Subgroups — C₂×C₆₇×#1561 #30696987

Order 17152.

1042 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#423×C₆₇	C₂	1	2	yes	maximal
3	C₆₇×#1561	C₂	1	8	yes	maximal, direct product factor
4	C₂×#332×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₆₇	—	2	2	no	—
6	#533×C₆₇	—	2	4	no	—
7	#533×C₆₇	C₄	1	4	yes	—
8	#533×C₆₇	C₂²	1	4	yes	—
9	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
10	C₂×#125×C₆₇	C₄	1	2	yes	—
11	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
12	C₂×C₄×Q₈×C₆₇	—	2	2	no	—
13	C₂×#117×C₆₇	C₄	1	2	yes	—
14	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
15	#423×C₆₇	—	2	4	no	—
16	#423×C₆₇	C₂²	1	8	yes	—
17	C₂×C₄⋊C₈×C₆₇	—	2	2	no	—
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
19	#332×C₆₇	C₂²	1	16	yes	complemented
20	C₂²×Q₈×C₆₇	—	2	2	no	—
21	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₆₇	D₄	1	2	yes	—
23	C₂×M₁₆×C₆₇	—	2	8	no	—
24	C₂×M₁₆×C₆₇	—	4	8	no	—
25	C₂²×C₈×C₆₇	—	2	4	no	—
26	C₄⋊Q₈×C₆₇	—	2	2	no	—
27	C₄⋊Q₈×C₆₇	—	2	2	no	—
28	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
29	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
30	#125×C₆₇	—	2	4	no	—
31	#125×C₆₇	C₂×C₄	1	8	yes	—
32	C₂²⋊Q₈×C₆₇	—	2	8	no	—
33	C₂²⋊Q₈×C₆₇	—	2	8	no	—
34	C₄×Q₈×C₆₇	—	2	8	no	—
35	C₄×Q₈×C₆₇	—	2	8	no	—
36	#117×C₆₇	—	2	4	no	—
37	#117×C₆₇	C₂×C₄	1	8	yes	—
38	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
39	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
41	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
42	C₂×C₄²×C₆₇	—	2	2	no	—
43	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₆₇	D₄	1	2	yes	—
46	C₂×C₄²×C₆₇	D₄	1	4	yes	—
47	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
49	C₄⋊C₈×C₆₇	—	2	8	no	—
50	C₄⋊C₈×C₆₇	—	2	4	no	—
51	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
52	C₂⁴×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×Q₈×C₆₇	—	2	4	no	—
54	C₂×Q₈×C₆₇	—	2	4	no	—
55	C₂×Q₈×C₆₇	—	4	4	no	—
56	C₂×Q₈×C₆₇	—	4	4	no	—
57	C₂²×C₄×C₆₇	—	2	4	no	—
58	C₂²×C₄×C₆₇	—	2	4	no	—
59	C₂²×C₄×C₆₇	—	2	4	no	—
60	C₂²×C₄×C₆₇	—	2	4	no	—
61	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
62	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
68	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
69	M₁₆×C₆₇	—	4	16	no	—
70	C₂×C₈×C₆₇	—	2	8	no	—
71	C₂×C₈×C₆₇	—	4	8	no	—
72	C₄⋊C₄×C₆₇	—	2	16	no	—
73	C₄⋊C₄×C₆₇	—	2	4	no	—
74	C₄⋊C₄×C₆₇	—	2	4	no	—
75	C₄⋊C₄×C₆₇	—	2	4	no	—
76	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
77	C₂²⋊C₄×C₆₇	—	2	8	no	—
78	C₄²×C₆₇	—	2	8	no	—
79	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
80	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
81	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
82	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₆₇	C₂×D₄	1	4	yes	—
84	C₄²×C₆₇	C₂×D₄	1	2	yes	—
85	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
86	C₂³×C₆₇	—	2	2	no	—
87	C₂³×C₆₇	—	2	4	no	—
88	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
89	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	Q₈×C₆₇	—	4	4	no	—
91	Q₈×C₆₇	—	4	4	no	—
92	Q₈×C₆₇	—	4	8	no	—
93	C₂×C₄×C₆₇	—	2	2	no	—
94	C₂×C₄×C₆₇	—	2	4	no	—
95	C₂×C₄×C₆₇	—	2	4	no	—
96	C₂×C₄×C₆₇	—	2	4	no	—
97	C₂×C₄×C₆₇	—	2	4	no	—
98	C₂×C₄×C₆₇	—	2	4	no	—
99	C₂×C₄×C₆₇	—	2	8	no	—
100	C₂×C₄×C₆₇	—	2	8	no	—
101	C₂×C₄×C₆₇	—	4	8	no	—
102	C₂×C₄×C₆₇	#101	1	4	yes	—
103	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
104	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
106	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
107	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
108	C₂×C₄×C₆₇	#137	1	4	yes	—
109	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
111	C₈×C₆₇	—	4	8	no	—
112	C₂²×C₆₇	—	2	2	no	—
113	C₂²×C₆₇	—	2	4	no	—
114	C₂²×C₆₇	—	4	4	no	—
115	C₂²×C₆₇	C₂×#101	1	2	yes	—
116	C₂²×C₆₇	#417	1	4	yes	—
117	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
118	C₄×C₆₇	—	2	1	no	—
119	C₄×C₆₇	—	2	1	no	—
120	C₄×C₆₇	—	2	4	no	—
121	C₄×C₆₇	—	4	8	no	—
122	C₄×C₆₇	C₂×#101	1	4	yes	—
123	C₄×C₆₇	#417	1	4	yes	—
124	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
125	C₄×C₆₇	C₂×#137	1	2	yes	—
126	C₂×#1561	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂×C₆₇	—	4	2	no	—
128	C₂×C₆₇	#1561	1	4	yes	direct product factor
129	C₂×C₆₇	C₂²×#101	1	1	yes	characteristic
130	C₂×C₆₇	C₂×#417	1	2	yes	—
131	C₂×#533	C₂×C₆₇	1	1	yes	characteristic
132	C₂×#423	C₂×C₆₇	1	2	yes	—
133	#1561	C₂×C₆₇	1	8	yes	direct product factor
134	C₂×#332	C₂×C₆₇	1	4	yes	complemented
135	C₆₇	C₂×#1561	1	1	yes	characteristic, direct product factor, Sylow
