Subgroups — C₆₇×#32946 #30712197

Order 17152.

1294 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#568×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#473×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#470×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3007	C₂	1	2	yes	maximal
5	C₆₇×#1908	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1901	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1815	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1811	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×#137×C₆₇	C₂²	1	4	yes	—
11	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
12	#568×C₆₇	—	2	8	no	—
13	C₂×#131×C₆₇	—	2	2	no	—
14	C₂×#131×C₆₇	C₂²	1	2	yes	—
15	C₂×#131×C₆₇	C₂²	1	2	yes	—
16	C₂²×M₁₆×C₆₇	—	2	2	no	—
17	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
20	#473×C₆₇	—	2	8	no	—
21	#470×C₆₇	—	2	2	no	—
22	#470×C₆₇	—	2	2	no	—
23	#452×C₆₇	—	2	4	no	—
24	#451×C₆₇	—	2	2	no	—
25	#450×C₆₇	—	2	2	no	—
26	C₂×#108×C₆₇	C₂²	1	2	yes	complemented
27	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic, complemented
28	#417×C₆₇	C₂²	1	4	yes	—
29	C₂×#103×C₆₇	C₂²	1	2	yes	complemented
30	C₂×#101×C₆₇	C₂²	1	4	yes	complemented
31	#407×C₆₇	C₂²	1	2	yes	—
32	#343×C₆₇	C₂²	1	4	yes	complemented
33	#341×C₆₇	C₂²	1	2	yes	complemented
34	C₂×#41×C₆₇	—	2	4	no	—
35	C₂×#41×C₆₇	—	2	4	no	—
36	C₂×#41×C₆₇	—	2	1	no	—
37	C₂×#41×C₆₇	—	2	1	no	—
38	C₂×#41×C₆₇	D₄	1	2	yes	—
39	C₂×#41×C₆₇	C₂³	1	2	yes	—
40	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
41	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
42	C₂²×D₄×C₆₇	—	2	1	no	—
43	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₆₇	—	2	1	no	—
45	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
46	#137×C₆₇	—	2	8	no	—
47	#137×C₆₇	—	2	8	no	—
48	C₂×Q₁₆×C₆₇	—	2	4	no	—
49	C₂×Q₁₆×C₆₇	—	2	2	no	—
50	C₂×Q₁₆×C₆₇	—	2	4	no	—
51	C₂×SD₁₆×C₆₇	—	2	4	no	—
52	#131×C₆₇	—	2	16	no	—
53	#131×C₆₇	—	2	4	no	—
54	#131×C₆₇	—	2	4	no	—
55	#131×C₆₇	—	2	4	no	—
56	#131×C₆₇	D₄	1	8	yes	—
57	C₂×M₁₆×C₆₇	—	2	16	no	—
58	C₂×M₁₆×C₆₇	—	2	1	no	—
59	C₂×M₁₆×C₆₇	—	2	4	no	—
60	C₂×M₁₆×C₆₇	—	2	2	no	—
61	C₂×M₁₆×C₆₇	—	2	2	no	—
62	C₂×M₁₆×C₆₇	—	2	4	no	—
63	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
64	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
65	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
66	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
67	C₂²×C₈×C₆₇	—	2	1	no	—
68	C₂²×C₈×C₆₇	—	2	2	no	—
69	C₂²×C₈×C₆₇	—	2	2	no	—
70	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₂²⋊Q₈×C₆₇	—	2	2	no	—
73	C₂²⋊Q₈×C₆₇	—	2	2	no	—
74	C₂²⋊Q₈×C₆₇	—	2	2	no	—
75	C₂²⋊Q₈×C₆₇	—	2	2	no	—
76	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
77	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
78	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
79	#108×C₆₇	—	2	4	no	—
80	#106×C₆₇	—	2	2	no	—
81	#103×C₆₇	—	2	4	no	—
82	#103×C₆₇	—	2	4	no	—
