Subgroups — C₂×C₆₇×#3518 #30742042

Order 17152.

7118 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3518	C₂	1	32	yes	maximal, direct product factor
2	C₂×Q₈²×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#556×C₆₇	C₂	1	6	yes	maximal, complemented
4	C₂×#556×C₆₇	C₂	1	3	yes	maximal, complemented
5	C₂×#555×C₆₇	C₂	1	3	yes	maximal, complemented
6	C₂×D₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#548×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#542×C₆₇	C₂	1	6	yes	maximal, complemented
9	C₂×#533×C₆₇	C₂	1	3	yes	maximal, complemented
10	C₂×#529×C₆₇	C₂	1	3	yes	maximal, complemented
11	C₂×#519×C₆₇	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×#41×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×Q₈×C₆₇	C₂²	1	2	yes	—
15	Q₈²×C₆₇	C₂²	1	16	yes	complemented
16	#556×C₆₇	C₂²	1	96	yes	complemented
17	#556×C₆₇	C₂²	1	48	yes	complemented
18	#555×C₆₇	C₂²	1	48	yes	complemented
19	D₄×Q₈×C₆₇	C₂²	1	16	yes	complemented
20	#548×C₆₇	C₂²	1	32	yes	complemented
21	#542×C₆₇	C₂²	1	96	yes	complemented
22	#533×C₆₇	C₂²	1	48	yes	complemented
23	C₂×C₄⋊Q₈×C₆₇	C₂²	1	3	yes	—
24	C₂×C₄⋊Q₈×C₆₇	C₂²	1	3	yes	—
25	C₂×C₄⋊Q₈×C₆₇	C₂²	1	6	yes	—
26	#529×C₆₇	C₂²	1	48	yes	complemented
27	C₂×#126×C₆₇	C₂²	1	6	yes	—
28	C₂×#126×C₆₇	C₂²	1	12	yes	—
29	C₂×#125×C₆₇	C₂²	1	3	yes	—
30	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×#124×C₆₇	C₂²	1	6	yes	—
32	C₂×#124×C₆₇	C₂²	1	12	yes	—
33	C₂×#123×C₆₇	C₂²	1	6	yes	—
34	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	3	yes	—
35	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	6	yes	—
36	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	6	yes	—
37	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	6	yes	—
38	C₂×#121×C₆₇	C₂²	1	3	yes	—
39	C₂×#121×C₆₇	C₂²	1	6	yes	—
40	#519×C₆₇	C₂²	1	32	yes	complemented
41	C₄×#41×C₆₇	C₂²	1	16	yes	complemented
42	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
43	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
44	C₂×C₄×Q₈×C₆₇	C₂²	1	6	yes	—
45	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
46	C₂×C₄×Q₈×C₆₇	C₂²	1	6	yes	—
47	C₂×C₄×Q₈×C₆₇	C₂²	1	3	yes	—
48	C₂×C₄×Q₈×C₆₇	C₂²	1	6	yes	—
49	C₂×C₄×D₄×C₆₇	C₂²	1	3	yes	—
50	C₂×C₄×D₄×C₆₇	C₂²	1	3	yes	—
51	C₂×C₄×D₄×C₆₇	C₂²	1	6	yes	—
52	C₂×C₄×D₄×C₆₇	C₂²	1	3	yes	—
53	C₂×#117×C₆₇	C₂²	1	3	yes	—
54	C₂×#117×C₆₇	C₂²	1	6	yes	—
55	C₂×#117×C₆₇	C₂²	1	6	yes	—
56	C₂×#117×C₆₇	C₂²	1	6	yes	—
57	C₂²×C₄²×C₆₇	C₂²	1	3	yes	—
58	C₂×#41×C₆₇	—	2	8	no	—
