Subgroups — C₆₇×#33317 #30712568

Order 17152.

822 conjugacy classes of subgroups.

The subgroup lattice has 190 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×#2862	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×#1846	C₂	1	4	yes	maximal
6	C₄×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
8	C₂×#125×C₆₇	C₂²	1	2	yes	—
9	#501×C₆₇	—	2	16	no	—
10	C₂×#107×C₆₇	C₂²	1	8	yes	—
11	C₂×#106×C₆₇	C₂²	1	8	yes	—
12	C₂×#97×C₆₇	C₂²	1	2	yes	—
13	C₂×#97×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
15	#391×C₆₇	—	2	4	no	—
16	#389×C₆₇	—	2	4	no	—
17	#384×C₆₇	C₂²	1	2	yes	—
18	#382×C₆₇	C₂²	1	2	yes	—
19	#382×C₆₇	C₂²	1	2	yes	—
20	#382×C₆₇	C₂²	1	2	yes	—
21	C₄³×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×Q₈×C₆₇	—	2	2	no	—
23	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
24	C₄⋊Q₈×C₆₇	—	2	8	no	—
25	#125×C₆₇	—	2	8	no	—
26	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
27	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
28	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
29	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
30	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
31	C₂×C₄²×C₆₇	D₄	1	4	yes	—
32	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
35	#107×C₆₇	—	2	16	no	—
36	#106×C₆₇	—	2	16	no	—
37	#97×C₆₇	D₄	1	8	yes	—
38	#97×C₆₇	Q₈	1	8	yes	complemented
39	C₄×C₈×C₆₇	Q₈	1	8	yes	complemented
40	#95×C₆₇	—	2	4	no	—
41	#95×C₆₇	—	2	4	no	—
42	#95×C₆₇	—	2	4	no	—
43	C₂×Q₈×C₆₇	—	4	8	no	—
44	C₂×Q₈×C₆₇	—	4	4	no	—
45	C₂²×C₄×C₆₇	—	2	4	no	—
46	C₂²×C₄×C₆₇	—	2	4	no	—
47	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
51	C₂×C₈×C₆₇	C₂×D₄	1	8	yes	—
52	C₂×C₈×C₆₇	C₂×Q₈	1	8	yes	—
53	C₂×C₈×C₆₇	C₂×Q₈	1	8	yes	—
54	C₄⋊C₄×C₆₇	—	2	4	no	—
55	C₄⋊C₄×C₆₇	—	2	8	no	—
56	C₄⋊C₄×C₆₇	—	2	4	no	—
57	C₄⋊C₄×C₆₇	—	4	8	no	—
58	C₄⋊C₄×C₆₇	—	4	8	no	—
59	C₄⋊C₄×C₆₇	—	4	4	no	—
60	C₄²×C₆₇	—	4	4	no	—
61	C₄²×C₆₇	C₂×D₄	1	4	yes	—
62	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
63	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
64	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
65	Q₈×C₆₇	—	8	8	no	—
66	C₂×C₄×C₆₇	—	2	4	no	—
67	C₂×C₄×C₆₇	—	2	4	no	—
68	C₂×C₄×C₆₇	—	2	4	no	—
69	C₂×C₄×C₆₇	—	4	4	no	—
70	C₂×C₄×C₆₇	—	4	4	no	—
71	C₂×C₄×C₆₇	—	4	8	no	—
72	C₂×C₄×C₆₇	—	4	8	no	—
73	C₂×C₄×C₆₇	#127	1	4	yes	—
74	C₂×C₄×C₆₇	C₄⋊Q₈	1	4	yes	—
75	C₂×C₄×C₆₇	C₄⋊Q₈	1	4	yes	—
76	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
77	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
78	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
79	C₈×C₆₇	C₄⋊Q₈	1	16	yes	—
80	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
81	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
82	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
83	C₂²×C₆₇	#575	1	4	yes	—
84	C₄×C₆₇	—	2	4	no	—
85	C₄×C₆₇	—	2	4	no	—
86	C₄×C₆₇	—	2	4	no	—
87	C₄×C₆₇	—	8	4	no	—
88	C₄×C₆₇	—	8	4	no	—
89	C₄×C₆₇	C₂×C₄⋊Q₈	1	4	yes	—
90	#33317	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂×C₆₇	#2868	1	1	yes	characteristic
92	C₂×C₆₇	#3153	1	2	yes	—
93	C₂×C₆₇	#3162	1	2	yes	—
94	C₂×C₆₇	#3166	1	2	yes	—
95	C₂×#501	C₂×C₆₇	1	4	yes	—
96	#2862	C₂×C₆₇	1	1	yes	characteristic
97	#2859	C₂×C₆₇	1	1	yes	characteristic
98	#1847	C₂×C₆₇	1	4	yes	—
99	#1846	C₂×C₆₇	1	4	yes	—
100	C₄×#97	C₂×C₆₇	1	1	yes	characteristic
101	C₆₇	#33317	1	1	yes	characteristic, direct product factor, Sylow
