Subgroups — C₆₇×#22717 #30701968

Order 17152.

822 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×M₁₆×C₆₇	C₂	1	2	yes	maximal
2	C₂×C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1950	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1855	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1837	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1810	C₂	1	2	yes	maximal, complemented
7	C₄×#108×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#341×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂²×M₁₆×C₆₇	C₂²	1	4	yes	—
10	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
11	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
12	#436×C₆₇	—	2	2	no	—
13	#435×C₆₇	—	2	4	no	—
14	C₈×D₄×C₆₇	—	2	2	no	—
15	C₈×D₄×C₆₇	C₄	1	4	yes	—
16	C₈×D₄×C₆₇	C₄	1	2	yes	—
17	C₈×D₄×C₆₇	C₄	1	2	yes	—
18	C₂×#108×C₆₇	C₂²	1	2	yes	—
19	C₂×#108×C₆₇	C₂²	1	4	yes	—
20	#423×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	#407×C₆₇	C₂²	1	4	yes	—
23	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
24	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	#341×C₆₇	C₂²	1	1	yes	characteristic
27	#341×C₆₇	C₂²	1	2	yes	—
28	#341×C₆₇	C₂²	1	4	yes	—
29	#341×C₆₇	C₂²	1	1	yes	characteristic
30	#341×C₆₇	C₂²	1	2	yes	—
31	C₂²×D₄×C₆₇	Q₈	1	1	yes	characteristic
32	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
33	C₂×M₁₆×C₆₇	—	2	4	no	—
34	C₂×M₁₆×C₆₇	—	2	8	no	—
35	C₂×M₁₆×C₆₇	—	4	4	no	—
36	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
37	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
38	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
39	C₂²×C₈×C₆₇	—	2	4	no	—
40	C₂²×C₈×C₆₇	—	4	4	no	—
41	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
42	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
43	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
45	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
46	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
47	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
50	C₄×D₄×C₆₇	D₄	1	2	yes	—
51	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₆₇	Q₈	1	1	yes	characteristic
53	C₂×C₂²⋊C₄×C₆₇	Q₈	1	2	yes	—
54	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
55	#108×C₆₇	—	2	4	no	—
56	#108×C₆₇	—	4	4	no	—
57	C₄⋊C₈×C₆₇	—	2	2	no	—
58	C₄⋊C₈×C₆₇	—	2	2	no	—
59	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
60	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
61	C₂²⋊C₈×C₆₇	—	2	4	no	—
62	C₂²⋊C₈×C₆₇	—	2	4	no	—
63	C₂²⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
64	C₂²⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
65	#97×C₆₇	—	2	2	no	—
66	C₄×C₈×C₆₇	—	2	2	no	—
67	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
69	C₂⁴×C₆₇	C₂×Q₈	1	2	yes	—
70	C₂×D₄×C₆₇	—	2	4	no	—
71	C₂×D₄×C₆₇	C₄⋊C₄	1	2	yes	—
72	C₂×D₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
73	C₂×D₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	—	2	4	no	—
75	C₂²×C₄×C₆₇	—	2	4	no	—
76	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
84	M₁₆×C₆₇	—	2	2	no	—
85	M₁₆×C₆₇	—	2	2	no	—
86	M₁₆×C₆₇	—	2	4	no	—
87	M₁₆×C₆₇	—	2	2	no	—
88	M₁₆×C₆₇	—	2	4	no	—
89	M₁₆×C₆₇	—	4	8	no	—
90	M₁₆×C₆₇	#41	1	4	yes	—
91	C₂×C₈×C₆₇	—	2	2	no	—
92	C₂×C₈×C₆₇	—	2	2	no	—
93	C₂×C₈×C₆₇	—	2	4	no	—
94	C₂×C₈×C₆₇	—	2	2	no	—
95	C₂×C₈×C₆₇	—	2	4	no	—
96	C₂×C₈×C₆₇	—	2	2	no	—
97	C₂×C₈×C₆₇	—	4	4	no	—
98	C₂×C₈×C₆₇	—	4	4	no	—
99	C₂×C₈×C₆₇	—	4	8	no	—
100	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
101	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
103	C₄⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
104	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
105	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
106	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
107	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
108	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
109	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
110	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₂³×C₆₇	—	2	4	no	—
115	C₂³×C₆₇	—	2	4	no	—
116	C₂³×C₆₇	—	4	2	no	—
117	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
119	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
