Subgroups — C₇×#19798 #13197014

Order 1792.

592 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×Q₈×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3375	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3361	C₂	1	1	yes	maximal, characteristic
4	C₇×#3299	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1720	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1704	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1678	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1677	C₂	1	1	yes	maximal, characteristic
9	C₇×#1672	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1640	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1633	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#1632	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#1627	C₂	1	1	yes	maximal, characteristic
14	C₇×#1582	C₂	1	1	yes	maximal, characteristic, complemented
15	C₇×#1579	C₂	1	1	yes	maximal, characteristic, complemented
16	C₇×#559	C₂²	1	1	yes	characteristic
17	C₇×Q₈²	C₂²	1	1	yes	characteristic
18	C₇×#554	C₂²	1	1	yes	characteristic
19	C₇×D₄×Q₈	C₂²	1	1	yes	characteristic
20	C₇×#498	—	2	1	no	—
21	C₇×#498	C₂²	1	1	yes	characteristic
22	C₇×#494	C₂²	1	1	yes	characteristic
23	C₇×#489	—	2	1	no	—
24	C₇×#488	—	2	1	no	—
25	C₇×#487	C₂²	1	1	yes	characteristic
26	C₇×#486	C₂²	1	1	yes	characteristic
27	C₇×#486	C₂²	1	1	yes	characteristic
28	C₇×#486	C₂²	1	1	yes	characteristic
29	C₇×#484	—	2	1	no	—
30	C₇×#478	—	2	1	no	—
31	C₇×#477	C₂²	1	1	yes	characteristic
32	C₇×#477	C₂²	1	1	yes	characteristic
33	C₇×#477	C₂²	1	1	yes	characteristic
34	C₇×#476	—	2	1	no	—
35	C₇×#476	C₂²	1	1	yes	characteristic
36	C₇×#475	—	2	1	no	—
37	C₇×#475	C₂²	1	1	yes	characteristic
38	C₇×#474	—	2	1	no	—
39	C₇×#467	—	2	1	no	—
40	C₇×#464	—	2	1	no	—
41	C₇×#463	—	2	1	no	—
42	C₇×#462	C₂²	1	1	yes	characteristic
43	C₇×#460	C₂²	1	1	yes	characteristic
44	C₇×#459	C₂²	1	1	yes	characteristic
45	C₇×#451	—	2	1	no	—
46	C₇×#446	C₂²	1	1	yes	characteristic
47	C₇×C₈×Q₈	C₂²	1	1	yes	characteristic
48	C₇×C₈×Q₈	C₂²	1	1	yes	characteristic
49	C₇×#440	—	2	1	no	—
50	C₄×C₇×Q₁₆	C₂²	1	1	yes	characteristic
51	C₄×C₇×SD₁₆	—	2	1	no	—
52	C₄×C₇×SD₁₆	C₂²	1	1	yes	characteristic
53	C₄×C₇×D₈	—	2	1	no	—
54	C₇×C₈×D₄	C₂²	1	1	yes	characteristic
55	C₂×C₇×#103	—	2	1	no	—
56	C₇×#334	C₂²	1	1	yes	characteristic
57	C₇×#334	C₂²	1	1	yes	characteristic
58	C₇×#333	C₂²	1	1	yes	characteristic
59	C₇×#333	C₂²	1	1	yes	characteristic
60	C₇×#332	C₂²	1	1	yes	characteristic
61	C₇×#331	C₂²	1	1	yes	characteristic
62	C₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic
63	C₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic
64	C₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic
