Subgroups — C₂×C₇×#3172 #13237891

Order 1792.

1986 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#529	C₂	1	1	yes	maximal, characteristic
3	C₇×#3172	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₇×#501	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₇×#497	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×#491	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₇×#489	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₇×#470	C₂	1	2	yes	maximal, complemented
9	C₂×C₇×#468	C₂	1	2	yes	maximal
10	C₂×C₄×C₇×Q₁₆	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×C₇×SD₁₆	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×C₇×Q₁₆	C₂²	1	2	yes	—
14	C₂²×C₇×SD₁₆	C₂²	1	2	yes	—
15	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
16	C₇×#533	—	2	4	no	—
17	C₇×#533	C₂²	1	8	yes	complemented
18	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₇×#529	—	2	4	no	—
21	C₇×#529	C₂²	1	8	yes	—
22	C₂×C₇×#126	—	2	1	no	—
23	C₂×C₇×#125	—	2	1	no	—
24	C₂×C₇×#125	C₂²	1	1	yes	characteristic
25	C₂×C₇×#124	C₂²	1	1	yes	characteristic
26	C₂×C₇×#123	—	2	1	no	—
27	C₂×C₇×C₂²⋊Q₈	—	2	1	no	—
28	C₂×C₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
29	C₂×C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
30	C₂×C₇×#121	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₇×Q₈	—	2	1	no	—
32	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	—
33	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₇×D₄	—	2	1	no	—
35	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
36	C₂×C₇×#117	—	2	1	no	—
37	C₂×C₇×#117	—	2	1	no	—
38	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
39	C₇×#501	C₂²	1	8	yes	complemented
40	C₇×#497	C₂²	1	8	yes	complemented
41	C₇×#491	C₂²	1	8	yes	complemented
42	C₇×#489	C₂²	1	8	yes	complemented
43	C₇×#470	C₂²	1	16	yes	complemented
44	C₇×#468	C₂²	1	16	yes	—
45	C₄×C₇×Q₁₆	C₂²	1	16	yes	complemented
46	C₄×C₇×SD₁₆	C₂²	1	16	yes	complemented
47	C₂×C₇×#107	C₂²	1	2	yes	—
48	C₂×C₇×#106	C₂²	1	2	yes	—
49	C₂×C₇×#103	C₂²	1	2	yes	—
50	C₂×C₇×#103	C₂²	1	2	yes	—
51	C₂×C₇×#103	C₂²	1	2	yes	—
52	C₂×C₇×#102	C₂²	1	2	yes	—
53	C₄×C₇×M₁₆	C₂²	1	8	yes	complemented
54	C₂×C₇×#97	C₂²	1	1	yes	characteristic
55	C₂×C₇×#97	C₂²	1	1	yes	characteristic
56	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
57	C₂×#3172	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂²×C₇×Q₈	—	2	1	no	—
59	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
60	C₂²×C₇×Q₈	C₂³	1	2	yes	—
61	C₂²×C₇×D₄	—	2	1	no	—
62	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₇	—	2	1	no	—
