Subgroups — C₇×#19683 #13196899

Order 1792.

724 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3209	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3206	C₂	1	1	yes	maximal, characteristic
3	C₇×#3167	C₂	1	1	yes	maximal, characteristic
4	C₇×#3165	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₇×#500	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#3141	C₂	1	1	yes	maximal, characteristic
7	C₇×#1723	C₂	1	2	yes	maximal, complemented
8	C₇×#1722	C₂	1	2	yes	maximal, complemented
9	C₇×#1718	C₂	1	2	yes	maximal, complemented
10	C₇×#1714	C₂	1	2	yes	maximal, complemented
11	C₇×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#533	C₂²	1	1	yes	characteristic
13	C₇×#532	C₂²	1	1	yes	characteristic
14	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₇×#501	—	2	2	no	—
17	C₇×#500	—	2	2	no	—
18	C₇×#500	C₂²	1	2	yes	—
19	C₇×#500	C₂²	1	2	yes	—
20	C₇×#500	C₂²	1	2	yes	—
21	C₇×#498	C₂²	1	2	yes	—
22	C₇×#498	C₂²	1	1	yes	characteristic
23	C₇×#494	C₂²	1	2	yes	—
24	C₇×#492	C₂²	1	1	yes	characteristic
25	C₇×#492	C₂²	1	2	yes	—
26	C₇×#489	—	2	2	no	—
27	C₇×#487	C₂²	1	2	yes	—
28	C₇×#487	C₂²	1	1	yes	characteristic
29	C₇×#487	C₂²	1	2	yes	—
30	C₇×#487	C₂²	1	2	yes	—
31	C₇×#486	C₂²	1	2	yes	—
32	C₇×#486	C₂²	1	1	yes	characteristic
33	C₇×#465	—	2	2	no	—
34	C₇×#462	—	2	2	no	—
35	C₇×#461	—	2	2	no	—
36	C₇×#451	—	2	2	no	—
37	C₇×#448	—	2	2	no	—
38	C₄×C₇×SD₁₆	—	2	2	no	—
39	C₇×#431	C₂²	1	1	yes	characteristic
40	C₇×#428	—	2	2	no	—
41	C₂×C₇×#107	—	2	2	no	—
42	C₇×#417	—	2	2	no	—
43	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
44	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
45	C₇×#322	C₂²	1	2	yes	—
46	C₇×#322	C₂²	1	2	yes	—
47	#19683	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂×C₇×#41	—	2	1	no	—
49	C₂²×C₇×Q₈	—	2	1	no	—
50	C₂²×C₇×Q₈	—	2	1	no	—
51	C₂×C₇×Q₁₆	—	2	4	no	—
52	C₂×C₇×SD₁₆	—	2	2	no	—
53	C₂×C₇×SD₁₆	—	2	4	no	—
54	C₂×C₇×M₁₆	D₄	1	2	yes	—
55	C₂²×C₇×C₈	D₄	1	2	yes	—
56	C₇×C₄⋊Q₈	—	2	1	no	—
57	C₇×C₄⋊Q₈	—	2	1	no	—
58	C₇×C₄⋊Q₈	—	2	1	no	—
59	C₇×C₄⋊Q₈	—	2	1	no	—
60	C₇×C₄⋊Q₈	—	2	1	no	—
61	C₇×C₄⋊Q₈	—	2	1	no	—
62	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
63	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
64	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
65	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
66	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
67	C₇×#127	C₂³	1	1	yes	characteristic
68	C₇×#125	—	2	1	no	—
69	C₇×#124	—	2	1	no	—
70	C₇×C₂²⋊Q₈	—	2	1	no	—
71	C₇×C₂²⋊Q₈	—	2	1	no	—
72	C₇×#121	—	2	1	no	—
73	C₇×#121	—	2	1	no	—
74	C₄×C₇×Q₈	—	2	1	no	—
75	C₄×C₇×Q₈	—	2	1	no	—
76	C₄×C₇×D₄	—	2	1	no	—
77	C₄×C₇×D₄	—	2	1	no	—
78	C₇×#117	—	2	1	no	—
79	C₂×C₇×C₄⋊C₄	—	2	1	no	—