65	C₇×D₄⋊C₈	C₂²	1	1	yes	characteristic
66	C₇×#322	C₂²	1	1	yes	characteristic
67	#19798	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂²×C₇×Q₈	—	2	1	no	—
69	C₂×C₇×Q₁₆	—	2	1	no	—
70	C₂×C₇×Q₁₆	—	4	1	no	—
71	C₂×C₇×Q₁₆	D₄	1	1	yes	characteristic
72	C₂×C₇×SD₁₆	—	2	1	no	—
73	C₂×C₇×SD₁₆	D₄	1	1	yes	characteristic
74	C₂×C₇×D₈	—	2	1	no	—
75	C₂²×C₇×C₈	—	2	1	no	—
76	C₇×C₄⋊Q₈	—	2	1	no	—
77	C₇×C₄⋊Q₈	—	2	1	no	—
78	C₇×C₄⋊Q₈	—	2	1	no	—
79	C₇×C₄⋊Q₈	—	4	1	no	—
80	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
81	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
82	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
83	C₇×#127	C₂³	1	1	yes	characteristic
84	C₇×#125	—	2	1	no	—
85	C₇×#124	—	2	1	no	—
86	C₇×#124	—	2	1	no	—
87	C₇×#123	—	4	1	no	—
88	C₇×C₂²⋊Q₈	—	2	1	no	—
89	C₇×C₂²⋊Q₈	—	2	1	no	—
90	C₇×C₂²⋊Q₈	—	4	1	no	—
91	C₇×C₂²⋊Q₈	—	4	1	no	—
92	C₄×C₇×Q₈	—	2	1	no	—
93	C₄×C₇×Q₈	—	2	1	no	—
94	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
95	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
96	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
97	C₄×C₇×D₄	—	2	1	no	—
98	C₄×C₇×D₄	—	2	1	no	—
99	C₄×C₇×D₄	—	2	1	no	—
100	C₄×C₇×D₄	—	2	1	no	—
101	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
102	C₂×C₇×C₄⋊C₄	—	2	1	no	—
103	C₇×#107	—	2	1	no	—
104	C₇×#107	—	2	1	no	—
105	C₇×#107	—	2	1	no	—
106	C₇×#107	D₄	1	1	yes	characteristic
107	C₇×#106	—	2	1	no	—
108	C₇×#106	—	2	1	no	—
109	C₇×#106	—	2	1	no	—
110	C₇×#106	—	2	1	no	—
111	C₇×#106	D₄	1	1	yes	characteristic
112	C₇×C₄⋊C₈	—	2	1	no	—
113	C₇×C₄⋊C₈	—	2	1	no	—
114	C₇×C₄⋊C₈	—	2	1	no	—
115	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
116	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
117	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
118	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
119	C₇×#103	—	2	1	no	—
120	C₇×#103	—	2	1	no	—
121	C₇×#103	—	2	1	no	—
122	C₇×#103	—	2	1	no	—
123	C₇×#103	—	2	1	no	—
124	C₇×#103	—	2	1	no	—
125	C₇×#103	—	2	1	no	—
126	C₇×#103	D₄	1	1	yes	characteristic
127	C₇×#103	D₄	1	1	yes	characteristic
128	C₇×#103	D₄	1	1	yes	characteristic
129	C₇×#102	—	2	1	no	—
130	C₇×#102	—	2	1	no	—
131	C₇×#102	—	2	1	no	—
132	C₇×#102	—	2	1	no	—
133	C₇×#102	—	2	1	no	—
134	C₇×#102	—	2	1	no	—
135	C₇×#102	—	2	1	no	—
136	C₇×#102	—	4	1	no	—
137	C₇×#102	D₄	1	1	yes	characteristic
138	C₇×C₂²⋊C₈	—	2	1	no	—
139	C₇×#97	—	2	1	no	—
140	C₄×C₇×C₈	—	2	1	no	—
141	C₄×C₇×C₈	—	2	1	no	—
142	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
143	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
144	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
145	Q₈×SD₁₆	C₂×C₇	1	1	yes	characteristic, complemented
