Subgroups — C₂×C₇×#2139 #13210595

Order 1792.

1214 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×#135	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₇×#107	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#2139	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₇×#358	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×#357	C₂	1	2	yes	maximal, complemented
7	C₂²×C₇×#41	—	2	1	no	—
8	C₂×C₇×#135	—	2	2	no	—
9	C₂×C₇×#135	—	2	4	no	—
10	C₂×C₇×#135	C₄	1	8	yes	complemented
11	C₂×C₇×#135	C₂²	1	4	yes	—
12	C₂²×C₇×Q₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₇×SD₁₆	—	2	1	no	—
14	C₂²×C₇×D₈	C₂²	1	1	yes	characteristic
15	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
16	C₂²×C₇×C₄⋊C₄	—	2	1	no	—
17	C₂×C₇×M₃₂	—	2	4	no	—
18	C₂×C₇×M₃₂	C₂²	1	4	yes	complemented
19	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
20	C₂×C₇×#107	—	2	2	no	—
21	C₂×C₇×#107	—	2	1	no	—
22	C₂×C₇×#107	C₂²	1	2	yes	—
23	C₂×C₇×#107	C₂²	1	4	yes	complemented
24	C₇×#358	C₂²	1	8	yes	complemented
25	C₇×#357	C₂²	1	8	yes	complemented
26	C₂×#2139	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂×C₇×#41	—	2	8	no	—
28	C₂×C₇×#41	—	2	4	no	—
29	C₂×C₇×#41	—	4	2	no	—
30	C₂×C₇×#41	—	4	4	no	—
31	C₂²×C₇×Q₈	—	2	1	no	—
32	C₂²×C₇×D₄	—	2	1	no	—
33	C₂²×C₇×D₄	—	4	1	no	—
34	C₂³×C₄×C₇	—	4	1	no	—
35	C₂³×C₄×C₇	—	4	1	no	—
36	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
37	C₇×#135	—	2	16	no	—
38	C₇×#135	—	2	8	no	—
39	C₇×#135	C₂×C₄	1	16	yes	complemented
40	C₂×C₇×Q₁₆	—	2	1	no	—
41	C₂×C₇×Q₁₆	—	2	2	no	—
42	C₂×C₇×Q₁₆	C₂×C₄	1	4	yes	complemented
43	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
44	C₂×C₇×SD₁₆	—	2	8	no	—
45	C₂×C₇×SD₁₆	—	2	4	no	—
46	C₂×C₇×D₈	—	2	1	no	—
47	C₂×C₇×D₈	—	2	2	no	—
48	C₂×C₇×D₈	C₂×C₄	1	4	yes	complemented
49	C₂×C₇×D₈	C₂³	1	2	yes	—
50	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
51	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
52	C₂²×C₇×C₈	D₄	1	2	yes	—
53	C₂²×C₇×C₈	D₄	1	4	yes	—
54	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
55	C₂²×C₇×C₈	C₂³	1	2	yes	—
56	C₂×C₇×C₄⋊C₄	—	2	2	no	—
57	C₂×C₇×C₄⋊C₄	—	2	4	no	—
58	C₂×C₇×C₄⋊C₄	—	4	2	no	—
59	C₂×C₇×C₄⋊C₄	—	4	1	no	—
60	C₇×M₃₂	—	2	8	no	—
61	C₂×C₇×C₁₆	—	2	4	no	—
62	C₂×C₇×C₁₆	C₂³	1	4	yes	—
63	C₇×#107	—	2	2	no	—
64	C₇×#107	—	2	4	no	—
65	C₇×#107	C₂³	1	4	yes	—
66	C₂²×#135	C₂×C₇	1	1	yes	characteristic
67	C₂²×M₃₂	C₂×C₇	1	1	yes	characteristic, complemented
68	C₂²×#107	C₂×C₇	1	1	yes	characteristic, complemented
69	#2139	C₂×C₇	1	8	yes	direct product factor
70	C₂×#358	C₂×C₇	1	2	yes	complemented
71	C₂×#357	C₂×C₇	1	2	yes	complemented
72	C₂⁴×C₇	—	4	1	no	—
73	C₂⁴×C₇	D₈	1	1	yes	characteristic
74	C₇×#41	—	2	16	no	—
75	C₇×#41	—	4	16	no	—
76	C₇×#41	—	4	8	no	—
77	C₂×C₇×Q₈	—	2	4	no	—
78	C₂×C₇×Q₈	—	2	2	no	—
79	C₂×C₇×Q₈	—	4	1	no	—
80	C₂×C₇×Q₈	—	4	2	no	—
81	C₂×C₇×D₄	—	2	4	no	—
82	C₂×C₇×D₄	—	2	2	no	—
83	C₂×C₇×D₄	—	4	8	no	—
84	C₂×C₇×D₄	—	4	4	no	—
85	C₂×C₇×D₄	—	4	1	no	—
86	C₂×C₇×D₄	—	4	2	no	—
87	C₂²×C₄×C₇	—	4	1	no	—
88	C₂²×C₄×C₇	—	4	8	no	—
89	C₂²×C₄×C₇	—	4	4	no	—
90	C₂²×C₄×C₇	—	4	4	no	—
91	C₂²×C₄×C₇	—	4	2	no	—
92	C₂²×C₄×C₇	—	8	4	no	—
93	C₂²×C₄×C₇	C₂²⋊C₄	1	4	yes	—
94	C₂²×C₄×C₇	D₈	1	2	yes	—