80	C₂×C₇×C₄⋊C₄	—	2	1	no	—
81	C₂×C₇×C₄⋊C₄	—	2	1	no	—
82	C₂×C₇×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
84	C₇×#107	—	2	4	no	—
85	C₇×#107	—	2	4	no	—
86	C₇×#107	—	2	4	no	—
87	C₇×#106	—	2	4	no	—
88	C₇×#106	—	2	2	no	—
89	C₇×C₄⋊C₈	D₄	1	2	yes	—
90	C₇×#103	—	2	4	no	—
91	C₇×#103	—	2	2	no	—
92	C₇×#103	—	2	4	no	—
93	C₇×#103	—	2	4	no	—
94	C₇×#102	—	2	4	no	—
95	C₇×#102	—	2	2	no	—
96	C₇×C₂²⋊C₈	D₄	1	2	yes	—
97	C₇×#97	D₄	1	2	yes	—
98	C₄×C₇×C₈	D₄	1	2	yes	—
99	C₄×C₇×C₈	C₂³	1	2	yes	—
100	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
101	C₄×C₇×C₈	C₂³	1	2	yes	—
102	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
103	#3209	C₂×C₇	1	1	yes	characteristic, complemented
104	#3206	C₂×C₇	1	1	yes	characteristic
105	#3167	C₂×C₇	1	1	yes	characteristic
106	#3165	C₂×C₇	1	1	yes	characteristic, complemented
107	C₂×#500	C₂×C₇	1	1	yes	characteristic, complemented
108	#3141	C₂×C₇	1	1	yes	characteristic
109	#1723	C₂×C₇	1	2	yes	complemented
110	#1722	C₂×C₇	1	2	yes	complemented
111	#1718	C₂×C₇	1	2	yes	complemented
112	#1714	C₂×C₇	1	2	yes	complemented
113	#1451	C₂×C₇	1	1	yes	characteristic, complemented
114	C₇×#41	—	4	4	no	—
115	C₂×C₇×Q₈	—	2	1	no	—
116	C₂×C₇×Q₈	—	2	1	no	—
117	C₂×C₇×Q₈	—	2	2	no	—
118	C₂×C₇×Q₈	—	2	2	no	—
119	C₂×C₇×Q₈	—	2	1	no	—
120	C₂×C₇×Q₈	—	4	1	no	—
121	C₂×C₇×Q₈	—	4	1	no	—
122	C₂×C₇×Q₈	—	8	2	no	—
123	C₂×C₇×Q₈	—	8	2	no	—
124	C₂×C₇×D₄	—	2	1	no	—
125	C₂×C₇×D₄	—	2	1	no	—
126	C₂×C₇×D₄	—	2	1	no	—
127	C₂×C₇×D₄	—	4	1	no	—
128	C₂²×C₄×C₇	—	4	1	no	—
129	C₂²×C₄×C₇	—	4	1	no	—
130	C₂²×C₄×C₇	—	4	1	no	—
131	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
134	C₇×Q₁₆	—	8	4	no	—
135	C₇×SD₁₆	—	4	4	no	—
136	C₇×SD₁₆	—	8	4	no	—
137	C₇×M₁₆	—	2	4	no	—
138	C₂×C₇×C₈	—	2	4	no	—
139	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
140	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
141	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
142	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
143	C₇×C₄⋊C₄	—	2	2	no	—
144	C₇×C₄⋊C₄	—	2	1	no	—
145	C₇×C₄⋊C₄	—	2	1	no	—
146	C₇×C₄⋊C₄	—	2	1	no	—
147	C₇×C₄⋊C₄	—	2	1	no	—
148	C₇×C₄⋊C₄	—	2	1	no	—
149	C₇×C₄⋊C₄	—	2	2	no	—
150	C₇×C₄⋊C₄	—	2	2	no	—
151	C₇×C₄⋊C₄	—	2	2	no	—
152	C₇×C₄⋊C₄	—	2	1	no	—
153	C₇×C₄⋊C₄	—	4	1	no	—
154	C₇×C₄⋊C₄	—	4	1	no	—
155	C₇×C₄⋊C₄	—	4	1	no	—
156	C₇×C₄⋊C₄	—	4	1	no	—
157	C₇×C₄⋊C₄	—	4	1	no	—
158	C₇×C₄⋊C₄	—	4	1	no	—
159	C₇×C₂²⋊C₄	—	4	1	no	—
160	C₇×C₂²⋊C₄	—	4	1	no	—
161	C₇×C₂²⋊C₄	—	4	1	no	—
162	C₄²×C₇	—	4	1	no	—
163	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
164	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
165	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
166	C₄²×C₇	C₂⁴	1	1	yes	characteristic