120	D₄×C₆₇	—	2	4	no	—
121	D₄×C₆₇	—	2	2	no	—
122	D₄×C₆₇	#108	1	4	yes	complemented
123	C₂×C₄×C₆₇	—	2	4	no	—
124	C₂×C₄×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	2	4	no	—
129	C₂×C₄×C₆₇	—	2	4	no	—
130	C₂×C₄×C₆₇	—	2	2	no	—
131	C₂×C₄×C₆₇	—	4	1	no	—
132	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
141	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
142	C₈×C₆₇	—	2	2	no	—
143	C₈×C₆₇	—	2	2	no	—
144	C₈×C₆₇	—	2	4	no	—
145	C₈×C₆₇	—	2	4	no	—
146	C₈×C₆₇	—	4	4	no	—
147	C₈×C₆₇	—	4	2	no	—
148	C₂²×C₆₇	—	2	2	no	—
149	C₂²×C₆₇	—	2	1	no	—
150	C₂²×C₆₇	—	2	1	no	—
151	C₂²×C₆₇	—	2	2	no	—
152	C₂²×C₆₇	—	2	2	no	—
153	C₂²×C₆₇	—	2	2	no	—
154	C₂²×C₆₇	—	2	2	no	—
155	C₂²×C₆₇	—	4	8	no	—
156	C₂²×C₆₇	C₂×#108	1	4	yes	—
157	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
158	C₂²×C₆₇	#554	1	2	yes	—
159	C₄×C₆₇	—	2	2	no	—
160	C₄×C₆₇	—	2	2	no	—
161	C₄×C₆₇	—	2	1	no	—
162	C₄×C₆₇	—	2	1	no	—
163	C₄×C₆₇	—	2	1	no	—
164	C₄×C₆₇	C₂×#108	1	2	yes	—
165	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
166	C₄×C₆₇	#518	1	1	yes	characteristic
167	C₄×C₆₇	#546	1	2	yes	—
168	#22717	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
169	C₂×C₆₇	—	2	2	no	—
170	C₂×C₆₇	—	2	4	no	—
171	C₂×C₆₇	—	4	2	no	—
172	C₂×C₆₇	#2349	1	1	yes	characteristic
173	C₂×C₆₇	C₂²×#108	1	1	yes	characteristic
174	C₂×C₆₇	#2916	1	1	yes	characteristic
175	D₄×M₁₆	C₂×C₆₇	1	2	yes	—
176	C₂×C₈×D₄	C₂×C₆₇	1	1	yes	characteristic
177	#1950	C₂×C₆₇	1	2	yes	complemented
178	#1855	C₂×C₆₇	1	4	yes	complemented
179	#1837	C₂×C₆₇	1	1	yes	characteristic, complemented
180	#1810	C₂×C₆₇	1	2	yes	complemented
181	C₄×#108	C₂×C₆₇	1	1	yes	characteristic, complemented
182	C₂×#341	C₂×C₆₇	1	2	yes	complemented
183	C₆₇	#22717	1	1	yes	characteristic, direct product factor, Sylow
184	C₂²×M₁₆	C₂²×C₆₇	1	4	yes	—
185	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
186	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
187	#436	—	2	2	no	—
188	#435	—	2	4	no	—
189	C₈×D₄	—	2	2	no	—
190	C₈×D₄	C₄×C₆₇	1	4	yes	—
191	C₈×D₄	C₄×C₆₇	1	2	yes	—
192	C₈×D₄	C₄×C₆₇	1	2	yes	—
193	C₂×#108	C₂²×C₆₇	1	2	yes	—
194	C₂×#108	C₂²×C₆₇	1	4	yes	—
195	#423	C₂²×C₆₇	1	2	yes	—
196	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
197	#407	C₂²×C₆₇	1	4	yes	—
198	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
199	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
200	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
201	#341	C₂²×C₆₇	1	1	yes	characteristic
202	#341	C₂²×C₆₇	1	2	yes	—
203	#341	C₂²×C₆₇	1	2	yes	—
204	#341	C₂²×C₆₇	1	4	yes	—
205	#341	C₂²×C₆₇	1	1	yes	characteristic
206	C₂²×D₄	Q₈×C₆₇	1	1	yes	characteristic
207	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
208	C₂×M₁₆	—	2	4	no	—
209	C₂×M₁₆	—	2	8	no	—
210	C₂×M₁₆	—	4	4	no	—
211	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
212	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
213	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
214	C₂²×C₈	—	2	4	no	—
215	C₂²×C₈	—	4	4	no	—
216	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
217	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
218	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
219	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
220	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
221	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
222	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
223	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
224	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
225	C₄×D₄	D₄×C₆₇	1	2	yes	—
226	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
227	C₂×C₄⋊C₄	Q₈×C₆₇	1	1	yes	characteristic
228	C₂×C₂²⋊C₄	Q₈×C₆₇	1	2	yes	—
229	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
230	#108	—	2	4	no	—
231	#108	—	4	4	no	—
232	C₄⋊C₈	—	2	2	no	—
233	C₄⋊C₈	—	2	2	no	—
234	C₄⋊C₈	C₂×C₄×C₆₇	1	2	yes	—
235	C₄⋊C₈	C₂×C₄×C₆₇	1	2	yes	—
236	C₂²⋊C₈	—	2	4	no	—
237	C₂²⋊C₈	—	2	4	no	—
238	C₂²⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
239	C₂²⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
240	#97	—	2	2	no	—
241	C₄×C₈	—	2	2	no	—
242	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
243	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