146	#3375	C₂×C₇	1	1	yes	characteristic, complemented
147	#3361	C₂×C₇	1	1	yes	characteristic
148	#3299	C₂×C₇	1	1	yes	characteristic, complemented
149	#1720	C₂×C₇	1	1	yes	characteristic, complemented
150	#1704	C₂×C₇	1	1	yes	characteristic, complemented
151	#1678	C₂×C₇	1	1	yes	characteristic, complemented
152	#1677	C₂×C₇	1	1	yes	characteristic
153	#1672	C₂×C₇	1	1	yes	characteristic, complemented
154	#1640	C₂×C₇	1	1	yes	characteristic, complemented
155	#1633	C₂×C₇	1	1	yes	characteristic, complemented
156	#1632	C₂×C₇	1	1	yes	characteristic, complemented
157	#1627	C₂×C₇	1	1	yes	characteristic
158	#1582	C₂×C₇	1	1	yes	characteristic, complemented
159	#1579	C₂×C₇	1	1	yes	characteristic, complemented
160	C₂×C₇×Q₈	—	2	1	no	—
161	C₂×C₇×Q₈	—	2	1	no	—
162	C₂×C₇×Q₈	—	2	1	no	—
163	C₂×C₇×Q₈	—	4	1	no	—
164	C₂×C₇×Q₈	—	4	1	no	—
165	C₂×C₇×Q₈	—	4	2	no	—
166	C₂×C₇×Q₈	—	8	1	no	—
167	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
168	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
169	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
170	C₂×C₇×D₄	—	2	1	no	—
171	C₂×C₇×D₄	—	2	1	no	—
172	C₂×C₇×D₄	—	2	1	no	—
173	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₇	—	2	1	no	—
175	C₂²×C₄×C₇	—	4	1	no	—
176	C₂²×C₄×C₇	—	4	1	no	—
177	C₂²×C₄×C₇	—	4	1	no	—
178	C₇×Q₁₆	—	2	2	no	—
179	C₇×Q₁₆	—	8	1	no	—
180	C₇×Q₁₆	—	8	2	no	—
181	C₇×SD₁₆	—	2	2	no	—
182	C₇×SD₁₆	—	4	2	no	—
183	C₇×D₈	—	4	2	no	—
184	C₂×C₇×C₈	—	2	1	no	—
185	C₂×C₇×C₈	—	2	1	no	—
186	C₂×C₇×C₈	—	2	1	no	—
187	C₂×C₇×C₈	—	2	1	no	—
188	C₂×C₇×C₈	—	2	1	no	—
189	C₂×C₇×C₈	—	8	1	no	—
190	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
191	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
192	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
193	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
194	C₇×C₄⋊C₄	—	2	1	no	—
195	C₇×C₄⋊C₄	—	2	1	no	—
196	C₇×C₄⋊C₄	—	2	1	no	—
197	C₇×C₄⋊C₄	—	2	1	no	—
198	C₇×C₄⋊C₄	—	2	1	no	—
199	C₇×C₄⋊C₄	—	2	1	no	—
200	C₇×C₄⋊C₄	—	2	1	no	—
201	C₇×C₄⋊C₄	—	2	1	no	—
202	C₇×C₄⋊C₄	—	2	1	no	—
203	C₇×C₄⋊C₄	—	4	1	no	—
204	C₇×C₄⋊C₄	—	4	1	no	—
205	C₇×C₄⋊C₄	—	4	1	no	—
206	C₇×C₄⋊C₄	—	4	1	no	—
207	C₇×C₄⋊C₄	—	4	1	no	—
208	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
209	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
210	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
211	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
212	C₇×C₂²⋊C₄	—	2	1	no	—
213	C₇×C₂²⋊C₄	—	4	1	no	—
214	C₇×C₂²⋊C₄	—	4	1	no	—
215	C₇×C₂²⋊C₄	—	4	1	no	—
216	C₇×C₂²⋊C₄	—	4	1	no	—
217	C₇×C₂²⋊C₄	—	4	1	no	—
218	C₄²×C₇	—	2	1	no	—
219	C₄²×C₇	—	2	1	no	—
220	C₄²×C₇	—	2	1	no	—
221	C₄²×C₇	—	4	1	no	—
222	C₄²×C₇	C₂⁴	1	1	yes	characteristic
223	#559	C₂²×C₇	1	1	yes	characteristic
