Subgroups — C₂²×C₇×#408 #13237277

Order 1792.

5326 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇×#41	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂⁴×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₇×#408	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₇×C₂²⋊C₈	C₂	1	4	yes	maximal, complemented
6	C₂²×C₇×#41	—	2	8	no	—
7	C₂²×C₇×#41	—	2	8	no	—
8	C₂²×C₇×#41	C₄	1	12	yes	—
9	C₂²×C₇×#41	C₂²	1	12	yes	—
10	C₂³×C₇×Q₈	C₄	1	1	yes	characteristic
11	C₂³×C₇×D₄	C₄	1	2	yes	—
12	C₂³×C₇×D₄	C₄	1	1	yes	characteristic
13	C₂⁴×C₄×C₇	C₂²	1	2	yes	—
14	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
15	C₂²×C₇×M₁₆	—	2	8	no	—
16	C₂²×C₇×M₁₆	C₂²	1	12	yes	complemented
17	C₂³×C₇×C₈	—	2	8	no	—
18	C₂³×C₇×C₈	C₂²	1	2	yes	—
19	C₂³×C₇×C₈	C₂²	1	2	yes	—
20	C₂³×C₇×C₈	C₂²	1	12	yes	complemented
21	C₇×#408	C₂²	1	64	yes	direct product factor
22	C₂×C₇×C₂²⋊C₈	C₂²	1	48	yes	complemented
23	C₂²×#408	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂⁵×C₇	C₂×C₄	1	2	yes	—
25	C₂⁵×C₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₇×#41	—	2	96	no	—
27	C₂×C₇×#41	—	2	48	no	—
28	C₂×C₇×#41	—	2	48	no	—
29	C₂×C₇×#41	C₂×C₄	1	48	yes	—
30	C₂×C₇×#41	C₂³	1	16	yes	—
31	C₂²×C₇×Q₈	—	2	8	no	—
32	C₂²×C₇×Q₈	C₂×C₄	1	12	yes	—
33	C₂²×C₇×D₄	—	2	16	no	—
34	C₂²×C₇×D₄	—	2	8	no	—
35	C₂²×C₇×D₄	C₂×C₄	1	24	yes	—
36	C₂²×C₇×D₄	C₂×C₄	1	12	yes	—
37	C₂³×C₄×C₇	—	2	8	no	—
38	C₂³×C₄×C₇	—	2	8	no	—
39	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
40	C₂³×C₄×C₇	C₂×C₄	1	12	yes	—
41	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₇	C₂×C₄	1	6	yes	—
43	C₂³×C₄×C₇	D₄	1	8	yes	—
44	C₂³×C₄×C₇	D₄	1	8	yes	—
45	C₂³×C₄×C₇	C₂³	1	12	yes	—
46	C₂³×C₄×C₇	C₂³	1	6	yes	—
47	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
48	C₂×C₇×M₁₆	—	2	48	no	—
49	C₂×C₇×M₁₆	C₂³	1	16	yes	complemented
50	C₂²×C₇×C₈	—	2	8	no	—
51	C₂²×C₇×C₈	—	2	48	no	—
52	C₂²×C₇×C₈	—	2	8	no	—
53	C₂²×C₇×C₈	C₂³	1	12	yes	—
54	C₂²×C₇×C₈	C₂³	1	12	yes	—
55	C₂²×C₇×C₈	C₂³	1	16	yes	complemented
56	C₇×C₂²⋊C₈	C₂³	1	64	yes	complemented
57	C₂³×#41	C₂×C₇	1	1	yes	characteristic
58	C₂³×M₁₆	C₂×C₇	1	1	yes	characteristic, complemented
59	C₂⁴×C₈	C₂×C₇	1	1	yes	characteristic, complemented
60	C₂×#408	C₂×C₇	1	24	yes	direct product factor
