Subgroups — C₂×C₆₇×#2399 #30713797

Order 17152.

2814 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₆₇	C₂	1	2	yes	maximal
2	C₆₇×#2399	C₂	1	16	yes	maximal, direct product factor
3	C₂×#398×C₆₇	C₂	1	4	yes	maximal
4	C₂×#395×C₆₇	C₂	1	4	yes	maximal
5	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	4	yes	maximal
6	C₂×#376×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₆₇	C₂²	1	2	yes	—
8	C₂×C₄×Q₈×C₆₇	—	2	16	no	—
9	C₂×C₄×Q₈×C₆₇	C₂²	1	16	yes	—
10	C₂²×C₄⋊C₄×C₆₇	C₂²	1	8	yes	—
11	C₂²×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
12	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
13	C₂²×C₄²×C₆₇	C₂²	1	4	yes	—
14	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
16	#398×C₆₇	C₂²	1	32	yes	—
17	#395×C₆₇	C₂²	1	32	yes	—
18	C₄×C₄⋊C₄×C₆₇	C₂²	1	32	yes	—
19	#376×C₆₇	C₂²	1	8	yes	—
20	C₂×#95×C₆₇	C₂²	1	4	yes	—
21	C₂×#95×C₆₇	C₂²	1	4	yes	—
22	C₂×#95×C₆₇	C₂²	1	4	yes	—
23	C₂²×Q₈×C₆₇	—	2	8	no	—
24	C₂²×Q₈×C₆₇	—	2	16	no	—
25	C₂²×Q₈×C₆₇	C₂³	1	8	yes	—
26	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
27	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
28	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
29	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
30	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
31	C₄×Q₈×C₆₇	—	2	64	no	—
32	C₂×C₄⋊C₄×C₆₇	—	2	16	no	—
33	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
34	C₂×C₄⋊C₄×C₆₇	—	2	32	no	—
35	C₂×C₄⋊C₄×C₆₇	C₂³	1	16	yes	—
36	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
37	C₂×C₄⋊C₄×C₆₇	C₂³	1	32	yes	—
38	C₂×C₄²×C₆₇	—	2	16	no	—
39	C₂×C₄²×C₆₇	D₄	1	8	yes	—
40	C₂×C₄²×C₆₇	Q₈	1	16	yes	complemented
41	C₂×C₄²×C₆₇	C₂³	1	16	yes	—
42	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
43	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
44	#95×C₆₇	C₂³	1	16	yes	—
45	#95×C₆₇	C₂³	1	16	yes	—
46	#95×C₆₇	C₂³	1	16	yes	—
47	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
48	C₂×Q₈×C₆₇	—	2	64	no	—
49	C₂×Q₈×C₆₇	—	2	16	no	—
50	C₂×Q₈×C₆₇	—	2	32	no	—
51	C₂²×C₄×C₆₇	—	2	8	no	—
52	C₂²×C₄×C₆₇	—	2	16	no	—
53	C₂²×C₄×C₆₇	—	2	8	no	—
54	C₂²×C₄×C₆₇	—	2	8	no	—
55	C₂²×C₄×C₆₇	—	2	8	no	—
56	C₂²×C₄×C₆₇	—	2	16	no	—
57	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
58	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
59	C₂²×C₄×C₆₇	C₂×Q₈	1	16	yes	—
60	C₂²×C₄×C₆₇	C₂×Q₈	1	8	yes	—
61	C₂²×C₄×C₆₇	#41	1	16	yes	—
62	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
63	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
64	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
65	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
66	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
67	C₄⋊C₄×C₆₇	—	2	32	no	—
68	C₄⋊C₄×C₆₇	—	2	16	no	—
69	C₄⋊C₄×C₆₇	—	2	64	no	—
70	C₄²×C₆₇	—	2	32	no	—
71	C₄²×C₆₇	C₂×D₄	1	16	yes	—
72	C₄²×C₆₇	C₂×Q₈	1	32	yes	complemented
73	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
74	C₂³×C₆₇	C₂²×Q₈	1	4	yes	—
75	C₂³×C₆₇	C₂×#41	1	8	yes	—
