Subgroups — C₆₇×#22166 #30701417

Order 17152.

752 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3035	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2931	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2922	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1980	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1975	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1950	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1938	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1876	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1866	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1806	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1770	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135×C₆₇	C₂²	1	2	yes	—
13	C₂×#131×C₆₇	C₂²	1	2	yes	—
14	#532×C₆₇	C₂²	1	1	yes	characteristic
15	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
16	#464×C₆₇	C₄	1	2	yes	complemented
17	#463×C₆₇	C₄	1	2	yes	complemented
18	#462×C₆₇	C₄	1	1	yes	characteristic, complemented
19	#461×C₆₇	C₄	1	1	yes	characteristic, complemented
20	#460×C₆₇	C₄	1	1	yes	characteristic, complemented
21	#459×C₆₇	C₄	1	1	yes	characteristic, complemented
22	#436×C₆₇	—	2	1	no	—
23	#436×C₆₇	—	2	1	no	—
24	C₈×D₄×C₆₇	—	2	1	no	—
25	C₈×D₄×C₆₇	—	2	1	no	—
26	#433×C₆₇	—	2	1	no	—
27	#433×C₆₇	—	2	1	no	—
28	#431×C₆₇	—	2	1	no	—
29	#431×C₆₇	—	2	1	no	—
30	C₂×#108×C₆₇	C₂²	1	2	yes	—
31	#424×C₆₇	C₂²	1	1	yes	characteristic
32	#424×C₆₇	C₂²	1	1	yes	characteristic
33	#423×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×#104×C₆₇	C₂²	1	2	yes	—
36	C₂×#104×C₆₇	C₂²	1	2	yes	—
37	C₂×#104×C₆₇	C₂²	1	2	yes	—
38	#416×C₆₇	C₂²	1	2	yes	—
39	#408×C₆₇	C₂²	1	1	yes	characteristic
40	#408×C₆₇	C₂²	1	1	yes	characteristic
41	#408×C₆₇	C₂²	1	2	yes	—
42	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
43	C₂×#97×C₆₇	C₂²	1	2	yes	—
44	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
45	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
46	#341×C₆₇	C₂²	1	2	yes	—
47	#339×C₆₇	C₂²	1	2	yes	—
48	#336×C₆₇	C₂²	1	2	yes	—
49	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
50	C₂×#41×C₆₇	C₂³	1	2	yes	—
51	#135×C₆₇	—	4	2	no	—
52	#135×C₆₇	—	4	2	no	—
53	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
54	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
55	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
56	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
57	#131×C₆₇	—	4	2	no	—
58	#131×C₆₇	—	4	2	no	—
59	C₂×M₁₆×C₆₇	—	2	2	no	—
60	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
61	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
62	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₆₇	—	2	2	no	—
64	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
65	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
68	#127×C₆₇	C₂×C₄	1	1	yes	characteristic
69	#124×C₆₇	C₂×C₄	1	2	yes	—
70	#121×C₆₇	—	2	1	no	—
71	#121×C₆₇	—	2	1	no	—
72	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
74	C₄×D₄×C₆₇	—	2	1	no	—
75	C₄×D₄×C₆₇	—	2	1	no	—
76	C₄×D₄×C₆₇	—	2	1	no	—
77	C₄×D₄×C₆₇	—	2	1	no	—
78	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
79	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
80	#117×C₆₇	—	2	1	no	—
81	#117×C₆₇	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
