Subgroups — C₆₇×#33316 #30712567

Order 17152.

894 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#501×C₆₇	C₂	1	4	yes	maximal
2	C₆₇×#2868	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2859	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1847	C₂	1	4	yes	maximal
5	C₆₇×#1845	C₂	1	2	yes	maximal
6	C₆₇×#1844	C₂	1	2	yes	maximal
7	C₄×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈×C₆₇	—	2	2	no	—
9	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
10	C₂×#125×C₆₇	C₂²	1	2	yes	—
11	#501×C₆₇	—	2	16	no	—
12	C₂×#107×C₆₇	C₂²	1	8	yes	—
13	C₂×#106×C₆₇	C₂²	1	8	yes	—
14	C₂×#97×C₆₇	C₂²	1	2	yes	—
15	C₂×#97×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
17	#389×C₆₇	—	2	2	no	—
18	#389×C₆₇	—	2	2	no	—
19	#384×C₆₇	—	2	2	no	—
20	#384×C₆₇	C₂²	1	1	yes	characteristic
21	#384×C₆₇	C₂²	1	1	yes	characteristic
22	#384×C₆₇	C₂²	1	2	yes	—
23	#384×C₆₇	C₂²	1	2	yes	—
24	#382×C₆₇	C₂²	1	2	yes	—
25	C₄³×C₆₇	C₂²	1	1	yes	characteristic
26	C₂²×Q₈×C₆₇	—	2	2	no	—
27	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
28	C₄⋊Q₈×C₆₇	—	2	8	no	—
29	C₄⋊Q₈×C₆₇	—	4	8	no	—
30	#125×C₆₇	—	2	8	no	—
31	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
32	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
33	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
34	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
35	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
39	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
41	C₂×C₄²×C₆₇	D₄	1	2	yes	—
42	C₂×C₄²×C₆₇	D₄	1	2	yes	—
43	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
46	#107×C₆₇	—	2	16	no	—
47	#106×C₆₇	—	2	16	no	—
48	#97×C₆₇	D₄	1	8	yes	—
49	#97×C₆₇	Q₈	1	8	yes	complemented
50	C₄×C₈×C₆₇	Q₈	1	8	yes	complemented
51	#95×C₆₇	—	2	4	no	—
52	C₂×Q₈×C₆₇	—	4	8	no	—
53	C₂×Q₈×C₆₇	—	4	4	no	—
54	C₂²×C₄×C₆₇	—	2	4	no	—
55	C₂²×C₄×C₆₇	—	2	2	no	—
56	C₂²×C₄×C₆₇	—	2	2	no	—
57	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
62	C₂×C₈×C₆₇	C₂×D₄	1	8	yes	—
63	C₂×C₈×C₆₇	C₂×Q₈	1	8	yes	—
64	C₂×C₈×C₆₇	C₂×Q₈	1	8	yes	—
65	C₄⋊C₄×C₆₇	—	2	4	no	—
66	C₄⋊C₄×C₆₇	—	2	8	no	—
67	C₄⋊C₄×C₆₇	—	2	4	no	—
68	C₄⋊C₄×C₆₇	—	4	8	no	—
69	C₄⋊C₄×C₆₇	—	4	4	no	—
70	C₄⋊C₄×C₆₇	—	4	4	no	—
71	C₄⋊C₄×C₆₇	—	4	4	no	—
72	C₄⋊C₄×C₆₇	—	4	4	no	—
73	C₄⋊C₄×C₆₇	—	4	8	no	—
74	C₄⋊C₄×C₆₇	—	4	4	no	—
75	C₄²×C₆₇	—	2	4	no	—
76	C₄²×C₆₇	—	2	4	no	—
77	C₄²×C₆₇	C₂×D₄	1	4	yes	—
78	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
79	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
80	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
81	Q₈×C₆₇	—	8	8	no	—
82	C₂×C₄×C₆₇	—	2	2	no	—
83	C₂×C₄×C₆₇	—	2	2	no	—
84	C₂×C₄×C₆₇	—	2	2	no	—
85	C₂×C₄×C₆₇	—	2	2	no	—
86	C₂×C₄×C₆₇	—	4	2	no	—
87	C₂×C₄×C₆₇	—	4	2	no	—
88	C₂×C₄×C₆₇	—	4	4	no	—
89	C₂×C₄×C₆₇	—	4	8	no	—
90	C₂×C₄×C₆₇	—	4	4	no	—
91	C₂×C₄×C₆₇	—	4	4	no	—
92	C₂×C₄×C₆₇	#127	1	2	yes	—
93	C₂×C₄×C₆₇	#127	1	2	yes	—
94	C₂×C₄×C₆₇	C₄⋊Q₈	1	4	yes	—
95	C₂×C₄×C₆₇	C₄⋊Q₈	1	4	yes	—
96	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
97	C₂×C₄×C₆₇	#137	1	4	yes	—
98	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
99	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
100	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
101	C₈×C₆₇	C₄⋊Q₈	1	16	yes	complemented
102	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
103	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
104	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
105	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
106	C₂²×C₆₇	C₂×#137	1	2	yes	—
107	C₄×C₆₇	—	2	2	no	—
108	C₄×C₆₇	—	2	2	no	—
109	C₄×C₆₇	—	2	4	no	—
110	C₄×C₆₇	—	8	2	no	—
111	C₄×C₆₇	—	8	4	no	—
112	C₄×C₆₇	—	8	2	no	—
113	C₄×C₆₇	#501	1	8	yes	complemented
114	C₄×C₆₇	C₂×C₄⋊Q₈	1	4	yes	—
115	#33316	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂×C₆₇	#2868	1	1	yes	characteristic
117	C₂×C₆₇	#3151	1	1	yes	characteristic
118	C₂×C₆₇	#3152	1	1	yes	characteristic
119	C₂×C₆₇	C₂×#501	1	2	yes	—
120	C₂×C₆₇	#3165	1	2	yes	—
121	C₂×#501	C₂×C₆₇	1	4	yes	—
122	#2868	C₂×C₆₇	1	1	yes	characteristic
123	#2859	C₂×C₆₇	1	1	yes	characteristic
124	#1847	C₂×C₆₇	1	4	yes	—
125	#1845	C₂×C₆₇	1	2	yes	—