167	#533	C₂²×C₇	1	1	yes	characteristic
168	#532	C₂²×C₇	1	1	yes	characteristic
169	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
170	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
171	#501	—	2	2	no	—
172	#500	—	2	2	no	—
173	#500	C₂²×C₇	1	2	yes	—
174	#500	C₂²×C₇	1	2	yes	—
175	#500	C₂²×C₇	1	2	yes	—
176	#498	C₂²×C₇	1	2	yes	—
177	#498	C₂²×C₇	1	1	yes	characteristic
178	#494	C₂²×C₇	1	2	yes	—
179	#492	C₂²×C₇	1	1	yes	characteristic
180	#492	C₂²×C₇	1	2	yes	—
181	#489	—	2	2	no	—
182	#487	C₂²×C₇	1	2	yes	—
183	#487	C₂²×C₇	1	1	yes	characteristic
184	#487	C₂²×C₇	1	2	yes	—
185	#487	C₂²×C₇	1	2	yes	—
186	#486	C₂²×C₇	1	2	yes	—
187	#486	C₂²×C₇	1	1	yes	characteristic
188	#465	—	2	2	no	—
189	#462	—	2	2	no	—
190	#461	—	2	2	no	—
191	#451	—	2	2	no	—
192	#448	—	2	2	no	—
193	C₄×SD₁₆	—	2	2	no	—
194	#431	C₂²×C₇	1	1	yes	characteristic
195	#428	—	2	2	no	—
196	C₂×#107	—	2	2	no	—
197	#417	—	2	2	no	—
198	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
199	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
200	#322	C₂²×C₇	1	2	yes	—
201	#322	C₂²×C₇	1	2	yes	—
202	C₂³×C₇	—	4	1	no	—
203	C₂³×C₇	#127	1	1	yes	characteristic
204	C₇×Q₈	—	4	2	no	—
205	C₇×Q₈	—	4	2	no	—
206	C₇×Q₈	—	4	4	no	—
207	C₇×Q₈	—	4	2	no	—
208	C₇×Q₈	—	4	4	no	—
209	C₇×Q₈	—	8	2	no	—
210	C₇×Q₈	—	8	2	no	—
211	C₇×D₄	—	4	2	no	—
212	C₇×D₄	—	4	2	no	—
213	C₇×D₄	—	4	2	no	—
214	C₇×D₄	—	8	2	no	—
215	C₂×C₄×C₇	—	2	2	no	—
216	C₂×C₄×C₇	—	2	2	no	—
217	C₂×C₄×C₇	—	4	1	no	—
218	C₂×C₄×C₇	—	4	1	no	—
219	C₂×C₄×C₇	—	4	1	no	—
220	C₂×C₄×C₇	—	4	2	no	—
221	C₂×C₄×C₇	—	4	2	no	—
222	C₂×C₄×C₇	—	8	2	no	—
223	C₂×C₄×C₇	—	8	2	no	—
224	C₂×C₄×C₇	—	8	2	no	—
225	C₂×C₄×C₇	#127	1	1	yes	characteristic
226	C₂×C₄×C₇	#127	1	1	yes	characteristic
227	C₂×C₄×C₇	#127	1	1	yes	characteristic
228	C₂×C₄×C₇	#137	1	4	yes	—
229	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
230	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
231	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
232	C₇×C₈	—	2	2	no	—
233	C₇×C₈	—	2	4	no	—
234	C₇×C₈	—	2	4	no	—
235	C₇×C₈	#137	1	4	yes	—
236	C₂×#41	—	2	1	no	—
237	C₂²×Q₈	—	2	1	no	—
238	C₂²×Q₈	—	2	1	no	—
239	C₂×Q₁₆	—	2	4	no	—
240	C₂×SD₁₆	—	2	2	no	—
241	C₂×SD₁₆	—	2	4	no	—
242	C₂×M₁₆	C₇×D₄	1	2	yes	—
243	C₂²×C₈	C₇×D₄	1	2	yes	—
244	C₄⋊Q₈	—	2	1	no	—
245	C₄⋊Q₈	—	2	1	no	—
246	C₄⋊Q₈	—	2	1	no	—
247	C₄⋊Q₈	—	2	1	no	—
248	C₄⋊Q₈	—	2	1	no	—
249	C₄⋊Q₈	—	2	1	no	—
250	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
251	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
252	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
253	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
254	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
255	#127	C₂³×C₇	1	1	yes	characteristic
