Subgroups — C₆₇×#33318 #30712569

Order 17152.

806 conjugacy classes of subgroups.

The subgroup lattice has 246 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	2	yes	maximal
2	C₂×#499×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#498×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2660	C₂	1	2	yes	maximal
5	C₆₇×#1847	C₂	1	4	yes	maximal
6	C₆₇×#1846	C₂	1	2	yes	maximal
7	C₆₇×#1844	C₂	1	2	yes	maximal
8	C₆₇×#1726	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
10	C₂×#125×C₆₇	C₂²	1	2	yes	—
11	#501×C₆₇	—	2	8	no	—
12	#499×C₆₇	—	2	4	no	—
13	#498×C₆₇	—	2	4	no	—
14	C₂×#107×C₆₇	C₂²	1	4	yes	—
15	C₂×#107×C₆₇	C₂²	1	2	yes	—
16	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#106×C₆₇	C₂²	1	2	yes	—
18	C₂×#106×C₆₇	C₂²	1	4	yes	—
19	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#106×C₆₇	C₂²	1	2	yes	—
21	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#97×C₆₇	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
25	#398×C₆₇	—	2	4	no	—
26	#395×C₆₇	—	2	4	no	—
27	#384×C₆₇	C₂²	1	4	yes	—
28	#382×C₆₇	C₂²	1	4	yes	—
29	#376×C₆₇	C₂²	1	1	yes	characteristic
30	C₂²×Q₈×C₆₇	—	2	2	no	—
31	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
34	C₄⋊Q₈×C₆₇	—	2	4	no	—
35	C₄⋊Q₈×C₆₇	—	2	4	no	—
36	#125×C₆₇	—	2	4	no	—
37	#125×C₆₇	—	2	4	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
42	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
44	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
45	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
47	#107×C₆₇	—	2	4	no	—
48	#107×C₆₇	—	2	8	no	—
49	#106×C₆₇	—	2	4	no	—
50	#106×C₆₇	—	2	4	no	—
51	#106×C₆₇	—	2	2	no	—
52	#106×C₆₇	—	2	2	no	—
53	#106×C₆₇	—	2	8	no	—
54	#97×C₆₇	D₄	1	4	yes	—
55	#97×C₆₇	Q₈	1	8	yes	complemented
56	C₄×C₈×C₆₇	D₄	1	4	yes	—
57	C₄×C₈×C₆₇	Q₈	1	8	yes	complemented
58	#95×C₆₇	—	2	8	no	—
59	#95×C₆₇	D₄	1	4	yes	—
60	C₂×Q₈×C₆₇	—	4	4	no	—
61	C₂×Q₈×C₆₇	—	8	4	no	—
62	C₂²×C₄×C₆₇	—	2	2	no	—
63	C₂²×C₄×C₆₇	—	2	4	no	—
64	C₂²×C₄×C₆₇	—	2	2	no	—
65	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
67	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
68	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
71	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
72	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
73	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
74	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
75	C₄⋊C₄×C₆₇	—	2	2	no	—
76	C₄⋊C₄×C₆₇	—	2	4	no	—
77	C₄⋊C₄×C₆₇	—	2	2	no	—
78	C₄⋊C₄×C₆₇	—	2	2	no	—
79	C₄⋊C₄×C₆₇	—	2	4	no	—
80	C₄⋊C₄×C₆₇	—	2	2	no	—
81	C₄⋊C₄×C₆₇	—	4	4	no	—
82	C₄⋊C₄×C₆₇	—	4	4	no	—
83	C₄⋊C₄×C₆₇	—	4	4	no	—
84	C₄⋊C₄×C₆₇	—	8	8	no	—
85	C₄²×C₆₇	C₂×D₄	1	2	yes	—
86	C₄²×C₆₇	C₂×D₄	1	2	yes	—
87	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
88	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
89	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
90	Q₈×C₆₇	—	8	8	no	—
91	C₂×C₄×C₆₇	—	2	8	no	—
92	C₂×C₄×C₆₇	—	2	4	no	—
93	C₂×C₄×C₆₇	—	4	2	no	—
94	C₂×C₄×C₆₇	—	4	2	no	—
95	C₂×C₄×C₆₇	—	4	4	no	—
96	C₂×C₄×C₆₇	—	4	4	no	—
97	C₂×C₄×C₆₇	—	4	8	no	—
98	C₂×C₄×C₆₇	—	4	4	no	—
99	C₂×C₄×C₆₇	#127	1	4	yes	—
100	C₂×C₄×C₆₇	C₄⋊Q₈	1	8	yes	—
101	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
104	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
105	C₈×C₆₇	C₄⋊Q₈	1	4	yes	—
106	C₈×C₆₇	C₄⋊Q₈	1	4	yes	—
107	C₈×C₆₇	C₄⋊Q₈	1	8	yes	—
108	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
109	C₂²×C₆₇	C₂×C₄⋊Q₈	1	2	yes	—
110	C₂²×C₆₇	#577	1	4	yes	—
111	C₄×C₆₇	—	2	4	no	—
112	C₄×C₆₇	—	2	8	no	—
113	C₄×C₆₇	—	8	2	no	—
114	C₄×C₆₇	—	8	2	no	—
115	C₄×C₆₇	—	8	4	no	—
116	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
117	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
118	C₄×C₆₇	C₂×C₄⋊Q₈	1	2	yes	—
119	#33318	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂×C₆₇	#2868	1	1	yes	characteristic
121	C₂×C₆₇	#3154	1	2	yes	—
122	C₂×C₆₇	#3163	1	4	yes	—
123	C₂×#501	C₂×C₆₇	1	2	yes	—
124	C₂×#499	C₂×C₆₇	1	1	yes	characteristic
125	C₂×#498	C₂×C₆₇	1	1	yes	characteristic
126	#2660	C₂×C₆₇	1	2	yes	—
127	#1847	C₂×C₆₇	1	4	yes	—
128	#1846	C₂×C₆₇	1	2	yes	—
129	#1844	C₂×C₆₇	1	2	yes	—
130	#1726	C₂×C₆₇	1	1	yes	characteristic