83	C₂×C₄²×C₆₇	—	2	1	no	—
84	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
85	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
86	#108×C₆₇	—	4	2	no	—
87	C₄⋊C₈×C₆₇	—	2	1	no	—
88	C₄⋊C₈×C₆₇	—	2	1	no	—
89	C₄⋊C₈×C₆₇	—	2	1	no	—
90	C₄⋊C₈×C₆₇	—	2	1	no	—
91	C₄⋊C₈×C₆₇	—	2	1	no	—
92	C₄⋊C₈×C₆₇	—	2	1	no	—
93	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
94	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
95	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
96	#104×C₆₇	—	4	2	no	—
97	#104×C₆₇	—	4	2	no	—
98	#104×C₆₇	—	4	2	no	—
99	#103×C₆₇	C₂×C₄	1	2	yes	—
100	#103×C₆₇	C₂×C₄	1	2	yes	—
101	#102×C₆₇	C₂×C₄	1	2	yes	—
102	#102×C₆₇	C₂×C₄	1	2	yes	—
103	C₂²⋊C₈×C₆₇	—	2	1	no	—
104	C₂²⋊C₈×C₆₇	—	2	1	no	—
105	C₂²⋊C₈×C₆₇	—	2	1	no	—
106	C₂²⋊C₈×C₆₇	—	2	1	no	—
107	C₂²⋊C₈×C₆₇	—	2	2	no	—
108	C₂²⋊C₈×C₆₇	—	2	2	no	—
109	#97×C₆₇	—	2	1	no	—
110	#97×C₆₇	—	4	2	no	—
111	C₄×C₈×C₆₇	—	2	1	no	—
112	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
113	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
114	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
115	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
116	#95×C₆₇	—	2	1	no	—
117	#41×C₆₇	—	2	1	no	—
118	#41×C₆₇	—	2	2	no	—
119	#41×C₆₇	—	2	2	no	—
120	#41×C₆₇	—	2	2	no	—
121	#41×C₆₇	—	2	1	no	—
122	#41×C₆₇	—	2	1	no	—
123	#41×C₆₇	—	2	1	no	—
124	#41×C₆₇	—	2	2	no	—
125	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
126	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
127	C₂×D₄×C₆₇	—	2	1	no	—
128	C₂×D₄×C₆₇	—	2	2	no	—
129	C₂×D₄×C₆₇	—	2	1	no	—
130	C₂×D₄×C₆₇	—	2	1	no	—
131	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
132	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₄×C₆₇	—	2	1	no	—
134	C₂²×C₄×C₆₇	—	2	1	no	—
135	C₂²×C₄×C₆₇	—	2	1	no	—
136	C₂²×C₄×C₆₇	—	2	2	no	—
137	C₂²×C₄×C₆₇	—	2	2	no	—
138	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
139	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
140	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
141	Q₁₆×C₆₇	—	4	2	no	—
142	SD₁₆×C₆₇	—	4	2	no	—
143	SD₁₆×C₆₇	—	4	2	no	—
144	D₈×C₆₇	—	4	2	no	—
145	M₁₆×C₆₇	—	2	1	no	—
146	M₁₆×C₆₇	—	2	1	no	—
147	M₁₆×C₆₇	—	2	2	no	—
148	M₁₆×C₆₇	—	2	1	no	—
149	M₁₆×C₆₇	—	2	2	no	—
150	M₁₆×C₆₇	—	2	1	no	—
151	M₁₆×C₆₇	—	4	4	no	—
152	C₂×C₈×C₆₇	—	2	2	no	—
153	C₂×C₈×C₆₇	—	2	1	no	—
154	C₂×C₈×C₆₇	—	2	1	no	—
155	C₂×C₈×C₆₇	—	2	2	no	—
156	C₂×C₈×C₆₇	—	2	1	no	—
157	C₂×C₈×C₆₇	—	2	1	no	—
158	C₂×C₈×C₆₇	—	2	2	no	—
159	C₂×C₈×C₆₇	—	2	1	no	—
160	C₂×C₈×C₆₇	—	2	1	no	—
161	C₂×C₈×C₆₇	—	4	4	no	—
162	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
163	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
164	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
165	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
166	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
167	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
168	C₄⋊C₄×C₆₇	—	2	1	no	—
169	C₄⋊C₄×C₆₇	—	2	1	no	—
170	C₄⋊C₄×C₆₇	—	2	1	no	—
171	C₄⋊C₄×C₆₇	—	2	1	no	—
172	C₄⋊C₄×C₆₇	—	4	2	no	—
173	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
174	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
175	C₂²⋊C₄×C₆₇	—	2	1	no	—
176	C₂²⋊C₄×C₆₇	—	2	1	no	—
177	C₂²⋊C₄×C₆₇	—	2	1	no	—
178	C₂²⋊C₄×C₆₇	—	2	2	no	—
179	C₂²⋊C₄×C₆₇	—	2	2	no	—
180	C₄²×C₆₇	—	2	1	no	—
