Subgroups — C₂×C₇×#1716 #13196823

Order 1792.

798 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#498	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#498	C₂	1	2	yes	maximal
3	C₂×C₇×#494	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×#487	C₂	1	2	yes	maximal
5	C₇×#1716	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₇×#322	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₇×Q₁₆	—	2	2	no	—
8	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₇×#498	—	2	4	no	—
12	C₇×#498	—	2	2	no	—
13	C₇×#498	C₂²	1	8	yes	—
14	C₇×#498	C₂²	1	4	yes	—
15	C₇×#494	—	2	2	no	—
16	C₇×#494	C₂²	1	4	yes	—
17	C₇×#487	—	2	4	no	—
18	C₇×#487	C₂²	1	8	yes	—
19	C₂×C₇×#106	—	2	4	no	—
20	C₂×C₇×#106	—	2	2	no	—
21	C₂×C₇×#103	—	2	4	no	—
22	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
23	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
24	C₇×#322	C₂²	1	4	yes	—
25	C₂×#1716	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂²×C₇×Q₈	—	2	2	no	—
27	C₂²×C₇×Q₈	—	2	1	no	—
28	C₂²×C₇×Q₈	—	2	1	no	—
29	C₂×C₇×Q₁₆	—	2	8	no	—
30	C₂×C₇×Q₁₆	—	4	8	no	—
31	C₂²×C₇×C₈	D₄	1	4	yes	—
32	C₂²×C₇×C₈	D₄	1	2	yes	—
33	C₇×C₄⋊Q₈	—	2	2	no	—
34	C₇×C₄⋊Q₈	—	2	2	no	—
35	C₇×C₄⋊Q₈	—	2	2	no	—
36	C₇×C₄⋊Q₈	—	2	2	no	—
37	C₇×C₄⋊Q₈	—	2	2	no	—
38	C₇×C₄⋊Q₈	—	2	1	no	—
39	C₇×C₄⋊Q₈	—	2	1	no	—
40	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
41	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
42	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
43	C₂×C₇×C₄⋊C₄	—	2	2	no	—
44	C₂×C₇×C₄⋊C₄	—	2	2	no	—
45	C₂×C₇×C₄⋊C₄	—	2	2	no	—
46	C₂×C₇×C₄⋊C₄	—	2	2	no	—
47	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
48	C₇×#106	—	2	16	no	—
49	C₇×#106	—	2	8	no	—
50	C₇×#103	—	2	16	no	—
51	C₄×C₇×C₈	D₄	1	4	yes	—
52	C₄×C₇×C₈	D₄	1	2	yes	—
53	C₄×C₇×C₈	C₂³	1	4	yes	—
54	C₄×C₇×C₈	C₂³	1	2	yes	—
55	C₂×#498	C₂×C₇	1	1	yes	characteristic
56	C₂×#498	C₂×C₇	1	2	yes	—
57	C₂×#494	C₂×C₇	1	1	yes	characteristic
58	C₂×#487	C₂×C₇	1	2	yes	—
59	#1716	C₂×C₇	1	8	yes	direct product factor
60	C₂×#322	C₂×C₇	1	1	yes	characteristic
61	C₂×C₇×Q₈	—	2	4	no	—
62	C₂×C₇×Q₈	—	2	2	no	—
63	C₂×C₇×Q₈	—	2	2	no	—
64	C₂×C₇×Q₈	—	4	1	no	—
65	C₂×C₇×Q₈	—	4	1	no	—
66	C₂×C₇×Q₈	—	4	2	no	—
67	C₂×C₇×Q₈	—	4	4	no	—
68	C₂×C₇×Q₈	—	4	4	no	—
69	C₂×C₇×Q₈	—	4	8	no	—
70	C₂²×C₄×C₇	—	4	1	no	—
71	C₂²×C₄×C₇	—	4	1	no	—
72	C₂²×C₄×C₇	—	4	2	no	—
73	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
75	C₇×Q₁₆	—	4	16	no	—