102	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
103	C₂×#125	C₂²×C₆₇	1	2	yes	—
104	#501	—	2	16	no	—
105	C₂×#107	C₂²×C₆₇	1	8	yes	—
106	C₂×#106	C₂²×C₆₇	1	8	yes	—
107	C₂×#97	C₂²×C₆₇	1	2	yes	—
108	C₂×#97	C₂²×C₆₇	1	2	yes	—
109	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
110	#391	—	2	4	no	—
111	#389	—	2	4	no	—
112	#384	C₂²×C₆₇	1	2	yes	—
113	#382	C₂²×C₆₇	1	2	yes	—
114	#382	C₂²×C₆₇	1	2	yes	—
115	#382	C₂²×C₆₇	1	2	yes	—
116	C₄³	C₂²×C₆₇	1	1	yes	characteristic
117	C₂²×Q₈	—	2	2	no	—
118	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
119	C₄⋊Q₈	—	2	8	no	—
120	#125	—	2	8	no	—
121	C₂×C₄⋊C₄	—	2	4	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	2	4	no	—
124	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
125	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
126	C₂×C₄²	D₄×C₆₇	1	4	yes	—
127	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
128	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
129	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
130	#107	—	2	16	no	—
131	#106	—	2	16	no	—
132	#97	D₄×C₆₇	1	8	yes	—
133	#97	Q₈×C₆₇	1	8	yes	complemented
134	C₄×C₈	Q₈×C₆₇	1	8	yes	complemented
135	#95	—	2	4	no	—
136	#95	—	2	4	no	—
137	#95	—	2	4	no	—
138	C₂×Q₈	—	4	8	no	—
139	C₂×Q₈	—	4	4	no	—
140	C₂²×C₄	—	2	4	no	—
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
143	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
144	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
145	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
146	C₂×C₈	C₂×D₄×C₆₇	1	8	yes	—
147	C₂×C₈	C₂×Q₈×C₆₇	1	8	yes	—
148	C₂×C₈	C₂×Q₈×C₆₇	1	8	yes	—
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	2	8	no	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	4	8	no	—
153	C₄⋊C₄	—	4	8	no	—
154	C₄⋊C₄	—	4	4	no	—
155	C₄²	—	4	4	no	—
156	C₄²	C₂×D₄×C₆₇	1	4	yes	—
157	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
158	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
159	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
160	Q₈	—	8	8	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	—	4	8	no	—
167	C₂×C₄	—	4	8	no	—
168	C₂×C₄	#127×C₆₇	1	4	yes	—
169	C₂×C₄	C₄⋊Q₈×C₆₇	1	4	yes	—
170	C₂×C₄	C₄⋊Q₈×C₆₇	1	4	yes	—
171	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
172	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
173	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
174	C₈	C₄⋊Q₈×C₆₇	1	16	yes	—
175	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
176	C₂²	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
177	C₂²	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
178	C₂²	#575×C₆₇	1	4	yes	—
179	C₄	—	2	4	no	—
180	C₄	—	2	4	no	—
181	C₄	—	2	4	no	—
182	C₄	—	8	4	no	—
183	C₄	—	8	4	no	—
184	C₄	C₂×C₄⋊Q₈×C₆₇	1	4	yes	—
185	C₂	C₆₇×#2868	1	1	yes	characteristic
186	C₂	C₆₇×#3153	1	2	yes	—
187	C₂	C₆₇×#3162	1	2	yes	—
188	C₂	C₆₇×#3166	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
38. #97×C₆₇	Q₈	semidirect
39. C₄×C₈×C₆₇	Q₈	semidirect
90. #33317	C₆₇	direct (N × H)
101. C₆₇	#33317	direct (N × H)
133. #97	Q₈×C₆₇	semidirect
134. 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
C₆₇×#2862	2
C₆₇×#2859	2
C₆₇×#1847	2
C₆₇×#1846	2
C₄×#97×C₆₇	2
#33317	67

Subgroups — C67×#33317 #30712568

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33317 #30712568