256	#125	—	2	1	no	—
257	#124	—	2	1	no	—
258	C₂²⋊Q₈	—	2	1	no	—
259	C₂²⋊Q₈	—	2	1	no	—
260	#121	—	2	1	no	—
261	#121	—	2	1	no	—
262	C₄×Q₈	—	2	1	no	—
263	C₄×Q₈	—	2	1	no	—
264	C₄×D₄	—	2	1	no	—
265	C₄×D₄	—	2	1	no	—
266	#117	—	2	1	no	—
267	C₂×C₄⋊C₄	—	2	1	no	—
268	C₂×C₄⋊C₄	—	2	1	no	—
269	C₂×C₄⋊C₄	—	2	1	no	—
270	C₂×C₄⋊C₄	—	2	1	no	—
271	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
272	#107	—	2	4	no	—
273	#107	—	2	4	no	—
274	#107	—	2	4	no	—
275	#106	—	2	4	no	—
276	#106	—	2	2	no	—
277	C₄⋊C₈	C₇×D₄	1	2	yes	—
278	#103	—	2	4	no	—
279	#103	—	2	2	no	—
280	#103	—	2	4	no	—
281	#103	—	2	4	no	—
282	#102	—	2	4	no	—
283	#102	—	2	2	no	—
284	C₂²⋊C₈	C₇×D₄	1	2	yes	—
285	#97	C₇×D₄	1	2	yes	—
286	C₄×C₈	C₇×D₄	1	2	yes	—
287	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
288	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
289	C₄×C₈	C₂³×C₇	1	2	yes	—
290	C₄×C₈	C₂³×C₇	1	2	yes	—
291	C₂²×C₇	—	2	1	no	—
292	C₂²×C₇	—	2	1	no	—
293	C₂²×C₇	—	8	1	no	—
294	C₂²×C₇	—	8	1	no	—
295	C₂²×C₇	—	8	1	no	—
296	C₂²×C₇	#497	1	2	yes	—
297	C₂²×C₇	C₂×#127	1	1	yes	characteristic
298	C₄×C₇	—	2	1	no	—
299	C₄×C₇	—	2	1	no	—
300	C₄×C₇	—	8	1	no	—
301	C₄×C₇	—	8	1	no	—
302	C₄×C₇	—	8	2	no	—
303	C₄×C₇	—	8	2	no	—
304	C₄×C₇	—	8	1	no	—
305	C₄×C₇	#497	1	2	yes	—
306	C₄×C₇	C₂×#137	1	2	yes	—
307	C₄×C₇	C₂×#137	1	2	yes	—
308	C₄×C₇	#576	1	2	yes	—
309	#41	—	4	4	no	—
310	C₂×Q₈	—	2	1	no	—
311	C₂×Q₈	—	2	1	no	—
312	C₂×Q₈	—	2	2	no	—
313	C₂×Q₈	—	2	2	no	—
314	C₂×Q₈	—	2	1	no	—
315	C₂×Q₈	—	4	1	no	—
316	C₂×Q₈	—	4	1	no	—
317	C₂×Q₈	—	8	2	no	—
318	C₂×Q₈	—	8	2	no	—
319	C₂×D₄	—	2	1	no	—
320	C₂×D₄	—	2	1	no	—
321	C₂×D₄	—	2	1	no	—
322	C₂×D₄	—	4	1	no	—
323	C₂²×C₄	—	4	1	no	—
324	C₂²×C₄	—	4	1	no	—
325	C₂²×C₄	—	4	1	no	—
326	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
327	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
328	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
329	Q₁₆	—	8	4	no	—
330	SD₁₆	—	4	4	no	—
331	SD₁₆	—	8	4	no	—
332	M₁₆	—	2	4	no	—
333	C₂×C₈	—	2	4	no	—
334	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
335	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
336	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
337	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
338	C₄⋊C₄	—	2	2	no	—
339	C₄⋊C₄	—	2	1	no	—
340	C₄⋊C₄	—	2	1	no	—
341	C₄⋊C₄	—	2	1	no	—
342	C₄⋊C₄	—	2	1	no	—
343	C₄⋊C₄	—	2	1	no	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	2	1	no	—
348	C₄⋊C₄	—	4	1	no	—
349	C₄⋊C₄	—	4	1	no	—
350	C₄⋊C₄	—	4	1	no	—
351	C₄⋊C₄	—	4	1	no	—
352	C₄⋊C₄	—	4	1	no	—
353	C₄⋊C₄	—	4	1	no	—
354	C₂²⋊C₄	—	4	1	no	—
355	C₂²⋊C₄	—	4	1	no	—
356	C₂²⋊C₄	—	4	1	no	—
357	C₄²	—	4	1	no	—
358	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
