Subgroups — C₂²×C₇×#418 #13237296

Order 1792.

7214 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₇×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₇×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₇×#418	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₇×#102	C₂	1	4	yes	maximal, complemented
6	C₂⁶×C₇	—	2	1	no	—
7	C₂³×C₇×D₄	—	2	6	no	—
8	C₂³×C₇×D₄	—	2	1	no	—
9	C₂³×C₇×D₄	—	2	8	no	—
10	C₂³×C₇×D₄	C₄	1	16	yes	complemented
11	C₂³×C₇×D₄	C₂²	1	2	yes	—
12	C₂³×C₇×D₄	C₂²	1	12	yes	—
13	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
14	C₂²×C₇×M₁₆	—	2	8	no	—
15	C₂²×C₇×M₁₆	C₂²	1	12	yes	complemented
16	C₂³×C₇×C₈	C₂²	1	2	yes	complemented
17	C₂×C₇×#117	—	2	6	no	—
18	C₂×C₇×#117	C₂²	1	12	yes	complemented
19	C₂²×C₇×C₄⋊C₄	C₂²	1	2	yes	complemented
20	C₂²×C₇×C₂²⋊C₄	—	2	1	no	—
21	C₂²×C₄²×C₇	—	2	1	no	—
22	C₇×#418	C₂²	1	64	yes	direct product factor
23	C₂×C₇×#102	C₂²	1	48	yes	complemented
24	C₂²×#418	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂⁵×C₇	—	2	16	no	—
26	C₂⁵×C₇	—	2	12	no	—
27	C₂⁵×C₇	—	2	2	no	—
28	C₂⁵×C₇	—	4	16	no	—
29	C₂⁵×C₇	D₄	1	1	yes	characteristic
30	C₂²×C₇×D₄	—	2	96	no	—
31	C₂²×C₇×D₄	—	2	16	no	—
32	C₂²×C₇×D₄	—	2	12	no	—
33	C₂²×C₇×D₄	—	2	6	no	—
34	C₂²×C₇×D₄	—	2	24	no	—
35	C₂²×C₇×D₄	—	2	8	no	—
36	C₂²×C₇×D₄	—	2	48	no	—
37	C₂²×C₇×D₄	C₂×C₄	1	16	yes	—
38	C₂²×C₇×D₄	C₂×C₄	1	96	yes	complemented
39	C₂²×C₇×D₄	C₂³	1	12	yes	—
40	C₂²×C₇×D₄	C₂³	1	16	yes	—
41	C₂³×C₄×C₇	—	2	2	no	—
42	C₂³×C₄×C₇	C₂×C₄	1	8	yes	—
43	C₂³×C₄×C₇	D₄	1	6	yes	—
44	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
45	C₂³×C₄×C₇	D₄	1	8	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	C₂³	1	12	yes	complemented
52	C₇×#117	—	2	24	no	—
53	C₇×#117	C₂³	1	16	yes	complemented
54	C₂×C₇×C₄⋊C₄	—	2	6	no	—
55	C₂×C₇×C₄⋊C₄	C₂³	1	12	yes	complemented
56	C₂×C₇×C₂²⋊C₄	—	2	12	no	—
57	C₂×C₄²×C₇	—	2	12	no	—
58	C₇×#102	C₂³	1	64	yes	complemented
59	C₂⁴×D₄	C₂×C₇	1	1	yes	characteristic
60	C₂³×M₁₆	C₂×C₇	1	1	yes	characteristic, complemented
61	C₂²×#117	C₂×C₇	1	1	yes	characteristic, complemented
62	C₂×#418	C₂×C₇	1	24	yes	direct product factor
63	C₂²×#102	C₂×C₇	1	4	yes	complemented
64	C₂⁴×C₇	—	2	8	no	—
65	C₂⁴×C₇	—	2	16	no	—
66	C₂⁴×C₇	—	2	96	no	—
67	C₂⁴×C₇	—	2	12	no	—
68	C₂⁴×C₇	—	2	16	no	—
69	C₂⁴×C₇	—	4	96	no	—
70	C₂⁴×C₇	—	4	128	no	—
71	C₂⁴×C₇	—	4	16	no	—
72	C₂⁴×C₇	C₂²⋊C₄	1	8	yes	—
73	C₂⁴×C₇	C₂×D₄	1	6	yes	—