83	#102×C₆₇	—	2	4	no	—
84	#101×C₆₇	—	2	8	no	—
85	C₂²⋊C₈×C₆₇	—	2	2	no	—
86	C₂²⋊C₈×C₆₇	—	2	2	no	—
87	C₂⁴×C₆₇	—	2	1	no	—
88	C₂⁴×C₆₇	#41	1	1	yes	characteristic
89	#41×C₆₇	—	2	4	no	—
90	#41×C₆₇	—	2	8	no	—
91	#41×C₆₇	—	2	4	no	—
92	#41×C₆₇	—	2	4	no	—
93	#41×C₆₇	—	2	4	no	—
94	#41×C₆₇	—	2	2	no	—
95	#41×C₆₇	—	2	2	no	—
96	#41×C₆₇	C₂×D₄	1	4	yes	—
97	#41×C₆₇	#41	1	4	yes	—
98	C₂×Q₈×C₆₇	—	2	4	no	—
99	C₂×Q₈×C₆₇	—	2	1	no	—
100	C₂×Q₈×C₆₇	—	2	4	no	—
101	C₂×Q₈×C₆₇	—	2	2	no	—
102	C₂×Q₈×C₆₇	—	2	2	no	—
103	C₂×Q₈×C₆₇	—	2	4	no	—
104	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
105	C₂×D₄×C₆₇	—	2	8	no	—
106	C₂×D₄×C₆₇	—	2	2	no	—
107	C₂×D₄×C₆₇	—	2	2	no	—
108	C₂×D₄×C₆₇	—	2	4	no	—
109	C₂×D₄×C₆₇	—	2	1	no	—
110	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₆₇	—	2	4	no	—
112	C₂²×C₄×C₆₇	—	2	4	no	—
113	C₂²×C₄×C₆₇	—	2	1	no	—
114	C₂²×C₄×C₆₇	—	2	2	no	—
115	C₂²×C₄×C₆₇	—	2	2	no	—
116	C₂²×C₄×C₆₇	—	2	2	no	—
117	C₂²×C₄×C₆₇	—	2	2	no	—
118	C₂²×C₄×C₆₇	—	2	2	no	—
119	C₂²×C₄×C₆₇	—	2	2	no	—
120	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
125	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
126	Q₁₆×C₆₇	—	4	8	no	—
127	Q₁₆×C₆₇	—	4	4	no	—
128	SD₁₆×C₆₇	—	4	8	no	—
129	M₁₆×C₆₇	—	2	4	no	—
130	M₁₆×C₆₇	—	2	8	no	—
131	M₁₆×C₆₇	—	2	2	no	—
132	M₁₆×C₆₇	—	2	4	no	—
133	M₁₆×C₆₇	—	2	8	no	—
134	M₁₆×C₆₇	—	2	4	no	—
135	M₁₆×C₆₇	—	2	4	no	—
136	M₁₆×C₆₇	—	2	4	no	—
137	M₁₆×C₆₇	C₂×D₄	1	4	yes	—
138	C₂×C₈×C₆₇	—	2	4	no	—
139	C₂×C₈×C₆₇	—	2	8	no	—
140	C₂×C₈×C₆₇	—	2	2	no	—
141	C₂×C₈×C₆₇	—	2	2	no	—
142	C₂×C₈×C₆₇	—	2	8	no	—
143	C₂×C₈×C₆₇	—	2	4	no	—
144	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
145	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
146	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
147	C₄⋊C₄×C₆₇	—	2	2	no	—
148	C₄⋊C₄×C₆₇	—	2	2	no	—
149	C₄⋊C₄×C₆₇	—	4	2	no	—
150	C₄⋊C₄×C₆₇	—	4	2	no	—
151	C₂²⋊C₄×C₆₇	—	4	2	no	—
152	C₂²⋊C₄×C₆₇	—	4	2	no	—
153	C₂³×C₆₇	—	2	4	no	—
154	C₂³×C₆₇	—	2	2	no	—
155	C₂³×C₆₇	—	2	2	no	—
156	C₂³×C₆₇	—	4	2	no	—
157	C₂³×C₆₇	—	8	2	no	—
158	C₂³×C₆₇	#121	1	1	yes	characteristic
159	C₂³×C₆₇	#121	1	1	yes	characteristic
160	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
161	Q₈×C₆₇	—	2	4	no	—
162	Q₈×C₆₇	—	2	2	no	—
163	Q₈×C₆₇	—	4	2	no	—
164	Q₈×C₆₇	—	4	4	no	—
165	Q₈×C₆₇	—	4	2	no	—
166	Q₈×C₆₇	#121	1	4	yes	—
167	D₄×C₆₇	—	2	8	no	—
168	D₄×C₆₇	—	2	4	no	—
169	D₄×C₆₇	—	2	4	no	—
170	D₄×C₆₇	—	2	4	no	—
171	D₄×C₆₇	—	2	2	no	—
172	D₄×C₆₇	#121	1	4	yes	—
173	C₂×C₄×C₆₇	—	2	4	no	—
174	C₂×C₄×C₆₇	—	2	8	no	—
175	C₂×C₄×C₆₇	—	2	2	no	—
176	C₂×C₄×C₆₇	—	2	2	no	—
177	C₂×C₄×C₆₇	—	2	4	no	—
178	C₂×C₄×C₆₇	—	2	1	no	—
179	C₂×C₄×C₆₇	—	2	1	no	—
180	C₂×C₄×C₆₇	—	2	2	no	—
181	C₂×C₄×C₆₇	—	2	2	no	—
182	C₂×C₄×C₆₇	—	2	1	no	—
183	C₂×C₄×C₆₇	—	2	4	no	—
184	C₂×C₄×C₆₇	—	2	1	no	—
185	C₂×C₄×C₆₇	—	4	2	no	—
186	C₂×C₄×C₆₇	—	4	2	no	—
187	C₂×C₄×C₆₇	—	4	4	no	—
188	C₂×C₄×C₆₇	—	8	2	no	—
189	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
