Subgroups — C₂×C₇×#1362 #13189144

Order 1792.

414 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#335	C₂	1	1	yes	maximal, characteristic
4	C₇×#1362	C₂	1	4	yes	maximal, direct product factor
5	C₂²×C₇×Q₁₆	—	2	1	no	—
6	C₂²×C₇×Q₁₆	C₄	1	2	yes	—
7	C₂×C₇×C₄⋊Q₈	—	2	1	no	—
8	C₂²×C₇×C₁₆	—	2	1	no	—
9	C₇×#494	—	2	1	no	—
10	C₇×#494	—	2	1	no	—
11	C₇×#494	C₄	1	2	yes	—
12	C₇×#494	C₂²	1	2	yes	—
13	C₂×C₇×C₄⋊C₈	—	2	1	no	—
14	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₇×C₄⋊C₁₆	—	2	1	no	—
16	C₇×C₄⋊C₁₆	C₂²	1	2	yes	—
17	C₇×#335	—	2	1	no	—
18	C₇×#335	C₂²	1	2	yes	—
19	C₂×#1362	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂²×C₇×Q₈	—	2	2	no	—
21	C₂×C₇×Q₁₆	—	2	2	no	—
22	C₂×C₇×Q₁₆	—	2	2	no	—
23	C₂×C₇×Q₁₆	—	2	2	no	—
24	C₂×C₇×Q₁₆	—	4	4	no	—
25	C₂×C₇×Q₁₆	—	4	2	no	—
26	C₂×C₇×Q₁₆	—	4	2	no	—
27	C₂×C₇×Q₁₆	C₂×C₄	1	4	yes	—
28	C₂²×C₇×C₈	—	4	1	no	—
29	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
31	C₇×C₄⋊Q₈	—	2	2	no	—
32	C₇×C₄⋊Q₈	—	2	2	no	—
33	C₇×C₄⋊Q₈	—	4	1	no	—
34	C₇×C₄⋊Q₈	—	4	1	no	—
35	C₂×C₇×C₄⋊C₄	—	4	1	no	—
36	C₂×C₇×C₄⋊C₄	—	4	1	no	—
37	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
38	C₂×C₇×C₁₆	—	2	2	no	—
39	C₂×C₇×C₁₆	—	4	2	no	—
40	C₇×C₄⋊C₈	—	2	2	no	—
41	C₇×C₄⋊C₈	—	4	1	no	—
42	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
43	C₄×C₇×C₈	D₄	1	1	yes	characteristic
44	C₄×C₇×C₈	D₄	1	1	yes	characteristic
45	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
46	C₂×#494	C₂×C₇	1	1	yes	characteristic
47	C₂×C₄⋊C₁₆	C₂×C₇	1	1	yes	characteristic
48	C₂×#335	C₂×C₇	1	1	yes	characteristic
49	#1362	C₂×C₇	1	4	yes	direct product factor
50	C₂×C₇×Q₈	—	2	4	no	—
51	C₂×C₇×Q₈	—	4	2	no	—
52	C₂×C₇×Q₈	—	4	4	no	—
53	C₂×C₇×Q₈	—	4	4	no	—
54	C₂²×C₄×C₇	—	4	2	no	—
55	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₇	SD₁₆	1	1	yes	characteristic
58	C₇×Q₁₆	—	4	4	no	—
59	C₇×Q₁₆	—	4	4	no	—
60	C₇×Q₁₆	—	4	4	no	—
61	C₇×Q₁₆	—	4	2	no	—
62	C₇×Q₁₆	—	4	2	no	—
63	C₂×C₇×C₈	—	2	2	no	—
64	C₂×C₇×C₈	—	4	2	no	—
65	C₂×C₇×C₈	—	8	2	no	—
66	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₇×C₈	D₈	1	2	yes	—
69	C₂×C₇×C₈	SD₁₆	1	2	yes	—
70	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
72	C₇×C₄⋊C₄	—	4	2	no	—
73	C₇×C₄⋊C₄	—	4	2	no	—
74	C₇×C₄⋊C₄	—	4	2	no	—
75	C₇×C₄⋊C₄	—	4	2	no	—
76	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₄²×C₇	D₈	1	1	yes	characteristic
78	C₄²×C₇	SD₁₆	1	1	yes	characteristic
79	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
80	C₇×C₁₆	—	4	2	no	—
81	C₂²×Q₁₆	—	2	1	no	—
82	C₂²×Q₁₆	C₄×C₇	1	2	yes	—
83	C₂×C₄⋊Q₈	—	2	1	no	—
84	C₂²×C₁₆	—	2	1	no	—