181	C₄²×C₆₇	—	2	1	no	—
182	C₄²×C₆₇	—	2	1	no	—
183	C₄²×C₆₇	—	2	2	no	—
184	C₄²×C₆₇	—	2	2	no	—
185	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
186	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
187	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
188	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
189	C₂³×C₆₇	—	2	1	no	—
190	C₂³×C₆₇	—	2	2	no	—
191	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
192	Q₈×C₆₇	—	2	1	no	—
193	Q₈×C₆₇	—	2	2	no	—
194	Q₈×C₆₇	—	2	2	no	—
195	Q₈×C₆₇	—	2	1	no	—
196	D₄×C₆₇	—	2	1	no	—
197	D₄×C₆₇	—	2	2	no	—
198	D₄×C₆₇	—	2	2	no	—
199	D₄×C₆₇	—	2	1	no	—
200	D₄×C₆₇	—	4	2	no	—
201	D₄×C₆₇	—	4	1	no	—
202	D₄×C₆₇	—	4	2	no	—
203	D₄×C₆₇	—	4	1	no	—
204	D₄×C₆₇	—	4	2	no	—
205	D₄×C₆₇	—	4	2	no	—
206	C₂×C₄×C₆₇	—	2	2	no	—
207	C₂×C₄×C₆₇	—	2	1	no	—
208	C₂×C₄×C₆₇	—	2	1	no	—
209	C₂×C₄×C₆₇	—	2	1	no	—
210	C₂×C₄×C₆₇	—	2	2	no	—
211	C₂×C₄×C₆₇	—	4	1	no	—
212	C₂×C₄×C₆₇	—	4	2	no	—
213	C₂×C₄×C₆₇	—	4	2	no	—
214	C₂×C₄×C₆₇	—	4	1	no	—
215	C₂×C₄×C₆₇	—	4	1	no	—
216	C₂×C₄×C₆₇	—	4	2	no	—
217	C₂×C₄×C₆₇	—	4	2	no	—
218	C₂×C₄×C₆₇	—	4	2	no	—
219	C₂×C₄×C₆₇	—	4	2	no	—
220	C₂×C₄×C₆₇	—	4	2	no	—
221	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
222	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
223	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
224	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
225	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
226	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
228	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
229	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
230	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
231	C₈×C₆₇	—	2	2	no	—
232	C₈×C₆₇	—	2	2	no	—
233	C₈×C₆₇	—	2	1	no	—
234	C₈×C₆₇	—	2	1	no	—
235	C₈×C₆₇	—	2	1	no	—
236	C₈×C₆₇	—	2	1	no	—
237	C₈×C₆₇	—	4	2	no	—
238	C₈×C₆₇	—	4	1	no	—
239	C₈×C₆₇	—	4	1	no	—
240	C₂²×C₆₇	—	2	1	no	—
241	C₂²×C₆₇	—	2	1	no	—
242	C₂²×C₆₇	—	4	2	no	—
243	C₂²×C₆₇	—	4	1	no	—
244	C₂²×C₆₇	—	4	1	no	—
245	C₂²×C₆₇	—	4	2	no	—
246	C₂²×C₆₇	—	4	1	no	—
247	C₂²×C₆₇	—	4	2	no	—
248	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
249	C₂²×C₆₇	#540	1	1	yes	characteristic
250	C₂²×C₆₇	#542	1	1	yes	characteristic
251	C₄×C₆₇	—	2	1	no	—
252	C₄×C₆₇	—	2	1	no	—
253	C₄×C₆₇	—	2	1	no	—
254	C₄×C₆₇	—	4	1	no	—
255	C₄×C₆₇	—	4	2	no	—
256	C₄×C₆₇	—	4	1	no	—
257	C₄×C₆₇	—	4	1	no	—
258	C₄×C₆₇	—	8	2	no	—
259	C₄×C₆₇	#444	1	2	yes	—
260	C₄×C₆₇	#518	1	1	yes	characteristic
261	C₄×C₆₇	#520	1	1	yes	characteristic
262	C₄×C₆₇	#535	1	1	yes	characteristic
263	C₄×C₆₇	#537	1	1	yes	characteristic
264	#22166	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
265	C₂×C₆₇	—	2	1	no	—
266	C₂×C₆₇	—	2	1	no	—
267	C₂×C₆₇	—	8	1	no	—
268	C₂×C₆₇	—	8	2	no	—
269	C₂×C₆₇	#2324	1	1	yes	characteristic
270	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
271	C₂×C₆₇	#2955	1	1	yes	characteristic
272	#3035	C₂×C₆₇	1	1	yes	characteristic
273	#2931	C₂×C₆₇	1	1	yes	characteristic, complemented
274	#2922	C₂×C₆₇	1	1	yes	characteristic, complemented
275	#1980	C₂×C₆₇	1	1	yes	characteristic, complemented
276	#1975	C₂×C₆₇	1	2	yes	complemented
277	#1950	C₂×C₆₇	1	1	yes	characteristic, complemented
278	#1938	C₂×C₆₇	1	2	yes	complemented
279	#1876	C₂×C₆₇	1	2	yes	complemented
280	#1866	C₂×C₆₇	1	2	yes	complemented
