Subgroups — C₆₇×#22451 #30701702

Order 17152.

1450 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#135×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2341	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1970	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1880	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1808	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1788	C₂	1	2	yes	maximal, complemented
8	C₈×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₆₇	C₂²	1	2	yes	—
10	C₂×#135×C₆₇	—	2	8	no	—
11	C₂×#135×C₆₇	C₄	1	8	yes	complemented
12	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	—
17	C₄×Q₁₆×C₆₇	—	2	8	no	—
18	C₂×#107×C₆₇	C₂²	1	2	yes	—
19	C₂×#103×C₆₇	C₂²	1	4	yes	—
20	C₂×#102×C₆₇	C₂²	1	4	yes	—
21	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
23	#390×C₆₇	—	2	4	no	—
24	#390×C₆₇	C₂²	1	4	yes	—
25	#380×C₆₇	—	2	2	no	—
26	#380×C₆₇	C₂²	1	2	yes	—
27	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
28	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
29	#340×C₆₇	C₂²	1	2	yes	—
30	#336×C₆₇	C₂²	1	2	yes	—
31	C₂×#41×C₆₇	—	2	4	no	—
32	C₂×#41×C₆₇	—	2	8	no	—
33	C₂×#41×C₆₇	—	4	4	no	—
34	C₂×#41×C₆₇	C₂×C₄	1	8	yes	—
35	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
36	C₂²×D₄×C₆₇	—	2	2	no	—
37	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
38	C₂³×C₄×C₆₇	—	2	2	no	—
39	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
40	#135×C₆₇	—	2	32	no	—
41	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	—
42	C₂×Q₁₆×C₆₇	D₄	1	8	yes	—
43	C₂×SD₁₆×C₆₇	—	2	8	no	—
44	C₂×SD₁₆×C₆₇	C₂×C₄	1	8	yes	—
45	C₂×D₈×C₆₇	—	2	4	no	—
46	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₆₇	—	2	4	no	—
48	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
50	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₄×Q₈×C₆₇	—	2	8	no	—
53	C₄×Q₈×C₆₇	—	4	4	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
55	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
56	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
57	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
58	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
59	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
60	C₂×C₄²×C₆₇	—	2	4	no	—
61	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
62	#107×C₆₇	—	2	4	no	—
63	#103×C₆₇	—	2	8	no	—
64	#102×C₆₇	—	4	4	no	—
65	C₂²⋊C₈×C₆₇	—	2	2	no	—
66	C₄×C₈×C₆₇	—	2	4	no	—
67	#95×C₆₇	—	2	2	no	—
68	#95×C₆₇	D₄	1	1	yes	characteristic
69	#95×C₆₇	D₄	1	1	yes	characteristic
70	C₂⁴×C₆₇	—	2	2	no	—
71	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
72	#41×C₆₇	—	2	16	no	—
73	#41×C₆₇	—	4	8	no	—
74	#41×C₆₇	—	4	16	no	—
75	C₂×Q₈×C₆₇	—	2	2	no	—
76	C₂×Q₈×C₆₇	—	2	4	no	—
77	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
78	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	—
79	C₂×D₄×C₆₇	—	2	8	no	—
80	C₂×D₄×C₆₇	—	2	4	no	—
81	C₂×D₄×C₆₇	—	2	2	no	—
82	C₂×D₄×C₆₇	—	2	4	no	—
83	C₂×D₄×C₆₇	—	4	8	no	—
84	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
85	C₂²×C₄×C₆₇	—	2	2	no	—
86	C₂²×C₄×C₆₇	—	2	2	no	—
87	C₂²×C₄×C₆₇	—	2	2	no	—
88	C₂²×C₄×C₆₇	—	2	8	no	—
89	C₂²×C₄×C₆₇	—	2	4	no	—
90	C₂²×C₄×C₆₇	—	4	8	no	—
91	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
95	C₂²×C₄×C₆₇	#41	1	2	yes	—
96	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