85	#494	—	2	1	no	—
86	#494	—	2	1	no	—
87	#494	C₄×C₇	1	2	yes	—
88	#494	C₂²×C₇	1	2	yes	—
89	C₂×C₄⋊C₈	—	2	1	no	—
90	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
91	C₄⋊C₁₆	—	2	1	no	—
92	C₄⋊C₁₆	C₂²×C₇	1	2	yes	—
93	#335	—	2	1	no	—
94	#335	C₂²×C₇	1	2	yes	—
95	C₂³×C₇	#102	1	1	yes	characteristic
96	C₇×Q₈	—	4	4	no	—
97	C₇×Q₈	—	4	4	no	—
98	C₇×Q₈	—	8	4	no	—
99	C₂×C₄×C₇	—	2	2	no	—
100	C₂×C₄×C₇	—	4	4	no	—
101	C₂×C₄×C₇	—	8	4	no	—
102	C₂×C₄×C₇	#100	1	2	yes	—
103	C₂×C₄×C₇	#102	1	1	yes	characteristic
104	C₂×C₄×C₇	#102	1	1	yes	characteristic
105	C₂×C₄×C₇	#102	1	2	yes	—
106	C₂×C₄×C₇	#102	1	1	yes	characteristic
107	C₂×C₄×C₇	SD₃₂	1	2	yes	—
108	C₂×C₄×C₇	Q₃₂	1	2	yes	—
109	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
112	C₇×C₈	—	2	1	no	—
113	C₇×C₈	—	2	1	no	—
114	C₇×C₈	—	8	2	no	—
115	C₇×C₈	#102	1	2	yes	—
116	C₇×C₈	C₂×D₈	1	1	yes	characteristic
117	C₇×C₈	C₂×SD₁₆	1	1	yes	characteristic
118	C₂²×Q₈	—	2	2	no	—
119	C₂×Q₁₆	—	2	2	no	—
120	C₂×Q₁₆	—	2	2	no	—
121	C₂×Q₁₆	—	2	2	no	—
122	C₂×Q₁₆	—	4	4	no	—
123	C₂×Q₁₆	—	4	2	no	—
124	C₂×Q₁₆	—	4	2	no	—
125	C₂×Q₁₆	C₂×C₄×C₇	1	4	yes	—
126	C₂²×C₈	—	4	1	no	—
127	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
128	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
129	C₄⋊Q₈	—	2	2	no	—
130	C₄⋊Q₈	—	2	2	no	—
131	C₄⋊Q₈	—	4	1	no	—
132	C₄⋊Q₈	—	4	1	no	—
133	C₂×C₄⋊C₄	—	4	1	no	—
134	C₂×C₄⋊C₄	—	4	1	no	—
135	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
136	C₂×C₁₆	—	2	2	no	—
137	C₂×C₁₆	—	4	2	no	—
138	C₄⋊C₈	—	2	2	no	—
139	C₄⋊C₈	—	4	1	no	—
140	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
141	C₄×C₈	C₇×D₄	1	1	yes	characteristic
142	C₄×C₈	C₇×D₄	1	1	yes	characteristic
143	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
144	C₂²×C₇	#331	1	2	yes	—
145	C₂²×C₇	#358	1	2	yes	—
146	C₂²×C₇	#362	1	2	yes	—
147	C₂²×C₇	C₂×#102	1	1	yes	characteristic
148	C₄×C₇	—	2	1	no	—
149	C₄×C₇	—	2	1	no	—
150	C₄×C₇	—	8	4	no	—
151	C₄×C₇	#331	1	2	yes	—
152	C₄×C₇	#358	1	2	yes	—
153	C₄×C₇	C₂×#100	1	1	yes	characteristic
154	C₄×C₇	C₂×#102	1	1	yes	characteristic
155	C₄×C₇	C₂×SD₃₂	1	1	yes	characteristic
156	C₄×C₇	C₂×Q₃₂	1	1	yes	characteristic
157	C₂×Q₈	—	2	4	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂×Q₈	—	4	4	no	—
160	C₂×Q₈	—	4	4	no	—
161	C₂²×C₄	—	4	2	no	—
162	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
163	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
164	C₂²×C₄	C₇×SD₁₆	1	1	yes	characteristic
165	Q₁₆	—	4	4	no	—
166	Q₁₆	—	4	4	no	—
167	Q₁₆	—	4	4	no	—
168	Q₁₆	—	4	2	no	—
169	Q₁₆	—	4	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	—	8	2	no	—