281	#1806	C₂×C₆₇	1	1	yes	characteristic, complemented
282	#1770	C₂×C₆₇	1	1	yes	characteristic, complemented
283	C₆₇	#22166	1	1	yes	characteristic, direct product factor, Sylow
284	C₂×#135	C₂²×C₆₇	1	2	yes	—
285	C₂×#131	C₂²×C₆₇	1	2	yes	—
286	#532	C₂²×C₆₇	1	1	yes	characteristic
287	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
288	#464	C₄×C₆₇	1	2	yes	complemented
289	#463	C₄×C₆₇	1	2	yes	complemented
290	#462	C₄×C₆₇	1	1	yes	characteristic, complemented
291	#461	C₄×C₆₇	1	1	yes	characteristic, complemented
292	#460	C₄×C₆₇	1	1	yes	characteristic, complemented
293	#459	C₄×C₆₇	1	1	yes	characteristic, complemented
294	#436	—	2	1	no	—
295	#436	—	2	1	no	—
296	C₈×D₄	—	2	1	no	—
297	C₈×D₄	—	2	1	no	—
298	#433	—	2	1	no	—
299	#433	—	2	1	no	—
300	#431	—	2	1	no	—
301	#431	—	2	1	no	—
302	C₂×#108	C₂²×C₆₇	1	2	yes	—
303	#424	C₂²×C₆₇	1	1	yes	characteristic
304	#424	C₂²×C₆₇	1	1	yes	characteristic
305	#423	C₂²×C₆₇	1	1	yes	characteristic
306	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
307	C₂×#104	C₂²×C₆₇	1	2	yes	—
308	C₂×#104	C₂²×C₆₇	1	2	yes	—
309	C₂×#104	C₂²×C₆₇	1	2	yes	—
310	#416	C₂²×C₆₇	1	2	yes	—
311	#408	C₂²×C₆₇	1	1	yes	characteristic
312	#408	C₂²×C₆₇	1	1	yes	characteristic
313	#408	C₂²×C₆₇	1	2	yes	—
314	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
315	C₂×#97	C₂²×C₆₇	1	2	yes	—
316	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
317	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
318	#341	C₂²×C₆₇	1	2	yes	—
319	#339	C₂²×C₆₇	1	2	yes	—
320	#336	C₂²×C₆₇	1	2	yes	—
321	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
322	C₂×#41	C₂³×C₆₇	1	2	yes	—
323	#135	—	4	2	no	—
324	#135	—	4	2	no	—
325	C₂×Q₁₆	C₂×C₄×C₆₇	1	2	yes	—
326	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
327	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
328	C₂×D₈	C₂×C₄×C₆₇	1	2	yes	—
329	#131	—	4	2	no	—
330	#131	—	4	2	no	—
331	C₂×M₁₆	—	2	2	no	—
332	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
333	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
334	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
335	C₂²×C₈	—	2	2	no	—
336	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
337	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
338	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
339	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
340	#127	C₂×C₄×C₆₇	1	1	yes	characteristic
341	#124	C₂×C₄×C₆₇	1	2	yes	—
342	#121	—	2	1	no	—
343	#121	—	2	1	no	—
344	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
345	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
346	C₄×D₄	—	2	1	no	—
347	C₄×D₄	—	2	1	no	—
348	C₄×D₄	—	2	1	no	—
349	C₄×D₄	—	2	1	no	—
350	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
351	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
352	#117	—	2	1	no	—
353	#117	C₂³×C₆₇	1	1	yes	characteristic
354	C₂×C₄⋊C₄	—	2	1	no	—
355	C₂×C₄²	—	2	1	no	—
356	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
357	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
358	#108	—	4	2	no	—
359	C₄⋊C₈	—	2	1	no	—
360	C₄⋊C₈	—	2	1	no	—
361	C₄⋊C₈	—	2	1	no	—
362	C₄⋊C₈	—	2	1	no	—
363	C₄⋊C₈	—	2	1	no	—
364	C₄⋊C₈	—	2	1	no	—
365	C₄⋊C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
366	C₄⋊C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