64	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
65	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
66	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
67	C₂×C₇×Q₁₆	—	2	8	no	—
68	C₂×C₇×Q₁₆	C₂³	1	8	yes	—
69	C₂×C₇×SD₁₆	—	2	8	no	—
70	C₂×C₇×SD₁₆	C₂³	1	8	yes	—
71	C₂×C₇×M₁₆	—	2	8	no	—
72	C₂×C₇×M₁₆	C₂³	1	8	yes	—
73	C₂²×C₇×C₈	C₂³	1	2	yes	—
74	C₂²×C₇×C₈	C₂³	1	2	yes	—
75	C₇×C₄⋊Q₈	—	2	2	no	—
76	C₇×C₄⋊Q₈	—	2	2	no	—
77	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
78	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
79	C₇×#126	—	2	8	no	—
80	C₇×#125	—	2	8	no	—
81	C₇×#125	—	2	2	no	—
82	C₇×#125	C₂³	1	4	yes	—
83	C₇×#124	—	2	2	no	—
84	C₇×#124	C₂³	1	4	yes	—
85	C₇×#123	—	2	8	no	—
86	C₇×C₂²⋊Q₈	—	2	8	no	—
87	C₇×C₂²⋊Q₈	—	2	2	no	—
88	C₇×C₂²⋊Q₈	—	2	4	no	—
89	C₇×C₂²⋊Q₈	C₂³	1	8	yes	—
90	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
91	C₇×#121	—	2	2	no	—
92	C₇×#121	C₂³	1	4	yes	—
93	C₄×C₇×Q₈	—	2	8	no	—
94	C₄×C₇×Q₈	—	2	2	no	—
95	C₄×C₇×Q₈	—	2	4	no	—
96	C₄×C₇×Q₈	C₂³	1	4	yes	—
97	C₄×C₇×Q₈	C₂³	1	8	yes	—
98	C₄×C₇×D₄	—	2	8	no	—
99	C₄×C₇×D₄	—	2	2	no	—
100	C₄×C₇×D₄	C₂³	1	4	yes	—
101	C₇×#117	—	2	8	no	—
102	C₇×#117	—	2	8	no	—
103	C₂×C₇×C₄⋊C₄	—	2	1	no	—
104	C₂×C₇×C₄⋊C₄	—	2	1	no	—
105	C₂×C₇×C₄⋊C₄	—	2	1	no	—
106	C₂×C₇×C₄⋊C₄	—	2	1	no	—
107	C₂×C₇×C₄⋊C₄	—	2	1	no	—
108	C₂×C₇×C₄⋊C₄	—	2	1	no	—
109	C₂×C₇×C₄⋊C₄	—	2	2	no	—
110	C₂×C₇×C₄⋊C₄	—	2	2	no	—
111	C₂×C₇×C₄⋊C₄	—	2	1	no	—
112	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
113	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
114	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
115	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
116	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
117	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
118	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
119	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
120	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
121	C₂×C₄²×C₇	—	2	1	no	—
122	C₂×C₄²×C₇	—	2	2	no	—
123	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
124	C₂×C₄²×C₇	D₄	1	4	yes	—
125	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
126	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₇	C₂³	1	4	yes	—
128	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
129	C₇×#107	C₂³	1	8	yes	—
130	C₇×#106	C₂³	1	8	yes	—
131	C₇×#103	C₂³	1	8	yes	—
132	C₇×#103	C₂³	1	8	yes	—
133	C₇×#103	C₂³	1	8	yes	—
134	C₇×#102	C₂³	1	8	yes	—
135	C₇×#97	C₂³	1	4	yes	—
136	C₇×#97	C₂³	1	4	yes	—
137	C₄×C₇×C₈	C₂³	1	8	yes	—
