Subgroups — C₂×C₇×#2138 #13210592

Order 1792.

1190 conjugacy classes of subgroups.

The subgroup lattice has 328 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	C₂³×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#427	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#2138	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	C₂×C₇×#117	—	2	1	no	—
17	C₂²×C₇×C₁₆	—	2	4	no	—
18	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
19	C₂²×C₇×C₁₆	C₂²	1	4	yes	complemented
20	C₇×#427	—	2	2	no	—
21	C₇×#427	C₂²	1	4	yes	complemented
22	C₂×C₇×#107	C₂²	1	2	yes	—
23	C₂×C₇×#106	—	2	1	no	—
24	C₇×#358	C₂²	1	8	yes	complemented
25	C₇×#357	C₂²	1	8	yes	complemented
26	C₂×#2138	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	D₄	1	1	yes	characteristic
36	C₇×#135	—	2	16	no	—
37	C₇×#135	—	2	8	no	—
38	C₇×#135	C₂×C₄	1	16	yes	complemented
39	C₂×C₇×Q₁₆	—	2	1	no	—
40	C₂×C₇×Q₁₆	—	2	2	no	—
41	C₂×C₇×Q₁₆	C₂×C₄	1	4	yes	—
42	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
43	C₂×C₇×SD₁₆	—	2	8	no	—
44	C₂×C₇×SD₁₆	—	2	4	no	—
45	C₂×C₇×D₈	—	2	1	no	—
46	C₂×C₇×D₈	—	2	2	no	—
47	C₂×C₇×D₈	C₂×C₄	1	4	yes	—
48	C₂×C₇×D₈	C₂³	1	2	yes	—
49	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
50	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
51	C₂²×C₇×C₈	D₄	1	2	yes	—
52	C₂²×C₇×C₈	D₄	1	4	yes	—
53	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
54	C₂²×C₇×C₈	C₂³	1	2	yes	—
55	C₇×#117	—	2	4	no	—
56	C₇×#117	—	4	2	no	—
57	C₂×C₇×C₄⋊C₄	—	2	2	no	—
58	C₂×C₇×C₂²⋊C₄	—	4	1	no	—
59	C₂×C₄²×C₇	—	4	1	no	—
60	C₂×C₇×C₁₆	—	2	4	no	—
61	C₂×C₇×C₁₆	—	2	8	no	—
62	C₂×C₇×C₁₆	C₂³	1	4	yes	—
63	C₇×#107	—	2	2	no	—
64	C₇×#107	C₂³	1	4	yes	—
65	C₇×#106	—	2	4	no	—
66	C₂²×#135	C₂×C₇	1	1	yes	characteristic
67	C₂³×C₁₆	C₂×C₇	1	1	yes	characteristic, complemented
68	C₂×#427	C₂×C₇	1	1	yes	characteristic, complemented
69	#2138	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	2	no	—
91	C₂²×C₄×C₇	C₂²⋊C₄	1	4	yes	—
92	C₂²×C₄×C₇	D₈	1	2	yes	—
93	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
94	C₂²×C₄×C₇	SD₁₆	1	4	yes	—
95	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
97	C₇×Q₁₆	—	2	4	no	—
98	C₇×Q₁₆	—	2	2	no	—
99	C₇×Q₁₆	C₂²×C₄	1	4	yes	—
100	C₇×SD₁₆	—	2	16	no	—
101	C₇×D₈	—	2	4	no	—
102	C₇×D₈	—	2	2	no	—
103	C₇×D₈	C₂²×C₄	1	4	yes	—
104	C₂×C₇×C₈	C₂²⋊C₄	1	4	yes	—
105	C₂×C₇×C₈	C₂²⋊C₄	1	8	yes	—
106	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
107	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
108	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
109	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
110	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
