Subgroups — C₄×#427×C₆₇ #30699700

Order 17152.

1182 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#427×C₆₇	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₄×#117×C₆₇	C₂	1	2	yes	maximal
4	C₄×#107×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₄×#106×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1735	C₂	1	2	yes	maximal
7	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×#117×C₆₇	C₂²	1	4	yes	—
10	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
11	#427×C₆₇	C₄	1	16	yes	direct product factor
12	C₂×#107×C₆₇	C₂²	1	4	yes	—
13	C₂×#106×C₆₇	C₂²	1	4	yes	—
14	C₂×C₄×C₈×C₆₇	C₄	1	8	yes	complemented
15	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
17	C₄×C₄⋊C₄×C₆₇	C₂²	1	8	yes	—
18	C₄×C₂²⋊C₄×C₆₇	—	2	4	no	—
19	#376×C₆₇	—	2	2	no	—
20	C₄³×C₆₇	—	2	2	no	—
21	#340×C₆₇	C₂²	1	8	yes	—
22	C₂³×C₄×C₆₇	D₄	1	2	yes	—
23	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₆₇	—	2	4	no	—
25	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	—
28	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
29	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
31	#117×C₆₇	—	2	8	no	—
32	#117×C₆₇	C₂×C₄	1	16	yes	—
33	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
34	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
35	C₂×C₄²×C₆₇	—	2	4	no	—
36	C₂×C₄²×C₆₇	—	2	8	no	—
37	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
41	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
42	#107×C₆₇	C₂×C₄	1	16	yes	complemented
43	#106×C₆₇	C₂×C₄	1	16	yes	complemented
44	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	—
45	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	—
46	#95×C₆₇	—	2	8	no	—
47	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	—	2	8	no	—
49	C₂²×C₄×C₆₇	—	2	8	no	—
50	C₂²×C₄×C₆₇	C₄⋊C₄	1	8	yes	—
51	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
53	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₆₇	#41	1	4	yes	—
58	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
59	C₂×C₈×C₆₇	—	2	8	no	—
60	C₂×C₈×C₆₇	—	2	4	no	—
61	C₂×C₈×C₆₇	C₄²	1	16	yes	complemented
62	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
63	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
65	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
67	C₄⋊C₄×C₆₇	—	2	8	no	—
68	C₄⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
69	C₂²⋊C₄×C₆₇	—	2	16	no	—
70	C₄²×C₆₇	—	2	16	no	—
71	C₄²×C₆₇	—	4	16	no	—
72	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
73	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
74	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
75	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
76	C₂³×C₆₇	—	2	4	no	—
77	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
78	C₂³×C₆₇	C₄×D₄	1	4	yes	—
79	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
80	C₂×C₄×C₆₇	—	2	4	no	—
81	C₂×C₄×C₆₇	—	2	16	no	—
82	C₂×C₄×C₆₇	—	2	16	no	—
83	C₂×C₄×C₆₇	—	4	16	no	—
84	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
85	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
86	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
87	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
88	C₂×C₄×C₆₇	#117	1	4	yes	—
89	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
90	C₂×C₄×C₆₇	C₄×Q₈	1	8	yes	—
91	C₂×C₄×C₆₇	#135	1	8	yes	—
92	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
94	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
96	C₈×C₆₇	—	2	4	no	—
97	C₈×C₆₇	C₂×C₄²	1	4	yes	—
98	C₈×C₆₇	C₂×C₄²	1	4	yes	—
99	C₂²×C₆₇	—	2	4	no	—
100	C₂²×C₆₇	—	2	8	no	—
101	C₂²×C₆₇	C₄×C₄⋊C₄	1	4	yes	—
102	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
103	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
104	C₂²×C₆₇	C₂×#135	1	4	yes	—
105	C₄×C₆₇	—	2	4	no	—
106	C₄×C₆₇	—	4	16	no	—
107	C₄×C₆₇	C₄×C₄⋊C₄	1	4	yes	—
108	C₄×C₆₇	#427	1	8	yes	direct product factor
109	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
110	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
111	C₄×C₆₇	C₂×C₄×Q₈	1	2	yes	—
112	C₄×#427	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂×C₆₇	—	2	4	no	—
114	C₂×C₆₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₆₇	C₂×#427	1	2	yes	—
116	C₂×C₆₇	C₄×#135	1	4	yes	—
117	C₂×#427	C₂×C₆₇	1	2	yes	—
118	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic
119	C₄×#117	C₂×C₆₇	1	2	yes	—
120	C₄×#107	C₂×C₆₇	1	4	yes	complemented
121	C₄×#106	C₂×C₆₇	1	4	yes	complemented
122	#1735	C₂×C₆₇	1	2	yes	—
123	C₆₇	C₄×#427	1	1	yes	characteristic, direct product factor, Sylow
124	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
125	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
126	C₂×#117	C₂²×C₆₇	1	4	yes	—
127	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
128	#427	C₄×C₆₇	1	16	yes	direct product factor
129	C₂×#107	C₂²×C₆₇	1	4	yes	—
130	C₂×#106	C₂²×C₆₇	1	4	yes	—
131	C₂×C₄×C₈	C₄×C₆₇	1	8	yes	complemented
132	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