59	C₂×#41×C₆₇	C₂³	1	8	yes	—
60	C₂²×Q₈×C₆₇	—	2	16	no	—
61	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₂²×Q₈×C₆₇	C₂³	1	6	yes	—
63	C₂²×Q₈×C₆₇	C₂³	1	16	yes	—
64	C₂²×Q₈×C₆₇	C₂³	1	6	yes	—
65	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
66	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
67	C₂²×D₄×C₆₇	C₂³	1	3	yes	—
68	C₂²×D₄×C₆₇	C₂³	1	6	yes	—
69	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
70	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
71	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
72	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
73	C₄⋊Q₈×C₆₇	C₂³	1	24	yes	—
74	C₄⋊Q₈×C₆₇	C₂³	1	24	yes	—
75	C₄⋊Q₈×C₆₇	C₂³	1	48	yes	—
76	#126×C₆₇	C₂³	1	48	yes	—
77	#126×C₆₇	C₂³	1	96	yes	—
78	#125×C₆₇	C₂³	1	24	yes	—
79	#124×C₆₇	C₂³	1	8	yes	—
80	#124×C₆₇	C₂³	1	48	yes	—
81	#124×C₆₇	C₂³	1	96	yes	—
82	#123×C₆₇	C₂³	1	48	yes	—
83	C₂²⋊Q₈×C₆₇	C₂³	1	24	yes	—
84	C₂²⋊Q₈×C₆₇	C₂³	1	48	yes	—
85	C₂²⋊Q₈×C₆₇	C₂³	1	48	yes	—
86	C₂²⋊Q₈×C₆₇	C₂³	1	48	yes	—
87	#121×C₆₇	C₂³	1	24	yes	—
88	#121×C₆₇	C₂³	1	48	yes	—
89	C₄×Q₈×C₆₇	C₂³	1	8	yes	—
90	C₄×Q₈×C₆₇	C₂³	1	8	yes	—
91	C₄×Q₈×C₆₇	C₂³	1	48	yes	—
92	C₄×Q₈×C₆₇	C₂³	1	16	yes	—
93	C₄×Q₈×C₆₇	C₂³	1	48	yes	—
94	C₄×Q₈×C₆₇	C₂³	1	24	yes	—
95	C₄×Q₈×C₆₇	C₂³	1	48	yes	—
96	C₄×D₄×C₆₇	C₂³	1	24	yes	—
97	C₄×D₄×C₆₇	C₂³	1	24	yes	—
98	C₄×D₄×C₆₇	C₂³	1	48	yes	—
99	C₄×D₄×C₆₇	C₂³	1	24	yes	—
100	#117×C₆₇	C₂³	1	24	yes	—
101	#117×C₆₇	C₂³	1	48	yes	—
102	#117×C₆₇	C₂³	1	48	yes	—
103	#117×C₆₇	C₂³	1	48	yes	—
104	C₂×C₄⋊C₄×C₆₇	C₂³	1	3	yes	—
105	C₂×C₄⋊C₄×C₆₇	C₂³	1	3	yes	—
106	C₂×C₄⋊C₄×C₆₇	C₂³	1	3	yes	—
107	C₂×C₄⋊C₄×C₆₇	C₂³	1	3	yes	—
108	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
109	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
110	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
111	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
112	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
113	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
114	C₂×C₂²⋊C₄×C₆₇	C₂³	1	6	yes	—
115	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
116	C₂×C₂²⋊C₄×C₆₇	C₂³	1	6	yes	—
117	C₂×C₂²⋊C₄×C₆₇	C₂³	1	12	yes	—
118	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
119	C₂×C₂²⋊C₄×C₆₇	C₂³	1	6	yes	—
120	C₂×C₂²⋊C₄×C₆₇	C₂³	1	6	yes	—
121	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
122	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
123	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
124	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