61	C₂²×C₂²⋊C₈	C₂×C₇	1	4	yes	complemented
62	C₂⁴×C₇	—	2	8	no	—
63	C₂⁴×C₇	—	2	8	no	—
64	C₂⁴×C₇	—	2	16	no	—
65	C₂⁴×C₇	C₂²⋊C₄	1	8	yes	—
66	C₂⁴×C₇	C₂²×C₄	1	12	yes	—
67	C₂⁴×C₇	C₂²×C₄	1	6	yes	—
68	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
69	C₇×#41	—	2	192	no	—
70	C₇×#41	—	2	32	no	—
71	C₇×#41	—	2	32	no	—
72	C₂×C₇×Q₈	—	2	48	no	—
73	C₂×C₇×Q₈	C₂²×C₄	1	16	yes	—
74	C₂×C₇×D₄	—	2	96	no	—
75	C₂×C₇×D₄	—	2	48	no	—
76	C₂×C₇×D₄	C₂²×C₄	1	32	yes	—
77	C₂×C₇×D₄	C₂²×C₄	1	16	yes	—
78	C₂²×C₄×C₇	—	2	8	no	—
79	C₂²×C₄×C₇	—	2	48	no	—
80	C₂²×C₄×C₇	—	2	24	no	—
81	C₂²×C₄×C₇	—	2	24	no	—
82	C₂²×C₄×C₇	C₂²⋊C₄	1	8	yes	—
83	C₂²×C₄×C₇	C₂²⋊C₄	1	48	yes	—
84	C₂²×C₄×C₇	C₂²×C₄	1	12	yes	—
85	C₂²×C₄×C₇	C₂²×C₄	1	24	yes	—
86	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
87	C₂²×C₄×C₇	C₂²×C₄	1	6	yes	—
88	C₂²×C₄×C₇	C₂²×C₄	1	12	yes	—
89	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
90	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
91	C₂²×C₄×C₇	C₂×D₄	1	24	yes	—
92	C₂²×C₄×C₇	C₂×D₄	1	24	yes	—
93	C₂²×C₄×C₇	C₂⁴	1	8	yes	—
94	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
95	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
96	C₇×M₁₆	—	2	32	no	—
97	C₂×C₇×C₈	—	2	24	no	—
98	C₂×C₇×C₈	—	2	32	no	—
99	C₂×C₇×C₈	—	2	24	no	—
100	C₂×C₇×C₈	C₂⁴	1	8	yes	—
101	C₂×C₇×C₈	C₂⁴	1	8	yes	—
102	C₂²×#41	—	2	8	no	—
103	C₂²×#41	—	2	8	no	—
104	C₂²×#41	C₄×C₇	1	12	yes	—
105	C₂²×#41	C₂²×C₇	1	12	yes	—
106	C₂³×Q₈	C₄×C₇	1	1	yes	characteristic
107	C₂³×D₄	C₄×C₇	1	2	yes	—
108	C₂³×D₄	C₄×C₇	1	1	yes	characteristic
109	C₂⁴×C₄	C₂²×C₇	1	2	yes	—
110	C₂⁴×C₄	C₂²×C₇	1	1	yes	characteristic
111	C₂²×M₁₆	—	2	8	no	—
112	C₂²×M₁₆	C₂²×C₇	1	12	yes	complemented
113	C₂³×C₈	—	2	8	no	—
114	C₂³×C₈	C₂²×C₇	1	2	yes	—
115	C₂³×C₈	C₂²×C₇	1	2	yes	—
116	C₂³×C₈	C₂²×C₇	1	12	yes	complemented
117	#408	C₂²×C₇	1	64	yes	direct product factor
118	C₂×C₂²⋊C₈	C₂²×C₇	1	48	yes	complemented
119	C₂³×C₇	—	2	24	no	—
120	C₂³×C₇	—	2	32	no	—
121	C₂³×C₇	—	2	24	no	—
122	C₂³×C₇	—	2	48	no	—
123	C₂³×C₇	—	4	32	no	—
124	C₂³×C₇	C₂×C₂²⋊C₄	1	24	yes	—
125	C₂³×C₇	C₂×C₂²⋊C₄	1	4	yes	—
126	C₂³×C₇	#131	1	8	yes	—
127	C₂³×C₇	C₂³×C₄	1	8	yes	—
128	C₂³×C₇	C₂³×C₄	1	4	yes	—
129	C₂³×C₇	C₂³×C₄	1	3	yes	—
130	C₇×Q₈	—	2	32	no	—