95	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
96	C₂²×C₄×C₇	SD₁₆	1	4	yes	—
97	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
99	C₇×Q₁₆	—	2	4	no	—
100	C₇×Q₁₆	—	2	2	no	—
101	C₇×Q₁₆	C₂²×C₄	1	4	yes	complemented
102	C₇×SD₁₆	—	2	16	no	—
103	C₇×D₈	—	2	4	no	—
104	C₇×D₈	—	2	2	no	—
105	C₇×D₈	C₂²×C₄	1	4	yes	complemented
106	C₂×C₇×C₈	C₂²⋊C₄	1	4	yes	—
107	C₂×C₇×C₈	C₂²⋊C₄	1	8	yes	—
108	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
109	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
110	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
111	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
112	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
113	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
114	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
115	C₇×C₄⋊C₄	—	2	4	no	—
116	C₇×C₄⋊C₄	—	4	4	no	—
117	C₇×C₄⋊C₄	—	4	2	no	—
118	C₇×C₁₆	—	2	4	no	—
119	C₂²×#41	—	2	1	no	—
120	C₂×#135	—	2	2	no	—
121	C₂×#135	—	2	4	no	—
122	C₂×#135	C₄×C₇	1	8	yes	complemented
123	C₂×#135	C₂²×C₇	1	4	yes	—
124	C₂²×Q₁₆	C₂²×C₇	1	1	yes	characteristic
125	C₂²×SD₁₆	—	2	1	no	—
126	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
127	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
128	C₂²×C₄⋊C₄	—	2	1	no	—
129	C₂×M₃₂	—	2	4	no	—
130	C₂×M₃₂	C₂²×C₇	1	4	yes	complemented
131	C₂²×C₁₆	C₂²×C₇	1	2	yes	—
132	C₂×#107	—	2	2	no	—
133	C₂×#107	—	2	1	no	—
134	C₂×#107	C₂²×C₇	1	2	yes	—
135	C₂×#107	C₂²×C₇	1	4	yes	complemented
136	#358	C₂²×C₇	1	8	yes	complemented
137	#357	C₂²×C₇	1	8	yes	complemented
138	C₂³×C₇	—	2	2	no	—
139	C₂³×C₇	—	2	2	no	—
140	C₂³×C₇	—	4	2	no	—
141	C₂³×C₇	—	4	4	no	—
142	C₂³×C₇	—	8	2	no	—
143	C₂³×C₇	—	8	2	no	—
144	C₂³×C₇	#102	1	4	yes	—
145	C₂³×C₇	C₂×D₈	1	2	yes	—
146	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
147	C₇×Q₈	—	2	4	no	—
148	C₇×Q₈	—	4	2	no	—
149	C₇×Q₈	—	4	4	no	—
150	C₇×D₄	—	2	4	no	—
151	C₇×D₄	—	4	16	no	—
152	C₇×D₄	—	4	2	no	—
153	C₇×D₄	—	4	4	no	—
154	C₂×C₄×C₇	—	4	2	no	—
155	C₂×C₄×C₇	—	4	4	no	—
156	C₂×C₄×C₇	—	4	16	no	—
157	C₂×C₄×C₇	—	4	4	no	—
158	C₂×C₄×C₇	—	8	8	no	—
159	C₂×C₄×C₇	—	8	4	no	—
160	C₂×C₄×C₇	#102	1	4	yes	—
161	C₂×C₄×C₇	#102	1	8	yes	—
162	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
163	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	4	yes	—
164	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
165	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
166	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
167	C₂×C₄×C₇	C₂×SD₁₆	1	4	yes	—
168	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
169	C₇×C₈	C₂×C₂²⋊C₄	1	2	yes	—
170	C₇×C₈	C₂×C₂²⋊C₄	1	4	yes	—
171	C₇×C₈	C₂³×C₄	1	1	yes	characteristic
172	C₇×C₈	C₂²×D₄	1	1	yes	characteristic
173	C₂×#41	—	2	8	no	—
174	C₂×#41	—	2	4	no	—
175	C₂×#41	—	4	2	no	—
176	C₂×#41	—	4	4	no	—
177	C₂²×Q₈	—	2	1	no	—
178	C₂²×D₄	—	2	1	no	—
179	C₂²×D₄	—	4	1	no	—
180	C₂³×C₄	—	4	1	no	—
181	C₂³×C₄	—	4	1	no	—
182	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
183	#135	—	2	16	no	—
184	#135	—	2	8	no	—
185	#135	C₂×C₄×C₇	1	16	yes	complemented
186	C₂×Q₁₆	—	2	1	no	—
187	C₂×Q₁₆	—	2	2	no	—
188	C₂×Q₁₆	C₂×C₄×C₇	1	4	yes	complemented
189	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
190	C₂×SD₁₆	—	2	8	no	—