125	C₂×C₄²×C₆₇	C₂³	1	24	yes	—
126	C₂×C₄²×C₆₇	C₂³	1	12	yes	—
127	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
128	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
129	C₂⁴×C₆₇	C₂⁴	1	3	yes	—
130	#41×C₆₇	—	2	32	no	—
131	C₂×Q₈×C₆₇	—	2	64	no	—
132	C₂×Q₈×C₆₇	—	2	24	no	—
133	C₂×Q₈×C₆₇	—	2	4	no	—
134	C₂×Q₈×C₆₇	—	4	12	no	—
135	C₂×Q₈×C₆₇	#41	1	8	yes	—
136	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
137	C₂×Q₈×C₆₇	C₂⁴	1	24	yes	—
138	C₂×Q₈×C₆₇	C₂⁴	1	24	yes	—
139	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
140	C₂×D₄×C₆₇	—	2	12	no	—
141	C₂×D₄×C₆₇	—	2	4	no	—
142	C₂×D₄×C₆₇	—	4	12	no	—
143	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
144	C₂×D₄×C₆₇	C₂⁴	1	12	yes	—
145	C₂×D₄×C₆₇	C₂⁴	1	24	yes	—
146	C₂²×C₄×C₆₇	—	2	24	no	—
147	C₂²×C₄×C₆₇	—	2	12	no	—
148	C₂²×C₄×C₆₇	—	2	12	no	—
149	C₂²×C₄×C₆₇	—	4	6	no	—
150	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
151	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
152	C₂²×C₄×C₆₇	C₂⁴	1	24	yes	—
153	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
154	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
155	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
156	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
157	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
158	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
159	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
160	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
161	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
162	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
163	C₄⋊C₄×C₆₇	C₂⁴	1	12	yes	—
164	C₄⋊C₄×C₆₇	C₂⁴	1	12	yes	—
165	C₄⋊C₄×C₆₇	C₂⁴	1	12	yes	—
166	C₄⋊C₄×C₆₇	C₂⁴	1	12	yes	—
167	C₄⋊C₄×C₆₇	C₂⁴	1	24	yes	—
168	C₄⋊C₄×C₆₇	C₂⁴	1	24	yes	—
169	C₄⋊C₄×C₆₇	C₂⁴	1	24	yes	—
170	C₄⋊C₄×C₆₇	C₂⁴	1	24	yes	—
171	C₄⋊C₄×C₆₇	C₂⁴	1	24	yes	—
172	C₄⋊C₄×C₆₇	C₂⁴	1	24	yes	—
173	C₂²⋊C₄×C₆₇	C₂⁴	1	24	yes	—
174	C₂²⋊C₄×C₆₇	C₂⁴	1	8	yes	—
175	C₂²⋊C₄×C₆₇	C₂⁴	1	24	yes	—
176	C₂²⋊C₄×C₆₇	C₂⁴	1	48	yes	—
177	C₂²⋊C₄×C₆₇	C₂⁴	1	8	yes	—
178	C₂²⋊C₄×C₆₇	C₂⁴	1	24	yes	—
179	C₂²⋊C₄×C₆₇	C₂⁴	1	24	yes	—
180	C₄²×C₆₇	C₂⁴	1	4	yes	—
181	C₄²×C₆₇	C₂⁴	1	12	yes	—
182	C₄²×C₆₇	C₂⁴	1	12	yes	—
183	C₄²×C₆₇	C₂⁴	1	12	yes	—
184	C₄²×C₆₇	C₂⁴	1	48	yes	—
185	C₄²×C₆₇	C₂⁴	1	24	yes	—
186	C₂³×C₆₇	—	2	4	no	—
187	C₂³×C₆₇	—	2	6	no	—
188	C₂³×C₆₇	—	2	6	no	—
189	C₂³×C₆₇	—	2	4	no	—
190	C₂³×C₆₇	—	2	6	no	—
191	C₂³×C₆₇	—	2	4	no	—
192	C₂³×C₆₇	C₂⁵	1	4	yes	—