367	C₄⋊C₈	C₂×C₄×C₆₇	1	2	yes	—
368	#104	—	4	2	no	—
369	#104	—	4	2	no	—
370	#104	—	4	2	no	—
371	#103	C₂×C₄×C₆₇	1	2	yes	—
372	#103	C₂×C₄×C₆₇	1	2	yes	—
373	#102	C₂×C₄×C₆₇	1	2	yes	—
374	#102	C₂×C₄×C₆₇	1	2	yes	—
375	C₂²⋊C₈	—	2	1	no	—
376	C₂²⋊C₈	—	2	1	no	—
377	C₂²⋊C₈	—	2	1	no	—
378	C₂²⋊C₈	—	2	1	no	—
379	C₂²⋊C₈	—	2	2	no	—
380	C₂²⋊C₈	—	2	2	no	—
381	#97	—	2	1	no	—
382	#97	—	4	2	no	—
383	C₄×C₈	—	2	1	no	—
384	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
385	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
386	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
387	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
388	#95	—	2	1	no	—
389	#41	—	2	1	no	—
390	#41	—	2	2	no	—
391	#41	—	2	2	no	—
392	#41	—	2	2	no	—
393	#41	—	2	1	no	—
394	#41	—	2	1	no	—
395	#41	—	2	1	no	—
396	#41	—	2	2	no	—
397	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
398	C₂×Q₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
399	C₂×D₄	—	2	1	no	—
400	C₂×D₄	—	2	2	no	—
401	C₂×D₄	—	2	1	no	—
402	C₂×D₄	—	2	1	no	—
403	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
404	C₂×D₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
405	C₂²×C₄	—	2	1	no	—
406	C₂²×C₄	—	2	1	no	—
407	C₂²×C₄	—	2	1	no	—
408	C₂²×C₄	—	2	2	no	—
409	C₂²×C₄	—	2	2	no	—
410	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
411	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
412	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
413	Q₁₆	—	4	2	no	—
414	SD₁₆	—	4	2	no	—
415	SD₁₆	—	4	2	no	—
416	D₈	—	4	2	no	—
417	M₁₆	—	2	1	no	—
418	M₁₆	—	2	1	no	—
419	M₁₆	—	2	2	no	—
420	M₁₆	—	2	1	no	—
421	M₁₆	—	2	2	no	—
422	M₁₆	—	2	1	no	—
423	M₁₆	—	4	4	no	—
424	C₂×C₈	—	2	1	no	—
425	C₂×C₈	—	2	2	no	—
426	C₂×C₈	—	2	2	no	—
427	C₂×C₈	—	2	1	no	—
428	C₂×C₈	—	2	1	no	—
429	C₂×C₈	—	2	1	no	—
430	C₂×C₈	—	2	1	no	—
431	C₂×C₈	—	2	1	no	—
432	C₂×C₈	—	2	2	no	—
433	C₂×C₈	—	4	4	no	—
434	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
435	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
436	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
437	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
438	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
439	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
440	C₄⋊C₄	—	2	1	no	—
441	C₄⋊C₄	—	2	1	no	—
442	C₄⋊C₄	—	2	1	no	—
443	C₄⋊C₄	—	2	1	no	—
444	C₄⋊C₄	—	4	2	no	—
445	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
446	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
447	C₂²⋊C₄	—	2	1	no	—
448	C₂²⋊C₄	—	2	1	no	—
449	C₂²⋊C₄	—	2	1	no	—
450	C₂²⋊C₄	—	2	2	no	—
451	C₂²⋊C₄	—	2	2	no	—
452	C₄²	—	2	1	no	—
453	C₄²	—	2	1	no	—
454	C₄²	—	2	1	no	—
455	C₄²	—	2	2	no	—
456	C₄²	—	2	2	no	—
457	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
458	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
459	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
460	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
461	C₂³	—	2	1	no	—
462	C₂³	—	2	2	no	—
463	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
464	Q₈	—	2	1	no	—
465	Q₈	—	2	2	no	—
466	Q₈	—	2	2	no	—
467	Q₈	—	2	1	no	—
468	D₄	—	2	1	no	—
469	D₄	—	2	2	no	—
470	D₄	—	2	2	no	—
471	D₄	—	2	1	no	—
472	D₄	—	4	2	no	—
473	D₄	—	4	1	no	—
474	D₄	—	4	2	no	—
475	D₄	—	4	1	no	—
476	D₄	—	4	2	no	—
477	D₄	—	4	2	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	1	no	—