111	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
112	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
113	C₇×C₄⋊C₄	—	2	4	no	—
114	C₇×C₄⋊C₄	—	4	2	no	—
115	C₇×C₂²⋊C₄	—	4	4	no	—
116	C₄²×C₇	—	4	4	no	—
117	C₇×C₁₆	—	2	4	no	—
118	C₂²×#41	—	2	1	no	—
119	C₂×#135	—	2	2	no	—
120	C₂×#135	—	2	4	no	—
121	C₂×#135	C₄×C₇	1	8	yes	complemented
122	C₂×#135	C₂²×C₇	1	4	yes	—
123	C₂²×Q₁₆	C₂²×C₇	1	1	yes	characteristic
124	C₂²×SD₁₆	—	2	1	no	—
125	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
126	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
127	C₂×#117	—	2	1	no	—
128	C₂²×C₁₆	—	2	4	no	—
129	C₂²×C₁₆	C₂²×C₇	1	2	yes	—
130	C₂²×C₁₆	C₂²×C₇	1	4	yes	complemented
131	#427	—	2	2	no	—
132	#427	C₂²×C₇	1	4	yes	complemented
133	C₂×#107	C₂²×C₇	1	2	yes	—
134	C₂×#106	—	2	1	no	—
135	#358	C₂²×C₇	1	8	yes	complemented
136	#357	C₂²×C₇	1	8	yes	complemented
137	C₂³×C₇	—	2	4	no	—
138	C₂³×C₇	—	4	2	no	—
139	C₂³×C₇	—	4	4	no	—
140	C₂³×C₇	—	8	4	no	—
141	C₂³×C₇	#102	1	4	yes	—
142	C₂³×C₇	C₂×D₈	1	2	yes	—
143	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
144	C₇×Q₈	—	2	4	no	—
145	C₇×Q₈	—	4	2	no	—
146	C₇×Q₈	—	4	4	no	—
147	C₇×D₄	—	2	4	no	—
148	C₇×D₄	—	4	16	no	—
149	C₇×D₄	—	4	2	no	—
150	C₇×D₄	—	4	4	no	—
151	C₂×C₄×C₇	—	4	2	no	—
152	C₂×C₄×C₇	—	4	4	no	—
153	C₂×C₄×C₇	—	4	16	no	—
154	C₂×C₄×C₇	—	4	4	no	—
155	C₂×C₄×C₇	—	8	4	no	—
156	C₂×C₄×C₇	#102	1	4	yes	—
157	C₂×C₄×C₇	#102	1	8	yes	—
158	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
159	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	4	yes	—
160	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
161	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
162	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
163	C₂×C₄×C₇	C₂×SD₁₆	1	4	yes	—
164	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
165	C₇×C₈	C₂×C₂²⋊C₄	1	2	yes	—
166	C₇×C₈	C₂×C₂²⋊C₄	1	4	yes	—
167	C₇×C₈	C₂³×C₄	1	1	yes	characteristic
168	C₇×C₈	C₂²×D₄	1	1	yes	characteristic
169	C₂×#41	—	2	8	no	—
170	C₂×#41	—	2	4	no	—
171	C₂×#41	—	4	2	no	—
172	C₂×#41	—	4	4	no	—
173	C₂²×Q₈	—	2	1	no	—
174	C₂²×D₄	—	2	1	no	—
175	C₂²×D₄	—	4	1	no	—
176	C₂³×C₄	—	4	1	no	—
177	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
178	#135	—	2	16	no	—
179	#135	—	2	8	no	—
180	#135	C₂×C₄×C₇	1	16	yes	complemented
181	C₂×Q₁₆	—	2	1	no	—
182	C₂×Q₁₆	—	2	2	no	—
183	C₂×Q₁₆	C₂×C₄×C₇	1	4	yes	—
184	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
185	C₂×SD₁₆	—	2	8	no	—
186	C₂×SD₁₆	—	2	4	no	—
187	C₂×D₈	—	2	1	no	—
188	C₂×D₈	—	2	2	no	—
189	C₂×D₈	C₂×C₄×C₇	1	4	yes	—
190	C₂×D₈	C₂³×C₇	1	2	yes	—
191	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
192	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
193	C₂²×C₈	C₇×D₄	1	2	yes	—