191	C₂×SD₁₆	—	2	4	no	—
192	C₂×D₈	—	2	1	no	—
193	C₂×D₈	—	2	2	no	—
194	C₂×D₈	C₂×C₄×C₇	1	4	yes	complemented
195	C₂×D₈	C₂³×C₇	1	2	yes	—
196	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
197	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
198	C₂²×C₈	C₇×D₄	1	2	yes	—
199	C₂²×C₈	C₇×D₄	1	4	yes	—
200	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
201	C₂²×C₈	C₂³×C₇	1	2	yes	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	4	no	—
204	C₂×C₄⋊C₄	—	4	1	no	—
205	C₂×C₄⋊C₄	—	4	2	no	—
206	M₃₂	—	2	8	no	—
207	C₂×C₁₆	—	2	4	no	—
208	C₂×C₁₆	C₂³×C₇	1	4	yes	—
209	#107	—	2	2	no	—
210	#107	—	2	4	no	—
211	#107	C₂³×C₇	1	4	yes	—
212	C₂²×C₇	—	2	2	no	—
213	C₂²×C₇	—	2	2	no	—
214	C₂²×C₇	—	2	8	no	—
215	C₂²×C₇	—	4	4	no	—
216	C₂²×C₇	—	8	2	no	—
217	C₂²×C₇	—	8	2	no	—
218	C₂²×C₇	—	8	8	no	—
219	C₂²×C₇	C₂×#102	1	4	yes	—
220	C₂²×C₇	C₂×#102	1	2	yes	—
221	C₂²×C₇	#509	1	2	yes	—
222	C₂²×C₇	#510	1	2	yes	—
223	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
224	C₄×C₇	—	4	4	no	—
225	C₄×C₇	—	8	4	no	—
226	C₄×C₇	C₂×#102	1	4	yes	—
227	C₄×C₇	C₂×#102	1	2	yes	—
228	C₄×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
229	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
230	C₂⁴	—	4	1	no	—
231	C₂⁴	C₇×D₈	1	1	yes	characteristic
232	#41	—	2	16	no	—
233	#41	—	4	16	no	—
234	#41	—	4	8	no	—
235	C₂×Q₈	—	2	4	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	4	1	no	—
238	C₂×Q₈	—	4	2	no	—
239	C₂×D₄	—	2	4	no	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	4	8	no	—
242	C₂×D₄	—	4	4	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂×D₄	—	4	2	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	—	4	8	no	—
247	C₂²×C₄	—	4	4	no	—
248	C₂²×C₄	—	4	4	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	—	8	4	no	—
251	C₂²×C₄	C₇×C₂²⋊C₄	1	4	yes	—
252	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
253	C₂²×C₄	C₇×D₈	1	2	yes	—
254	C₂²×C₄	C₇×SD₁₆	1	4	yes	—
255	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
257	Q₁₆	—	2	4	no	—
258	Q₁₆	—	2	2	no	—
259	Q₁₆	C₂²×C₄×C₇	1	4	yes	complemented
260	SD₁₆	—	2	16	no	—
261	D₈	—	2	4	no	—
262	D₈	—	2	2	no	—
263	D₈	C₂²×C₄×C₇	1	4	yes	complemented
264	C₂×C₈	C₇×C₂²⋊C₄	1	4	yes	—
265	C₂×C₈	C₇×C₂²⋊C₄	1	8	yes	—
266	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
267	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
268	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
269	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
270	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
271	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
272	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
273	C₄⋊C₄	—	2	4	no	—
274	C₄⋊C₄	—	4	4	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₁₆	—	2	4	no	—
277	C₂×C₇	—	2	4	no	—
278	C₂×C₇	—	8	4	no	—
279	C₂×C₇	#2139	1	4	yes	direct product factor
280	C₂×C₇	C₂²×#102	1	1	yes	characteristic
281	C₂×C₇	C₂×#509	1	1	yes	characteristic
282	C₂×C₇	C₂×#510	1	1	yes	characteristic
283	C₂³	—	2	2	no	—
284	C₂³	—	2	2	no	—
285	C₂³	—	4	4	no	—
286	C₂³	—	4	2	no	—
287	C₂³	—	8	2	no	—
288	C₂³	—	8	2	no	—
289	C₂³	C₇×#102	1	4	yes	—
290	C₂³	C₂×C₇×D₈	1	2	yes	—
291	C₂³	C₂×C₇×D₈	1	1	yes	characteristic
292	Q₈	—	2	4	no	—