138	C₂×#533	C₂×C₇	1	1	yes	characteristic, complemented
139	C₂×#529	C₂×C₇	1	1	yes	characteristic
140	#3172	C₂×C₇	1	16	yes	direct product factor
141	C₂×#501	C₂×C₇	1	1	yes	characteristic, complemented
142	C₂×#497	C₂×C₇	1	1	yes	characteristic, complemented
143	C₂×#491	C₂×C₇	1	1	yes	characteristic, complemented
144	C₂×#489	C₂×C₇	1	1	yes	characteristic, complemented
145	C₂×#470	C₂×C₇	1	2	yes	complemented
146	C₂×#468	C₂×C₇	1	2	yes	—
147	C₂×C₄×Q₁₆	C₂×C₇	1	2	yes	complemented
148	C₂×C₄×SD₁₆	C₂×C₇	1	2	yes	complemented
149	C₂×C₄×M₁₆	C₂×C₇	1	1	yes	characteristic, complemented
150	C₂⁴×C₇	—	2	1	no	—
151	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
152	C₂×C₇×Q₈	—	2	1	no	—
153	C₂×C₇×Q₈	—	2	2	no	—
154	C₂×C₇×Q₈	—	2	4	no	—
155	C₂×C₇×Q₈	—	2	4	no	—
156	C₂×C₇×Q₈	—	2	4	no	—
157	C₂×C₇×Q₈	—	2	4	no	—
158	C₂×C₇×Q₈	—	4	4	no	—
159	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
160	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
161	C₂×C₇×D₄	—	2	4	no	—
162	C₂×C₇×D₄	—	2	1	no	—
163	C₂×C₇×D₄	—	2	4	no	—
164	C₂×C₇×D₄	—	4	4	no	—
165	C₂×C₇×D₄	C₂⁴	1	2	yes	—
166	C₂²×C₄×C₇	—	2	4	no	—
167	C₂²×C₄×C₇	—	2	4	no	—
168	C₂²×C₄×C₇	—	2	1	no	—
169	C₂²×C₄×C₇	—	2	2	no	—
170	C₂²×C₄×C₇	—	2	1	no	—
171	C₂²×C₄×C₇	—	2	2	no	—
172	C₂²×C₄×C₇	—	2	1	no	—
173	C₂²×C₄×C₇	—	2	4	no	—
174	C₂²×C₄×C₇	—	4	2	no	—
175	C₂²×C₄×C₇	—	4	4	no	—
176	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
178	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
182	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
183	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
184	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
185	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
186	C₇×Q₁₆	—	2	16	no	—
187	C₇×SD₁₆	—	2	16	no	—
188	C₇×M₁₆	—	2	16	no	—
189	C₂×C₇×C₈	—	2	4	no	—
190	C₂×C₇×C₈	#41	1	8	yes	—
191	C₂×C₇×C₈	C₂⁴	1	4	yes	—
192	C₂×C₇×C₈	C₂⁴	1	4	yes	—
193	C₇×C₄⋊C₄	—	2	4	no	—
194	C₇×C₄⋊C₄	—	2	4	no	—
195	C₇×C₄⋊C₄	—	2	4	no	—
196	C₇×C₄⋊C₄	—	2	4	no	—
197	C₇×C₄⋊C₄	—	2	4	no	—
198	C₇×C₄⋊C₄	—	2	4	no	—
199	C₇×C₄⋊C₄	—	2	8	no	—
200	C₇×C₄⋊C₄	—	2	8	no	—
201	C₇×C₄⋊C₄	—	2	4	no	—
202	C₇×C₄⋊C₄	—	2	1	no	—
203	C₇×C₄⋊C₄	—	2	1	no	—
204	C₇×C₄⋊C₄	—	2	2	no	—
205	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
206	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
207	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
208	C₇×C₂²⋊C₄	—	2	8	no	—
209	C₇×C₂²⋊C₄	—	2	4	no	—
210	C₇×C₂²⋊C₄	—	2	4	no	—
211	C₇×C₂²⋊C₄	—	2	4	no	—
212	C₇×C₂²⋊C₄	—	2	4	no	—
213	C₇×C₂²⋊C₄	—	2	4	no	—
214	C₄²×C₇	—	2	4	no	—