76	C₂×C₇×C₈	—	2	8	no	—
77	C₂×C₇×C₈	SD₁₆	1	8	yes	—
78	C₂×C₇×C₈	C₂×D₄	1	8	yes	—
79	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
80	C₇×C₄⋊C₄	—	2	4	no	—
81	C₇×C₄⋊C₄	—	2	4	no	—
82	C₇×C₄⋊C₄	—	2	4	no	—
83	C₇×C₄⋊C₄	—	2	4	no	—
84	C₇×C₄⋊C₄	—	4	2	no	—
85	C₇×C₄⋊C₄	—	4	2	no	—
86	C₇×C₄⋊C₄	—	4	2	no	—
87	C₇×C₄⋊C₄	—	4	2	no	—
88	C₄²×C₇	C₂×D₄	1	2	yes	—
89	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₇	C₂⁴	1	1	yes	characteristic
91	C₂²×Q₁₆	—	2	2	no	—
92	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
93	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
94	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
95	#498	—	2	4	no	—
96	#498	—	2	2	no	—
97	#498	C₂²×C₇	1	8	yes	—
98	#498	C₂²×C₇	1	4	yes	—
99	#494	—	2	2	no	—
100	#494	C₂²×C₇	1	4	yes	—
101	#487	—	2	4	no	—
102	#487	C₂²×C₇	1	8	yes	—
103	C₂×#106	—	2	4	no	—
104	C₂×#106	—	2	2	no	—
105	C₂×#103	—	2	4	no	—
106	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
107	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
108	#322	C₂²×C₇	1	4	yes	—
109	C₂³×C₇	#127	1	1	yes	characteristic
110	C₇×Q₈	—	4	4	no	—
111	C₇×Q₈	—	4	4	no	—
112	C₇×Q₈	—	4	8	no	—
113	C₇×Q₈	—	8	2	no	—
114	C₇×Q₈	—	8	2	no	—
115	C₇×Q₈	—	8	4	no	—
116	C₂×C₄×C₇	—	4	2	no	—
117	C₂×C₄×C₇	—	4	2	no	—
118	C₂×C₄×C₇	—	4	4	no	—
119	C₂×C₄×C₇	—	8	2	no	—
120	C₂×C₄×C₇	—	8	2	no	—
121	C₂×C₄×C₇	—	8	4	no	—
122	C₂×C₄×C₇	#127	1	1	yes	characteristic
123	C₂×C₄×C₇	#127	1	2	yes	—
124	C₂×C₄×C₇	C₂×SD₁₆	1	4	yes	—
125	C₂×C₄×C₇	#137	1	8	yes	—
126	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
128	C₇×C₈	—	2	8	no	—
129	C₇×C₈	C₂×SD₁₆	1	8	yes	—
130	C₂²×Q₈	—	2	1	no	—
131	C₂²×Q₈	—	2	1	no	—
132	C₂²×Q₈	—	2	2	no	—
133	C₂×Q₁₆	—	2	8	no	—
134	C₂×Q₁₆	—	4	8	no	—
135	C₂²×C₈	C₇×D₄	1	4	yes	—
136	C₂²×C₈	C₇×D₄	1	2	yes	—
137	C₄⋊Q₈	—	2	2	no	—
138	C₄⋊Q₈	—	2	2	no	—
139	C₄⋊Q₈	—	2	2	no	—
140	C₄⋊Q₈	—	2	2	no	—
141	C₄⋊Q₈	—	2	2	no	—
142	C₄⋊Q₈	—	2	1	no	—
143	C₄⋊Q₈	—	2	1	no	—
144	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
145	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
146	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	2	2	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
152	#106	—	2	16	no	—
153	#106	—	2	8	no	—
154	#103	—	2	16	no	—
155	C₄×C₈	C₇×D₄	1	4	yes	—
156	C₄×C₈	C₇×D₄	1	2	yes	—
157	C₄×C₈	C₂³×C₇	1	4	yes	—
158	C₄×C₈	C₂³×C₇	1	2	yes	—
159	C₂²×C₇	#492	1	2	yes	—
160	C₂²×C₇	#497	1	4	yes	—
161	C₂²×C₇	C₂×#127	1	1	yes	characteristic
