Subgroups — C₇×#22687 #13199903

Order 1792.

858 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#444	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#443	C₂	1	1	yes	maximal, characteristic
3	C₇×#2943	C₂	1	1	yes	maximal, characteristic
4	C₇×#2906	C₂	1	2	yes	maximal, complemented
5	C₇×#1946	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1945	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1866	C₂	1	2	yes	maximal, complemented
8	C₇×#1866	C₂	1	2	yes	maximal, complemented
9	C₇×#1849	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1806	C₂	1	2	yes	maximal, complemented
11	C₇×#1771	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×#135	C₂²	1	1	yes	characteristic
13	C₂×C₇×#131	C₂²	1	2	yes	—
14	C₂×C₇×#131	C₂²	1	2	yes	—
15	C₄×C₇×#41	C₂²	1	2	yes	—
16	C₇×#444	—	2	4	no	—
17	C₇×#443	—	2	4	no	—
18	C₇×#442	C₄	1	2	yes	complemented
19	C₇×#441	C₄	1	2	yes	complemented
20	C₇×#440	C₄	1	2	yes	complemented
21	C₇×#440	C₄	1	2	yes	complemented
22	C₂×C₇×#108	C₂²	1	1	yes	characteristic
23	C₂×C₇×#108	C₂²	1	1	yes	characteristic
24	C₂×C₇×#108	C₂²	1	2	yes	—
25	C₇×#427	C₂²	1	1	yes	characteristic
26	C₇×#424	—	2	2	no	—
27	C₇×#424	—	2	2	no	—
28	C₇×#424	C₂²	1	2	yes	—
29	C₇×#424	C₂²	1	2	yes	—
30	C₇×#423	—	2	2	no	—
31	C₇×#423	C₂²	1	2	yes	—
32	C₂×C₇×C₄⋊C₈	—	2	2	no	—
33	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₂×C₇×#104	C₂²	1	2	yes	—
35	C₂×C₇×#104	C₂²	1	2	yes	—
36	C₇×#416	C₂²	1	2	yes	—
37	C₂×C₇×#97	C₂²	1	1	yes	characteristic
38	C₂×C₇×#97	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
40	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
41	C₇×#339	C₂²	1	2	yes	—
42	C₇×#339	C₂²	1	2	yes	—
43	C₇×#336	C₂²	1	1	yes	characteristic
44	C₇×#336	C₂²	1	1	yes	characteristic
45	#22687	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂×C₇×#41	C₂³	1	2	yes	—
47	C₇×#135	D₄	1	4	yes	—
48	C₇×#135	Q₈	1	4	yes	—
49	C₂×C₇×Q₁₆	C₂×C₄	1	1	yes	characteristic
50	C₂×C₇×SD₁₆	C₂×C₄	1	2	yes	—
51	C₂×C₇×D₈	C₂×C₄	1	1	yes	characteristic
52	C₇×#131	—	2	4	no	—
53	C₇×#131	—	2	4	no	—
54	C₇×#131	—	4	2	no	—
55	C₇×#131	—	4	2	no	—
56	C₂×C₇×M₁₆	—	2	2	no	—
57	C₂×C₇×M₁₆	—	2	2	no	—
58	C₂×C₇×M₁₆	C₂³	1	2	yes	—
59	C₂×C₇×M₁₆	C₂³	1	2	yes	—
60	C₂²×C₇×C₈	—	2	2	no	—
61	C₂²×C₇×C₈	—	2	2	no	—
62	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
63	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
64	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
65	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
66	C₄×C₇×Q₈	C₂×C₄	1	2	yes	—
67	C₄×C₇×Q₈	C₂×C₄	1	2	yes	—
68	C₄×C₇×D₄	—	2	2	no	—
69	C₄×C₇×D₄	—	2	2	no	—
70	C₄×C₇×D₄	C₂×C₄	1	2	yes	—
71	C₄×C₇×D₄	C₂×C₄	1	2	yes	—