215	C₄²×C₇	—	2	8	no	—
216	C₄²×C₇	C₂×D₄	1	4	yes	—
217	C₄²×C₇	C₂×D₄	1	2	yes	—
218	C₄²×C₇	C₂×D₄	1	2	yes	—
219	C₄²×C₇	C₂×D₄	1	4	yes	—
220	C₄²×C₇	C₂⁴	1	2	yes	—
221	C₄²×C₇	C₂⁴	1	2	yes	—
222	C₂²×Q₁₆	C₂²×C₇	1	2	yes	—
223	C₂²×SD₁₆	C₂²×C₇	1	2	yes	—
224	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
225	#533	—	2	4	no	—
226	#533	C₂²×C₇	1	8	yes	complemented
227	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
228	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
229	#529	—	2	4	no	—
230	#529	C₂²×C₇	1	8	yes	—
231	C₂×#126	—	2	1	no	—
232	C₂×#125	—	2	1	no	—
233	C₂×#125	C₂²×C₇	1	1	yes	characteristic
234	C₂×#124	C₂²×C₇	1	1	yes	characteristic
235	C₂×#123	—	2	1	no	—
236	C₂×C₂²⋊Q₈	—	2	1	no	—
237	C₂×C₂²⋊Q₈	C₂²×C₇	1	1	yes	characteristic
238	C₂×C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
239	C₂×#121	C₂²×C₇	1	1	yes	characteristic
240	C₂×C₄×Q₈	—	2	1	no	—
241	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
242	C₂×C₄×Q₈	C₂²×C₇	1	2	yes	—
243	C₂×C₄×D₄	—	2	1	no	—
244	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
245	C₂×#117	—	2	1	no	—
246	C₂×#117	—	2	1	no	—
247	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
248	#501	C₂²×C₇	1	8	yes	complemented
249	#497	C₂²×C₇	1	8	yes	complemented
250	#491	C₂²×C₇	1	8	yes	complemented
251	#489	C₂²×C₇	1	8	yes	complemented
252	#470	C₂²×C₇	1	16	yes	complemented
253	#468	C₂²×C₇	1	16	yes	—
254	C₄×Q₁₆	C₂²×C₇	1	16	yes	complemented
255	C₄×SD₁₆	C₂²×C₇	1	16	yes	complemented
256	C₂×#107	C₂²×C₇	1	2	yes	—
257	C₂×#106	C₂²×C₇	1	2	yes	—
258	C₂×#103	C₂²×C₇	1	2	yes	—
259	C₂×#103	C₂²×C₇	1	2	yes	—
260	C₂×#103	C₂²×C₇	1	2	yes	—
261	C₂×#102	C₂²×C₇	1	2	yes	—
262	C₄×M₁₆	C₂²×C₇	1	8	yes	complemented
263	C₂×#97	C₂²×C₇	1	1	yes	characteristic
264	C₂×#97	C₂²×C₇	1	1	yes	characteristic
265	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
266	C₂³×C₇	—	2	2	no	—
267	C₂³×C₇	—	2	2	no	—
268	C₂³×C₇	—	2	2	no	—
269	C₂³×C₇	—	2	2	no	—
270	C₂³×C₇	—	4	2	no	—
271	C₂³×C₇	—	4	2	no	—
272	C₂³×C₇	—	4	2	no	—
273	C₂³×C₇	C₂²×D₄	1	2	yes	—
274	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
275	C₇×Q₈	—	2	4	no	—
276	C₇×Q₈	—	2	4	no	—
277	C₇×Q₈	—	2	4	no	—
278	C₇×Q₈	—	4	8	no	—
279	C₇×Q₈	—	4	2	no	—
280	C₇×Q₈	—	4	4	no	—
281	C₇×D₄	—	2	4	no	—
282	C₇×D₄	—	4	8	no	—
283	C₇×D₄	—	4	2	no	—
284	C₂×C₄×C₇	—	2	2	no	—
285	C₂×C₄×C₇	—	2	2	no	—
286	C₂×C₄×C₇	—	2	2	no	—
287	C₂×C₄×C₇	—	2	2	no	—
288	C₂×C₄×C₇	—	2	4	no	—
289	C₂×C₄×C₇	—	2	8	no	—
290	C₂×C₄×C₇	—	2	4	no	—
291	C₂×C₄×C₇	—	2	2	no	—
292	C₂×C₄×C₇	—	2	4	no	—
293	C₂×C₄×C₇	—	2	2	no	—
294	C₂×C₄×C₇	—	2	4	no	—