162	C₄×C₇	—	8	2	no	—
163	C₄×C₇	—	8	2	no	—
164	C₄×C₇	—	8	4	no	—
165	C₄×C₇	#492	1	2	yes	—
166	C₄×C₇	#497	1	4	yes	—
167	C₄×C₇	C₂²×SD₁₆	1	2	yes	—
168	C₄×C₇	C₂×#137	1	4	yes	—
169	C₂×Q₈	—	2	4	no	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	2	2	no	—
172	C₂×Q₈	—	4	1	no	—
173	C₂×Q₈	—	4	1	no	—
174	C₂×Q₈	—	4	2	no	—
175	C₂×Q₈	—	4	4	no	—
176	C₂×Q₈	—	4	4	no	—
177	C₂×Q₈	—	4	8	no	—
178	C₂²×C₄	—	4	1	no	—
179	C₂²×C₄	—	4	1	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
183	Q₁₆	—	4	16	no	—
184	C₂×C₈	—	2	8	no	—
185	C₂×C₈	C₇×SD₁₆	1	8	yes	—
186	C₂×C₈	C₂×C₇×D₄	1	8	yes	—
187	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	2	4	no	—
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄²	C₂×C₇×D₄	1	2	yes	—
197	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
198	C₄²	C₂⁴×C₇	1	1	yes	characteristic
199	C₂×C₇	#1716	1	4	yes	direct product factor
200	C₂×C₇	C₂×#492	1	1	yes	characteristic
201	C₂×C₇	C₂×#497	1	2	yes	—
202	C₂³	C₇×#127	1	1	yes	characteristic
203	Q₈	—	4	4	no	—
204	Q₈	—	4	4	no	—
205	Q₈	—	4	8	no	—
206	Q₈	—	8	2	no	—
207	Q₈	—	8	2	no	—
208	Q₈	—	8	4	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	8	2	no	—
213	C₂×C₄	—	8	2	no	—
214	C₂×C₄	—	8	4	no	—
215	C₂×C₄	C₇×#127	1	1	yes	characteristic
216	C₂×C₄	C₇×#127	1	2	yes	—
217	C₂×C₄	C₂×C₇×SD₁₆	1	4	yes	—
218	C₂×C₄	C₇×#137	1	8	yes	—
219	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
220	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
221	C₈	—	2	8	no	—
222	C₈	C₂×C₇×SD₁₆	1	8	yes	—
223	C₇	C₂×#1716	1	1	yes	characteristic, direct product factor, Sylow
224	C₂²	C₇×#492	1	2	yes	—
225	C₂²	C₇×#497	1	4	yes	—
226	C₂²	C₂×C₇×#127	1	1	yes	characteristic
227	C₄	—	8	2	no	—
228	C₄	—	8	2	no	—
229	C₄	—	8	4	no	—
230	C₄	C₇×#492	1	2	yes	—
231	C₄	C₇×#497	1	4	yes	—
232	C₄	C₂²×C₇×SD₁₆	1	2	yes	—
233	C₄	C₂×C₇×#137	1	4	yes	—
234	C₂	C₇×#1716	1	4	yes	direct product factor
235	C₂	C₂×C₇×#492	1	1	yes	characteristic
236	C₂	C₂×C₇×#497	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₇×#1716	C₂	direct (N × H)
25. C₂×#1716	C₇	direct (N × H)
59. #1716	C₂×C₇	direct (N × H)
199. C₂×C₇	#1716	direct (N × H)
223. C₇	C₂×#1716	direct (N × H)
234. C₂	C₇×#1716	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×#498	2
C₂×C₇×#494	2
C₂×C₇×#487	2
C₇×#1716	2
C₂×C₇×#322	2
C₂×#1716	7

Subgroups — C2×C7×#1716 #13196823

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1716 #13196823