72	C₇×#117	—	2	2	no	—
73	C₇×#117	C₂³	1	2	yes	—
74	C₂×C₇×C₄⋊C₄	—	2	1	no	—
75	C₂×C₄²×C₇	—	2	2	no	—
76	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
79	C₇×#108	—	2	2	no	—
80	C₇×#108	—	2	2	no	—
81	C₇×#108	—	4	2	no	—
82	C₇×#107	C₂×C₄	1	1	yes	characteristic
83	C₇×#107	C₂×C₄	1	1	yes	characteristic
84	C₇×#106	C₂×C₄	1	2	yes	—
85	C₇×C₄⋊C₈	—	2	2	no	—
86	C₇×C₄⋊C₈	—	2	2	no	—
87	C₇×C₄⋊C₈	—	2	2	no	—
88	C₇×C₄⋊C₈	—	2	2	no	—
89	C₇×C₄⋊C₈	—	2	2	no	—
90	C₇×C₄⋊C₈	—	2	2	no	—
91	C₇×C₄⋊C₈	D₄	1	2	yes	—
92	C₇×C₄⋊C₈	D₄	1	2	yes	—
93	C₇×#104	—	2	4	no	—
94	C₇×#104	—	2	4	no	—
95	C₇×#103	C₂×C₄	1	2	yes	—
96	C₇×#103	C₂×C₄	1	2	yes	—
97	C₇×#102	C₂×C₄	1	2	yes	—
98	C₇×#102	C₂×C₄	1	2	yes	—
99	C₇×#97	—	2	2	no	—
100	C₇×#97	C₂×C₄	1	2	yes	—
101	C₇×#97	C₂×C₄	1	2	yes	—
102	C₄×C₇×C₈	—	2	2	no	—
103	C₇×#95	—	2	1	no	—
104	C₂×#444	C₂×C₇	1	1	yes	characteristic
105	C₂×#443	C₂×C₇	1	1	yes	characteristic
106	#2943	C₂×C₇	1	1	yes	characteristic
107	#2906	C₂×C₇	1	2	yes	complemented
108	#1946	C₂×C₇	1	1	yes	characteristic, complemented
109	#1945	C₂×C₇	1	1	yes	characteristic, complemented
110	#1866	C₂×C₇	1	2	yes	complemented
111	#1866	C₂×C₇	1	2	yes	complemented
112	#1849	C₂×C₇	1	1	yes	characteristic, complemented
113	#1806	C₂×C₇	1	2	yes	complemented
114	#1771	C₂×C₇	1	1	yes	characteristic, complemented
115	C₇×#41	—	2	2	no	—
116	C₇×#41	—	2	2	no	—
117	C₇×#41	C₂×D₄	1	4	yes	—
118	C₇×#41	C₂×Q₈	1	4	yes	—
119	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
120	C₂×C₇×D₄	—	2	2	no	—
121	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
122	C₂²×C₄×C₇	—	2	2	no	—
123	C₂²×C₄×C₇	—	2	1	no	—
124	C₂²×C₄×C₇	—	2	2	no	—
125	C₂²×C₄×C₇	—	2	2	no	—
126	C₂²×C₄×C₇	—	2	1	no	—
127	C₂²×C₄×C₇	#41	1	1	yes	characteristic
128	C₂²×C₄×C₇	#41	1	1	yes	characteristic
129	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
130	C₇×Q₁₆	C₄⋊C₄	1	4	yes	complemented
131	C₇×SD₁₆	C₄⋊C₄	1	8	yes	complemented
132	C₇×D₈	C₄⋊C₄	1	4	yes	complemented
133	C₇×M₁₆	—	2	2	no	—
134	C₇×M₁₆	—	2	2	no	—
135	C₇×M₁₆	—	2	2	no	—
136	C₇×M₁₆	—	2	2	no	—
137	C₇×M₁₆	—	4	4	no	—
138	C₇×M₁₆	—	4	4	no	—
139	C₂×C₇×C₈	—	2	1	no	—
140	C₂×C₇×C₈	—	2	1	no	—
141	C₂×C₇×C₈	—	2	1	no	—
142	C₂×C₇×C₈	—	2	1	no	—
143	C₂×C₇×C₈	—	2	1	no	—
144	C₂×C₇×C₈	—	2	1	no	—
145	C₂×C₇×C₈	—	2	2	no	—
146	C₂×C₇×C₈	—	2	2	no	—
147	C₂×C₇×C₈	—	4	4	no	—
148	C₂×C₇×C₈	—	4	4	no	—
149	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
150	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
151	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
152	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
153	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
