Subgroups — C₂×C₆₇×#1835 #30700010

Order 17152.

1006 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#382×C₆₇	C₂	1	2	yes	maximal
3	C₆₇×#1835	C₂	1	8	yes	maximal, direct product factor
4	C₂×#340×C₆₇	C₂	1	4	yes	maximal
5	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
6	C₂²×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
7	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₆₇	C₄	1	8	yes	complemented
9	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	—
10	#382×C₆₇	—	2	4	no	—
11	#382×C₆₇	C₂²	1	8	yes	—
12	C₂×#95×C₆₇	—	2	4	no	—
13	#340×C₆₇	C₂²	1	16	yes	—
14	C₂³×C₄×C₆₇	—	2	4	no	—
15	C₂³×C₄×C₆₇	D₄	1	2	yes	—
16	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
17	C₂²×C₈×C₆₇	—	2	8	no	—
18	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	—
19	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
20	C₂×C₄⋊C₄×C₆₇	—	2	16	no	—
21	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
22	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
23	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₆₇	D₄	1	2	yes	—
25	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
26	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
27	C₄×C₈×C₆₇	C₂×C₄	1	16	yes	complemented
28	#95×C₆₇	—	2	16	no	—
29	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₄×C₆₇	—	2	8	no	—
31	C₂²×C₄×C₆₇	—	2	8	no	—
32	C₂²×C₄×C₆₇	—	4	8	no	—
33	C₂²×C₄×C₆₇	—	4	16	no	—
34	C₂²×C₄×C₆₇	C₄⋊C₄	1	8	yes	—
35	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
36	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
37	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
39	C₂²×C₄×C₆₇	#41	1	8	yes	—
40	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
41	C₂×C₈×C₆₇	—	2	16	no	—
42	C₂×C₈×C₆₇	—	2	8	no	—
43	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
44	C₄⋊C₄×C₆₇	—	2	16	no	—
45	C₄⋊C₄×C₆₇	—	4	8	no	—
46	C₄²×C₆₇	C₄⋊C₄	1	8	yes	—
47	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
48	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
49	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
50	C₂³×C₆₇	#124	1	4	yes	—
51	C₂³×C₆₇	#135	1	8	yes	—
52	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
53	C₂×C₄×C₆₇	—	2	8	no	—
54	C₂×C₄×C₆₇	—	2	16	no	—
55	C₂×C₄×C₆₇	—	4	8	no	—
56	C₂×C₄×C₆₇	—	4	16	no	—
57	C₂×C₄×C₆₇	—	8	16	no	—
58	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	8	yes	—
59	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
60	C₂×C₄×C₆₇	#117	1	8	yes	—
61	C₂×C₄×C₆₇	#124	1	4	yes	—
62	C₂×C₄×C₆₇	#125	1	8	yes	—
63	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
65	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
66	C₈×C₆₇	—	2	8	no	—
67	C₂²×C₆₇	#382	1	4	yes	—
68	C₂²×C₆₇	#427	1	8	yes	—
69	C₂²×C₆₇	#488	1	16	yes	—
70	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₆₇	C₂×#124	1	2	yes	—
72	C₂²×C₆₇	C₂×#135	1	4	yes	—
73	C₄×C₆₇	—	2	8	no	—
74	C₄×C₆₇	—	8	8	no	—
75	C₄×C₆₇	#382	1	4	yes	—
76	C₄×C₆₇	C₂×#117	1	2	yes	—
77	C₄×C₆₇	C₂×#125	1	2	yes	—
78	C₂×#1835	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂×C₆₇	#1835	1	8	yes	direct product factor
80	C₂×C₆₇	C₂×#382	1	1	yes	characteristic
81	C₂×C₆₇	C₂×#427	1	2	yes	—
82	C₂×C₆₇	C₂×#488	1	4	yes	—
83	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic
84	C₂×#382	C₂×C₆₇	1	2	yes	—
85	#1835	C₂×C₆₇	1	8	yes	direct product factor
86	C₂×#340	C₂×C₆₇	1	4	yes	—
87	C₆₇	C₂×#1835	1	1	yes	characteristic, direct product factor, Sylow
88	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