74	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₇×D₄	—	2	48	no	—
76	C₂×C₇×D₄	—	2	192	no	—
77	C₂×C₇×D₄	—	2	96	no	—
78	C₂×C₇×D₄	—	2	16	no	—
79	C₂×C₇×D₄	—	2	12	no	—
80	C₂×C₇×D₄	—	2	24	no	—
81	C₂×C₇×D₄	—	2	32	no	—
82	C₂×C₇×D₄	C₂²×C₄	1	48	yes	—
83	C₂×C₇×D₄	C₂²×C₄	1	64	yes	complemented
84	C₂×C₇×D₄	C₂⁴	1	8	yes	—
85	C₂²×C₄×C₇	—	2	12	no	—
86	C₂²×C₄×C₇	—	4	8	no	—
87	C₂²×C₄×C₇	C₂²⋊C₄	1	48	yes	—
88	C₂²×C₄×C₇	C₂²⋊C₄	1	8	yes	—
89	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
90	C₂²×C₄×C₇	C₂²×C₄	1	24	yes	—
91	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
92	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
93	C₂²×C₄×C₇	C₂×D₄	1	12	yes	—
94	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
95	C₂²×C₄×C₇	C₂×D₄	1	24	yes	—
96	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
97	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
98	C₇×M₁₆	—	2	32	no	—
99	C₂×C₇×C₈	—	2	24	no	—
100	C₂×C₇×C₈	C₂⁴	1	8	yes	complemented
101	C₇×C₄⋊C₄	—	2	12	no	—
102	C₇×C₄⋊C₄	C₂⁴	1	8	yes	complemented
103	C₇×C₂²⋊C₄	—	2	16	no	—
104	C₄²×C₇	—	2	16	no	—
105	C₂⁶	—	2	1	no	—
106	C₂³×D₄	—	2	6	no	—
107	C₂³×D₄	—	2	1	no	—
108	C₂³×D₄	—	2	8	no	—
109	C₂³×D₄	C₄×C₇	1	16	yes	complemented
110	C₂³×D₄	C₂²×C₇	1	2	yes	—
111	C₂³×D₄	C₂²×C₇	1	12	yes	—
112	C₂⁴×C₄	C₂²×C₇	1	1	yes	characteristic
113	C₂²×M₁₆	—	2	8	no	—
114	C₂²×M₁₆	C₂²×C₇	1	12	yes	complemented
115	C₂³×C₈	C₂²×C₇	1	2	yes	complemented
116	C₂×#117	—	2	6	no	—
117	C₂×#117	C₂²×C₇	1	12	yes	complemented
118	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	complemented
119	C₂²×C₂²⋊C₄	—	2	1	no	—
120	C₂²×C₄²	—	2	1	no	—
121	#418	C₂²×C₇	1	64	yes	direct product factor
122	C₂×#102	C₂²×C₇	1	48	yes	complemented
123	C₂³×C₇	—	2	24	no	—
124	C₂³×C₇	—	2	32	no	—
125	C₂³×C₇	—	2	48	no	—
126	C₂³×C₇	—	2	64	no	—
127	C₂³×C₇	—	2	8	no	—
128	C₂³×C₇	—	4	384	no	—
129	C₂³×C₇	—	4	64	no	—
130	C₂³×C₇	—	4	48	no	—
131	C₂³×C₇	—	4	64	no	—
132	C₂³×C₇	C₂×C₂²⋊C₄	1	24	yes	—
133	C₂³×C₇	C₂×C₂²⋊C₄	1	4	yes	—
134	C₂³×C₇	AGL(1,ℤ/8)	1	8	yes	—
135	C₂³×C₇	C₂²×D₄	1	4	yes	—
136	C₂³×C₇	C₂²×D₄	1	3	yes	—
137	C₇×D₄	—	2	48	no	—
138	C₇×D₄	—	2	64	no	—
139	C₇×D₄	—	2	8	no	—
140	C₇×D₄	C₂³×C₄	1	16	yes	—
141	C₂×C₄×C₇	—	2	8	no	—
142	C₂×C₄×C₇	—	4	24	no	—
143	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	48	yes	—
144	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	12	yes	—