76	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
77	Q₈×C₆₇	—	2	64	no	—
78	C₂×C₄×C₆₇	—	2	16	no	—
79	C₂×C₄×C₆₇	—	2	32	no	—
80	C₂×C₄×C₆₇	—	2	32	no	—
81	C₂×C₄×C₆₇	—	2	8	no	—
82	C₂×C₄×C₆₇	—	2	16	no	—
83	C₂×C₄×C₆₇	—	2	8	no	—
84	C₂×C₄×C₆₇	—	2	8	no	—
85	C₂×C₄×C₆₇	—	2	8	no	—
86	C₂×C₄×C₆₇	—	2	16	no	—
87	C₂×C₄×C₆₇	—	4	16	no	—
88	C₂×C₄×C₆₇	C₄⋊Q₈	1	16	yes	—
89	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
90	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
91	C₂×C₄×C₆₇	C₂²×Q₈	1	16	yes	—
92	C₂×C₄×C₆₇	C₂²×Q₈	1	8	yes	—
93	C₂×C₄×C₆₇	C₂×#41	1	16	yes	—
94	C₂²×C₆₇	#525	1	8	yes	—
95	C₂²×C₆₇	C₂×C₄⋊Q₈	1	4	yes	—
96	C₂²×C₆₇	#533	1	16	yes	—
97	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
98	C₂²×C₆₇	C₂³×Q₈	1	2	yes	—
99	C₂²×C₆₇	C₂²×#41	1	4	yes	—
100	C₄×C₆₇	—	2	8	no	—
101	C₄×C₆₇	—	2	16	no	—
102	C₄×C₆₇	—	2	16	no	—
103	C₄×C₆₇	—	4	8	no	—
104	C₄×C₆₇	C₂×C₄⋊Q₈	1	8	yes	—
105	C₂×#2399	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂×C₆₇	#2399	1	8	yes	direct product factor
107	C₂×C₆₇	C₂×#525	1	2	yes	—
108	C₂×C₆₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
109	C₂×C₆₇	C₂×#533	1	4	yes	—
110	C₂²×C₄×Q₈	C₂×C₆₇	1	2	yes	—
111	#2399	C₂×C₆₇	1	16	yes	direct product factor
112	C₂×#398	C₂×C₆₇	1	4	yes	—
113	C₂×#395	C₂×C₆₇	1	4	yes	—
114	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	4	yes	—
115	C₂×#376	C₂×C₆₇	1	1	yes	characteristic
116	C₆₇	C₂×#2399	1	1	yes	characteristic, direct product factor, Sylow
117	C₂³×Q₈	C₂²×C₆₇	1	2	yes	—
118	C₂×C₄×Q₈	—	2	16	no	—
119	C₂×C₄×Q₈	C₂²×C₆₇	1	16	yes	—
120	C₂²×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
121	C₂²×C₄⋊C₄	C₂²×C₆₇	1	8	yes	—
122	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
123	C₂²×C₄²	C₂²×C₆₇	1	4	yes	—
124	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
125	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
126	#398	C₂²×C₆₇	1	32	yes	—
127	#395	C₂²×C₆₇	1	32	yes	—
128	C₄×C₄⋊C₄	C₂²×C₆₇	1	32	yes	—
129	#376	C₂²×C₆₇	1	8	yes	—
130	C₂×#95	C₂²×C₆₇	1	4	yes	—
131	C₂×#95	C₂²×C₆₇	1	4	yes	—
132	C₂×#95	C₂²×C₆₇	1	4	yes	—
133	C₂²×Q₈	—	2	8	no	—
134	C₂²×Q₈	—	2	16	no	—
135	C₂²×Q₈	C₂³×C₆₇	1	8	yes	—
136	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
137	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
138	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
139	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
140	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
141	C₄×Q₈	—	2	64	no	—
142	C₂×C₄⋊C₄	—	2	16	no	—
143	C₂×C₄⋊C₄	—	2	8	no	—
144	C₂×C₄⋊C₄	—	2	32	no	—
145	C₂×C₄⋊C₄	C₂³×C₆₇	1	16	yes	—
146	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
147	C₂×C₄⋊C₄	C₂³×C₆₇	1	32	yes	—
148	C₂×C₄²	—	2	16	no	—
149	C₂×C₄²	D₄×C₆₇	1	8	yes	—
150	C₂×C₄²	Q₈×C₆₇	1	16	yes	complemented
151	C₂×C₄²	C₂³×C₆₇	1	16	yes	—
152	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
153	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
154	#95	C₂³×C₆₇	1	16	yes	—