224	Q₈²	C₂²×C₇	1	1	yes	characteristic
225	#554	C₂²×C₇	1	1	yes	characteristic
226	D₄×Q₈	C₂²×C₇	1	1	yes	characteristic
227	#498	—	2	1	no	—
228	#498	C₂²×C₇	1	1	yes	characteristic
229	#494	C₂²×C₇	1	1	yes	characteristic
230	#489	—	2	1	no	—
231	#488	—	2	1	no	—
232	#487	C₂²×C₇	1	1	yes	characteristic
233	#486	C₂²×C₇	1	1	yes	characteristic
234	#486	C₂²×C₇	1	1	yes	characteristic
235	#486	C₂²×C₇	1	1	yes	characteristic
236	#484	—	2	1	no	—
237	#478	—	2	1	no	—
238	#477	C₂²×C₇	1	1	yes	characteristic
239	#477	C₂²×C₇	1	1	yes	characteristic
240	#477	C₂²×C₇	1	1	yes	characteristic
241	#476	—	2	1	no	—
242	#476	C₂²×C₇	1	1	yes	characteristic
243	#475	—	2	1	no	—
244	#475	C₂²×C₇	1	1	yes	characteristic
245	#474	—	2	1	no	—
246	#467	—	2	1	no	—
247	#464	—	2	1	no	—
248	#463	—	2	1	no	—
249	#462	C₂²×C₇	1	1	yes	characteristic
250	#460	C₂²×C₇	1	1	yes	characteristic
251	#459	C₂²×C₇	1	1	yes	characteristic
252	#451	—	2	1	no	—
253	#446	C₂²×C₇	1	1	yes	characteristic
254	C₈×Q₈	C₂²×C₇	1	1	yes	characteristic
255	C₈×Q₈	C₂²×C₇	1	1	yes	characteristic
256	#440	—	2	1	no	—
257	C₄×Q₁₆	C₂²×C₇	1	1	yes	characteristic
258	C₄×SD₁₆	—	2	1	no	—
259	C₄×SD₁₆	C₂²×C₇	1	1	yes	characteristic
260	C₄×D₈	—	2	1	no	—
261	C₈×D₄	C₂²×C₇	1	1	yes	characteristic
262	C₂×#103	—	2	1	no	—
263	#334	C₂²×C₇	1	1	yes	characteristic
264	#334	C₂²×C₇	1	1	yes	characteristic
265	#333	C₂²×C₇	1	1	yes	characteristic
266	#333	C₂²×C₇	1	1	yes	characteristic
267	#332	C₂²×C₇	1	1	yes	characteristic
268	#331	C₂²×C₇	1	1	yes	characteristic
269	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
270	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
271	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
272	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
273	#322	C₂²×C₇	1	1	yes	characteristic
274	C₂³×C₇	—	2	1	no	—
275	C₂³×C₇	—	4	1	no	—
276	C₇×Q₈	—	2	1	no	—
277	C₇×Q₈	—	2	1	no	—
278	C₇×Q₈	—	2	1	no	—
279	C₇×Q₈	—	2	1	no	—
280	C₇×Q₈	—	2	1	no	—
281	C₇×Q₈	—	4	1	no	—
282	C₇×Q₈	—	4	1	no	—
283	C₇×Q₈	—	8	2	no	—
284	C₇×Q₈	—	8	1	no	—
285	C₇×Q₈	—	8	1	no	—
286	C₇×Q₈	—	16	1	no	—
287	C₇×Q₈	#135	1	2	yes	—
288	C₇×D₄	—	2	1	no	—
289	C₇×D₄	—	4	1	no	—
290	C₇×D₄	—	4	1	no	—
291	C₇×D₄	—	4	1	no	—
292	C₇×D₄	—	4	1	no	—
293	C₇×D₄	—	8	1	no	—
294	C₇×D₄	#137	1	2	yes	—
295	C₂×C₄×C₇	—	2	1	no	—
296	C₂×C₄×C₇	—	2	1	no	—
297	C₂×C₄×C₇	—	2	1	no	—
298	C₂×C₄×C₇	—	2	1	no	—
299	C₂×C₄×C₇	—	2	1	no	—
300	C₂×C₄×C₇	—	2	1	no	—
301	C₂×C₄×C₇	—	2	1	no	—
302	C₂×C₄×C₇	—	4	1	no	—
303	C₂×C₄×C₇	—	4	1	no	—
304	C₂×C₄×C₇	—	4	1	no	—
305	C₂×C₄×C₇	—	4	1	no	—
306	C₂×C₄×C₇	—	4	1	no	—
307	C₂×C₄×C₇	—	8	2	no	—
308	C₂×C₄×C₇	—	16	1	no	—
309	C₂×C₄×C₇	—	16	1	no	—
310	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