97	Q₁₆×C₆₇	C₂²⋊C₄	1	16	yes	complemented
98	SD₁₆×C₆₇	—	2	16	no	—
99	D₈×C₆₇	—	2	8	no	—
100	C₂×C₈×C₆₇	—	2	2	no	—
101	C₂×C₈×C₆₇	—	2	8	no	—
102	C₂×C₈×C₆₇	—	2	2	no	—
103	C₂×C₈×C₆₇	—	2	2	no	—
104	C₂×C₈×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
106	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
107	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
108	C₄⋊C₄×C₆₇	—	2	4	no	—
109	C₄⋊C₄×C₆₇	—	2	4	no	—
110	C₄⋊C₄×C₆₇	—	4	8	no	—
111	C₂²⋊C₄×C₆₇	—	2	4	no	—
112	C₂²⋊C₄×C₆₇	—	4	8	no	—
113	C₂²⋊C₄×C₆₇	—	8	4	no	—
114	C₂²⋊C₄×C₆₇	—	8	2	no	—
115	C₄²×C₆₇	—	2	2	no	—
116	C₄²×C₆₇	—	2	2	no	—
117	C₄²×C₆₇	—	4	8	no	—
118	C₂³×C₆₇	—	2	2	no	—
119	C₂³×C₆₇	—	2	2	no	—
120	C₂³×C₆₇	—	2	1	no	—
121	C₂³×C₆₇	—	2	1	no	—
122	C₂³×C₆₇	—	2	4	no	—
123	C₂³×C₆₇	—	4	4	no	—
124	C₂³×C₆₇	—	4	4	no	—
125	C₂³×C₆₇	—	4	2	no	—
126	C₂³×C₆₇	—	4	2	no	—
127	C₂³×C₆₇	C₄×D₄	1	2	yes	—
128	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
129	Q₈×C₆₇	—	2	4	no	—
130	Q₈×C₆₇	—	2	8	no	—
131	Q₈×C₆₇	C₂×C₂²⋊C₄	1	8	yes	—
132	D₄×C₆₇	—	2	4	no	—
133	D₄×C₆₇	—	2	4	no	—
134	D₄×C₆₇	—	4	16	no	—
135	D₄×C₆₇	—	4	4	no	—
136	C₂×C₄×C₆₇	—	2	4	no	—
137	C₂×C₄×C₆₇	—	2	2	no	—
138	C₂×C₄×C₆₇	—	2	8	no	—
139	C₂×C₄×C₆₇	—	2	2	no	—
140	C₂×C₄×C₆₇	—	2	2	no	—
141	C₂×C₄×C₆₇	—	2	2	no	—
142	C₂×C₄×C₆₇	—	2	2	no	—
143	C₂×C₄×C₆₇	—	2	4	no	—
144	C₂×C₄×C₆₇	—	2	2	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	4	16	no	—
148	C₂×C₄×C₆₇	—	4	4	no	—
149	C₂×C₄×C₆₇	—	4	4	no	—
150	C₂×C₄×C₆₇	—	4	4	no	—
151	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
152	C₂×C₄×C₆₇	#135	1	4	yes	complemented
153	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
154	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
155	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
156	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
157	C₈×C₆₇	—	2	2	no	—
158	C₈×C₆₇	—	4	2	no	—
159	C₈×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
160	C₂²×C₆₇	—	2	2	no	—
161	C₂²×C₆₇	—	2	2	no	—
162	C₂²×C₆₇	—	2	2	no	—
163	C₂²×C₆₇	—	4	2	no	—
164	C₂²×C₆₇	—	4	2	no	—
165	C₂²×C₆₇	—	4	4	no	—
166	C₂²×C₆₇	—	4	4	no	—
167	C₂²×C₆₇	—	4	4	no	—
168	C₂²×C₆₇	—	8	4	no	—
169	C₂²×C₆₇	—	8	4	no	—
170	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂²×C₆₇	D₄²	1	2	yes	—
172	C₂²×C₆₇	C₂×#135	1	2	yes	—
173	C₂²×C₆₇	#576	1	1	yes	characteristic
174	C₂²×C₆₇	#578	1	1	yes	characteristic
175	C₄×C₆₇	—	2	2	no	—
176	C₄×C₆₇	—	2	8	no	—
177	C₄×C₆₇	—	2	4	no	—
178	C₄×C₆₇	—	4	2	no	—
179	C₄×C₆₇	—	8	4	no	—
180	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
181	C₄×C₆₇	#518	1	1	yes	characteristic
182	C₄×C₆₇	#546	1	2	yes	—
183	#22451	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
184	C₂×C₆₇	—	4	2	no	—
185	C₂×C₆₇	—	8	4	no	—
186	C₂×C₆₇	D₄×C₂²⋊C₄	1	1	yes	characteristic
187	C₂×C₆₇	C₄×#135	1	1	yes	characteristic
188	C₂×C₆₇	#2951	1	1	yes	characteristic
189	C₂×C₆₇	#3285	1	2	yes	—
190	C₂×C₆₇	#3287	1	2	yes	—
191	C₂²×#135	C₂×C₆₇	1	1	yes	characteristic
192	C₂×C₄×Q₁₆	C₂×C₆₇	1	2	yes	complemented
193	#2341	C₂×C₆₇	1	2	yes	complemented
194	#1970	C₂×C₆₇	1	2	yes	complemented
195	#1880	C₂×C₆₇	1	4	yes	complemented
196	#1808	C₂×C₆₇	1	1	yes	characteristic, complemented
197	#1788	C₂×C₆₇	1	2	yes	complemented
198	C₈×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
199	C₆₇	#22451	1	1	yes	characteristic, direct product factor, Sylow
200	C₂²×#41	C₂²×C₆₇	1	2	yes	—