295	C₂×C₄×C₇	—	2	2	no	—
296	C₂×C₄×C₇	—	4	4	no	—
297	C₂×C₄×C₇	—	4	2	no	—
298	C₂×C₄×C₇	—	4	4	no	—
299	C₂×C₄×C₇	—	4	2	no	—
300	C₂×C₄×C₇	—	4	2	no	—
301	C₂×C₄×C₇	—	4	4	no	—
302	C₂×C₄×C₇	—	4	8	no	—
303	C₂×C₄×C₇	#137	1	4	yes	—
304	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
305	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
306	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
307	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
308	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
309	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
310	C₂×C₄×C₇	C₂×#41	1	4	yes	—
311	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
312	C₇×C₈	—	2	4	no	—
313	C₇×C₈	C₂×#41	1	8	yes	—
314	C₂²×Q₈	—	2	1	no	—
315	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
316	C₂²×Q₈	C₂³×C₇	1	2	yes	—
317	C₂²×D₄	—	2	1	no	—
318	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
319	C₂³×C₄	—	2	1	no	—
320	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
321	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
322	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
323	C₂×Q₁₆	—	2	8	no	—
324	C₂×Q₁₆	C₂³×C₇	1	8	yes	—
325	C₂×SD₁₆	—	2	8	no	—
326	C₂×SD₁₆	C₂³×C₇	1	8	yes	—
327	C₂×M₁₆	—	2	8	no	—
328	C₂×M₁₆	C₂³×C₇	1	8	yes	—
329	C₂²×C₈	C₂³×C₇	1	2	yes	—
330	C₂²×C₈	C₂³×C₇	1	2	yes	—
331	C₄⋊Q₈	—	2	2	no	—
332	C₄⋊Q₈	—	2	2	no	—
333	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
334	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
335	#126	—	2	8	no	—
336	#125	—	2	8	no	—
337	#125	—	2	2	no	—
338	#125	C₂³×C₇	1	4	yes	—
339	#124	—	2	2	no	—
340	#124	C₂³×C₇	1	4	yes	—
341	#123	—	2	8	no	—
342	C₂²⋊Q₈	—	2	8	no	—
343	C₂²⋊Q₈	—	2	2	no	—
344	C₂²⋊Q₈	—	2	4	no	—
345	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
346	C₂²⋊Q₈	C₂³×C₇	1	8	yes	—
347	#121	—	2	2	no	—
348	#121	C₂³×C₇	1	4	yes	—
349	C₄×Q₈	—	2	8	no	—
350	C₄×Q₈	—	2	2	no	—
351	C₄×Q₈	—	2	4	no	—
352	C₄×Q₈	C₂³×C₇	1	4	yes	—
353	C₄×Q₈	C₂³×C₇	1	8	yes	—
354	C₄×D₄	—	2	8	no	—
355	C₄×D₄	—	2	2	no	—
356	C₄×D₄	C₂³×C₇	1	4	yes	—
357	#117	—	2	8	no	—
358	#117	—	2	8	no	—
359	C₂×C₄⋊C₄	—	2	1	no	—
360	C₂×C₄⋊C₄	—	2	1	no	—
361	C₂×C₄⋊C₄	—	2	1	no	—
362	C₂×C₄⋊C₄	—	2	1	no	—
363	C₂×C₄⋊C₄	—	2	1	no	—
364	C₂×C₄⋊C₄	—	2	1	no	—
365	C₂×C₄⋊C₄	—	2	2	no	—
366	C₂×C₄⋊C₄	—	2	2	no	—
367	C₂×C₄⋊C₄	—	2	1	no	—
368	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
369	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
370	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
371	C₂×C₂²⋊C₄	—	2	2	no	—
372	C₂×C₂²⋊C₄	—	2	1	no	—