311	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
312	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
313	C₇×C₈	—	2	1	no	—
314	C₇×C₈	—	2	1	no	—
315	C₇×C₈	—	2	1	no	—
316	C₇×C₈	—	2	1	no	—
317	C₇×C₈	—	4	1	no	—
318	C₇×C₈	—	4	1	no	—
319	C₇×C₈	—	4	1	no	—
320	C₇×C₈	—	4	1	no	—
321	C₇×C₈	—	4	1	no	—
322	C₂²×Q₈	—	2	1	no	—
323	C₂×Q₁₆	—	2	1	no	—
324	C₂×Q₁₆	—	4	1	no	—
325	C₂×Q₁₆	C₇×D₄	1	1	yes	characteristic
326	C₂×SD₁₆	—	2	1	no	—
327	C₂×SD₁₆	C₇×D₄	1	1	yes	characteristic
328	C₂×D₈	—	2	1	no	—
329	C₂²×C₈	—	2	1	no	—
330	C₄⋊Q₈	—	2	1	no	—
331	C₄⋊Q₈	—	2	1	no	—
332	C₄⋊Q₈	—	2	1	no	—
333	C₄⋊Q₈	—	4	1	no	—
334	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
335	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
336	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
337	#127	C₂³×C₇	1	1	yes	characteristic
338	#125	—	2	1	no	—
339	#124	—	2	1	no	—
340	#124	—	2	1	no	—
341	#123	—	4	1	no	—
342	C₂²⋊Q₈	—	2	1	no	—
343	C₂²⋊Q₈	—	2	1	no	—
344	C₂²⋊Q₈	—	4	1	no	—
345	C₂²⋊Q₈	—	4	1	no	—
346	C₄×Q₈	—	2	1	no	—
347	C₄×Q₈	—	2	1	no	—
348	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
349	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
350	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
351	C₄×D₄	—	2	1	no	—
352	C₄×D₄	—	2	1	no	—
353	C₄×D₄	—	2	1	no	—
354	C₄×D₄	—	2	1	no	—
355	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
356	C₂×C₄⋊C₄	—	2	1	no	—
357	#107	—	2	1	no	—
358	#107	—	2	1	no	—
359	#107	—	2	1	no	—
360	#107	C₇×D₄	1	1	yes	characteristic
361	#106	—	2	1	no	—
362	#106	—	2	1	no	—
363	#106	—	2	1	no	—
364	#106	—	2	1	no	—
365	#106	C₇×D₄	1	1	yes	characteristic
366	C₄⋊C₈	—	2	1	no	—
367	C₄⋊C₈	—	2	1	no	—
368	C₄⋊C₈	—	2	1	no	—
369	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
370	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
371	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
372	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
373	#103	—	2	1	no	—
374	#103	—	2	1	no	—
375	#103	—	2	1	no	—
376	#103	—	2	1	no	—
377	#103	—	2	1	no	—
378	#103	—	2	1	no	—
379	#103	—	2	1	no	—
380	#103	C₇×D₄	1	1	yes	characteristic
381	#103	C₇×D₄	1	1	yes	characteristic
382	#103	C₇×D₄	1	1	yes	characteristic
383	#102	—	2	1	no	—
384	#102	—	2	1	no	—
385	#102	—	2	1	no	—
386	#102	—	2	1	no	—
387	#102	—	2	1	no	—
388	#102	—	2	1	no	—
389	#102	—	2	1	no	—
390	#102	—	4	1	no	—
391	#102	C₇×D₄	1	1	yes	characteristic
392	C₂²⋊C₈	—	2	1	no	—
393	#97	—	2	1	no	—
394	C₄×C₈	—	2	1	no	—
395	C₄×C₈	—	2	1	no	—
396	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
397	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
398	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
399	C₂²×C₇	—	2	2	no	—
400	C₂²×C₇	—	4	1	no	—