193	C₂³×C₆₇	C₂⁵	1	6	yes	—
194	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
195	Q₈×C₆₇	—	2	48	no	—
196	Q₈×C₆₇	—	2	8	no	—
197	Q₈×C₆₇	—	4	24	no	—
198	Q₈×C₆₇	C₂×#41	1	16	yes	—
199	D₄×C₆₇	—	2	24	no	—
200	D₄×C₆₇	—	2	8	no	—
201	D₄×C₆₇	—	4	24	no	—
202	C₂×C₄×C₆₇	—	2	2	no	—
203	C₂×C₄×C₆₇	—	2	12	no	—
204	C₂×C₄×C₆₇	—	2	6	no	—
205	C₂×C₄×C₆₇	—	2	4	no	—
206	C₂×C₄×C₆₇	—	2	2	no	—
207	C₂×C₄×C₆₇	—	2	6	no	—
208	C₂×C₄×C₆₇	—	2	12	no	—
209	C₂×C₄×C₆₇	—	2	48	no	—
210	C₂×C₄×C₆₇	—	2	24	no	—
211	C₂×C₄×C₆₇	—	2	24	no	—
212	C₂×C₄×C₆₇	—	4	12	no	—
213	C₂×C₄×C₆₇	C₂×#41	1	12	yes	—
214	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
215	C₂×C₄×C₆₇	C₂⁵	1	6	yes	—
216	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
217	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
218	C₂×C₄×C₆₇	C₂⁵	1	4	yes	—
219	C₂×C₄×C₆₇	C₂⁵	1	12	yes	—
220	C₂×C₄×C₆₇	C₂⁵	1	12	yes	—
221	C₂×C₄×C₆₇	C₂⁵	1	6	yes	—
222	C₂×C₄×C₆₇	C₂⁵	1	6	yes	—
223	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
224	C₂²×C₆₇	—	2	4	no	—
225	C₂²×C₆₇	—	2	6	no	—
226	C₂²×C₆₇	—	2	6	no	—
227	C₂²×C₆₇	—	2	4	no	—
228	C₂²×C₆₇	—	2	6	no	—
229	C₂²×C₆₇	—	2	4	no	—
230	C₂²×C₆₇	—	4	8	no	—
231	C₂²×C₆₇	—	4	12	no	—
232	C₂²×C₆₇	C₂²×#41	1	2	yes	—
233	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
234	C₂²×C₆₇	#585	1	2	yes	—
235	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
236	C₄×C₆₇	—	2	2	no	—
237	C₄×C₆₇	—	2	12	no	—
238	C₄×C₆₇	—	2	6	no	—
239	C₄×C₆₇	—	2	4	no	—
240	C₄×C₆₇	—	2	2	no	—
241	C₄×C₆₇	—	2	6	no	—
242	C₄×C₆₇	—	2	12	no	—
243	C₄×C₆₇	C₂²×#41	1	12	yes	—
244	C₄×C₆₇	C₂×2₋¹⁺⁴	1	4	yes	—
245	C₂×#3518	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
246	C₂×C₆₇	—	4	4	no	—
247	C₂×C₆₇	—	4	6	no	—
248	C₂×C₆₇	#3518	1	4	yes	direct product factor
249	C₂×C₆₇	C₂³×#41	1	1	yes	characteristic
250	C₂×C₆₇	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
251	C₂×C₆₇	C₂×#585	1	1	yes	characteristic
252	#3518	C₂×C₆₇	1	32	yes	direct product factor
253	C₂×Q₈²	C₂×C₆₇	1	1	yes	characteristic, complemented
254	C₂×#556	C₂×C₆₇	1	6	yes	complemented
255	C₂×#556	C₂×C₆₇	1	3	yes	complemented
256	C₂×#555	C₂×C₆₇	1	3	yes	complemented
257	C₂×D₄×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
258	C₂×#548	C₂×C₆₇	1	2	yes	complemented
259	C₂×#542	C₂×C₆₇	1	6	yes	complemented
260	C₂×#533	C₂×C₆₇	1	3	yes	complemented
261	C₂×#529	C₂×C₆₇	1	3	yes	complemented
262	C₂×#519	C₂×C₆₇	1	2	yes	complemented
263	C₂×C₄×#41	C₂×C₆₇	1	1	yes	characteristic, complemented
