C₂×Q₁₆	C₂×C₄×C₇	1	12	yes	complemented
171	C₂×SD₁₆	C₂×C₄×C₇	1	24	yes	complemented
172	C₂×D₈	C₂×C₄×C₇	1	12	yes	complemented
173	#131	—	2	48	no	—
174	#131	C₂³×C₇	1	32	yes	complemented
175	C₂×M₁₆	—	2	6	no	—
176	C₂×M₁₆	—	2	24	no	—
177	C₂×M₁₆	C₂³×C₇	1	12	yes	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	2	24	no	—
180	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
181	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
182	C₂²×C₈	C₂×C₄×C₇	1	6	yes	—
183	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
184	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
185	C₂²×C₈	C₇×D₄	1	6	yes	—
186	C₂²×C₈	C₂³×C₇	1	6	yes	—
187	C₂²×C₈	C₂³×C₇	1	6	yes	—
188	C₂×C₄²	—	2	3	no	—
189	C₂×C₄²	C₂³×C₇	1	6	yes	—
190	#108	C₂³×C₇	1	16	yes	complemented
191	#104	C₂³×C₇	1	32	yes	complemented
192	C₄×C₈	C₂³×C₇	1	16	yes	complemented
193	C₂²×C₇	—	2	6	no	—
194	C₂²×C₇	—	2	6	no	—
195	C₂²×C₇	—	2	8	no	—
196	C₂²×C₇	—	4	24	no	—
197	C₂²×C₇	#443	1	4	yes	direct product factor
198	C₂²×C₇	C₂×C₄×D₄	1	3	yes	—
199	C₂²×C₇	C₂×C₄×D₄	1	3	yes	—
200	C₂²×C₇	C₂²×#41	1	1	yes	characteristic
201	C₄×C₇	—	2	8	no	—
202	C₄×C₇	—	2	8	no	—
203	C₄×C₇	C₂×C₄×D₄	1	3	yes	—
204	C₄×C₇	C₂×C₄×D₄	1	3	yes	—
205	C₄×C₇	C₂⁴×C₄	1	1	yes	characteristic
206	C₄×C₇	C₂³×D₄	1	1	yes	characteristic
207	C₂⁴	—	2	2	no	—
208	C₂⁴	C₇×#41	1	1	yes	characteristic
209	#41	—	2	24	no	—
210	#41	—	2	12	no	—
211	#41	—	2	12	no	—
212	#41	C₂²×C₄×C₇	1	24	yes	—
213	#41	C₂⁴×C₇	1	8	yes	—
214	C₂×Q₈	—	2	6	no	—
215	C₂×Q₈	C₂²×C₄×C₇	1	12	yes	—
216	C₂×D₄	—	2	12	no	—
217	C₂×D₄	—	2	12	no	—
218	C₂×D₄	—	2	6	no	—
219	C₂×D₄	C₂²×C₄×C₇	1	12	yes	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	6	no	—
223	C₂²×C₄	—	2	12	no	—
224	C₂²×C₄	—	2	12	no	—
225	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
226	C₂²×C₄	C₂²×C₄×C₇	1	3	yes	—
227	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
229	C₂²×C₄	C₇×#41	1	3	yes	—
230	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
231	Q₁₆	C₂²×C₄×C₇	1	16	yes	complemented
232	SD₁₆	C₂²×C₄×C₇	1	32	yes	complemented
233	D₈	C₂²×C₄×C₇	1	16	yes	complemented
234	M₁₆	—	2	12	no	—
235	M₁₆	—	2	32	no	—
236	M₁₆	C₂⁴×C₇	1	8	yes	—
237	C₂×C₈	—	2	12	no	—
238	C₂×C₈	—	2	32	no	—
239	C₂×C₈	C₂²×C₄×C₇	1	6	yes	—
240	C₂×C₈	C₂²×C₄×C₇	1	6	yes	—
241	C₂×C₈	C₂²×C₄×C₇	1	12	yes	—
242	C₂×C₈	C₂×C₇×D₄	1	6	yes	—
243	C₂×C₈	C₂×C₇×D₄	1	6	yes	—
244	C₂×C₈	C₂×C₇×D₄	1	12	yes	—
245	C₂×C₈	C₂⁴×C₇	1	4	yes	—
246	C₂×C₈	C₂⁴×C₇	1	4	yes	—
247	C₄²	—	2	6	no	—
248	C₄²	C₂⁴×C₇	1	4	yes	—
249	C₂×C₇	—	2	1	no	—
250	C₂×C₇	—	2	3	no	—
251	C₂×C₇	—	4	8	no	—
252	C₂×C₇	C₂×#443	1	6	yes	direct product factor
253	C₂×C₇	C₂²×C₄×D₄	1	1	yes	characteristic
254	C₂³	—	2	4	no	—
255	C₂³	—	2	12	no	—
256	C₂³	—	4	8	no	—
257	C₂³	C₄×C₇×D₄	1	3	yes	—
258	C₂³	C₄×C₇×D₄	1	1	yes	characteristic
259	C₂³	C₂×C₇×#41	1	3	yes	—
260	Q₈	—	2	12	no	—
261	Q₈	C₂³×C₄×C₇	1	8	yes	—
262	D₄	—	2	8	no	—
263	D₄	—	2	24	no	—
264	D₄	—	2	12	no	—
265	D₄	C₂³×C₄×C₇	1	8	yes	—
266	C₂×C₄	—	2	24	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	12	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	2	24	no	—
271	C₂×C₄	C₄×C₇×D₄	1	3	yes	—
272	C₂×C₄	C₄×C₇×D₄	1	3	yes	—
273	C₂×C₄	C₄×C₇×D₄	1	6	yes	—
274	C₂×C₄	C₂³×C₄×C₇	1	3	yes	—
275	C₂×C₄	C₂³×C₄×C₇	1	3	yes	—