480	C₂×C₄	—	2	1	no	—
481	C₂×C₄	—	2	1	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	4	1	no	—
484	C₂×C₄	—	4	2	no	—
485	C₂×C₄	—	4	2	no	—
486	C₂×C₄	—	4	1	no	—
487	C₂×C₄	—	4	1	no	—
488	C₂×C₄	—	4	2	no	—
489	C₂×C₄	—	4	2	no	—
490	C₂×C₄	—	4	2	no	—
491	C₂×C₄	—	4	2	no	—
492	C₂×C₄	—	4	2	no	—
493	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
494	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
495	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
496	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
497	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
498	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
499	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
500	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
501	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
502	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
503	C₈	—	2	2	no	—
504	C₈	—	2	2	no	—
505	C₈	—	2	1	no	—
506	C₈	—	2	1	no	—
507	C₈	—	2	1	no	—
508	C₈	—	2	1	no	—
509	C₈	—	4	2	no	—
510	C₈	—	4	1	no	—
511	C₈	—	4	1	no	—
512	C₂²	—	2	1	no	—
513	C₂²	—	2	1	no	—
514	C₂²	—	4	2	no	—
515	C₂²	—	4	1	no	—
516	C₂²	—	4	1	no	—
517	C₂²	—	4	2	no	—
518	C₂²	—	4	1	no	—
519	C₂²	—	4	2	no	—
520	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
521	C₂²	#540×C₆₇	1	1	yes	characteristic
522	C₂²	#542×C₆₇	1	1	yes	characteristic
523	C₄	—	2	1	no	—
524	C₄	—	2	1	no	—
525	C₄	—	2	1	no	—
526	C₄	—	4	1	no	—
527	C₄	—	4	2	no	—
528	C₄	—	4	1	no	—
529	C₄	—	4	1	no	—
530	C₄	—	8	2	no	—
531	C₄	#444×C₆₇	1	2	yes	—
532	C₄	#518×C₆₇	1	1	yes	characteristic
533	C₄	#520×C₆₇	1	1	yes	characteristic
534	C₄	#535×C₆₇	1	1	yes	characteristic
535	C₄	#537×C₆₇	1	1	yes	characteristic
536	C₂	—	2	1	no	—
537	C₂	—	2	1	no	—
538	C₂	—	8	1	no	—
539	C₂	—	8	2	no	—
540	C₂	C₆₇×#2324	1	1	yes	characteristic
541	C₂	C₂×#444×C₆₇	1	1	yes	characteristic
542	C₂	C₆₇×#2955	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	C₂×C₄×D₄
prime radical	C₆₇×#3035	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2931	C₂	semidirect
3. C₆₇×#2922	C₂	semidirect
4. C₆₇×#1980	C₂	semidirect
5. C₆₇×#1975	C₂	semidirect
6. C₆₇×#1950	C₂	semidirect
7. C₆₇×#1938	C₂	semidirect
8. C₆₇×#1876	C₂	semidirect
9. C₆₇×#1866	C₂	semidirect
10. C₆₇×#1806	C₂	semidirect
11. C₆₇×#1770	C₂	semidirect
16. #464×C₆₇	C₄	semidirect
17. #463×C₆₇	C₄	semidirect
18. #462×C₆₇	C₄	semidirect
19. #461×C₆₇	C₄	semidirect
20. #460×C₆₇	C₄	semidirect
21. #459×C₆₇	C₄	semidirect
264. #22166	C₆₇	direct (N × H)
273. #2931	C₂×C₆₇	semidirect
274. #2922	C₂×C₆₇	semidirect
275. #1980	C₂×C₆₇	semidirect
276. #1975	C₂×C₆₇	semidirect
277. #1950	C₂×C₆₇	semidirect
278. #1938	C₂×C₆₇	semidirect
279. #1876	C₂×C₆₇	semidirect
280. #1866	C₂×C₆₇	semidirect
281. #1806	C₂×C₆₇	semidirect
282. #1770	C₂×C₆₇	semidirect
283. C₆₇	#22166	direct (N × H)
288. #464	C₄×C₆₇	semidirect
289. #463	C₄×C₆₇	semidirect
290. #462	C₄×C₆₇	semidirect
291. #461	C₄×C₆₇	semidirect
292. #460	C₄×C₆₇	semidirect
293. #459	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3035	2
C₆₇×#2931	2
C₆₇×#2922	2
C₆₇×#1980	2
C₆₇×#1975	2
C₆₇×#1950	2
C₆₇×#1938	2
C₆₇×#1876	2
C₆₇×#1866	2
C₆₇×#1806	2
C₆₇×#1770	2
#22166	67

Subgroups — C67×#22166 #30701417

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22166 #30701417