264	C₆₇	C₂×#3518	1	1	yes	characteristic, direct product factor, Sylow
265	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
266	C₂³×Q₈	C₂²×C₆₇	1	2	yes	—
267	Q₈²	C₂²×C₆₇	1	16	yes	complemented
268	#556	C₂²×C₆₇	1	96	yes	complemented
269	#556	C₂²×C₆₇	1	48	yes	complemented
270	#555	C₂²×C₆₇	1	48	yes	complemented
271	D₄×Q₈	C₂²×C₆₇	1	16	yes	complemented
272	#548	C₂²×C₆₇	1	32	yes	complemented
273	#542	C₂²×C₆₇	1	96	yes	complemented
274	#533	C₂²×C₆₇	1	48	yes	complemented
275	C₂×C₄⋊Q₈	C₂²×C₆₇	1	3	yes	—
276	C₂×C₄⋊Q₈	C₂²×C₆₇	1	3	yes	—
277	C₂×C₄⋊Q₈	C₂²×C₆₇	1	6	yes	—
278	#529	C₂²×C₆₇	1	48	yes	complemented
279	C₂×#126	C₂²×C₆₇	1	6	yes	—
280	C₂×#126	C₂²×C₆₇	1	12	yes	—
281	C₂×#125	C₂²×C₆₇	1	3	yes	—
282	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
283	C₂×#124	C₂²×C₆₇	1	6	yes	—
284	C₂×#124	C₂²×C₆₇	1	12	yes	—
285	C₂×#123	C₂²×C₆₇	1	6	yes	—
286	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	3	yes	—
287	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	6	yes	—
288	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	6	yes	—
289	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	6	yes	—
290	C₂×#121	C₂²×C₆₇	1	3	yes	—
291	C₂×#121	C₂²×C₆₇	1	6	yes	—
292	#519	C₂²×C₆₇	1	32	yes	complemented
293	C₄×#41	C₂²×C₆₇	1	16	yes	complemented
294	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
295	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
296	C₂×C₄×Q₈	C₂²×C₆₇	1	6	yes	—
297	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
298	C₂×C₄×Q₈	C₂²×C₆₇	1	6	yes	—
299	C₂×C₄×Q₈	C₂²×C₆₇	1	3	yes	—
300	C₂×C₄×Q₈	C₂²×C₆₇	1	6	yes	—
301	C₂×C₄×D₄	C₂²×C₆₇	1	3	yes	—
302	C₂×C₄×D₄	C₂²×C₆₇	1	3	yes	—
303	C₂×C₄×D₄	C₂²×C₆₇	1	6	yes	—
304	C₂×C₄×D₄	C₂²×C₆₇	1	3	yes	—
305	C₂×#117	C₂²×C₆₇	1	3	yes	—
306	C₂×#117	C₂²×C₆₇	1	6	yes	—
307	C₂×#117	C₂²×C₆₇	1	6	yes	—
308	C₂×#117	C₂²×C₆₇	1	6	yes	—
309	C₂²×C₄²	C₂²×C₆₇	1	3	yes	—
310	C₂×#41	—	2	8	no	—
311	C₂×#41	C₂³×C₆₇	1	8	yes	—
312	C₂²×Q₈	—	2	16	no	—
313	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
314	C₂²×Q₈	C₂³×C₆₇	1	6	yes	—
315	C₂²×Q₈	C₂³×C₆₇	1	16	yes	—
316	C₂²×Q₈	C₂³×C₆₇	1	6	yes	—
317	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
318	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
319	C₂²×D₄	C₂³×C₆₇	1	3	yes	—
320	C₂²×D₄	C₂³×C₆₇	1	6	yes	—
321	C₂³×C₄	C₂³×C₆₇	1	6	yes	—
322	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
323	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
324	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
325	C₄⋊Q₈	C₂³×C₆₇	1	24	yes	—