145	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	24	yes	—
146	C₂×C₄×C₇	C₂³×C₄	1	6	yes	—
147	C₂×C₄×C₇	C₂³×C₄	1	8	yes	—
148	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
149	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
150	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
151	C₂×C₄×C₇	C₂²×D₄	1	8	yes	—
152	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
153	C₇×C₈	—	2	8	no	—
154	C₂⁵	—	2	16	no	—
155	C₂⁵	—	2	2	no	—
156	C₂⁵	—	2	12	no	—
157	C₂⁵	—	4	16	no	—
158	C₂⁵	C₇×D₄	1	1	yes	characteristic
159	C₂²×D₄	—	2	96	no	—
160	C₂²×D₄	—	2	16	no	—
161	C₂²×D₄	—	2	12	no	—
162	C₂²×D₄	—	2	6	no	—
163	C₂²×D₄	—	2	24	no	—
164	C₂²×D₄	—	2	8	no	—
165	C₂²×D₄	—	2	48	no	—
166	C₂²×D₄	C₂×C₄×C₇	1	16	yes	—
167	C₂²×D₄	C₂×C₄×C₇	1	96	yes	complemented
168	C₂²×D₄	C₂³×C₇	1	12	yes	—
169	C₂²×D₄	C₂³×C₇	1	16	yes	—
170	C₂³×C₄	—	2	2	no	—
171	C₂³×C₄	C₂×C₄×C₇	1	8	yes	—
172	C₂³×C₄	C₇×D₄	1	6	yes	—
173	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
174	C₂³×C₄	C₇×D₄	1	8	yes	—
175	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
176	C₂³×C₄	C₂³×C₇	1	6	yes	—
177	C₂×M₁₆	—	2	48	no	—
178	C₂×M₁₆	C₂³×C₇	1	16	yes	complemented
179	C₂²×C₈	—	2	8	no	—
180	C₂²×C₈	C₂³×C₇	1	12	yes	complemented
181	#117	—	2	24	no	—
182	#117	C₂³×C₇	1	16	yes	complemented
183	C₂×C₄⋊C₄	—	2	6	no	—
184	C₂×C₄⋊C₄	C₂³×C₇	1	12	yes	complemented
185	C₂×C₂²⋊C₄	—	2	12	no	—
186	C₂×C₄²	—	2	12	no	—
187	#102	C₂³×C₇	1	64	yes	complemented
188	C₂²×C₇	—	2	48	no	—
189	C₂²×C₇	—	2	8	no	—
190	C₂²×C₇	—	2	16	no	—
191	C₂²×C₇	—	4	128	no	—
192	C₂²×C₇	—	4	96	no	—
193	C₂²×C₇	—	4	16	no	—
194	C₂²×C₇	#418	1	16	yes	direct product factor
195	C₂²×C₇	C₂²×C₂²⋊C₄	1	8	yes	—
196	C₂²×C₇	C₂²×C₂²⋊C₄	1	6	yes	—
197	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	12	yes	—
198	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
199	C₄×C₇	—	4	8	no	—
200	C₄×C₇	C₂²×C₂²⋊C₄	1	6	yes	—
201	C₄×C₇	C₂²×C₂²⋊C₄	1	8	yes	—
202	C₄×C₇	C₂⁴×C₄	1	1	yes	characteristic
203	C₄×C₇	C₂³×D₄	1	1	yes	characteristic
204	C₂⁴	—	2	8	no	—
205	C₂⁴	—	2	16	no	—
206	C₂⁴	—	2	96	no	—
207	C₂⁴	—	2	12	no	—
208	C₂⁴	—	2	16	no	—
209	C₂⁴	—	4	128	no	—
210	C₂⁴	—	4	96	no	—
211	C₂⁴	—	4	16	no	—
212	C₂⁴	C₇×C₂²⋊C₄	1	8	yes	—
213	C₂⁴	C₂×C₇×D₄	1	6	yes	—
214	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
215	C₂×D₄	—	2	48	no	—
216	C₂×D₄	—	2	192	no	—
217	C₂×D₄	—	2	96	no	—
218	C₂×D₄	—	2	24	no	—
219	C₂×D₄	—	2	32	no	—