126	#1844	C₂×C₆₇	1	2	yes	—
127	C₄×#97	C₂×C₆₇	1	1	yes	characteristic
128	C₆₇	#33316	1	1	yes	characteristic, direct product factor, Sylow
129	C₂×C₄⋊Q₈	—	2	2	no	—
130	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
131	C₂×#125	C₂²×C₆₇	1	2	yes	—
132	#501	—	2	16	no	—
133	C₂×#107	C₂²×C₆₇	1	8	yes	—
134	C₂×#106	C₂²×C₆₇	1	8	yes	—
135	C₂×#97	C₂²×C₆₇	1	2	yes	—
136	C₂×#97	C₂²×C₆₇	1	2	yes	—
137	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
138	#389	—	2	2	no	—
139	#389	—	2	2	no	—
140	#384	—	2	2	no	—
141	#384	C₂²×C₆₇	1	1	yes	characteristic
142	#384	C₂²×C₆₇	1	1	yes	characteristic
143	#384	C₂²×C₆₇	1	2	yes	—
144	#384	C₂²×C₆₇	1	2	yes	—
145	#382	C₂²×C₆₇	1	2	yes	—
146	C₄³	C₂²×C₆₇	1	1	yes	characteristic
147	C₂²×Q₈	—	2	2	no	—
148	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
149	C₄⋊Q₈	—	2	8	no	—
150	C₄⋊Q₈	—	4	8	no	—
151	#125	—	2	8	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	2	4	no	—
155	C₂×C₄⋊C₄	—	2	4	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
160	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
161	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
162	C₂×C₄²	D₄×C₆₇	1	2	yes	—
163	C₂×C₄²	D₄×C₆₇	1	2	yes	—
164	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
165	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
166	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
167	#107	—	2	16	no	—
168	#106	—	2	16	no	—
169	#97	D₄×C₆₇	1	8	yes	—
170	#97	Q₈×C₆₇	1	8	yes	complemented
171	C₄×C₈	Q₈×C₆₇	1	8	yes	complemented
172	#95	—	2	4	no	—
173	C₂×Q₈	—	4	8	no	—
174	C₂×Q₈	—	4	4	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
181	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
182	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
183	C₂×C₈	C₂×D₄×C₆₇	1	8	yes	—
184	C₂×C₈	C₂×Q₈×C₆₇	1	8	yes	—
185	C₂×C₈	C₂×Q₈×C₆₇	1	8	yes	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	2	8	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	4	8	no	—
190	C₄⋊C₄	—	4	4	no	—
191	C₄⋊C₄	—	4	4	no	—
192	C₄⋊C₄	—	4	4	no	—
193	C₄⋊C₄	—	4	8	no	—
194	C₄⋊C₄	—	4	4	no	—
195	C₄⋊C₄	—	4	4	no	—
196	C₄²	—	2	4	no	—
197	C₄²	—	2	4	no	—
198	C₄²	C₂×D₄×C₆₇	1	4	yes	—
199	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
200	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
201	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
202	Q₈	—	8	8	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	4	8	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	#127×C₆₇	1	2	yes	—
214	C₂×C₄	#127×C₆₇	1	2	yes	—
215	C₂×C₄	C₄⋊Q₈×C₆₇	1	4	yes	—
216	C₂×C₄	C₄⋊Q₈×C₆₇	1	4	yes	—
217	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
218	C₂×C₄	#137×C₆₇	1	4	yes	—
219	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
220	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
221	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
222	C₈	C₄⋊Q₈×C₆₇	1	16	yes	complemented
223	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
224	C₂²	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
225	C₂²	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
226	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
227	C₂²	C₂×#137×C₆₇	1	2	yes	—
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	4	no	—
231	C₄	—	8	4	no	—
232	C₄	—	8	2	no	—
233	C₄	—	8	2	no	—
234	C₄	#501×C₆₇	1	8	yes	complemented
235	C₄	C₂×C₄⋊Q₈×C₆₇	1	4	yes	—
236	C₂	C₆₇×#2868	1	1	yes	characteristic
237	C₂	C₆₇×#3151	1	1	yes	characteristic
238	C₂	C₆₇×#3152	1	1	yes	characteristic
239	C₂	C₂×#501×C₆₇	1	2	yes	—
240	C₂	C₆₇×#3165	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
49. #97×C₆₇	Q₈	semidirect
50. C₄×C₈×C₆₇	Q₈	semidirect
101. C₈×C₆₇	C₄⋊Q₈	semidirect
113. C₄×C₆₇	#501	semidirect
115. #33316	C₆₇	direct (N × H)
128. C₆₇	#33316	direct (N × H)
170. #97	Q₈×C₆₇	semidirect
171. C₄×C₈	Q₈×C₆₇	semidirect
222. C₈	C₄⋊Q₈×C₆₇	semidirect
234. C₄	#501×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#501×C₆₇	2
C₆₇×#2868	2
C₆₇×#2859	2
C₆₇×#1847	2
C₆₇×#1845	2
C₆₇×#1844	2
C₄×#97×C₆₇	2
#33316	67

Subgroups — C67×#33316 #30712567

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33316 #30712567