136	C₂²×M₁₆	—	2	2	no	—
137	#533	—	2	4	no	—
138	#533	C₄×C₆₇	1	4	yes	—
139	#533	C₂²×C₆₇	1	4	yes	—
140	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
141	C₂×#125	C₄×C₆₇	1	2	yes	—
142	C₂×C₂²⋊Q₈	—	2	2	no	—
143	C₂×C₄×Q₈	—	2	2	no	—
144	C₂×#117	C₄×C₆₇	1	2	yes	—
145	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
146	#423	—	2	4	no	—
147	#423	C₂²×C₆₇	1	8	yes	—
148	C₂×C₄⋊C₈	—	2	2	no	—
149	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
150	#332	C₂²×C₆₇	1	16	yes	complemented
151	C₂²×Q₈	—	2	2	no	—
152	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
153	C₂³×C₄	D₄×C₆₇	1	2	yes	—
154	C₂×M₁₆	—	2	8	no	—
155	C₂×M₁₆	—	4	8	no	—
156	C₂²×C₈	—	2	4	no	—
157	C₄⋊Q₈	—	2	2	no	—
158	C₄⋊Q₈	—	2	2	no	—
159	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
160	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
161	#125	—	2	4	no	—
162	#125	C₂×C₄×C₆₇	1	8	yes	—
163	C₂²⋊Q₈	—	2	8	no	—
164	C₂²⋊Q₈	—	2	8	no	—
165	C₄×Q₈	—	2	8	no	—
166	C₄×Q₈	—	2	8	no	—
167	#117	—	2	4	no	—
168	#117	C₂×C₄×C₆₇	1	8	yes	—
169	C₂×C₄⋊C₄	—	2	4	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
172	C₂×C₂²⋊C₄	—	2	2	no	—
173	C₂×C₄²	—	2	2	no	—
174	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
175	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
176	C₂×C₄²	D₄×C₆₇	1	2	yes	—
177	C₂×C₄²	D₄×C₆₇	1	4	yes	—
178	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
179	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
180	C₄⋊C₈	—	2	8	no	—
181	C₄⋊C₈	—	2	4	no	—
182	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
183	C₂⁴	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
184	C₂×Q₈	—	2	4	no	—
185	C₂×Q₈	—	2	4	no	—
186	C₂×Q₈	—	4	4	no	—
187	C₂×Q₈	—	4	4	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
193	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
194	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
195	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
196	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
197	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
198	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
199	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
200	M₁₆	—	4	16	no	—
201	C₂×C₈	—	2	8	no	—
202	C₂×C₈	—	4	8	no	—
203	C₄⋊C₄	—	2	16	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
208	C₂²⋊C₄	—	2	8	no	—
209	C₄²	—	2	8	no	—
210	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
211	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
212	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
213	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
214	C₄²	C₂×D₄×C₆₇	1	4	yes	—
215	C₄²	C₂×D₄×C₆₇	1	2	yes	—
216	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
217	C₂³	—	2	2	no	—
218	C₂³	—	2	4	no	—
219	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
220	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
221	Q₈	—	4	4	no	—
222	Q₈	—	4	4	no	—
223	Q₈	—	4	8	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	4	8	no	—
233	C₂×C₄	#101×C₆₇	1	4	yes	—
234	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
235	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
237	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
238	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
239	C₂×C₄	#137×C₆₇	1	4	yes	—
240	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
241	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
242	C₈	—	4	8	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	4	no	—
245	C₂²	—	4	4	no	—
246	C₂²	C₂×#101×C₆₇	1	2	yes	—
247	C₂²	#417×C₆₇	1	4	yes	—
248	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	4	no	—
252	C₄	—	4	8	no	—
253	C₄	C₂×#101×C₆₇	1	4	yes	—
254	C₄	#417×C₆₇	1	4	yes	—
255	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
256	C₄	C₂×#137×C₆₇	1	2	yes	—
257	C₂	—	4	2	no	—
258	C₂	C₆₇×#1561	1	4	yes	direct product factor
259	C₂	C₂²×#101×C₆₇	1	1	yes	characteristic
260	C₂	C₂×#417×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1561	C₂	direct (N × H)
4. C₂×#332×C₆₇	C₂	semidirect
19. #332×C₆₇	C₂²	semidirect
126. C₂×#1561	C₆₇	direct (N × H)
128. C₂×C₆₇	#1561	direct (N × H)
133. #1561	C₂×C₆₇	direct (N × H)
134. C₂×#332	C₂×C₆₇	semidirect
135. C₆₇	C₂×#1561	direct (N × H)
150. #332	C₂²×C₆₇	semidirect
258. C₂	C₆₇×#1561	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×#533×C₆₇	2
C₂×#423×C₆₇	2
C₆₇×#1561	2
C₂×#332×C₆₇	2
C₂×#1561	67

Subgroups — C2×C67×#1561 #30696987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1561 #30696987