373	C₂×C₂²⋊C₄	—	2	1	no	—
374	C₂×C₂²⋊C₄	—	2	1	no	—
375	C₂×C₂²⋊C₄	—	2	1	no	—
376	C₂×C₂²⋊C₄	—	2	1	no	—
377	C₂×C₄²	—	2	1	no	—
378	C₂×C₄²	—	2	2	no	—
379	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
380	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
381	C₂×C₄²	C₇×D₄	1	4	yes	—
382	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
383	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
384	C₂×C₄²	C₂³×C₇	1	4	yes	—
385	#107	C₂³×C₇	1	8	yes	—
386	#106	C₂³×C₇	1	8	yes	—
387	#103	C₂³×C₇	1	8	yes	—
388	#103	C₂³×C₇	1	8	yes	—
389	#103	C₂³×C₇	1	8	yes	—
390	#102	C₂³×C₇	1	8	yes	—
391	#97	C₂³×C₇	1	4	yes	—
392	#97	C₂³×C₇	1	4	yes	—
393	C₄×C₈	C₂³×C₇	1	8	yes	—
394	C₂²×C₇	—	2	2	no	—
395	C₂²×C₇	—	2	2	no	—
396	C₂²×C₇	—	2	2	no	—
397	C₂²×C₇	—	4	4	no	—
398	C₂²×C₇	—	4	2	no	—
399	C₂²×C₇	—	4	2	no	—
400	C₂²×C₇	—	4	2	no	—
401	C₂²×C₇	—	8	4	no	—
402	C₂²×C₇	#532	1	2	yes	—
403	C₂²×C₇	C₂×#137	1	2	yes	—
404	C₂²×C₇	#575	1	2	yes	—
405	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
406	C₄×C₇	—	2	2	no	—
407	C₄×C₇	—	2	2	no	—
408	C₄×C₇	—	2	2	no	—
409	C₄×C₇	—	2	2	no	—
410	C₄×C₇	—	4	2	no	—
411	C₄×C₇	—	4	4	no	—
412	C₄×C₇	—	4	2	no	—
413	C₄×C₇	—	4	2	no	—
414	C₄×C₇	—	4	4	no	—
415	C₄×C₇	#532	1	2	yes	—
416	C₄×C₇	C₂×#137	1	4	yes	—
417	C₄×C₇	C₂²×#41	1	2	yes	—
418	C₂⁴	—	2	1	no	—
419	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
420	C₂×Q₈	—	2	1	no	—
421	C₂×Q₈	—	2	2	no	—
422	C₂×Q₈	—	2	4	no	—
423	C₂×Q₈	—	2	4	no	—
424	C₂×Q₈	—	2	4	no	—
425	C₂×Q₈	—	2	4	no	—
426	C₂×Q₈	—	4	4	no	—
427	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
428	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
429	C₂×D₄	—	2	4	no	—
430	C₂×D₄	—	2	1	no	—
431	C₂×D₄	—	2	4	no	—
432	C₂×D₄	—	4	4	no	—
433	C₂×D₄	C₂⁴×C₇	1	2	yes	—
434	C₂²×C₄	—	2	4	no	—
435	C₂²×C₄	—	2	4	no	—
436	C₂²×C₄	—	2	1	no	—
437	C₂²×C₄	—	2	2	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	2	no	—
440	C₂²×C₄	—	2	1	no	—
441	C₂²×C₄	—	2	4	no	—
442	C₂²×C₄	—	4	2	no	—
443	C₂²×C₄	—	4	4	no	—
444	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
445	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
446	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
447	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
448	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
449	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
450	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
451	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
452	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
453	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