326	C₄⋊Q₈	C₂³×C₆₇	1	24	yes	—
327	C₄⋊Q₈	C₂³×C₆₇	1	48	yes	—
328	#126	C₂³×C₆₇	1	48	yes	—
329	#126	C₂³×C₆₇	1	96	yes	—
330	#125	C₂³×C₆₇	1	24	yes	—
331	#124	C₂³×C₆₇	1	8	yes	—
332	#124	C₂³×C₆₇	1	48	yes	—
333	#124	C₂³×C₆₇	1	96	yes	—
334	#123	C₂³×C₆₇	1	48	yes	—
335	C₂²⋊Q₈	C₂³×C₆₇	1	24	yes	—
336	C₂²⋊Q₈	C₂³×C₆₇	1	48	yes	—
337	C₂²⋊Q₈	C₂³×C₆₇	1	48	yes	—
338	C₂²⋊Q₈	C₂³×C₆₇	1	48	yes	—
339	#121	C₂³×C₆₇	1	24	yes	—
340	#121	C₂³×C₆₇	1	48	yes	—
341	C₄×Q₈	C₂³×C₆₇	1	8	yes	—
342	C₄×Q₈	C₂³×C₆₇	1	8	yes	—
343	C₄×Q₈	C₂³×C₆₇	1	48	yes	—
344	C₄×Q₈	C₂³×C₆₇	1	16	yes	—
345	C₄×Q₈	C₂³×C₆₇	1	48	yes	—
346	C₄×Q₈	C₂³×C₆₇	1	24	yes	—
347	C₄×Q₈	C₂³×C₆₇	1	48	yes	—
348	C₄×D₄	C₂³×C₆₇	1	24	yes	—
349	C₄×D₄	C₂³×C₆₇	1	24	yes	—
350	C₄×D₄	C₂³×C₆₇	1	48	yes	—
351	C₄×D₄	C₂³×C₆₇	1	24	yes	—
352	#117	C₂³×C₆₇	1	24	yes	—
353	#117	C₂³×C₆₇	1	48	yes	—
354	#117	C₂³×C₆₇	1	48	yes	—
355	#117	C₂³×C₆₇	1	48	yes	—
356	C₂×C₄⋊C₄	C₂³×C₆₇	1	3	yes	—
357	C₂×C₄⋊C₄	C₂³×C₆₇	1	3	yes	—
358	C₂×C₄⋊C₄	C₂³×C₆₇	1	3	yes	—
359	C₂×C₄⋊C₄	C₂³×C₆₇	1	3	yes	—
360	C₂×C₄⋊C₄	C₂³×C₆₇	1	6	yes	—
361	C₂×C₄⋊C₄	C₂³×C₆₇	1	6	yes	—
362	C₂×C₄⋊C₄	C₂³×C₆₇	1	6	yes	—
363	C₂×C₄⋊C₄	C₂³×C₆₇	1	6	yes	—
364	C₂×C₄⋊C₄	C₂³×C₆₇	1	6	yes	—
365	C₂×C₄⋊C₄	C₂³×C₆₇	1	6	yes	—
366	C₂×C₂²⋊C₄	C₂³×C₆₇	1	6	yes	—
367	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
368	C₂×C₂²⋊C₄	C₂³×C₆₇	1	6	yes	—
369	C₂×C₂²⋊C₄	C₂³×C₆₇	1	12	yes	—
370	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
371	C₂×C₂²⋊C₄	C₂³×C₆₇	1	6	yes	—
372	C₂×C₂²⋊C₄	C₂³×C₆₇	1	6	yes	—
373	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
374	C₂×C₄²	C₂³×C₆₇	1	12	yes	—
375	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
376	C₂×C₄²	C₂³×C₆₇	1	6	yes	—
377	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
378	C₂×C₄²	C₂³×C₆₇	1	24	yes	—
379	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
380	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
381	C₂⁴	C₂⁴×C₆₇	1	3	yes	—
382	#41	—	2	32	no	—
383	C₂×Q₈	—	2	64	no	—
384	C₂×Q₈	—	2	24	no	—
385	C₂×Q₈	—	2	4	no	—
386	C₂×Q₈	—	4	12	no	—
387	C₂×Q₈	#41×C₆₇	1	8	yes	—
388	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
389	C₂×Q₈	C₂⁴×C₆₇	1	24	yes	—
390	C₂×Q₈	C₂⁴×C₆₇	1	24	yes	—
391	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
392	C₂×D₄	—	2	12	no	—
393	C₂×D₄	—	2	4	no	—
394	C₂×D₄	—	4	12	no	—
395	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
396	C₂×D₄	C₂⁴×C₆₇	1	12	yes	—
397	C₂×D₄	C₂⁴×C₆₇	1	24	yes	—
398	C₂²×C₄	—	2	24	no	—
399	C₂²×C₄	—	2	12	no	—