190	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
191	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
192	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
195	C₈×C₆₇	—	2	4	no	—
196	C₈×C₆₇	—	2	2	no	—
197	C₈×C₆₇	—	4	4	no	—
198	C₈×C₆₇	#120	1	4	yes	—
199	C₂²×C₆₇	—	2	2	no	—
200	C₂²×C₆₇	—	2	1	no	—
201	C₂²×C₆₇	—	2	1	no	—
202	C₂²×C₆₇	—	2	4	no	—
203	C₂²×C₆₇	—	2	2	no	—
204	C₂²×C₆₇	—	2	2	no	—
205	C₂²×C₆₇	—	4	4	no	—
206	C₂²×C₆₇	—	8	2	no	—
207	C₂²×C₆₇	—	8	4	no	—
208	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
209	C₂²×C₆₇	#525	1	2	yes	—
210	C₄×C₆₇	—	2	1	no	—
211	C₄×C₆₇	—	2	4	no	—
212	C₄×C₆₇	—	2	1	no	—
213	C₄×C₆₇	—	8	2	no	—
214	C₄×C₆₇	—	8	2	no	—
215	C₄×C₆₇	C₂×#120	1	1	yes	characteristic
216	C₄×C₆₇	C₂×#121	1	2	yes	—
217	C₄×C₆₇	#525	1	1	yes	characteristic
218	#32946	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
219	C₂×C₆₇	—	2	2	no	—
220	C₂×C₆₇	—	4	1	no	—
221	C₂×C₆₇	—	4	1	no	—
222	C₂×C₆₇	—	4	4	no	—
223	C₂×C₆₇	#2847	1	1	yes	characteristic
224	C₂×C₆₇	#3069	1	2	yes	—
225	C₂×#568	C₂×C₆₇	1	1	yes	characteristic
226	C₂×#473	C₂×C₆₇	1	2	yes	complemented
227	C₂×#470	C₂×C₆₇	1	1	yes	characteristic, complemented
228	#3007	C₂×C₆₇	1	2	yes	—
229	#1908	C₂×C₆₇	1	4	yes	complemented
230	#1901	C₂×C₆₇	1	2	yes	complemented
231	#1815	C₂×C₆₇	1	2	yes	complemented
232	#1811	C₂×C₆₇	1	1	yes	characteristic, complemented
233	C₆₇	#32946	1	1	yes	characteristic, direct product factor, Sylow
234	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
235	C₂×#137	C₂²×C₆₇	1	4	yes	—
236	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
237	#568	—	2	8	no	—
238	C₂×#131	—	2	2	no	—
239	C₂×#131	C₂²×C₆₇	1	2	yes	—
240	C₂×#131	C₂²×C₆₇	1	2	yes	—
241	C₂²×M₁₆	—	2	2	no	—
242	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
243	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
244	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
245	#473	—	2	8	no	—
246	#470	—	2	2	no	—
247	#470	—	2	2	no	—
248	#452	—	2	4	no	—
249	#451	—	2	2	no	—
250	#450	—	2	2	no	—
251	C₂×#108	C₂²×C₆₇	1	2	yes	complemented
252	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic, complemented
253	#417	C₂²×C₆₇	1	4	yes	—
254	C₂×#103	C₂²×C₆₇	1	2	yes	complemented
255	C₂×#101	C₂²×C₆₇	1	4	yes	complemented
256	#407	C₂²×C₆₇	1	2	yes	—
257	#343	C₂²×C₆₇	1	4	yes	complemented
258	#341	C₂²×C₆₇	1	2	yes	complemented
259	C₂×#41	—	2	4	no	—
260	C₂×#41	—	2	4	no	—
261	C₂×#41	—	2	1	no	—
262	C₂×#41	—	2	1	no	—
263	C₂×#41	D₄×C₆₇	1	2	yes	—
264	C₂×#41	C₂³×C₆₇	1	2	yes	—
265	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
266	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
267	C₂²×D₄	—	2	1	no	—
268	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
269	C₂³×C₄	—	2	1	no	—
270	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
271	#137	—	2	8	no	—
272	#137	—	2	8	no	—
273	C₂×Q₁₆	—	2	4	no	—
274	C₂×Q₁₆	—	2	2	no	—
275	C₂×Q₁₆	—	2	4	no	—