131	C₇×D₄	—	2	64	no	—
132	C₇×D₄	—	2	32	no	—
133	C₂×C₄×C₇	—	2	24	no	—
134	C₂×C₄×C₇	—	2	48	no	—
135	C₂×C₄×C₇	—	2	8	no	—
136	C₂×C₄×C₇	—	2	8	no	—
137	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	12	yes	—
138	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	24	yes	—
139	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	48	yes	—
140	C₂×C₄×C₇	C₂³×C₄	1	8	yes	—
141	C₂×C₄×C₇	C₂³×C₄	1	3	yes	—
142	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
143	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
144	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
145	C₂×C₄×C₇	C₂²×D₄	1	8	yes	—
146	C₂×C₄×C₇	C₂²×D₄	1	8	yes	—
147	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
148	C₇×C₈	—	2	8	no	—
149	C₇×C₈	—	2	8	no	—
150	C₂⁵	C₂×C₄×C₇	1	2	yes	—
151	C₂⁵	C₂×C₄×C₇	1	1	yes	characteristic
152	C₂×#41	—	2	96	no	—
153	C₂×#41	—	2	48	no	—
154	C₂×#41	—	2	48	no	—
155	C₂×#41	C₂×C₄×C₇	1	48	yes	—
156	C₂×#41	C₂³×C₇	1	16	yes	—
157	C₂²×Q₈	—	2	8	no	—
158	C₂²×Q₈	C₂×C₄×C₇	1	12	yes	—
159	C₂²×D₄	—	2	16	no	—
160	C₂²×D₄	—	2	8	no	—
161	C₂²×D₄	C₂×C₄×C₇	1	24	yes	—
162	C₂²×D₄	C₂×C₄×C₇	1	12	yes	—
163	C₂³×C₄	—	2	8	no	—
164	C₂³×C₄	—	2	8	no	—
165	C₂³×C₄	C₂×C₄×C₇	1	2	yes	—
166	C₂³×C₄	C₂×C₄×C₇	1	12	yes	—
167	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
168	C₂³×C₄	C₂×C₄×C₇	1	6	yes	—
169	C₂³×C₄	C₇×D₄	1	8	yes	—
170	C₂³×C₄	C₇×D₄	1	8	yes	—
171	C₂³×C₄	C₂³×C₇	1	12	yes	—
172	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
173	C₂³×C₄	C₂³×C₇	1	6	yes	—
174	C₂×M₁₆	—	2	48	no	—
175	C₂×M₁₆	C₂³×C₇	1	16	yes	complemented
176	C₂²×C₈	—	2	8	no	—
177	C₂²×C₈	—	2	48	no	—
178	C₂²×C₈	—	2	8	no	—
179	C₂²×C₈	C₂³×C₇	1	12	yes	—
180	C₂²×C₈	C₂³×C₇	1	12	yes	—
181	C₂²×C₈	C₂³×C₇	1	16	yes	complemented
182	C₂²⋊C₈	C₂³×C₇	1	64	yes	complemented
183	C₂²×C₇	—	2	48	no	—
184	C₂²×C₇	—	2	8	no	—
185	C₂²×C₇	—	2	8	no	—
186	C₂²×C₇	—	2	16	no	—
187	C₂²×C₇	—	4	48	no	—
188	C₂²×C₇	#408	1	16	yes	direct product factor
189	C₂²×C₇	C₂²×C₂²⋊C₄	1	8	yes	—
190	C₂²×C₇	C₂²×C₂²⋊C₄	1	6	yes	—
191	C₂²×C₇	C₂×#131	1	12	yes	—
192	C₂²×C₇	C₂⁴×C₄	1	1	yes	characteristic
193	C₄×C₇	—	2	8	no	—
194	C₄×C₇	C₂²×C₂²⋊C₄	1	6	yes	—
195	C₄×C₇	C₂²×C₂²⋊C₄	1	8	yes	—
196	C₄×C₇	C₂³×D₄	1	1	yes	characteristic