359	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
360	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
361	C₄²	C₂⁴×C₇	1	1	yes	characteristic
362	C₂×C₇	—	2	2	no	—
363	C₂×C₇	—	16	1	no	—
364	C₂×C₇	C₂×#497	1	1	yes	characteristic
365	C₂×C₇	#3152	1	1	yes	characteristic
366	C₂×C₇	#3155	1	1	yes	characteristic
367	C₂³	—	4	1	no	—
368	C₂³	C₇×#127	1	1	yes	characteristic
369	Q₈	—	4	2	no	—
370	Q₈	—	4	2	no	—
371	Q₈	—	4	4	no	—
372	Q₈	—	4	2	no	—
373	Q₈	—	4	4	no	—
374	Q₈	—	8	2	no	—
375	Q₈	—	8	2	no	—
376	D₄	—	4	2	no	—
377	D₄	—	4	2	no	—
378	D₄	—	4	2	no	—
379	D₄	—	8	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	4	1	no	—
383	C₂×C₄	—	4	1	no	—
384	C₂×C₄	—	4	1	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	8	2	no	—
388	C₂×C₄	—	8	2	no	—
389	C₂×C₄	—	8	2	no	—
390	C₂×C₄	C₇×#127	1	1	yes	characteristic
391	C₂×C₄	C₇×#127	1	1	yes	characteristic
392	C₂×C₄	C₇×#127	1	1	yes	characteristic
393	C₂×C₄	C₇×#137	1	4	yes	—
394	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
395	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
396	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
397	C₈	—	2	2	no	—
398	C₈	—	2	4	no	—
399	C₈	—	2	4	no	—
400	C₈	C₇×#137	1	4	yes	—
401	C₇	#19683	1	1	yes	characteristic, direct product factor, Sylow
402	C₂²	—	2	1	no	—
403	C₂²	—	2	1	no	—
404	C₂²	—	8	1	no	—
405	C₂²	—	8	1	no	—
406	C₂²	—	8	1	no	—
407	C₂²	C₇×#497	1	2	yes	—
408	C₂²	C₂×C₇×#127	1	1	yes	characteristic
409	C₄	—	2	1	no	—
410	C₄	—	2	1	no	—
411	C₄	—	8	2	no	—
412	C₄	—	8	2	no	—
413	C₄	—	8	1	no	—
414	C₄	—	8	1	no	—
415	C₄	—	8	1	no	—
416	C₄	C₇×#497	1	2	yes	—
417	C₄	C₂×C₇×#137	1	2	yes	—
418	C₄	C₂×C₇×#137	1	2	yes	—
419	C₄	C₇×#576	1	2	yes	—
420	C₂	—	2	2	no	—
421	C₂	—	16	1	no	—
422	C₂	C₂×C₇×#497	1	1	yes	characteristic
423	C₂	C₇×#3152	1	1	yes	characteristic
424	C₂	C₇×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₇×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3209	C₂	semidirect
4. C₇×#3165	C₂	semidirect
5. C₂×C₇×#500	C₂	semidirect
7. C₇×#1723	C₂	semidirect
8. C₇×#1722	C₂	semidirect
9. C₇×#1718	C₂	semidirect
10. C₇×#1714	C₂	semidirect
11. C₇×#1451	C₂	semidirect
47. #19683	C₇	direct (N × H)
103. #3209	C₂×C₇	semidirect
106. #3165	C₂×C₇	semidirect
107. C₂×#500	C₂×C₇	semidirect
109. #1723	C₂×C₇	semidirect
110. #1722	C₂×C₇	semidirect
111. #1718	C₂×C₇	semidirect
112. #1714	C₂×C₇	semidirect
113. #1451	C₂×C₇	semidirect
401. C₇	#19683	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×#3209	2
C₇×#3206	2
C₇×#3167	2
C₇×#3165	2
C₂×C₇×#500	2
C₇×#3141	2
C₇×#1723	2
C₇×#1722	2
C₇×#1718	2
C₇×#1714	2
C₇×#1451	2
#19683	7

Subgroups — C7×#19683 #13196899

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19683 #13196899