244	C₂⁴	C₂×Q₈×C₆₇	1	2	yes	—
245	C₂×D₄	—	2	4	no	—
246	C₂×D₄	C₄⋊C₄×C₆₇	1	2	yes	—
247	C₂×D₄	C₄⋊C₄×C₆₇	1	1	yes	characteristic
248	C₂×D₄	C₄⋊C₄×C₆₇	1	1	yes	characteristic
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	2	4	no	—
251	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
252	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
253	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
254	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
255	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
256	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
257	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
258	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
259	M₁₆	—	2	2	no	—
260	M₁₆	—	2	2	no	—
261	M₁₆	—	2	4	no	—
262	M₁₆	—	2	2	no	—
263	M₁₆	—	2	4	no	—
264	M₁₆	—	4	8	no	—
265	M₁₆	#41×C₆₇	1	4	yes	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	4	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	4	no	—
272	C₂×C₈	—	4	4	no	—
273	C₂×C₈	—	4	4	no	—
274	C₂×C₈	—	4	8	no	—
275	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
276	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
277	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
278	C₄⋊C₄	C₄⋊C₄×C₆₇	1	2	yes	—
279	C₄⋊C₄	C₄⋊C₄×C₆₇	1	1	yes	characteristic
280	C₄⋊C₄	C₄⋊C₄×C₆₇	1	1	yes	characteristic
281	C₂²⋊C₄	C₄⋊C₄×C₆₇	1	2	yes	—
282	C₂²⋊C₄	C₄⋊C₄×C₆₇	1	2	yes	—
283	C₂²⋊C₄	C₄⋊C₄×C₆₇	1	2	yes	—
284	C₂²⋊C₄	C₄⋊C₄×C₆₇	1	2	yes	—
285	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
286	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
287	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
288	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
289	C₂³	—	2	4	no	—
290	C₂³	—	2	4	no	—
291	C₂³	—	4	2	no	—
292	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
293	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
294	C₂³	C₂²×Q₈×C₆₇	1	1	yes	characteristic
295	D₄	—	2	4	no	—
296	D₄	—	2	2	no	—
297	D₄	#108×C₆₇	1	4	yes	complemented
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
308	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
309	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
310	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
311	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
312	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
313	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
314	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
315	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
316	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
317	C₈	—	2	2	no	—
318	C₈	—	2	2	no	—
319	C₈	—	2	4	no	—
320	C₈	—	2	4	no	—
321	C₈	—	4	4	no	—
322	C₈	—	4	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	2	2	no	—
330	C₂²	—	4	8	no	—
331	C₂²	C₂×#108×C₆₇	1	4	yes	—
332	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
333	C₂²	#554×C₆₇	1	2	yes	—
334	C₄	—	2	2	no	—
335	C₄	—	2	2	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	C₂×#108×C₆₇	1	2	yes	—
340	C₄	C₄×#41×C₆₇	1	1	yes	characteristic
341	C₄	#518×C₆₇	1	1	yes	characteristic
342	C₄	#546×C₆₇	1	2	yes	—
343	C₂	—	2	2	no	—
344	C₂	—	2	4	no	—
345	C₂	—	4	2	no	—
346	C₂	C₆₇×#2349	1	1	yes	characteristic
347	C₂	C₂²×#108×C₆₇	1	1	yes	characteristic
348	C₂	C₆₇×#2916	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂²×D₄×C₆₇	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄×D₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1950	C₂	semidirect
4. C₆₇×#1855	C₂	semidirect
5. C₆₇×#1837	C₂	semidirect
6. C₆₇×#1810	C₂	semidirect
7. C₄×#108×C₆₇	C₂	semidirect
8. C₂×#341×C₆₇	C₂	semidirect
122. D₄×C₆₇	#108	semidirect
168. #22717	C₆₇	direct (N × H)
177. #1950	C₂×C₆₇	semidirect
178. #1855	C₂×C₆₇	semidirect
179. #1837	C₂×C₆₇	semidirect
180. #1810	C₂×C₆₇	semidirect
181. C₄×#108	C₂×C₆₇	semidirect
182. C₂×#341	C₂×C₆₇	semidirect
183. C₆₇	#22717	direct (N × H)
297. D₄	#108×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
D₄×M₁₆×C₆₇	2
C₂×C₈×D₄×C₆₇	2
C₆₇×#1950	2
C₆₇×#1855	2
C₆₇×#1837	2
C₆₇×#1810	2
C₄×#108×C₆₇	2
C₂×#341×C₆₇	2
#22717	67

Subgroups — C67×#22717 #30701968

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22717 #30701968