220	C₂×D₄	—	2	16	no	—
221	C₂×D₄	—	2	12	no	—
222	C₂×D₄	C₂²×C₄×C₇	1	48	yes	—
223	C₂×D₄	C₂²×C₄×C₇	1	64	yes	complemented
224	C₂×D₄	C₂⁴×C₇	1	8	yes	—
225	C₂²×C₄	—	2	12	no	—
226	C₂²×C₄	—	4	8	no	—
227	C₂²×C₄	C₇×C₂²⋊C₄	1	48	yes	—
228	C₂²×C₄	C₇×C₂²⋊C₄	1	8	yes	—
229	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
230	C₂²×C₄	C₂²×C₄×C₇	1	24	yes	—
231	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
232	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
233	C₂²×C₄	C₂×C₇×D₄	1	12	yes	—
234	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
235	C₂²×C₄	C₂×C₇×D₄	1	24	yes	—
236	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
237	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
238	M₁₆	—	2	32	no	—
239	C₂×C₈	—	2	24	no	—
240	C₂×C₈	C₂⁴×C₇	1	8	yes	complemented
241	C₄⋊C₄	—	2	12	no	—
242	C₄⋊C₄	C₂⁴×C₇	1	8	yes	complemented
243	C₂²⋊C₄	—	2	16	no	—
244	C₄²	—	2	16	no	—
245	C₂×C₇	—	2	8	no	—
246	C₂×C₇	—	4	16	no	—
247	C₂×C₇	C₂×#418	1	12	yes	direct product factor
248	C₂×C₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
249	C₂×C₇	C₂²×AGL(1,ℤ/8)	1	2	yes	—
250	C₂³	—	2	32	no	—
251	C₂³	—	2	24	no	—
252	C₂³	—	2	8	no	—
253	C₂³	—	2	48	no	—
254	C₂³	—	2	64	no	—
255	C₂³	—	4	64	no	—
256	C₂³	—	4	384	no	—
257	C₂³	—	4	64	no	—
258	C₂³	—	4	48	no	—
259	C₂³	C₂×C₇×C₂²⋊C₄	1	24	yes	—
260	C₂³	C₂×C₇×C₂²⋊C₄	1	4	yes	—
261	C₂³	C₇×AGL(1,ℤ/8)	1	8	yes	—
262	C₂³	C₂²×C₇×D₄	1	4	yes	—
263	C₂³	C₂²×C₇×D₄	1	3	yes	—
264	D₄	—	2	8	no	—
265	D₄	—	2	48	no	—
266	D₄	—	2	64	no	—
267	D₄	C₂³×C₄×C₇	1	16	yes	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	4	24	no	—
270	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	48	yes	—
271	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	12	yes	—
272	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	24	yes	—
273	C₂×C₄	C₂³×C₄×C₇	1	6	yes	—
274	C₂×C₄	C₂³×C₄×C₇	1	8	yes	—
275	C₂×C₄	C₂²×C₇×D₄	1	6	yes	—
276	C₂×C₄	C₂²×C₇×D₄	1	8	yes	—
277	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
278	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
279	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
280	C₈	—	2	8	no	—
281	C₇	C₂²×#418	1	1	yes	characteristic, direct product factor, Sylow
282	C₂²	—	2	8	no	—
283	C₂²	—	2	48	no	—
284	C₂²	—	2	16	no	—
285	C₂²	—	4	128	no	—
286	C₂²	—	4	16	no	—
287	C₂²	—	4	96	no	—
288	C₂²	C₇×#418	1	16	yes	direct product factor
289	C₂²	C₂²×C₇×C₂²⋊C₄	1	8	yes	—