454	Q₁₆	—	2	16	no	—
455	SD₁₆	—	2	16	no	—
456	M₁₆	—	2	16	no	—
457	C₂×C₈	—	2	4	no	—
458	C₂×C₈	C₇×#41	1	8	yes	—
459	C₂×C₈	C₂⁴×C₇	1	4	yes	—
460	C₂×C₈	C₂⁴×C₇	1	4	yes	—
461	C₄⋊C₄	—	2	4	no	—
462	C₄⋊C₄	—	2	4	no	—
463	C₄⋊C₄	—	2	4	no	—
464	C₄⋊C₄	—	2	4	no	—
465	C₄⋊C₄	—	2	4	no	—
466	C₄⋊C₄	—	2	4	no	—
467	C₄⋊C₄	—	2	8	no	—
468	C₄⋊C₄	—	2	8	no	—
469	C₄⋊C₄	—	2	4	no	—
470	C₄⋊C₄	—	2	1	no	—
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	—	2	2	no	—
473	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
474	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
475	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
476	C₂²⋊C₄	—	2	8	no	—
477	C₂²⋊C₄	—	2	4	no	—
478	C₂²⋊C₄	—	2	4	no	—
479	C₂²⋊C₄	—	2	4	no	—
480	C₂²⋊C₄	—	2	4	no	—
481	C₂²⋊C₄	—	2	4	no	—
482	C₄²	—	2	4	no	—
483	C₄²	—	2	8	no	—
484	C₄²	C₂×C₇×D₄	1	4	yes	—
485	C₄²	C₂×C₇×D₄	1	2	yes	—
486	C₄²	C₂×C₇×D₄	1	2	yes	—
487	C₄²	C₂×C₇×D₄	1	4	yes	—
488	C₄²	C₂⁴×C₇	1	2	yes	—
489	C₄²	C₂⁴×C₇	1	2	yes	—
490	C₂×C₇	—	4	2	no	—
491	C₂×C₇	—	8	2	no	—
492	C₂×C₇	#3172	1	4	yes	direct product factor
493	C₂×C₇	C₂×#532	1	1	yes	characteristic
494	C₂×C₇	C₂²×#137	1	1	yes	characteristic
495	C₂×C₇	C₂×#575	1	1	yes	characteristic
496	C₂³	—	2	2	no	—
497	C₂³	—	2	2	no	—
498	C₂³	—	2	2	no	—
499	C₂³	—	2	2	no	—
500	C₂³	—	4	2	no	—
501	C₂³	—	4	2	no	—
502	C₂³	—	4	2	no	—
503	C₂³	C₂²×C₇×D₄	1	2	yes	—
504	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
505	Q₈	—	2	4	no	—
506	Q₈	—	2	4	no	—
507	Q₈	—	2	4	no	—
508	Q₈	—	4	8	no	—
509	Q₈	—	4	2	no	—
510	Q₈	—	4	4	no	—
511	D₄	—	2	4	no	—
512	D₄	—	4	8	no	—
513	D₄	—	4	2	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	4	no	—
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	2	8	no	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	2	4	no	—
523	C₂×C₄	—	2	2	no	—
524	C₂×C₄	—	2	4	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	4	4	no	—
527	C₂×C₄	—	4	2	no	—
528	C₂×C₄	—	4	4	no	—
529	C₂×C₄	—	4	2	no	—
530	C₂×C₄	—	4	2	no	—
531	C₂×C₄	—	4	4	no	—
532	C₂×C₄	—	4	8	no	—
533	C₂×C₄	C₇×#137	1	4	yes	—
534	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
535	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
536	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
537	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
538	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
539	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
540	C₂×C₄	C₂×C₇×#41	1	4	yes	—
541	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
542	C₈	—	2	4	no	—
543	C₈	C₂×C₇×#41	1	8	yes	—