194	C₂²×C₈	C₇×D₄	1	4	yes	—
195	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
196	C₂²×C₈	C₂³×C₇	1	2	yes	—
197	#117	—	2	4	no	—
198	#117	—	4	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₂²⋊C₄	—	4	1	no	—
201	C₂×C₄²	—	4	1	no	—
202	C₂×C₁₆	—	2	4	no	—
203	C₂×C₁₆	—	2	8	no	—
204	C₂×C₁₆	C₂³×C₇	1	4	yes	—
205	#107	—	2	2	no	—
206	#107	C₂³×C₇	1	4	yes	—
207	#106	—	2	4	no	—
208	C₂²×C₇	—	2	4	no	—
209	C₂²×C₇	—	2	8	no	—
210	C₂²×C₇	—	4	4	no	—
211	C₂²×C₇	—	8	4	no	—
212	C₂²×C₇	—	8	8	no	—
213	C₂²×C₇	C₂×#102	1	4	yes	—
214	C₂²×C₇	C₂×#102	1	2	yes	—
215	C₂²×C₇	#508	1	4	yes	—
216	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
217	C₄×C₇	—	4	4	no	—
218	C₄×C₇	—	8	4	no	—
219	C₄×C₇	C₂×#102	1	4	yes	—
220	C₄×C₇	C₂×#102	1	2	yes	—
221	C₄×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
222	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
223	C₂⁴	—	4	1	no	—
224	C₂⁴	C₇×D₈	1	1	yes	characteristic
225	#41	—	2	16	no	—
226	#41	—	4	16	no	—
227	#41	—	4	8	no	—
228	C₂×Q₈	—	2	4	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	4	1	no	—
231	C₂×Q₈	—	4	2	no	—
232	C₂×D₄	—	2	4	no	—
233	C₂×D₄	—	2	2	no	—
234	C₂×D₄	—	4	8	no	—
235	C₂×D₄	—	4	4	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	—	4	8	no	—
240	C₂²×C₄	—	4	4	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	C₇×C₂²⋊C₄	1	4	yes	—
243	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
244	C₂²×C₄	C₇×D₈	1	2	yes	—
245	C₂²×C₄	C₇×SD₁₆	1	4	yes	—
246	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
248	Q₁₆	—	2	4	no	—
249	Q₁₆	—	2	2	no	—
250	Q₁₆	C₂²×C₄×C₇	1	4	yes	—
251	SD₁₆	—	2	16	no	—
252	D₈	—	2	4	no	—
253	D₈	—	2	2	no	—
254	D₈	C₂²×C₄×C₇	1	4	yes	—
255	C₂×C₈	C₇×C₂²⋊C₄	1	4	yes	—
256	C₂×C₈	C₇×C₂²⋊C₄	1	8	yes	—
257	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
258	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
259	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
260	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
261	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
262	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
263	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₂²⋊C₄	—	4	4	no	—
267	C₄²	—	4	4	no	—
268	C₁₆	—	2	4	no	—
269	C₂×C₇	—	2	4	no	—
270	C₂×C₇	—	8	4	no	—
271	C₂×C₇	#2138	1	4	yes	direct product factor
272	C₂×C₇	C₂²×#102	1	1	yes	characteristic
273	C₂×C₇	C₂×#508	1	2	yes	—
274	C₂³	—	2	4	no	—
275	C₂³	—	4	4	no	—
276	C₂³	—	4	2	no	—
277	C₂³	—	8	4	no	—
278	C₂³	C₇×#102	1	4	yes	—
279	C₂³	C₂×C₇×D₈	1	2	yes	—
280	C₂³	C₂×C₇×D₈	1	1	yes	characteristic
281	Q₈	—	2	4	no	—
282	Q₈	—	4	4	no	—
283	Q₈	—	4	2	no	—
284	D₄	—	2	4	no	—
285	D₄	—	4	16	no	—
286	D₄	—	4	4	no	—