154	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
155	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
156	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
157	C₇×C₄⋊C₄	—	2	2	no	—
158	C₇×C₄⋊C₄	—	2	2	no	—
159	C₇×C₄⋊C₄	—	4	2	no	—
160	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
161	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
162	C₇×C₂²⋊C₄	—	2	2	no	—
163	C₇×C₂²⋊C₄	—	2	2	no	—
164	C₇×C₂²⋊C₄	—	2	2	no	—
165	C₄²×C₇	—	2	2	no	—
166	C₄²×C₇	—	2	2	no	—
167	C₄²×C₇	—	2	2	no	—
168	C₄²×C₇	—	2	1	no	—
169	C₄²×C₇	—	2	1	no	—
170	C₄²×C₇	—	2	1	no	—
171	C₄²×C₇	—	2	1	no	—
172	C₄²×C₇	C₂²×C₄	1	2	yes	—
173	C₄²×C₇	C₂×D₄	1	2	yes	—
174	C₂×#135	C₂²×C₇	1	1	yes	characteristic
175	C₂×#131	C₂²×C₇	1	2	yes	—
176	C₂×#131	C₂²×C₇	1	2	yes	—
177	C₄×#41	C₂²×C₇	1	2	yes	—
178	#444	—	2	4	no	—
179	#443	—	2	4	no	—
180	#442	C₄×C₇	1	2	yes	complemented
181	#441	C₄×C₇	1	2	yes	complemented
182	#440	C₄×C₇	1	2	yes	complemented
183	#440	C₄×C₇	1	2	yes	complemented
184	C₂×#108	C₂²×C₇	1	1	yes	characteristic
185	C₂×#108	C₂²×C₇	1	1	yes	characteristic
186	C₂×#108	C₂²×C₇	1	2	yes	—
187	#427	C₂²×C₇	1	1	yes	characteristic
188	#424	—	2	2	no	—
189	#424	—	2	2	no	—
190	#424	C₂²×C₇	1	2	yes	—
191	#424	C₂²×C₇	1	2	yes	—
192	#423	—	2	2	no	—
193	#423	C₂²×C₇	1	2	yes	—
194	C₂×C₄⋊C₈	—	2	2	no	—
195	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
196	C₂×#104	C₂²×C₇	1	2	yes	—
197	C₂×#104	C₂²×C₇	1	2	yes	—
198	#416	C₂²×C₇	1	2	yes	—
199	C₂×#97	C₂²×C₇	1	1	yes	characteristic
200	C₂×#97	C₂²×C₇	1	1	yes	characteristic
201	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
202	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
203	#339	C₂²×C₇	1	2	yes	—
204	#339	C₂²×C₇	1	2	yes	—
205	#336	C₂²×C₇	1	1	yes	characteristic
206	#336	C₂²×C₇	1	1	yes	characteristic
207	C₂³×C₇	—	2	2	no	—
208	C₂³×C₇	C₂×#41	1	1	yes	characteristic
209	C₇×Q₈	—	2	2	no	—
210	C₇×Q₈	C₂×C₄⋊C₄	1	4	yes	—
211	C₇×D₄	—	2	4	no	—
212	C₇×D₄	—	2	4	no	—
213	C₇×D₄	—	2	2	no	—
214	C₇×D₄	C₂×C₄⋊C₄	1	4	yes	—
215	C₂×C₄×C₇	—	2	2	no	—
216	C₂×C₄×C₇	—	2	4	no	—
217	C₂×C₄×C₇	—	2	4	no	—
218	C₂×C₄×C₇	—	2	2	no	—
219	C₂×C₄×C₇	—	2	2	no	—
220	C₂×C₄×C₇	—	2	2	no	—
221	C₂×C₄×C₇	—	4	1	no	—
222	C₂×C₄×C₇	—	4	1	no	—
223	C₂×C₄×C₇	—	4	4	no	—
224	C₂×C₄×C₇	—	4	1	no	—
225	C₂×C₄×C₇	—	4	1	no	—
226	C₂×C₄×C₇	—	4	1	no	—
227	C₂×C₄×C₇	—	4	1	no	—
228	C₂×C₄×C₇	—	4	1	no	—
229	C₂×C₄×C₇	—	8	2	no	—
230	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
231	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
232	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
233	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
234	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