201	C₂×#135	—	2	8	no	—
202	C₂×#135	C₄×C₆₇	1	8	yes	complemented
203	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
204	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
205	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
206	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
207	C₂×C₄×Q₈	C₂²×C₆₇	1	4	yes	—
208	C₄×Q₁₆	—	2	8	no	—
209	C₂×#107	C₂²×C₆₇	1	2	yes	—
210	C₂×#103	C₂²×C₆₇	1	4	yes	—
211	C₂×#102	C₂²×C₆₇	1	4	yes	—
212	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
213	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
214	#390	—	2	4	no	—
215	#390	C₂²×C₆₇	1	4	yes	—
216	#380	—	2	2	no	—
217	#380	C₂²×C₆₇	1	2	yes	—
218	C₄×C₂²⋊C₄	—	2	2	no	—
219	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
220	#340	C₂²×C₆₇	1	2	yes	—
221	#336	C₂²×C₆₇	1	2	yes	—
222	C₂×#41	—	2	4	no	—
223	C₂×#41	—	2	8	no	—
224	C₂×#41	—	4	4	no	—
225	C₂×#41	C₂×C₄×C₆₇	1	8	yes	—
226	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
227	C₂²×D₄	—	2	2	no	—
228	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
229	C₂³×C₄	—	2	2	no	—
230	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
231	#135	—	2	32	no	—
232	C₂×Q₁₆	C₂×C₄×C₆₇	1	4	yes	—
233	C₂×Q₁₆	D₄×C₆₇	1	8	yes	—
234	C₂×SD₁₆	—	2	8	no	—
235	C₂×SD₁₆	C₂×C₄×C₆₇	1	8	yes	—
236	C₂×D₈	—	2	4	no	—
237	C₂×D₈	C₂×C₄×C₆₇	1	4	yes	—
238	C₂²×C₈	—	2	4	no	—
239	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
240	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
241	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
242	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
243	C₄×Q₈	—	2	8	no	—
244	C₄×Q₈	—	4	4	no	—
245	C₂×C₄⋊C₄	—	2	4	no	—
246	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
247	C₂×C₂²⋊C₄	—	2	4	no	—
248	C₂×C₂²⋊C₄	—	2	2	no	—
249	C₂×C₂²⋊C₄	—	2	4	no	—
250	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
251	C₂×C₄²	—	2	4	no	—
252	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
253	#107	—	2	4	no	—
254	#103	—	2	8	no	—
255	#102	—	4	4	no	—
256	C₂²⋊C₈	—	2	2	no	—
257	C₄×C₈	—	2	4	no	—
258	#95	—	2	2	no	—
259	#95	D₄×C₆₇	1	1	yes	characteristic
260	#95	D₄×C₆₇	1	1	yes	characteristic
261	C₂⁴	—	2	2	no	—
262	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
263	#41	—	2	16	no	—
264	#41	—	4	8	no	—
265	#41	—	4	16	no	—
266	C₂×Q₈	—	2	2	no	—
267	C₂×Q₈	—	2	4	no	—
268	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
269	C₂×Q₈	C₂×D₄×C₆₇	1	8	yes	—
270	C₂×D₄	—	2	8	no	—
271	C₂×D₄	—	2	4	no	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	—	2	4	no	—
274	C₂×D₄	—	4	8	no	—
275	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
276	C₂²×C₄	—	2	2	no	—
277	C₂²×C₄	—	2	2	no	—
278	C₂²×C₄	—	2	2	no	—
279	C₂²×C₄	—	2	8	no	—
280	C₂²×C₄	—	2	4	no	—
281	C₂²×C₄	—	4	8	no	—
282	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
283	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
284	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
285	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
286	C₂²×C₄	#41×C₆₇	1	2	yes	—
287	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
288	Q₁₆	C₂²⋊C₄×C₆₇	1	16	yes	complemented
289	SD₁₆	—	2	16	no	—
290	D₈	—	2	8	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	8	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
297	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