133	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
134	C₄×C₄⋊C₄	C₂²×C₆₇	1	8	yes	—
135	C₄×C₂²⋊C₄	—	2	4	no	—
136	#376	—	2	2	no	—
137	C₄³	—	2	2	no	—
138	#340	C₂²×C₆₇	1	8	yes	—
139	C₂³×C₄	D₄×C₆₇	1	2	yes	—
140	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
141	C₂²×C₈	—	2	4	no	—
142	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
143	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
144	C₂²×C₈	C₂×C₄×C₆₇	1	8	yes	—
145	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
146	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
147	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
148	#117	—	2	8	no	—
149	#117	C₂×C₄×C₆₇	1	16	yes	—
150	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
151	C₂×C₂²⋊C₄	—	2	4	no	—
152	C₂×C₄²	—	2	4	no	—
153	C₂×C₄²	—	2	8	no	—
154	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
155	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
156	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
157	C₂×C₄²	Q₈×C₆₇	1	4	yes	—
158	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
159	#107	C₂×C₄×C₆₇	1	16	yes	complemented
160	#106	C₂×C₄×C₆₇	1	16	yes	complemented
161	C₄×C₈	C₂×C₄×C₆₇	1	8	yes	—
162	C₄×C₈	C₂×C₄×C₆₇	1	8	yes	—
163	#95	—	2	8	no	—
164	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
165	C₂²×C₄	—	2	8	no	—
166	C₂²×C₄	—	2	8	no	—
167	C₂²×C₄	C₄⋊C₄×C₆₇	1	8	yes	—
168	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
169	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
170	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
171	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
172	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
173	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
174	C₂²×C₄	#41×C₆₇	1	4	yes	—
175	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
176	C₂×C₈	—	2	8	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	C₄²×C₆₇	1	16	yes	complemented
179	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
180	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
181	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
182	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
183	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
184	C₄⋊C₄	—	2	8	no	—
185	C₄⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
186	C₂²⋊C₄	—	2	16	no	—
187	C₄²	—	2	16	no	—
188	C₄²	—	4	16	no	—
189	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
190	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
191	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
192	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
193	C₂³	—	2	4	no	—
194	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
195	C₂³	C₄×D₄×C₆₇	1	4	yes	—
196	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	16	no	—
199	C₂×C₄	—	2	16	no	—
200	C₂×C₄	—	4	16	no	—
201	C₂×C₄	C₂×C₄²×C₆₇	1	4	yes	—
202	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
203	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
204	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
205	C₂×C₄	#117×C₆₇	1	4	yes	—
206	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
207	C₂×C₄	C₄×Q₈×C₆₇	1	8	yes	—
208	C₂×C₄	#135×C₆₇	1	8	yes	—
209	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
210	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
211	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
212	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
213	C₈	—	2	4	no	—
214	C₈	C₂×C₄²×C₆₇	1	4	yes	—
215	C₈	C₂×C₄²×C₆₇	1	4	yes	—
216	C₂²	—	2	4	no	—
217	C₂²	—	2	8	no	—
218	C₂²	C₄×C₄⋊C₄×C₆₇	1	4	yes	—
219	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
220	C₂²	C₂×C₄×D₄×C₆₇	1	2	yes	—
221	C₂²	C₂×#135×C₆₇	1	4	yes	—
222	C₄	—	2	4	no	—
223	C₄	—	4	16	no	—
224	C₄	C₄×C₄⋊C₄×C₆₇	1	4	yes	—
225	C₄	#427×C₆₇	1	8	yes	direct product factor
226	C₄	C₂²×C₄²×C₆₇	1	1	yes	characteristic
227	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
228	C₄	C₂×C₄×Q₈×C₆₇	1	2	yes	—
229	C₂	—	2	4	no	—
230	C₂	C₂×C₄×C₄⋊C₄×C₆₇	1	1	yes	characteristic
231	C₂	C₂×#427×C₆₇	1	2	yes	—
232	C₂	C₄×#135×C₆₇	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₄×#107×C₆₇	C₂	semidirect
5. C₄×#106×C₆₇	C₂	semidirect
11. #427×C₆₇	C₄	direct (N × H)
14. C₂×C₄×C₈×C₆₇	C₄	semidirect
42. #107×C₆₇	C₂×C₄	semidirect
43. #106×C₆₇	C₂×C₄	semidirect
61. C₂×C₈×C₆₇	C₄²	semidirect
108. C₄×C₆₇	#427	direct (N × H)
112. C₄×#427	C₆₇	direct (N × H)
120. C₄×#107	C₂×C₆₇	semidirect
121. C₄×#106	C₂×C₆₇	semidirect
123. C₆₇	C₄×#427	direct (N × H)
128. #427	C₄×C₆₇	direct (N × H)
131. C₂×C₄×C₈	C₄×C₆₇	semidirect
159. #107	C₂×C₄×C₆₇	semidirect
160. #106	C₂×C₄×C₆₇	semidirect
178. C₂×C₈	C₄²×C₆₇	semidirect
225. C₄	#427×C₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#427×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₄×#117×C₆₇	2
C₄×#107×C₆₇	2
C₄×#106×C₆₇	2
C₆₇×#1735	2
C₄×#427	67

Subgroups — C4×#427×C67 #30699700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#427×C₆₇ #30699700