197	C₄×C₇	C₂³×D₄	1	1	yes	characteristic
198	C₂⁴	—	2	8	no	—
199	C₂⁴	—	2	8	no	—
200	C₂⁴	—	2	16	no	—
201	C₂⁴	C₇×C₂²⋊C₄	1	8	yes	—
202	C₂⁴	C₂²×C₄×C₇	1	12	yes	—
203	C₂⁴	C₂²×C₄×C₇	1	6	yes	—
204	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
205	#41	—	2	192	no	—
206	#41	—	2	32	no	—
207	#41	—	2	32	no	—
208	C₂×Q₈	—	2	48	no	—
209	C₂×Q₈	C₂²×C₄×C₇	1	16	yes	—
210	C₂×D₄	—	2	96	no	—
211	C₂×D₄	—	2	48	no	—
212	C₂×D₄	C₂²×C₄×C₇	1	32	yes	—
213	C₂×D₄	C₂²×C₄×C₇	1	16	yes	—
214	C₂²×C₄	—	2	8	no	—
215	C₂²×C₄	—	2	48	no	—
216	C₂²×C₄	—	2	24	no	—
217	C₂²×C₄	—	2	24	no	—
218	C₂²×C₄	C₇×C₂²⋊C₄	1	8	yes	—
219	C₂²×C₄	C₇×C₂²⋊C₄	1	48	yes	—
220	C₂²×C₄	C₂²×C₄×C₇	1	12	yes	—
221	C₂²×C₄	C₂²×C₄×C₇	1	24	yes	—
222	C₂²×C₄	C₂²×C₄×C₇	1	3	yes	—
223	C₂²×C₄	C₂²×C₄×C₇	1	6	yes	—
224	C₂²×C₄	C₂²×C₄×C₇	1	12	yes	—
225	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
226	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
227	C₂²×C₄	C₂×C₇×D₄	1	24	yes	—
228	C₂²×C₄	C₂×C₇×D₄	1	24	yes	—
229	C₂²×C₄	C₂⁴×C₇	1	8	yes	—
230	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
231	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
232	M₁₆	—	2	32	no	—
233	C₂×C₈	—	2	24	no	—
234	C₂×C₈	—	2	32	no	—
235	C₂×C₈	—	2	24	no	—
236	C₂×C₈	C₂⁴×C₇	1	8	yes	—
237	C₂×C₈	C₂⁴×C₇	1	8	yes	—
238	C₂×C₇	—	2	8	no	—
239	C₂×C₇	—	4	8	no	—
240	C₂×C₇	C₂×#408	1	12	yes	direct product factor
241	C₂×C₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
242	C₂×C₇	C₂²×#131	1	2	yes	—
243	C₂³	—	2	32	no	—
244	C₂³	—	2	24	no	—
245	C₂³	—	2	24	no	—
246	C₂³	—	2	48	no	—
247	C₂³	—	4	32	no	—
248	C₂³	C₂×C₇×C₂²⋊C₄	1	24	yes	—
249	C₂³	C₂×C₇×C₂²⋊C₄	1	4	yes	—
250	C₂³	C₇×#131	1	8	yes	—
251	C₂³	C₂³×C₄×C₇	1	8	yes	—
252	C₂³	C₂³×C₄×C₇	1	4	yes	—
253	C₂³	C₂³×C₄×C₇	1	3	yes	—
254	Q₈	—	2	32	no	—
255	D₄	—	2	64	no	—
256	D₄	—	2	32	no	—
257	C₂×C₄	—	2	24	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	—	2	48	no	—
261	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	12	yes	—
262	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	24	yes	—
263	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	48	yes	—
264	C₂×C₄	C₂³×C₄×C₇	1	8	yes	—
265	C₂×C₄	C₂³×C₄×C₇	1	3	yes	—
266	C₂×C₄	C₂³×C₄×C₇	1	4	yes	—
267	C₂×C₄	C₂²×C₇×D₄	1	6	yes	—