276	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
277	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
278	C₂×C₄	C₂×C₇×#41	1	3	yes	—
279	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
280	C₈	—	2	8	no	—
281	C₈	C₂³×C₄×C₇	1	4	yes	—
282	C₈	C₂³×C₄×C₇	1	4	yes	—
283	C₈	C₂²×C₇×D₄	1	4	yes	—
284	C₈	C₂²×C₇×D₄	1	4	yes	—
285	C₇	C₂²×#443	1	1	yes	characteristic, direct product factor, Sylow
286	C₂²	—	2	6	no	—
287	C₂²	—	2	6	no	—
288	C₂²	—	2	8	no	—
289	C₂²	—	4	24	no	—
290	C₂²	C₇×#443	1	4	yes	direct product factor
291	C₂²	C₂×C₄×C₇×D₄	1	3	yes	—
292	C₂²	C₂×C₄×C₇×D₄	1	3	yes	—
293	C₂²	C₂²×C₇×#41	1	1	yes	characteristic
294	C₄	—	2	8	no	—
295	C₄	—	2	8	no	—
296	C₄	C₂×C₄×C₇×D₄	1	3	yes	—
297	C₄	C₂×C₄×C₇×D₄	1	3	yes	—
298	C₄	C₂⁴×C₄×C₇	1	1	yes	characteristic
299	C₄	C₂³×C₇×D₄	1	1	yes	characteristic
300	C₂	—	2	1	no	—
301	C₂	—	2	3	no	—
302	C₂	—	4	8	no	—
303	C₂	C₂×C₇×#443	1	6	yes	direct product factor
304	C₂	C₂²×C₄×C₇×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₇×C₈	D₄
commutator	C₄	C₂⁴×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₇
socle	C₂³×C₇	C₄×D₄
prime radical	C₂²×C₇×#135	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇×C₈ ◁ C₂³×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₇×#131	C₂	semidirect
3. C₂×C₇×#443	C₂	direct (N × H)
4. C₂²×C₇×#108	C₂	semidirect
5. C₂²×C₇×#104	C₂	semidirect
6. C₂²×C₄×C₇×C₈	C₂	semidirect
8. C₂×C₇×#135	C₄	semidirect
10. C₂²×C₇×Q₁₆	C₄	semidirect
11. C₂²×C₇×SD₁₆	C₄	semidirect
12. C₂²×C₇×D₈	C₄	semidirect
14. C₂×C₇×#131	C₂²	semidirect
21. C₇×#443	C₂²	direct (N × H)
22. C₂×C₇×#108	C₂²	semidirect
23. C₂×C₇×#104	C₂²	semidirect
24. C₂×C₄×C₇×C₈	C₂²	semidirect
25. C₂²×#443	C₇	direct (N × H)
36. C₇×#135	C₂×C₄	semidirect
38. C₂×C₇×Q₁₆	C₂×C₄	semidirect
39. C₂×C₇×SD₁₆	C₂×C₄	semidirect
40. C₂×C₇×D₈	C₂×C₄	semidirect
42. C₇×#131	C₂³	semidirect
58. C₇×#108	C₂³	semidirect
59. C₇×#104	C₂³	semidirect
60. C₄×C₇×C₈	C₂³	semidirect
62. C₂²×#131	C₂×C₇	semidirect
63. C₂×#443	C₂×C₇	direct (N × H)
64. C₂²×#108	C₂×C₇	semidirect
65. C₂²×#104	C₂×C₇	semidirect
66. C₂²×C₄×C₈	C₂×C₇	semidirect
91. C₇×Q₁₆	C₂²×C₄	semidirect
92. C₇×SD₁₆	C₂²×C₄	semidirect
93. C₇×D₈	C₂²×C₄	semidirect
110. C₂×#135	C₄×C₇	semidirect
112. C₂²×Q₁₆	C₄×C₇	semidirect
113. C₂²×SD₁₆	C₄×C₇	semidirect
114. C₂²×D₈	C₄×C₇	semidirect
116. C₂×#131	C₂²×C₇	semidirect
123. #443	C₂²×C₇	direct (N × H)
124. C₂×#108	C₂²×C₇	semidirect
125. C₂×#104	C₂²×C₇	semidirect
126. C₂×C₄×C₈	C₂²×C₇	semidirect
168. #135	C₂×C₄×C₇	semidirect
170. C₂×Q₁₆	C₂×C₄×C₇	semidirect
171. C₂×SD₁₆	C₂×C₄×C₇	semidirect
172. C₂×D₈	C₂×C₄×C₇	semidirect
174. #131	C₂³×C₇	semidirect
190. #108	C₂³×C₇	semidirect
191. #104	C₂³×C₇	semidirect
192. C₄×C₈	C₂³×C₇	semidirect
197. C₂²×C₇	#443	direct (N × H)
231. Q₁₆	C₂²×C₄×C₇	semidirect
232. SD₁₆	C₂²×C₄×C₇	semidirect
233. D₈	C₂²×C₄×C₇	semidirect
252. C₂×C₇	C₂×#443	direct (N × H)
285. C₇	C₂²×#443	direct (N × H)
290. C₂²	C₇×#443	direct (N × H)
303. C₂	C₂×C₇×#443	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₇×#135	2
C₂²×C₇×#131	2
C₂×C₇×#443	2
C₂²×C₇×#108	2
C₂²×C₇×#104	2
C₂²×C₄×C₇×C₈	2
C₂²×#443	7

Subgroups — C22×C7×#443 #13237393

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×#443 #13237393