298	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
299	C₄⋊C₄	—	2	4	no	—
300	C₄⋊C₄	—	2	4	no	—
301	C₄⋊C₄	—	4	8	no	—
302	C₂²⋊C₄	—	2	4	no	—
303	C₂²⋊C₄	—	4	8	no	—
304	C₂²⋊C₄	—	8	4	no	—
305	C₂²⋊C₄	—	8	2	no	—
306	C₄²	—	2	2	no	—
307	C₄²	—	2	2	no	—
308	C₄²	—	4	8	no	—
309	C₂³	—	2	2	no	—
310	C₂³	—	2	2	no	—
311	C₂³	—	2	1	no	—
312	C₂³	—	2	1	no	—
313	C₂³	—	2	4	no	—
314	C₂³	—	4	4	no	—
315	C₂³	—	4	4	no	—
316	C₂³	—	4	2	no	—
317	C₂³	—	4	2	no	—
318	C₂³	C₄×D₄×C₆₇	1	2	yes	—
319	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
320	Q₈	—	2	4	no	—
321	Q₈	—	2	8	no	—
322	Q₈	C₂×C₂²⋊C₄×C₆₇	1	8	yes	—
323	D₄	—	2	4	no	—
324	D₄	—	2	4	no	—
325	D₄	—	4	16	no	—
326	D₄	—	4	4	no	—
327	C₂×C₄	—	2	4	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	8	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	4	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	4	16	no	—
339	C₂×C₄	—	4	4	no	—
340	C₂×C₄	—	4	4	no	—
341	C₂×C₄	—	4	4	no	—
342	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
343	C₂×C₄	#135×C₆₇	1	4	yes	complemented
344	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
345	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
346	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
347	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
348	C₈	—	2	2	no	—
349	C₈	—	4	2	no	—
350	C₈	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
351	C₂²	—	2	2	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	2	2	no	—
354	C₂²	—	4	2	no	—
355	C₂²	—	4	2	no	—
356	C₂²	—	4	4	no	—
357	C₂²	—	4	4	no	—
358	C₂²	—	4	4	no	—
359	C₂²	—	8	4	no	—
360	C₂²	—	8	4	no	—
361	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
362	C₂²	D₄²×C₆₇	1	2	yes	—
363	C₂²	C₂×#135×C₆₇	1	2	yes	—
364	C₂²	#576×C₆₇	1	1	yes	characteristic
365	C₂²	#578×C₆₇	1	1	yes	characteristic
366	C₄	—	2	4	no	—
367	C₄	—	2	2	no	—
368	C₄	—	2	8	no	—
369	C₄	—	4	2	no	—
370	C₄	—	8	4	no	—
371	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
372	C₄	#518×C₆₇	1	1	yes	characteristic
373	C₄	#546×C₆₇	1	2	yes	—
374	C₂	—	4	2	no	—
375	C₂	—	8	4	no	—
376	C₂	D₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
377	C₂	C₄×#135×C₆₇	1	1	yes	characteristic
378	C₂	C₆₇×#2951	1	1	yes	characteristic
379	C₂	C₆₇×#3285	1	2	yes	—
380	C₂	C₆₇×#3287	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×#135×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄×C₂²⋊C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Q₁₆×C₆₇	C₂	semidirect
3. C₆₇×#2341	C₂	semidirect
4. C₆₇×#1970	C₂	semidirect
5. C₆₇×#1880	C₂	semidirect
6. C₆₇×#1808	C₂	semidirect
7. C₆₇×#1788	C₂	semidirect
8. C₈×C₂²⋊C₄×C₆₇	C₂	semidirect
11. C₂×#135×C₆₇	C₄	semidirect
97. Q₁₆×C₆₇	C₂²⋊C₄	semidirect
152. C₂×C₄×C₆₇	#135	semidirect
183. #22451	C₆₇	direct (N × H)
192. C₂×C₄×Q₁₆	C₂×C₆₇	semidirect
193. #2341	C₂×C₆₇	semidirect
194. #1970	C₂×C₆₇	semidirect
195. #1880	C₂×C₆₇	semidirect
196. #1808	C₂×C₆₇	semidirect
197. #1788	C₂×C₆₇	semidirect
198. C₈×C₂²⋊C₄	C₂×C₆₇	semidirect
199. C₆₇	#22451	direct (N × H)
202. C₂×#135	C₄×C₆₇	semidirect
288. Q₁₆	C₂²⋊C₄×C₆₇	semidirect
343. C₂×C₄	#135×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#135×C₆₇	2
C₂×C₄×Q₁₆×C₆₇	2
C₆₇×#2341	2
C₆₇×#1970	2
C₆₇×#1880	2
C₆₇×#1808	2
C₆₇×#1788	2
C₈×C₂²⋊C₄×C₆₇	2
#22451	67

Subgroups — C67×#22451 #30701702

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22451 #30701702