290	C₂²	C₂²×C₇×C₂²⋊C₄	1	6	yes	—
291	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	12	yes	—
292	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
293	C₄	—	4	8	no	—
294	C₄	C₂²×C₇×C₂²⋊C₄	1	6	yes	—
295	C₄	C₂²×C₇×C₂²⋊C₄	1	8	yes	—
296	C₄	C₂⁴×C₄×C₇	1	1	yes	characteristic
297	C₄	C₂³×C₇×D₄	1	1	yes	characteristic
298	C₂	—	2	8	no	—
299	C₂	—	4	16	no	—
300	C₂	C₂×C₇×#418	1	12	yes	direct product factor
301	C₂	C₂³×C₇×C₂²⋊C₄	1	1	yes	characteristic
302	C₂	C₂²×C₇×AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₇	C₂×D₄
commutator	C₄	C₂⁴×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₂⁴×C₇×D₄	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ C₂⁴×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. C₂²×C₇×#117	C₂	semidirect
4. C₂×C₇×#418	C₂	direct (N × H)
5. C₂²×C₇×#102	C₂	semidirect
10. C₂³×C₇×D₄	C₄	semidirect
15. C₂²×C₇×M₁₆	C₂²	semidirect
16. C₂³×C₇×C₈	C₂²	semidirect
18. C₂×C₇×#117	C₂²	semidirect
19. C₂²×C₇×C₄⋊C₄	C₂²	semidirect
22. C₇×#418	C₂²	direct (N × H)
23. C₂×C₇×#102	C₂²	semidirect
24. C₂²×#418	C₇	direct (N × H)
38. C₂²×C₇×D₄	C₂×C₄	semidirect
49. C₂×C₇×M₁₆	C₂³	semidirect
51. C₂²×C₇×C₈	C₂³	semidirect
53. C₇×#117	C₂³	semidirect
55. C₂×C₇×C₄⋊C₄	C₂³	semidirect
58. C₇×#102	C₂³	semidirect
60. C₂³×M₁₆	C₂×C₇	semidirect
61. C₂²×#117	C₂×C₇	semidirect
62. C₂×#418	C₂×C₇	direct (N × H)
63. C₂²×#102	C₂×C₇	semidirect
83. C₂×C₇×D₄	C₂²×C₄	semidirect
100. C₂×C₇×C₈	C₂⁴	semidirect
102. C₇×C₄⋊C₄	C₂⁴	semidirect
109. C₂³×D₄	C₄×C₇	semidirect
114. C₂²×M₁₆	C₂²×C₇	semidirect
115. C₂³×C₈	C₂²×C₇	semidirect
117. C₂×#117	C₂²×C₇	semidirect
118. C₂²×C₄⋊C₄	C₂²×C₇	semidirect
121. #418	C₂²×C₇	direct (N × H)
122. C₂×#102	C₂²×C₇	semidirect
167. C₂²×D₄	C₂×C₄×C₇	semidirect
178. C₂×M₁₆	C₂³×C₇	semidirect
180. C₂²×C₈	C₂³×C₇	semidirect
182. #117	C₂³×C₇	semidirect
184. C₂×C₄⋊C₄	C₂³×C₇	semidirect
187. #102	C₂³×C₇	semidirect
194. C₂²×C₇	#418	direct (N × H)
223. C₂×D₄	C₂²×C₄×C₇	semidirect
240. C₂×C₈	C₂⁴×C₇	semidirect
242. C₄⋊C₄	C₂⁴×C₇	semidirect
247. C₂×C₇	C₂×#418	direct (N × H)
281. C₇	C₂²×#418	direct (N × H)
288. C₂²	C₇×#418	direct (N × H)
300. C₂	C₂×C₇×#418	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₇×D₄	2
C₂³×C₇×M₁₆	2
C₂²×C₇×#117	2
C₂×C₇×#418	2
C₂²×C₇×#102	2
C₂²×#418	7

Subgroups — C22×C7×#418 #13237296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×#418 #13237296