276	C₂×SD₁₆	—	2	4	no	—
277	#131	—	2	16	no	—
278	#131	—	2	4	no	—
279	#131	—	2	4	no	—
280	#131	—	2	4	no	—
281	#131	D₄×C₆₇	1	8	yes	—
282	C₂×M₁₆	—	2	16	no	—
283	C₂×M₁₆	—	2	1	no	—
284	C₂×M₁₆	—	2	4	no	—
285	C₂×M₁₆	—	2	2	no	—
286	C₂×M₁₆	—	2	2	no	—
287	C₂×M₁₆	—	2	4	no	—
288	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
289	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
290	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
291	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
292	C₂²×C₈	—	2	1	no	—
293	C₂²×C₈	—	2	2	no	—
294	C₂²×C₈	—	2	2	no	—
295	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
296	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
297	C₂²⋊Q₈	—	2	2	no	—
298	C₂²⋊Q₈	—	2	2	no	—
299	C₂²⋊Q₈	—	2	2	no	—
300	C₂²⋊Q₈	—	2	2	no	—
301	C₂×C₄⋊C₄	—	2	2	no	—
302	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
303	C₂×C₂²⋊C₄	—	2	2	no	—
304	#108	—	2	4	no	—
305	#106	—	2	2	no	—
306	#103	—	2	4	no	—
307	#103	—	2	4	no	—
308	#102	—	2	4	no	—
309	#101	—	2	8	no	—
310	C₂²⋊C₈	—	2	2	no	—
311	C₂²⋊C₈	—	2	2	no	—
312	C₂⁴	—	2	1	no	—
313	C₂⁴	#41×C₆₇	1	1	yes	characteristic
314	#41	—	2	4	no	—
315	#41	—	2	8	no	—
316	#41	—	2	4	no	—
317	#41	—	2	4	no	—
318	#41	—	2	4	no	—
319	#41	—	2	2	no	—
320	#41	—	2	2	no	—
321	#41	C₂×D₄×C₆₇	1	4	yes	—
322	#41	#41×C₆₇	1	4	yes	—
323	C₂×Q₈	—	2	4	no	—
324	C₂×Q₈	—	2	1	no	—
325	C₂×Q₈	—	2	4	no	—
326	C₂×Q₈	—	2	2	no	—
327	C₂×Q₈	—	2	2	no	—
328	C₂×Q₈	—	2	4	no	—
329	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
330	C₂×D₄	—	2	8	no	—
331	C₂×D₄	—	2	2	no	—
332	C₂×D₄	—	2	2	no	—
333	C₂×D₄	—	2	4	no	—
334	C₂×D₄	—	2	1	no	—
335	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
336	C₂²×C₄	—	2	4	no	—
337	C₂²×C₄	—	2	4	no	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	2	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	2	2	no	—
345	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
346	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
347	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
348	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
349	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
350	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
351	Q₁₆	—	4	8	no	—
352	Q₁₆	—	4	4	no	—
353	SD₁₆	—	4	8	no	—
354	M₁₆	—	2	4	no	—
355	M₁₆	—	2	8	no	—
356	M₁₆	—	2	2	no	—
357	M₁₆	—	2	4	no	—
358	M₁₆	—	2	8	no	—
359	M₁₆	—	2	4	no	—
360	M₁₆	—	2	4	no	—
361	M₁₆	—	2	4	no	—
362	M₁₆	C₂×D₄×C₆₇	1	4	yes	—
363	C₂×C₈	—	2	4	no	—
364	C₂×C₈	—	2	8	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	2	8	no	—
368	C₂×C₈	—	2	4	no	—
369	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
370	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
371	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
372	C₄⋊C₄	—	2	2	no	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	4	2	no	—
375	C₄⋊C₄	—	4	2	no	—
376	C₂²⋊C₄	—	4	2	no	—
377	C₂²⋊C₄	—	4	2	no	—
378	C₂³	—	2	4	no	—