235	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
236	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
237	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
238	C₇×C₈	—	2	2	no	—
239	C₇×C₈	—	2	1	no	—
240	C₇×C₈	—	2	2	no	—
241	C₇×C₈	—	2	1	no	—
242	C₇×C₈	—	4	2	no	—
243	C₇×C₈	—	4	2	no	—
244	C₇×C₈	C₂×C₄⋊C₄	1	2	yes	—
245	C₇×C₈	C₂×C₄⋊C₄	1	2	yes	—
246	C₂×#41	C₂³×C₇	1	2	yes	—
247	#135	C₇×D₄	1	4	yes	—
248	#135	C₇×Q₈	1	4	yes	—
249	C₂×Q₁₆	C₂×C₄×C₇	1	1	yes	characteristic
250	C₂×SD₁₆	C₂×C₄×C₇	1	2	yes	—
251	C₂×D₈	C₂×C₄×C₇	1	1	yes	characteristic
252	#131	—	2	4	no	—
253	#131	—	2	4	no	—
254	#131	—	4	2	no	—
255	#131	—	4	2	no	—
256	C₂×M₁₆	—	2	2	no	—
257	C₂×M₁₆	—	2	2	no	—
258	C₂×M₁₆	C₂³×C₇	1	2	yes	—
259	C₂×M₁₆	C₂³×C₇	1	2	yes	—
260	C₂²×C₈	—	2	2	no	—
261	C₂²×C₈	—	2	2	no	—
262	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
263	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
264	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
265	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
266	C₄×Q₈	C₂×C₄×C₇	1	2	yes	—
267	C₄×Q₈	C₂×C₄×C₇	1	2	yes	—
268	C₄×D₄	—	2	2	no	—
269	C₄×D₄	—	2	2	no	—
270	C₄×D₄	C₂×C₄×C₇	1	2	yes	—
271	C₄×D₄	C₂×C₄×C₇	1	2	yes	—
272	#117	—	2	2	no	—
273	#117	C₂³×C₇	1	2	yes	—
274	C₂×C₄⋊C₄	—	2	1	no	—
275	C₂×C₄²	—	2	2	no	—
276	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
277	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
278	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
279	#108	—	2	2	no	—
280	#108	—	2	2	no	—
281	#108	—	4	2	no	—
282	#107	C₂×C₄×C₇	1	1	yes	characteristic
283	#107	C₂×C₄×C₇	1	1	yes	characteristic
284	#106	C₂×C₄×C₇	1	2	yes	—
285	C₄⋊C₈	—	2	2	no	—
286	C₄⋊C₈	—	2	2	no	—
287	C₄⋊C₈	—	2	2	no	—
288	C₄⋊C₈	—	2	2	no	—
289	C₄⋊C₈	—	2	2	no	—
290	C₄⋊C₈	—	2	2	no	—
291	C₄⋊C₈	C₇×D₄	1	2	yes	—
292	C₄⋊C₈	C₇×D₄	1	2	yes	—
293	#104	—	2	4	no	—
294	#104	—	2	4	no	—
295	#103	C₂×C₄×C₇	1	2	yes	—
296	#103	C₂×C₄×C₇	1	2	yes	—
297	#102	C₂×C₄×C₇	1	2	yes	—
298	#102	C₂×C₄×C₇	1	2	yes	—
299	#97	—	2	2	no	—
300	#97	C₂×C₄×C₇	1	2	yes	—
301	#97	C₂×C₄×C₇	1	2	yes	—
302	C₄×C₈	—	2	2	no	—
303	#95	—	2	1	no	—
304	C₂²×C₇	—	2	1	no	—
305	C₂²×C₇	—	2	1	no	—
306	C₂²×C₇	—	2	4	no	—
307	C₂²×C₇	—	4	2	no	—
308	C₂²×C₇	—	4	2	no	—
309	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
310	C₂²×C₇	#547	1	1	yes	characteristic
311	C₂²×C₇	#554	1	1	yes	characteristic
312	C₄×C₇	—	2	4	no	—
313	C₄×C₇	—	2	2	no	—
314	C₄×C₇	—	2	2	no	—
315	C₄×C₇	—	4	2	no	—
316	C₄×C₇	—	4	2	no	—
317	C₄×C₇	—	4	2	no	—
318	C₄×C₇	—	8	1	no	—
319	C₄×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
320	C₄×C₇	#520	1	1	yes	characteristic