268	C₂×C₄	C₂²×C₇×D₄	1	6	yes	—
269	C₂×C₄	C₂²×C₇×D₄	1	8	yes	—
270	C₂×C₄	C₂²×C₇×D₄	1	8	yes	—
271	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
272	C₈	—	2	8	no	—
273	C₈	—	2	8	no	—
274	C₇	C₂²×#408	1	1	yes	characteristic, direct product factor, Sylow
275	C₂²	—	2	8	no	—
276	C₂²	—	2	48	no	—
277	C₂²	—	2	16	no	—
278	C₂²	—	2	8	no	—
279	C₂²	—	4	48	no	—
280	C₂²	C₇×#408	1	16	yes	direct product factor
281	C₂²	C₂²×C₇×C₂²⋊C₄	1	8	yes	—
282	C₂²	C₂²×C₇×C₂²⋊C₄	1	6	yes	—
283	C₂²	C₂×C₇×#131	1	12	yes	—
284	C₂²	C₂⁴×C₄×C₇	1	1	yes	characteristic
285	C₄	—	2	8	no	—
286	C₄	C₂²×C₇×C₂²⋊C₄	1	6	yes	—
287	C₄	C₂²×C₇×C₂²⋊C₄	1	8	yes	—
288	C₄	C₂³×C₇×D₄	1	1	yes	characteristic
289	C₄	C₂³×C₇×D₄	1	1	yes	characteristic
290	C₂	—	2	8	no	—
291	C₂	—	4	8	no	—
292	C₂	C₂×C₇×#408	1	12	yes	direct product factor
293	C₂	C₂³×C₇×C₂²⋊C₄	1	1	yes	characteristic
294	C₂	C₂²×C₇×#131	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₇	C₂³
commutator	C₂²	C₂⁴×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₇
socle	C₂⁴×C₇	C₂²×C₄
prime radical	C₂³×C₇×#41	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₇×M₁₆	C₂	semidirect
3. C₂⁴×C₇×C₈	C₂	semidirect
4. C₂×C₇×#408	C₂	direct (N × H)
5. C₂²×C₇×C₂²⋊C₈	C₂	semidirect
16. C₂²×C₇×M₁₆	C₂²	semidirect
20. C₂³×C₇×C₈	C₂²	semidirect
21. C₇×#408	C₂²	direct (N × H)
22. C₂×C₇×C₂²⋊C₈	C₂²	semidirect
23. C₂²×#408	C₇	direct (N × H)
49. C₂×C₇×M₁₆	C₂³	semidirect
55. C₂²×C₇×C₈	C₂³	semidirect
56. C₇×C₂²⋊C₈	C₂³	semidirect
58. C₂³×M₁₆	C₂×C₇	semidirect
59. C₂⁴×C₈	C₂×C₇	semidirect
60. C₂×#408	C₂×C₇	direct (N × H)
61. C₂²×C₂²⋊C₈	C₂×C₇	semidirect
112. C₂²×M₁₆	C₂²×C₇	semidirect
116. C₂³×C₈	C₂²×C₇	semidirect
117. #408	C₂²×C₇	direct (N × H)
118. C₂×C₂²⋊C₈	C₂²×C₇	semidirect
175. C₂×M₁₆	C₂³×C₇	semidirect
181. C₂²×C₈	C₂³×C₇	semidirect
182. C₂²⋊C₈	C₂³×C₇	semidirect
188. C₂²×C₇	#408	direct (N × H)
240. C₂×C₇	C₂×#408	direct (N × H)
274. C₇	C₂²×#408	direct (N × H)
280. C₂²	C₇×#408	direct (N × H)
292. C₂	C₂×C₇×#408	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₇×#41	2
C₂³×C₇×M₁₆	2
C₂⁴×C₇×C₈	2
C₂×C₇×#408	2
C₂²×C₇×C₂²⋊C₈	2
C₂²×#408	7

Subgroups — C22×C7×#408 #13237277

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×#408 #13237277