400	C₂²×C₄	—	2	12	no	—
401	C₂²×C₄	—	4	6	no	—
402	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
403	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
404	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
405	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
406	C₂²×C₄	C₂⁴×C₆₇	1	6	yes	—
407	C₂²×C₄	C₂⁴×C₆₇	1	6	yes	—
408	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
409	C₂²×C₄	C₂⁴×C₆₇	1	12	yes	—
410	C₂²×C₄	C₂⁴×C₆₇	1	12	yes	—
411	C₂²×C₄	C₂⁴×C₆₇	1	24	yes	—
412	C₂²×C₄	C₂⁴×C₆₇	1	12	yes	—
413	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
414	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
415	C₄⋊C₄	C₂⁴×C₆₇	1	12	yes	—
416	C₄⋊C₄	C₂⁴×C₆₇	1	12	yes	—
417	C₄⋊C₄	C₂⁴×C₆₇	1	12	yes	—
418	C₄⋊C₄	C₂⁴×C₆₇	1	12	yes	—
419	C₄⋊C₄	C₂⁴×C₆₇	1	24	yes	—
420	C₄⋊C₄	C₂⁴×C₆₇	1	24	yes	—
421	C₄⋊C₄	C₂⁴×C₆₇	1	24	yes	—
422	C₄⋊C₄	C₂⁴×C₆₇	1	24	yes	—
423	C₄⋊C₄	C₂⁴×C₆₇	1	24	yes	—
424	C₄⋊C₄	C₂⁴×C₆₇	1	24	yes	—
425	C₂²⋊C₄	C₂⁴×C₆₇	1	24	yes	—
426	C₂²⋊C₄	C₂⁴×C₆₇	1	8	yes	—
427	C₂²⋊C₄	C₂⁴×C₆₇	1	24	yes	—
428	C₂²⋊C₄	C₂⁴×C₆₇	1	48	yes	—
429	C₂²⋊C₄	C₂⁴×C₆₇	1	8	yes	—
430	C₂²⋊C₄	C₂⁴×C₆₇	1	24	yes	—
431	C₂²⋊C₄	C₂⁴×C₆₇	1	24	yes	—
432	C₄²	C₂⁴×C₆₇	1	4	yes	—
433	C₄²	C₂⁴×C₆₇	1	12	yes	—
434	C₄²	C₂⁴×C₆₇	1	12	yes	—
435	C₄²	C₂⁴×C₆₇	1	12	yes	—
436	C₄²	C₂⁴×C₆₇	1	48	yes	—
437	C₄²	C₂⁴×C₆₇	1	24	yes	—
438	C₂³	—	2	4	no	—
439	C₂³	—	2	6	no	—
440	C₂³	—	2	6	no	—
441	C₂³	—	2	4	no	—
442	C₂³	—	2	6	no	—
443	C₂³	—	2	4	no	—
444	C₂³	C₂⁵×C₆₇	1	4	yes	—
445	C₂³	C₂⁵×C₆₇	1	6	yes	—
446	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
447	Q₈	—	2	48	no	—
448	Q₈	—	2	8	no	—
449	Q₈	—	4	24	no	—
450	Q₈	C₂×#41×C₆₇	1	16	yes	—
451	D₄	—	2	24	no	—
452	D₄	—	2	8	no	—
453	D₄	—	4	24	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	12	no	—
456	C₂×C₄	—	2	6	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	6	no	—
460	C₂×C₄	—	2	12	no	—
461	C₂×C₄	—	2	48	no	—
462	C₂×C₄	—	2	24	no	—
463	C₂×C₄	—	2	24	no	—
464	C₂×C₄	—	4	12	no	—
465	C₂×C₄	C₂×#41×C₆₇	1	12	yes	—
466	C₂×C₄	2₋¹⁺⁴×C₆₇	1	4	yes	—
467	C₂×C₄	C₂⁵×C₆₇	1	6	yes	—
468	C₂×C₄	C₂⁵×C₆₇	1	2	yes	—
469	C₂×C₄	C₂⁵×C₆₇	1	2	yes	—
470	C₂×C₄	C₂⁵×C₆₇	1	4	yes	—
471	C₂×C₄	C₂⁵×C₆₇	1	12	yes	—
472	C₂×C₄	C₂⁵×C₆₇	1	12	yes	—
473	C₂×C₄	C₂⁵×C₆₇	1	6	yes	—
474	C₂×C₄	C₂⁵×C₆₇	1	6	yes	—
475	C₂×C₄	C₂⁵×C₆₇	1	2	yes	—
476	C₂²	—	2	4	no	—
477	C₂²	—	2	6	no	—
478	C₂²	—	2	6	no	—
479	C₂²	—	2	4	no	—
480	C₂²	—	2	6	no	—
481	C₂²	—	2	4	no	—
482	C₂²	—	4	8	no	—
483	C₂²	—	4	12	no	—
484	C₂²	C₂²×#41×C₆₇	1	2	yes	—