89	C₂²×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
90	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
91	C₂×C₄×C₈	C₄×C₆₇	1	8	yes	complemented
92	C₂×C₄×C₈	C₂²×C₆₇	1	4	yes	—
93	#382	—	2	4	no	—
94	#382	C₂²×C₆₇	1	8	yes	—
95	C₂×#95	—	2	4	no	—
96	#340	C₂²×C₆₇	1	16	yes	—
97	C₂³×C₄	—	2	4	no	—
98	C₂³×C₄	D₄×C₆₇	1	2	yes	—
99	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
100	C₂²×C₈	—	2	8	no	—
101	C₂²×C₈	C₂×C₄×C₆₇	1	8	yes	—
102	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
103	C₂×C₄⋊C₄	—	2	16	no	—
104	C₂×C₄⋊C₄	—	2	4	no	—
105	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
106	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
107	C₂×C₄²	D₄×C₆₇	1	2	yes	—
108	C₂×C₄²	Q₈×C₆₇	1	4	yes	—
109	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
110	C₄×C₈	C₂×C₄×C₆₇	1	16	yes	complemented
111	#95	—	2	16	no	—
112	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
113	C₂²×C₄	—	2	8	no	—
114	C₂²×C₄	—	2	8	no	—
115	C₂²×C₄	—	4	8	no	—
116	C₂²×C₄	—	4	16	no	—
117	C₂²×C₄	C₄⋊C₄×C₆₇	1	8	yes	—
118	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
119	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
120	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
121	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
122	C₂²×C₄	#41×C₆₇	1	8	yes	—
123	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
124	C₂×C₈	—	2	16	no	—
125	C₂×C₈	—	2	8	no	—
126	C₂×C₈	C₂²×C₄×C₆₇	1	8	yes	—
127	C₄⋊C₄	—	2	16	no	—
128	C₄⋊C₄	—	4	8	no	—
129	C₄²	C₄⋊C₄×C₆₇	1	8	yes	—
130	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
131	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
132	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
133	C₂³	#124×C₆₇	1	4	yes	—
134	C₂³	#135×C₆₇	1	8	yes	—
135	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
136	C₂×C₄	—	2	8	no	—
137	C₂×C₄	—	2	16	no	—
138	C₂×C₄	—	4	8	no	—
139	C₂×C₄	—	4	16	no	—
140	C₂×C₄	—	8	16	no	—
141	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	8	yes	—
142	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
143	C₂×C₄	#117×C₆₇	1	8	yes	—
144	C₂×C₄	#124×C₆₇	1	4	yes	—
145	C₂×C₄	#125×C₆₇	1	8	yes	—
146	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
147	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
148	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
149	C₈	—	2	8	no	—
150	C₂²	#382×C₆₇	1	4	yes	—
151	C₂²	#427×C₆₇	1	8	yes	—
152	C₂²	#488×C₆₇	1	16	yes	—
153	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
154	C₂²	C₂×#124×C₆₇	1	2	yes	—
155	C₂²	C₂×#135×C₆₇	1	4	yes	—
156	C₄	—	2	8	no	—
157	C₄	—	8	8	no	—
158	C₄	#382×C₆₇	1	4	yes	—
159	C₄	C₂×#117×C₆₇	1	2	yes	—
160	C₄	C₂×#125×C₆₇	1	2	yes	—
161	C₂	C₆₇×#1835	1	8	yes	direct product factor
162	C₂	C₂×#382×C₆₇	1	1	yes	characteristic
163	C₂	C₂×#427×C₆₇	1	2	yes	—
164	C₂	C₂×#488×C₆₇	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₆₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1835	C₂	direct (N × H)
8. C₂×C₄×C₈×C₆₇	C₄	semidirect
27. C₄×C₈×C₆₇	C₂×C₄	semidirect
78. C₂×#1835	C₆₇	direct (N × H)
79. C₂×C₆₇	#1835	direct (N × H)
85. #1835	C₂×C₆₇	direct (N × H)
87. C₆₇	C₂×#1835	direct (N × H)
91. C₂×C₄×C₈	C₄×C₆₇	semidirect
110. C₄×C₈	C₂×C₄×C₆₇	semidirect
161. C₂	C₆₇×#1835	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₈×C₆₇	2
C₂×#382×C₆₇	2
C₆₇×#1835	2
C₂×#340×C₆₇	2
C₂×#1835	67

Subgroups — C2×C67×#1835 #30700010

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1835 #30700010