293	Q₈	—	4	4	no	—
294	Q₈	—	4	2	no	—
295	D₄	—	2	4	no	—
296	D₄	—	4	16	no	—
297	D₄	—	4	4	no	—
298	D₄	—	4	2	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	16	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	8	8	no	—
304	C₂×C₄	—	8	4	no	—
305	C₂×C₄	C₇×#102	1	4	yes	—
306	C₂×C₄	C₇×#102	1	8	yes	—
307	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
308	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	4	yes	—
309	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
310	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
311	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
312	C₂×C₄	C₂×C₇×SD₁₆	1	4	yes	—
313	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
314	C₈	C₂×C₇×C₂²⋊C₄	1	2	yes	—
315	C₈	C₂×C₇×C₂²⋊C₄	1	4	yes	—
316	C₈	C₂³×C₄×C₇	1	1	yes	characteristic
317	C₈	C₂²×C₇×D₄	1	1	yes	characteristic
318	C₇	C₂×#2139	1	1	yes	characteristic, direct product factor, Sylow
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	8	no	—
322	C₂²	—	4	4	no	—
323	C₂²	—	8	2	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	8	8	no	—
326	C₂²	C₂×C₇×#102	1	4	yes	—
327	C₂²	C₂×C₇×#102	1	2	yes	—
328	C₂²	C₇×#509	1	2	yes	—
329	C₂²	C₇×#510	1	2	yes	—
330	C₂²	C₂²×C₇×D₈	1	1	yes	characteristic
331	C₄	—	4	4	no	—
332	C₄	—	8	4	no	—
333	C₄	C₂×C₇×#102	1	4	yes	—
334	C₄	C₂×C₇×#102	1	2	yes	—
335	C₄	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
336	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
337	C₂	—	2	4	no	—
338	C₂	—	8	4	no	—
339	C₂	C₇×#2139	1	4	yes	direct product factor
340	C₂	C₂²×C₇×#102	1	1	yes	characteristic
341	C₂	C₂×C₇×#509	1	1	yes	characteristic
342	C₂	C₂×C₇×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂×D₈
commutator	C₈	C₂³×C₄×C₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₇
prime radical	C₂²×C₇×#135	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂³×C₄×C₇ ◁ C₂³×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₇×M₃₂	C₂	semidirect
3. C₂²×C₇×#107	C₂	semidirect
4. C₇×#2139	C₂	direct (N × H)
5. C₂×C₇×#358	C₂	semidirect
6. C₂×C₇×#357	C₂	semidirect
10. C₂×C₇×#135	C₄	semidirect
18. C₂×C₇×M₃₂	C₂²	semidirect
23. C₂×C₇×#107	C₂²	semidirect
24. C₇×#358	C₂²	semidirect
25. C₇×#357	C₂²	semidirect
26. C₂×#2139	C₇	direct (N × H)
39. C₇×#135	C₂×C₄	semidirect
42. C₂×C₇×Q₁₆	C₂×C₄	semidirect
48. C₂×C₇×D₈	C₂×C₄	semidirect
67. C₂²×M₃₂	C₂×C₇	semidirect
68. C₂²×#107	C₂×C₇	semidirect
69. #2139	C₂×C₇	direct (N × H)
70. C₂×#358	C₂×C₇	semidirect
71. C₂×#357	C₂×C₇	semidirect
101. C₇×Q₁₆	C₂²×C₄	semidirect
105. C₇×D₈	C₂²×C₄	semidirect
122. C₂×#135	C₄×C₇	semidirect
130. C₂×M₃₂	C₂²×C₇	semidirect
135. C₂×#107	C₂²×C₇	semidirect
136. #358	C₂²×C₇	semidirect
137. #357	C₂²×C₇	semidirect
185. #135	C₂×C₄×C₇	semidirect
188. C₂×Q₁₆	C₂×C₄×C₇	semidirect
194. C₂×D₈	C₂×C₄×C₇	semidirect
259. Q₁₆	C₂²×C₄×C₇	semidirect
263. D₈	C₂²×C₄×C₇	semidirect
279. C₂×C₇	#2139	direct (N × H)
318. C₇	C₂×#2139	direct (N × H)
339. C₂	C₇×#2139	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₇×#135	2
C₂²×C₇×M₃₂	2
C₂²×C₇×#107	2
C₇×#2139	2
C₂×C₇×#358	2
C₂×C₇×#357	2
C₂×#2139	7

Subgroups — C2×C7×#2139 #13210595

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#2139 #13210595