379	C₂³	—	2	2	no	—
380	C₂³	—	2	2	no	—
381	C₂³	—	4	2	no	—
382	C₂³	—	8	2	no	—
383	C₂³	#121×C₆₇	1	1	yes	characteristic
384	C₂³	#121×C₆₇	1	1	yes	characteristic
385	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
386	Q₈	—	2	4	no	—
387	Q₈	—	2	2	no	—
388	Q₈	—	4	2	no	—
389	Q₈	—	4	4	no	—
390	Q₈	—	4	2	no	—
391	Q₈	#121×C₆₇	1	4	yes	—
392	D₄	—	2	8	no	—
393	D₄	—	2	4	no	—
394	D₄	—	2	4	no	—
395	D₄	—	2	4	no	—
396	D₄	—	2	2	no	—
397	D₄	#121×C₆₇	1	4	yes	—
398	C₂×C₄	—	2	4	no	—
399	C₂×C₄	—	2	8	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	4	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	4	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	4	no	—
413	C₂×C₄	—	8	2	no	—
414	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
415	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
416	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
417	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
418	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
419	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
420	C₈	—	2	4	no	—
421	C₈	—	2	2	no	—
422	C₈	—	4	4	no	—
423	C₈	#120×C₆₇	1	4	yes	—
424	C₂²	—	2	2	no	—
425	C₂²	—	2	2	no	—
426	C₂²	—	2	2	no	—
427	C₂²	—	2	1	no	—
428	C₂²	—	2	1	no	—
429	C₂²	—	2	4	no	—
430	C₂²	—	4	4	no	—
431	C₂²	—	8	2	no	—
432	C₂²	—	8	4	no	—
433	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
434	C₂²	#525×C₆₇	1	2	yes	—
435	C₄	—	2	1	no	—
436	C₄	—	2	4	no	—
437	C₄	—	2	1	no	—
438	C₄	—	8	2	no	—
439	C₄	—	8	2	no	—
440	C₄	C₂×#120×C₆₇	1	1	yes	characteristic
441	C₄	C₂×#121×C₆₇	1	2	yes	—
442	C₄	#525×C₆₇	1	1	yes	characteristic
443	C₂	—	2	2	no	—
444	C₂	—	4	1	no	—
445	C₂	—	4	1	no	—
446	C₂	—	4	4	no	—
447	C₂	C₆₇×#2847	1	1	yes	characteristic
448	C₂	C₆₇×#3069	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#473×C₆₇	C₂	semidirect
3. C₂×#470×C₆₇	C₂	semidirect
5. C₆₇×#1908	C₂	semidirect
6. C₆₇×#1901	C₂	semidirect
7. C₆₇×#1815	C₂	semidirect
8. C₆₇×#1811	C₂	semidirect
11. C₂²×Q₁₆×C₆₇	C₂²	semidirect
26. C₂×#108×C₆₇	C₂²	semidirect
27. C₂×#106×C₆₇	C₂²	semidirect
29. C₂×#103×C₆₇	C₂²	semidirect
30. C₂×#101×C₆₇	C₂²	semidirect
32. #343×C₆₇	C₂²	semidirect
33. #341×C₆₇	C₂²	semidirect
218. #32946	C₆₇	direct (N × H)
226. C₂×#473	C₂×C₆₇	semidirect
227. C₂×#470	C₂×C₆₇	semidirect
229. #1908	C₂×C₆₇	semidirect
230. #1901	C₂×C₆₇	semidirect
231. #1815	C₂×C₆₇	semidirect
232. #1811	C₂×C₆₇	semidirect
233. C₆₇	#32946	direct (N × H)
236. C₂²×Q₁₆	C₂²×C₆₇	semidirect
251. C₂×#108	C₂²×C₆₇	semidirect
252. C₂×#106	C₂²×C₆₇	semidirect
254. C₂×#103	C₂²×C₆₇	semidirect
255. C₂×#101	C₂²×C₆₇	semidirect
257. #343	C₂²×C₆₇	semidirect
258. #341	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#568×C₆₇	2
C₂×#473×C₆₇	2
C₂×#470×C₆₇	2
C₆₇×#3007	2
C₆₇×#1908	2
C₆₇×#1901	2
C₆₇×#1815	2
C₆₇×#1811	2
#32946	67

Subgroups — C67×#32946 #30712197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32946 #30712197