401	C₂²×C₇	—	4	1	no	—
402	C₂²×C₇	—	8	1	no	—
403	C₂²×C₇	—	8	1	no	—
404	C₂²×C₇	—	8	1	no	—
405	C₂²×C₇	D₄²	1	1	yes	characteristic
406	C₄×C₇	—	2	2	no	—
407	C₄×C₇	—	2	1	no	—
408	C₄×C₇	—	4	1	no	—
409	C₄×C₇	—	4	1	no	—
410	C₄×C₇	—	4	1	no	—
411	C₄×C₇	—	4	1	no	—
412	C₄×C₇	—	4	1	no	—
413	C₄×C₇	—	4	1	no	—
414	C₄×C₇	—	8	1	no	—
415	C₄×C₇	—	8	1	no	—
416	C₄×C₇	—	8	1	no	—
417	C₄×C₇	C₂×#135	1	1	yes	characteristic
418	C₄×C₇	C₂×#137	1	1	yes	characteristic
419	C₄×C₇	#577	1	1	yes	characteristic
420	C₄×C₇	#578	1	1	yes	characteristic
421	C₂×Q₈	—	2	1	no	—
422	C₂×Q₈	—	2	1	no	—
423	C₂×Q₈	—	2	1	no	—
424	C₂×Q₈	—	4	1	no	—
425	C₂×Q₈	—	4	1	no	—
426	C₂×Q₈	—	4	2	no	—
427	C₂×Q₈	—	8	1	no	—
428	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
429	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
430	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
431	C₂×D₄	—	2	1	no	—
432	C₂×D₄	—	2	1	no	—
433	C₂×D₄	—	2	1	no	—
434	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
435	C₂²×C₄	—	2	1	no	—
436	C₂²×C₄	—	4	1	no	—
437	C₂²×C₄	—	4	1	no	—
438	C₂²×C₄	—	4	1	no	—
439	Q₁₆	—	2	2	no	—
440	Q₁₆	—	8	1	no	—
441	Q₁₆	—	8	2	no	—
442	SD₁₆	—	2	2	no	—
443	SD₁₆	—	4	2	no	—
444	D₈	—	4	2	no	—
445	C₂×C₈	—	2	1	no	—
446	C₂×C₈	—	2	1	no	—
447	C₂×C₈	—	2	1	no	—
448	C₂×C₈	—	2	1	no	—
449	C₂×C₈	—	2	1	no	—
450	C₂×C₈	—	8	1	no	—
451	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
452	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
453	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
454	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
455	C₄⋊C₄	—	2	1	no	—
456	C₄⋊C₄	—	2	1	no	—
457	C₄⋊C₄	—	2	1	no	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	—	2	1	no	—
462	C₄⋊C₄	—	2	1	no	—
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	—	4	1	no	—
465	C₄⋊C₄	—	4	1	no	—
466	C₄⋊C₄	—	4	1	no	—
467	C₄⋊C₄	—	4	1	no	—
468	C₄⋊C₄	—	4	1	no	—
469	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
470	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
471	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
472	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
473	C₂²⋊C₄	—	2	1	no	—
474	C₂²⋊C₄	—	4	1	no	—
475	C₂²⋊C₄	—	4	1	no	—
476	C₂²⋊C₄	—	4	1	no	—
477	C₂²⋊C₄	—	4	1	no	—
478	C₂²⋊C₄	—	4	1	no	—
479	C₄²	—	2	1	no	—
480	C₄²	—	2	1	no	—
481	C₄²	—	2	1	no	—
482	C₄²	—	4	1	no	—
483	C₄²	C₂⁴×C₇	1	1	yes	characteristic
484	C₂×C₇	—	4	2	no	—
485	C₂×C₇	—	16	1	no	—
486	C₂×C₇	#3270	1	1	yes	characteristic
487	C₂×C₇	#3282	1	1	yes	characteristic
488	C₂×C₇	#3288	1	1	yes	characteristic
489	C₂³	—	2	1	no	—
490	C₂³	—	4	1	no	—
491	Q₈	—	2	1	no	—
492	Q₈	—	2	1	no	—
493	Q₈	—	2	1	no	—
494	Q₈	—	2	1	no	—
495	Q₈	—	2	1	no	—
496	Q₈	—	4	1	no	—
497	Q₈	—	4	1	no	—