131	C₆₇	#33318	1	1	yes	characteristic, direct product factor, Sylow
132	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
133	C₂×#125	C₂²×C₆₇	1	2	yes	—
134	#501	—	2	8	no	—
135	#499	—	2	4	no	—
136	#498	—	2	4	no	—
137	C₂×#107	C₂²×C₆₇	1	4	yes	—
138	C₂×#107	C₂²×C₆₇	1	2	yes	—
139	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
140	C₂×#106	C₂²×C₆₇	1	2	yes	—
141	C₂×#106	C₂²×C₆₇	1	4	yes	—
142	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
143	C₂×#106	C₂²×C₆₇	1	2	yes	—
144	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
145	C₂×#97	C₂²×C₆₇	1	2	yes	—
146	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
148	#398	—	2	4	no	—
149	#395	—	2	4	no	—
150	#384	C₂²×C₆₇	1	4	yes	—
151	#382	C₂²×C₆₇	1	4	yes	—
152	#376	C₂²×C₆₇	1	1	yes	characteristic
153	C₂²×Q₈	—	2	2	no	—
154	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
155	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
156	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
157	C₄⋊Q₈	—	2	4	no	—
158	C₄⋊Q₈	—	2	4	no	—
159	#125	—	2	4	no	—
160	#125	—	2	4	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	2	8	no	—
165	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
166	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
167	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
168	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
169	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
170	#107	—	2	4	no	—
171	#107	—	2	8	no	—
172	#106	—	2	4	no	—
173	#106	—	2	4	no	—
174	#106	—	2	2	no	—
175	#106	—	2	2	no	—
176	#106	—	2	8	no	—
177	#97	D₄×C₆₇	1	4	yes	—
178	#97	Q₈×C₆₇	1	8	yes	complemented
179	C₄×C₈	D₄×C₆₇	1	4	yes	—
180	C₄×C₈	Q₈×C₆₇	1	8	yes	complemented
181	#95	—	2	8	no	—
182	#95	D₄×C₆₇	1	4	yes	—
183	C₂×Q₈	—	4	4	no	—
184	C₂×Q₈	—	8	4	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
189	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
190	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
191	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
192	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
193	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
194	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
195	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
196	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
197	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₄⋊C₄	—	4	4	no	—
207	C₄⋊C₄	—	8	8	no	—
208	C₄²	C₂×D₄×C₆₇	1	2	yes	—
209	C₄²	C₂×D₄×C₆₇	1	2	yes	—
210	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
211	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
212	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
213	Q₈	—	8	8	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	8	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	#127×C₆₇	1	4	yes	—
223	C₂×C₄	C₄⋊Q₈×C₆₇	1	8	yes	—
224	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
225	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
226	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
227	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
228	C₈	C₄⋊Q₈×C₆₇	1	4	yes	—
229	C₈	C₄⋊Q₈×C₆₇	1	4	yes	—
230	C₈	C₄⋊Q₈×C₆₇	1	8	yes	—
231	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
232	C₂²	C₂×C₄⋊Q₈×C₆₇	1	2	yes	—
233	C₂²	#577×C₆₇	1	4	yes	—
234	C₄	—	2	4	no	—
235	C₄	—	2	8	no	—
236	C₄	—	8	4	no	—
237	C₄	—	8	2	no	—
238	C₄	—	8	2	no	—
239	C₄	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
240	C₄	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
241	C₄	C₂×C₄⋊Q₈×C₆₇	1	2	yes	—
242	C₂	C₆₇×#2868	1	1	yes	characteristic
243	C₂	C₆₇×#3154	1	2	yes	—
244	C₂	C₆₇×#3163	1	4	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 ◁ #376×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
55. #97×C₆₇	Q₈	semidirect
57. C₄×C₈×C₆₇	Q₈	semidirect
119. #33318	C₆₇	direct (N × H)
131. C₆₇	#33318	direct (N × H)
178. #97	Q₈×C₆₇	semidirect
180. C₄×C₈	Q₈×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
C₂×#499×C₆₇	2
C₂×#498×C₆₇	2
C₆₇×#2660	2
C₆₇×#1847	2
C₆₇×#1846	2
C₆₇×#1844	2
C₆₇×#1726	2
#33318	67

Subgroups — C67×#33318 #30712569

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33318 #30712569