155	#95	C₂³×C₆₇	1	16	yes	—
156	#95	C₂³×C₆₇	1	16	yes	—
157	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
158	C₂×Q₈	—	2	64	no	—
159	C₂×Q₈	—	2	16	no	—
160	C₂×Q₈	—	2	32	no	—
161	C₂²×C₄	—	2	8	no	—
162	C₂²×C₄	—	2	16	no	—
163	C₂²×C₄	—	2	8	no	—
164	C₂²×C₄	—	2	8	no	—
165	C₂²×C₄	—	2	8	no	—
166	C₂²×C₄	—	2	16	no	—
167	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
168	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
169	C₂²×C₄	C₂×Q₈×C₆₇	1	16	yes	—
170	C₂²×C₄	C₂×Q₈×C₆₇	1	8	yes	—
171	C₂²×C₄	#41×C₆₇	1	16	yes	—
172	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
173	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
174	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
175	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
176	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
177	C₄⋊C₄	—	2	32	no	—
178	C₄⋊C₄	—	2	16	no	—
179	C₄⋊C₄	—	2	64	no	—
180	C₄²	—	2	32	no	—
181	C₄²	C₂×D₄×C₆₇	1	16	yes	—
182	C₄²	C₂×Q₈×C₆₇	1	32	yes	complemented
183	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
184	C₂³	C₂²×Q₈×C₆₇	1	4	yes	—
185	C₂³	C₂×#41×C₆₇	1	8	yes	—
186	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
187	Q₈	—	2	64	no	—
188	C₂×C₄	—	2	16	no	—
189	C₂×C₄	—	2	32	no	—
190	C₂×C₄	—	2	32	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	2	16	no	—
193	C₂×C₄	—	2	8	no	—
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	—	2	8	no	—
196	C₂×C₄	—	2	16	no	—
197	C₂×C₄	—	4	16	no	—
198	C₂×C₄	C₄⋊Q₈×C₆₇	1	16	yes	—
199	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
200	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
201	C₂×C₄	C₂²×Q₈×C₆₇	1	16	yes	—
202	C₂×C₄	C₂²×Q₈×C₆₇	1	8	yes	—
203	C₂×C₄	C₂×#41×C₆₇	1	16	yes	—
204	C₂²	#525×C₆₇	1	8	yes	—
205	C₂²	C₂×C₄⋊Q₈×C₆₇	1	4	yes	—
206	C₂²	#533×C₆₇	1	16	yes	—
207	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
208	C₂²	C₂³×Q₈×C₆₇	1	2	yes	—
209	C₂²	C₂²×#41×C₆₇	1	4	yes	—
210	C₄	—	2	8	no	—
211	C₄	—	2	16	no	—
212	C₄	—	2	16	no	—
213	C₄	—	4	8	no	—
214	C₄	C₂×C₄⋊Q₈×C₆₇	1	8	yes	—
215	C₂	C₆₇×#2399	1	8	yes	direct product factor
216	C₂	C₂×#525×C₆₇	1	2	yes	—
217	C₂	C₂²×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
218	C₂	C₂×#533×C₆₇	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄×C₆₇	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle, prime radical	C₂⁴×C₆₇	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2399	C₂	direct (N × H)
40. C₂×C₄²×C₆₇	Q₈	semidirect
72. C₄²×C₆₇	C₂×Q₈	semidirect
105. C₂×#2399	C₆₇	direct (N × H)
106. C₂×C₆₇	#2399	direct (N × H)
111. #2399	C₂×C₆₇	direct (N × H)
116. C₆₇	C₂×#2399	direct (N × H)
150. C₂×C₄²	Q₈×C₆₇	semidirect
182. C₄²	C₂×Q₈×C₆₇	semidirect
215. C₂	C₆₇×#2399	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×Q₈×C₆₇	2
C₆₇×#2399	2
C₂×#398×C₆₇	2
C₂×#395×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₂×#376×C₆₇	2
C₂×#2399	67

Subgroups — C2×C67×#2399 #30713797

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2399 #30713797