485	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
486	C₂²	#585×C₆₇	1	2	yes	—
487	C₂²	C₂⁶×C₆₇	1	1	yes	characteristic
488	C₄	—	2	2	no	—
489	C₄	—	2	12	no	—
490	C₄	—	2	6	no	—
491	C₄	—	2	4	no	—
492	C₄	—	2	2	no	—
493	C₄	—	2	6	no	—
494	C₄	—	2	12	no	—
495	C₄	C₂²×#41×C₆₇	1	12	yes	—
496	C₄	C₂×2₋¹⁺⁴×C₆₇	1	4	yes	—
497	C₂	—	4	4	no	—
498	C₂	—	4	6	no	—
499	C₂	C₆₇×#3518	1	4	yes	direct product factor
500	C₂	C₂³×#41×C₆₇	1	1	yes	characteristic
501	C₂	C₂²×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
502	C₂	C₂×#585×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶×C₆₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3518	C₂	direct (N × H)
2. C₂×Q₈²×C₆₇	C₂	semidirect
3. C₂×#556×C₆₇	C₂	semidirect
4. C₂×#556×C₆₇	C₂	semidirect
5. C₂×#555×C₆₇	C₂	semidirect
6. C₂×D₄×Q₈×C₆₇	C₂	semidirect
7. C₂×#548×C₆₇	C₂	semidirect
8. C₂×#542×C₆₇	C₂	semidirect
9. C₂×#533×C₆₇	C₂	semidirect
10. C₂×#529×C₆₇	C₂	semidirect
11. C₂×#519×C₆₇	C₂	semidirect
12. C₂×C₄×#41×C₆₇	C₂	semidirect
15. Q₈²×C₆₇	C₂²	semidirect
16. #556×C₆₇	C₂²	semidirect
17. #556×C₆₇	C₂²	semidirect
18. #555×C₆₇	C₂²	semidirect
19. D₄×Q₈×C₆₇	C₂²	semidirect
20. #548×C₆₇	C₂²	semidirect
21. #542×C₆₇	C₂²	semidirect
22. #533×C₆₇	C₂²	semidirect
26. #529×C₆₇	C₂²	semidirect
40. #519×C₆₇	C₂²	semidirect
41. C₄×#41×C₆₇	C₂²	semidirect
245. C₂×#3518	C₆₇	direct (N × H)
248. C₂×C₆₇	#3518	direct (N × H)
252. #3518	C₂×C₆₇	direct (N × H)
253. C₂×Q₈²	C₂×C₆₇	semidirect
254. C₂×#556	C₂×C₆₇	semidirect
255. C₂×#556	C₂×C₆₇	semidirect
256. C₂×#555	C₂×C₆₇	semidirect
257. C₂×D₄×Q₈	C₂×C₆₇	semidirect
258. C₂×#548	C₂×C₆₇	semidirect
259. C₂×#542	C₂×C₆₇	semidirect
260. C₂×#533	C₂×C₆₇	semidirect
261. C₂×#529	C₂×C₆₇	semidirect
262. C₂×#519	C₂×C₆₇	semidirect
263. C₂×C₄×#41	C₂×C₆₇	semidirect
264. C₆₇	C₂×#3518	direct (N × H)
267. Q₈²	C₂²×C₆₇	semidirect
268. #556	C₂²×C₆₇	semidirect
269. #556	C₂²×C₆₇	semidirect
270. #555	C₂²×C₆₇	semidirect
271. D₄×Q₈	C₂²×C₆₇	semidirect
272. #548	C₂²×C₆₇	semidirect
273. #542	C₂²×C₆₇	semidirect
274. #533	C₂²×C₆₇	semidirect
278. #529	C₂²×C₆₇	semidirect
292. #519	C₂²×C₆₇	semidirect
293. C₄×#41	C₂²×C₆₇	semidirect
499. C₂	C₆₇×#3518	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#3518	2
C₂×Q₈²×C₆₇	2
C₂×#556×C₆₇	2
C₂×#555×C₆₇	2
C₂×D₄×Q₈×C₆₇	2
C₂×#548×C₆₇	2
C₂×#542×C₆₇	2
C₂×#533×C₆₇	2
C₂×#529×C₆₇	2
C₂×#519×C₆₇	2
C₂×C₄×#41×C₆₇	2
C₂×#3518	67

Subgroups — C2×C67×#3518 #30742042

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3518 #30742042