544	C₇	C₂×#3172	1	1	yes	characteristic, direct product factor, Sylow
545	C₂²	—	2	2	no	—
546	C₂²	—	2	2	no	—
547	C₂²	—	2	2	no	—
548	C₂²	—	4	4	no	—
549	C₂²	—	4	2	no	—
550	C₂²	—	4	2	no	—
551	C₂²	—	4	2	no	—
552	C₂²	—	8	4	no	—
553	C₂²	C₇×#532	1	2	yes	—
554	C₂²	C₂×C₇×#137	1	2	yes	—
555	C₂²	C₇×#575	1	2	yes	—
556	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
557	C₄	—	2	2	no	—
558	C₄	—	2	2	no	—
559	C₄	—	2	2	no	—
560	C₄	—	2	2	no	—
561	C₄	—	4	2	no	—
562	C₄	—	4	2	no	—
563	C₄	—	4	4	no	—
564	C₄	—	4	2	no	—
565	C₄	—	4	4	no	—
566	C₄	C₇×#532	1	2	yes	—
567	C₄	C₂×C₇×#137	1	4	yes	—
568	C₄	C₂²×C₇×#41	1	2	yes	—
569	C₂	—	4	2	no	—
570	C₂	—	8	2	no	—
571	C₂	C₇×#3172	1	4	yes	direct product factor
572	C₂	C₂×C₇×#532	1	1	yes	characteristic
573	C₂	C₂²×C₇×#137	1	1	yes	characteristic
574	C₂	C₂×C₇×#575	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
socle	C₂³×C₇	C₂²×D₄
prime radical	C₂×C₇×#121	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#533	C₂	semidirect
3. C₇×#3172	C₂	direct (N × H)
4. C₂×C₇×#501	C₂	semidirect
5. C₂×C₇×#497	C₂	semidirect
6. C₂×C₇×#491	C₂	semidirect
7. C₂×C₇×#489	C₂	semidirect
8. C₂×C₇×#470	C₂	semidirect
10. C₂×C₄×C₇×Q₁₆	C₂	semidirect
11. C₂×C₄×C₇×SD₁₆	C₂	semidirect
12. C₂×C₄×C₇×M₁₆	C₂	semidirect
17. C₇×#533	C₂²	semidirect
39. C₇×#501	C₂²	semidirect
40. C₇×#497	C₂²	semidirect
41. C₇×#491	C₂²	semidirect
42. C₇×#489	C₂²	semidirect
43. C₇×#470	C₂²	semidirect
45. C₄×C₇×Q₁₆	C₂²	semidirect
46. C₄×C₇×SD₁₆	C₂²	semidirect
53. C₄×C₇×M₁₆	C₂²	semidirect
57. C₂×#3172	C₇	direct (N × H)
138. C₂×#533	C₂×C₇	semidirect
140. #3172	C₂×C₇	direct (N × H)
141. C₂×#501	C₂×C₇	semidirect
142. C₂×#497	C₂×C₇	semidirect
143. C₂×#491	C₂×C₇	semidirect
144. C₂×#489	C₂×C₇	semidirect
145. C₂×#470	C₂×C₇	semidirect
147. C₂×C₄×Q₁₆	C₂×C₇	semidirect
148. C₂×C₄×SD₁₆	C₂×C₇	semidirect
149. C₂×C₄×M₁₆	C₂×C₇	semidirect
226. #533	C₂²×C₇	semidirect
248. #501	C₂²×C₇	semidirect
249. #497	C₂²×C₇	semidirect
250. #491	C₂²×C₇	semidirect
251. #489	C₂²×C₇	semidirect
252. #470	C₂²×C₇	semidirect
254. C₄×Q₁₆	C₂²×C₇	semidirect
255. C₄×SD₁₆	C₂²×C₇	semidirect
262. C₄×M₁₆	C₂²×C₇	semidirect
492. C₂×C₇	#3172	direct (N × H)
544. C₇	C₂×#3172	direct (N × H)
571. C₂	C₇×#3172	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×C₇×#533	2
C₂×C₇×#529	2
C₇×#3172	2
C₂×C₇×#501	2
C₂×C₇×#497	2
C₂×C₇×#491	2
C₂×C₇×#489	2
C₂×C₇×#470	2
C₂×C₇×#468	2
C₂×C₄×C₇×Q₁₆	2
C₂×C₄×C₇×SD₁₆	2
C₂×C₄×C₇×M₁₆	2
C₂×#3172	7

Subgroups — C2×C7×#3172 #13237891

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#3172 #13237891