321	C₄×C₇	D₄²	1	1	yes	characteristic
322	C₄×C₇	D₄×Q₈	1	1	yes	characteristic
323	#41	—	2	2	no	—
324	#41	—	2	2	no	—
325	#41	C₂×C₇×D₄	1	4	yes	—
326	#41	C₂×C₇×Q₈	1	4	yes	—
327	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
328	C₂×D₄	—	2	2	no	—
329	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
330	C₂²×C₄	—	2	2	no	—
331	C₂²×C₄	—	2	1	no	—
332	C₂²×C₄	—	2	2	no	—
333	C₂²×C₄	—	2	2	no	—
334	C₂²×C₄	—	2	1	no	—
335	C₂²×C₄	C₇×#41	1	1	yes	characteristic
336	C₂²×C₄	C₇×#41	1	1	yes	characteristic
337	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
338	Q₁₆	C₇×C₄⋊C₄	1	4	yes	complemented
339	SD₁₆	C₇×C₄⋊C₄	1	8	yes	complemented
340	D₈	C₇×C₄⋊C₄	1	4	yes	complemented
341	M₁₆	—	2	2	no	—
342	M₁₆	—	2	2	no	—
343	M₁₆	—	2	2	no	—
344	M₁₆	—	2	2	no	—
345	M₁₆	—	4	4	no	—
346	M₁₆	—	4	4	no	—
347	C₂×C₈	—	2	1	no	—
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	2	1	no	—
350	C₂×C₈	—	2	1	no	—
351	C₂×C₈	—	2	1	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	—	2	2	no	—
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	—	4	4	no	—
356	C₂×C₈	—	4	4	no	—
357	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
358	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
359	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
360	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
361	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
362	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
363	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
364	C₂×C₈	C₂×C₇×Q₈	1	2	yes	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	4	2	no	—
368	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
369	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
370	C₂²⋊C₄	—	2	2	no	—
371	C₂²⋊C₄	—	2	2	no	—
372	C₂²⋊C₄	—	2	2	no	—
373	C₄²	—	2	2	no	—
374	C₄²	—	2	2	no	—
375	C₄²	—	2	2	no	—
376	C₄²	—	2	1	no	—
377	C₄²	—	2	1	no	—
378	C₄²	—	2	1	no	—
379	C₄²	—	2	1	no	—
380	C₄²	C₂²×C₄×C₇	1	2	yes	—
381	C₄²	C₂×C₇×D₄	1	2	yes	—
382	C₂×C₇	—	2	1	no	—
383	C₂×C₇	—	2	1	no	—
384	C₂×C₇	—	4	4	no	—
385	C₂×C₇	D₄×C₄⋊C₄	1	1	yes	characteristic
386	C₂×C₇	#2955	1	1	yes	characteristic
387	C₂×C₇	#2956	1	1	yes	characteristic
388	C₂³	—	2	2	no	—
389	C₂³	C₂×C₇×#41	1	1	yes	characteristic
390	Q₈	—	2	2	no	—
391	Q₈	C₂×C₇×C₄⋊C₄	1	4	yes	—
392	D₄	—	2	4	no	—
393	D₄	—	2	4	no	—
394	D₄	—	2	2	no	—
395	D₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	4	no	—
398	C₂×C₄	—	2	4	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	4	1	no	—
403	C₂×C₄	—	4	1	no	—
404	C₂×C₄	—	4	4	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	1	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	8	2	no	—