498	Q₈	—	8	2	no	—
499	Q₈	—	8	1	no	—
500	Q₈	—	8	1	no	—
501	Q₈	—	16	1	no	—
502	Q₈	C₇×#135	1	2	yes	—
503	D₄	—	2	1	no	—
504	D₄	—	4	1	no	—
505	D₄	—	4	1	no	—
506	D₄	—	4	1	no	—
507	D₄	—	4	1	no	—
508	D₄	—	8	1	no	—
509	D₄	C₇×#137	1	2	yes	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	4	1	no	—
518	C₂×C₄	—	4	1	no	—
519	C₂×C₄	—	4	1	no	—
520	C₂×C₄	—	4	1	no	—
521	C₂×C₄	—	4	1	no	—
522	C₂×C₄	—	8	2	no	—
523	C₂×C₄	—	16	1	no	—
524	C₂×C₄	—	16	1	no	—
525	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
526	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
527	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
528	C₈	—	2	1	no	—
529	C₈	—	2	1	no	—
530	C₈	—	2	1	no	—
531	C₈	—	2	1	no	—
532	C₈	—	4	1	no	—
533	C₈	—	4	1	no	—
534	C₈	—	4	1	no	—
535	C₈	—	4	1	no	—
536	C₈	—	4	1	no	—
537	C₇	#19798	1	1	yes	characteristic, direct product factor, Sylow
538	C₂²	—	2	2	no	—
539	C₂²	—	4	1	no	—
540	C₂²	—	4	1	no	—
541	C₂²	—	8	1	no	—
542	C₂²	—	8	1	no	—
543	C₂²	—	8	1	no	—
544	C₂²	C₇×D₄²	1	1	yes	characteristic
545	C₄	—	2	2	no	—
546	C₄	—	2	1	no	—
547	C₄	—	4	1	no	—
548	C₄	—	4	1	no	—
549	C₄	—	4	1	no	—
550	C₄	—	4	1	no	—
551	C₄	—	4	1	no	—
552	C₄	—	4	1	no	—
553	C₄	—	8	1	no	—
554	C₄	—	8	1	no	—
555	C₄	—	8	1	no	—
556	C₄	C₂×C₇×#135	1	1	yes	characteristic
557	C₄	C₂×C₇×#137	1	1	yes	characteristic
558	C₄	C₇×#577	1	1	yes	characteristic
559	C₄	C₇×#578	1	1	yes	characteristic
560	C₂	—	4	2	no	—
561	C₂	—	16	1	no	—
562	C₂	C₇×#3270	1	1	yes	characteristic
563	C₂	C₇×#3282	1	1	yes	characteristic
564	C₂	C₇×#3288	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₇×#459	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×Q₈×SD₁₆	C₂	semidirect
2. C₇×#3375	C₂	semidirect
4. C₇×#3299	C₂	semidirect
5. C₇×#1720	C₂	semidirect
6. C₇×#1704	C₂	semidirect
7. C₇×#1678	C₂	semidirect
9. C₇×#1672	C₂	semidirect
10. C₇×#1640	C₂	semidirect
11. C₇×#1633	C₂	semidirect
12. C₇×#1632	C₂	semidirect
14. C₇×#1582	C₂	semidirect
15. C₇×#1579	C₂	semidirect
67. #19798	C₇	direct (N × H)
145. Q₈×SD₁₆	C₂×C₇	semidirect
146. #3375	C₂×C₇	semidirect
148. #3299	C₂×C₇	semidirect
149. #1720	C₂×C₇	semidirect
150. #1704	C₂×C₇	semidirect
151. #1678	C₂×C₇	semidirect
153. #1672	C₂×C₇	semidirect
154. #1640	C₂×C₇	semidirect
155. #1633	C₂×C₇	semidirect
156. #1632	C₂×C₇	semidirect
158. #1582	C₂×C₇	semidirect
159. #1579	C₂×C₇	semidirect
537. C₇	#19798	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×Q₈×SD₁₆	2
C₇×#3375	2
C₇×#3361	2
C₇×#3299	2
C₇×#1720	2
C₇×#1704	2
C₇×#1678	2
C₇×#1677	2
C₇×#1672	2
C₇×#1640	2
C₇×#1633	2
C₇×#1632	2
C₇×#1627	2
C₇×#1582	2
C₇×#1579	2
#19798	7

Subgroups — C7×#19798 #13197014

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19798 #13197014