173	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
174	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
175	C₂×C₈	C₇×D₈	1	2	yes	—
176	C₂×C₈	C₇×SD₁₆	1	2	yes	—
177	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
178	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
184	C₄²	C₇×D₈	1	1	yes	characteristic
185	C₄²	C₇×SD₁₆	1	1	yes	characteristic
186	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
187	C₁₆	—	4	2	no	—
188	C₂×C₇	#1362	1	4	yes	direct product factor
189	C₂×C₇	C₂×#331	1	1	yes	characteristic
190	C₂×C₇	C₂×#358	1	1	yes	characteristic
191	C₂×C₇	C₂×#362	1	1	yes	characteristic
192	C₂³	C₇×#102	1	1	yes	characteristic
193	Q₈	—	4	4	no	—
194	Q₈	—	4	4	no	—
195	Q₈	—	8	4	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	—	8	4	no	—
199	C₂×C₄	C₇×#100	1	2	yes	—
200	C₂×C₄	C₇×#102	1	1	yes	characteristic
201	C₂×C₄	C₇×#102	1	1	yes	characteristic
202	C₂×C₄	C₇×#102	1	2	yes	—
203	C₂×C₄	C₇×#102	1	1	yes	characteristic
204	C₂×C₄	C₇×SD₃₂	1	2	yes	—
205	C₂×C₄	C₇×Q₃₂	1	2	yes	—
206	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
207	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
208	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	—	8	2	no	—
212	C₈	C₇×#102	1	2	yes	—
213	C₈	C₂×C₇×D₈	1	1	yes	characteristic
214	C₈	C₂×C₇×SD₁₆	1	1	yes	characteristic
215	C₇	C₂×#1362	1	1	yes	characteristic, direct product factor, Sylow
216	C₂²	C₇×#331	1	2	yes	—
217	C₂²	C₇×#358	1	2	yes	—
218	C₂²	C₇×#362	1	2	yes	—
219	C₂²	C₂×C₇×#102	1	1	yes	characteristic
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	8	4	no	—
223	C₄	C₇×#331	1	2	yes	—
224	C₄	C₇×#358	1	2	yes	—
225	C₄	C₂×C₇×#100	1	1	yes	characteristic
226	C₄	C₂×C₇×#102	1	1	yes	characteristic
227	C₄	C₂×C₇×SD₃₂	1	1	yes	characteristic
228	C₄	C₂×C₇×Q₃₂	1	1	yes	characteristic
229	C₂	C₇×#1362	1	4	yes	direct product factor
230	C₂	C₂×C₇×#331	1	1	yes	characteristic
231	C₂	C₂×C₇×#358	1	1	yes	characteristic
232	C₂	C₂×C₇×#362	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	#102
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₄×C₈	C₂³×C₇
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇×#1362	C₂	direct (N × H)
19. C₂×#1362	C₇	direct (N × H)
49. #1362	C₂×C₇	direct (N × H)
188. C₂×C₇	#1362	direct (N × H)
215. C₇	C₂×#1362	direct (N × H)
229. C₂	C₇×#1362	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×#494	2
C₂×C₇×C₄⋊C₁₆	2
C₂×C₇×#335	2
C₇×#1362	2
C₂×#1362	7

Subgroups — C2×C7×#1362 #13189144

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1362 #13189144