411	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
412	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
413	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
414	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
415	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
416	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
417	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
418	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
419	C₈	—	2	2	no	—
420	C₈	—	2	1	no	—
421	C₈	—	2	2	no	—
422	C₈	—	2	1	no	—
423	C₈	—	4	2	no	—
424	C₈	—	4	2	no	—
425	C₈	C₂×C₇×C₄⋊C₄	1	2	yes	—
426	C₈	C₂×C₇×C₄⋊C₄	1	2	yes	—
427	C₇	#22687	1	1	yes	characteristic, direct product factor, Sylow
428	C₂²	—	2	1	no	—
429	C₂²	—	2	1	no	—
430	C₂²	—	2	4	no	—
431	C₂²	—	4	2	no	—
432	C₂²	—	4	2	no	—
433	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
434	C₂²	C₇×#547	1	1	yes	characteristic
435	C₂²	C₇×#554	1	1	yes	characteristic
436	C₄	—	2	2	no	—
437	C₄	—	2	2	no	—
438	C₄	—	2	4	no	—
439	C₄	—	4	2	no	—
440	C₄	—	4	2	no	—
441	C₄	—	4	2	no	—
442	C₄	—	8	1	no	—
443	C₄	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
444	C₄	C₇×#520	1	1	yes	characteristic
445	C₄	C₇×D₄²	1	1	yes	characteristic
446	C₄	C₇×D₄×Q₈	1	1	yes	characteristic
447	C₂	—	2	1	no	—
448	C₂	—	2	1	no	—
449	C₂	—	4	4	no	—
450	C₂	C₇×D₄×C₄⋊C₄	1	1	yes	characteristic
451	C₂	C₇×#2955	1	1	yes	characteristic
452	C₂	C₇×#2956	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₇×#2906	C₂	semidirect
5. C₇×#1946	C₂	semidirect
6. C₇×#1945	C₂	semidirect
7. C₇×#1866	C₂	semidirect
8. C₇×#1866	C₂	semidirect
9. C₇×#1849	C₂	semidirect
10. C₇×#1806	C₂	semidirect
11. C₇×#1771	C₂	semidirect
18. C₇×#442	C₄	semidirect
19. C₇×#441	C₄	semidirect
20. C₇×#440	C₄	semidirect
21. C₇×#440	C₄	semidirect
45. #22687	C₇	direct (N × H)
107. #2906	C₂×C₇	semidirect
108. #1946	C₂×C₇	semidirect
109. #1945	C₂×C₇	semidirect
110. #1866	C₂×C₇	semidirect
111. #1866	C₂×C₇	semidirect
112. #1849	C₂×C₇	semidirect
113. #1806	C₂×C₇	semidirect
114. #1771	C₂×C₇	semidirect
130. C₇×Q₁₆	C₄⋊C₄	semidirect
131. C₇×SD₁₆	C₄⋊C₄	semidirect
132. C₇×D₈	C₄⋊C₄	semidirect
180. #442	C₄×C₇	semidirect
181. #441	C₄×C₇	semidirect
182. #440	C₄×C₇	semidirect
183. #440	C₄×C₇	semidirect
338. Q₁₆	C₇×C₄⋊C₄	semidirect
339. SD₁₆	C₇×C₄⋊C₄	semidirect
340. D₈	C₇×C₄⋊C₄	semidirect
427. C₇	#22687	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×#444	2
C₂×C₇×#443	2
C₇×#2943	2
C₇×#2906	2
C₇×#1946	2
C₇×#1945	2
C₇×#1866	2
C₇×#1849	2
C₇×#1806	2
C₇×#1771	2
#22687	7

Subgroups — C7×#22687 #13199903

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22687 #13199903