287	D₄	—	4	2	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	16	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	8	4	no	—
293	C₂×C₄	C₇×#102	1	4	yes	—
294	C₂×C₄	C₇×#102	1	8	yes	—
295	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
296	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	4	yes	—
297	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
298	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
299	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
300	C₂×C₄	C₂×C₇×SD₁₆	1	4	yes	—
301	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
302	C₈	C₂×C₇×C₂²⋊C₄	1	2	yes	—
303	C₈	C₂×C₇×C₂²⋊C₄	1	4	yes	—
304	C₈	C₂³×C₄×C₇	1	1	yes	characteristic
305	C₈	C₂²×C₇×D₄	1	1	yes	characteristic
306	C₇	C₂×#2138	1	1	yes	characteristic, direct product factor, Sylow
307	C₂²	—	2	4	no	—
308	C₂²	—	2	8	no	—
309	C₂²	—	4	4	no	—
310	C₂²	—	8	4	no	—
311	C₂²	—	8	8	no	—
312	C₂²	C₂×C₇×#102	1	4	yes	—
313	C₂²	C₂×C₇×#102	1	2	yes	—
314	C₂²	C₇×#508	1	4	yes	—
315	C₂²	C₂²×C₇×D₈	1	1	yes	characteristic
316	C₄	—	4	4	no	—
317	C₄	—	8	4	no	—
318	C₄	C₂×C₇×#102	1	4	yes	—
319	C₄	C₂×C₇×#102	1	2	yes	—
320	C₄	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
321	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
322	C₂	—	2	4	no	—
323	C₂	—	8	4	no	—
324	C₂	C₇×#2138	1	4	yes	direct product factor
325	C₂	C₂²×C₇×#102	1	1	yes	characteristic
326	C₂	C₂×C₇×#508	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₇	D₈
commutator	C₈	C₂³×C₄×C₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₇
socle	C₂³×C₇	C₂×D₈
prime radical	C₂²×C₇×#135	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×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. C₂³×C₇×C₁₆	C₂	semidirect
3. C₂×C₇×#427	C₂	semidirect
4. C₇×#2138	C₂	direct (N × H)
5. C₂×C₇×#358	C₂	semidirect
6. C₂×C₇×#357	C₂	semidirect
10. C₂×C₇×#135	C₄	semidirect
19. C₂²×C₇×C₁₆	C₂²	semidirect
21. C₇×#427	C₂²	semidirect
24. C₇×#358	C₂²	semidirect
25. C₇×#357	C₂²	semidirect
26. C₂×#2138	C₇	direct (N × H)
38. C₇×#135	C₂×C₄	semidirect
67. C₂³×C₁₆	C₂×C₇	semidirect
68. C₂×#427	C₂×C₇	semidirect
69. #2138	C₂×C₇	direct (N × H)
70. C₂×#358	C₂×C₇	semidirect
71. C₂×#357	C₂×C₇	semidirect
121. C₂×#135	C₄×C₇	semidirect
130. C₂²×C₁₆	C₂²×C₇	semidirect
132. #427	C₂²×C₇	semidirect
135. #358	C₂²×C₇	semidirect
136. #357	C₂²×C₇	semidirect
180. #135	C₂×C₄×C₇	semidirect
271. C₂×C₇	#2138	direct (N × H)
306. C₇	C₂×#2138	direct (N × H)
324. C₂	C₇×#2138	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
C₂³×C₇×C₁₆	2
C₂×C₇×#427	2
C₇×#2138	2
C₂×C₇×#358	2
C₂×C₇×#357	2
C₂×#2138	7

Subgroups — C2×C7×#2138 #13210592

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#2138 #13210592